Properties of Topological Spaces

Properties of topological spacesFrom Wikipedia, the free encyclopedia

Contents

1 a-paracompact space 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Alexandrov topology 22.1 Characterizations of Alexandrov topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Duality with preordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 The Alexandrov topology on a preordered set . . . . . . . . . . . . . . . . . . . . . . . . 32.2.2 The specialization preorder on a topological space . . . . . . . . . . . . . . . . . . . . . . 32.2.3 Equivalence between preorders and Alexandrov topologies . . . . . . . . . . . . . . . . . 32.2.4 Equivalence between monotony and continuity . . . . . . . . . . . . . . . . . . . . . . . . 42.2.5 Category theoretic description of the duality . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.6 Relationship to the construction of modal algebras from modal frames . . . . . . . . . . . 5

2.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Baire space 73.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2.1 Modern definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.2 Historical definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Baire category theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.8 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Collectionwise Hausdorff space 104.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Collectionwise normal space 115.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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6 Compact space 126.1 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.3.1 Open cover definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3.2 Equivalent definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.3.3 Compactness of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.4 Properties of compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4.1 Functions and compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4.2 Compact spaces and set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4.3 Ordered compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.5.1 Algebraic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7 Connected space 217.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.1.1 Connected components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.1.2 Disconnected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.3 Path connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.4 Arc connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.5 Local connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.6 Set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.7 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.8 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.9 Stronger forms of connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7.11.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.11.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8 Contractible space 298.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.2 Locally contractible spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9 Countably compact space 31

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9.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10 Door space 3210.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

11 Dowker space 3311.1 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

12 Dyadic space 3412.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

13 End (topology) 3513.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.4 Ends of graphs and groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.5 Ends of a CW complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

14 Extremally disconnected space 3714.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3714.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

15 Feebly compact space 38

16 First-countable space 3916.1 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3916.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3916.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4016.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

17 Glossary of topology 4117.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4217.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4317.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4317.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4517.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4517.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4517.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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17.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4617.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4717.10K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4717.11L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4817.12M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4817.13N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4917.14O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5017.15P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5017.16Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5117.17R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5217.18S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5217.19T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.20U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5417.21W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.22Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.24External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

18 H-closed space 5718.1 Examples and equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

19 Heine–Borel theorem 5819.1 History and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5819.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5819.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5919.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6019.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6019.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6019.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

20 Hemicompact space 6220.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6220.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6220.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6220.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

21 Hyperconnected space 6421.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.2 Hyperconnectedness vs. connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.4 Irreducible components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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21.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6521.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

22 Kolmogorov space 6622.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.2 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

22.2.1 Spaces which are not T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.2.2 Spaces which are T0 but not T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

22.3 Operating with T0 spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6722.4 The Kolmogorov quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6722.5 Removing T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6822.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

23 Limit point compact 6923.1 Properties and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7023.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

24 Lindelöf space 7124.1 Properties of Lindelöf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7124.2 Properties of strongly Lindelöf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7124.3 Product of Lindelöf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7124.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

25 Locally compact space 7325.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7325.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

25.2.1 Compact Hausdorff spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7425.2.2 Locally compact Hausdorff spaces that are not compact . . . . . . . . . . . . . . . . . . . 7425.2.3 Hausdorff spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 7425.2.4 Non-Hausdorff examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

25.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.3.1 The point at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

25.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7625.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

26 Locally connected space 7726.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

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26.2 Definitions and first examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7826.2.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

26.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7926.4 Components and path components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

26.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8026.5 Quasicomponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

26.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8026.6 More on local connectedness versus weak local connectedness . . . . . . . . . . . . . . . . . . . . 8126.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8126.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8126.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

27 Locally finite collection 8327.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

27.1.1 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.1.2 Second countable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

27.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8427.3 Countably locally finite collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8427.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

28 Locally finite space 8528.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

29 Locally Hausdorff space 8629.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

30 Locally normal space 8730.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8730.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8730.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8730.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8730.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

31 Locally regular space 8931.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8931.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8931.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8931.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

32 Locally simply connected space 9032.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

33 Luzin space 92

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33.1 In real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9233.2 Example of a Luzin set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9233.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

34 Mesocompact space 9434.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9434.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

35 Metacompact space 9535.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9535.2 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9535.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9535.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

36 Michael selection theorem 9736.1 Other selection theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9736.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

37 Monotonically normal space 9937.1 Equivalent definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

37.1.1 Definition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9937.1.2 Definition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9937.1.3 Definition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

37.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10037.3 Some discussion links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

38 n-connected 10138.1 n-connected space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

38.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10138.2 n-connected map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

38.2.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10238.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10338.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

39 Negative-dimensional space 10439.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10439.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10439.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10439.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10439.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

40 Noetherian topological space 10640.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10640.2 Relation to compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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40.3 Noetherian topological spaces from algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . 10640.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10740.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

41 Normal space 10841.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10841.2 Examples of normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10941.3 Examples of non-normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10941.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11041.5 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11041.6 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11041.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

42 Orthocompact space 11142.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

43 P-space 11243.1 Generic use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11243.2 P-spaces in the sense of Gillman–Henriksen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11243.3 P-spaces in the sense of Morita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11243.4 p-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11243.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11243.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11343.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

44 Paracompact space 11444.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11444.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11444.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11544.4 Paracompact Hausdorff Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

44.4.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11644.5 Relationship with compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

44.5.1 Comparison of properties with compactness . . . . . . . . . . . . . . . . . . . . . . . . . 11744.6 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

44.6.1 Definition of relevant terms for the variations . . . . . . . . . . . . . . . . . . . . . . . . . 11844.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11844.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11844.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11944.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

45 Paranormal space 12045.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12045.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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46 Perfect set 12146.1 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12146.2 Imperfection of a space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12146.3 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12146.4 Connection with other topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12246.5 Perfect spaces in descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12246.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12246.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

47 Polyadic space 12347.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12347.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12347.3 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12347.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12347.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

47.5.1 Ramsey’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12447.5.2 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

47.6 Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12547.6.1 Centred space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12547.6.2 AD-compact space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12547.6.3 ξ-adic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12547.6.4 Hyadic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

47.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12647.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

48 Pseudocompact space 12848.1 Properties related to pseudocompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12848.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12848.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

49 Pseudometric space 13049.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13049.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13049.3 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13149.4 Metric identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13149.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13149.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

50 Pseudonormal space 13350.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

51 Realcompact space 13451.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

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51.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13451.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

52 Regular space 13652.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13652.2 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13752.3 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13752.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13852.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

53 Relatively compact subspace 13953.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13953.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

54 Resolvable space 14054.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14054.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14054.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

55 Rickart space 14155.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

56 Second-countable space 14256.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

56.1.1 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14256.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14356.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

57 Semi-locally simply connected 14457.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14457.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14457.3 Topology of fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14557.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14557.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

58 Separable space 14658.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14658.2 Separability versus second countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14658.3 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14758.4 Constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14758.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

58.5.1 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14758.5.2 Non-separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

58.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

CONTENTS xi

58.6.1 Embedding separable metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14858.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

59 Sequential space 15059.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15059.2 Sequential closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15059.3 Fréchet–Urysohn space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15159.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15159.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15159.6 Equivalent conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15259.7 Categorical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15259.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15259.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

60 Shrinking space 15460.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

61 Simply connected at infinity 15561.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

62 Simply connected space 15662.1 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15662.2 Formal definition and equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15762.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15762.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15962.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15962.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

63 Sub-Stonean space 16063.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16063.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16063.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

64 Supercompact space 16164.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16164.2 Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16164.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

65 T1 space 16365.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16365.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16365.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16465.4 Generalisations to other kinds of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16565.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

xii CONTENTS

66 Topological manifold 16666.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16666.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16666.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

66.3.1 The Hausdorff axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16766.3.2 Compactness and countability axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16766.3.3 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

66.4 Coordinate charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16866.5 Classification of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16866.6 Manifolds with boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16966.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16966.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16966.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

67 Topological property 17067.1 Common topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

67.1.1 Cardinal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17067.1.2 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17067.1.3 Countability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17167.1.4 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17167.1.5 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17267.1.6 Metrizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17267.1.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

67.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17367.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17367.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

68 Toronto space 17468.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

69 Totally disconnected space 17569.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17569.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17569.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17669.4 Constructing a disconnected space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17669.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17669.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

70 Ultraconnected space 17770.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17770.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17770.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

CONTENTS xiii

71 Uniformizable space 17871.1 Induced uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17871.2 Fine uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17871.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

72 Volterra space 18072.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

73 Weak Hausdorff space 18173.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

74 Zero-dimensional space 18274.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18274.2 Properties of spaces with covering dimension zero . . . . . . . . . . . . . . . . . . . . . . . . . . 18274.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18274.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

75 σ-compact space 18475.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18475.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18475.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18575.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

76 ω-bounded space 18676.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18676.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 187

76.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18776.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19176.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Chapter 1

a-paracompact space

In mathematics, in the field of topology, a topological space is said to be a-paracompact if every open cover of thespace has a locally finite refinement. In contrast to the definition of paracompactness, the refinement is not requiredto be open.Every paracompact space is a-paracompact, and in regular spaces the two notions coincide.

1.1 References• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

1

Chapter 2

Alexandrov topology

In topology, an Alexandrov space (or Alexandrov-discrete space) is a topological space in which the intersectionof any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets isopen. In an Alexandrov space the finite restriction is dropped.Alexandrov topologies are uniquely determined by their specialization preorders. Indeed, given any preorder ≤ on aset X, there is a unique Alexandrov topology on X for which the specialization preorder is ≤. The open sets are justthe upper sets with respect to ≤. Thus, Alexandrov topologies on X are in one-to-one correspondence with preorderson X.Alexandrov spaces are also called finitely generated spaces since their topology is uniquely determined by the familyof all finite subspaces. Alexandrov spaces can be viewed as a generalization of finite topological spaces.

2.1 Characterizations of Alexandrov topologies

Alexandrov topologies have numerous characterizations. Let X = <X, T> be a topological space. Then the followingare equivalent:

• Open and closed set characterizations:

• Open set. An arbitrary intersection of open sets in X is open.• Closed set. An arbitrary union of closed sets in X is closed.

• Neighbourhood characterizations:

• Smallest neighbourhood. Every point of X has a smallest neighbourhood.• Neighbourhood filter. The neighbourhood filter of every point in X is closed under arbitrary intersec-tions.

• Interior and closure algebraic characterizations:

• Interior operator. The interior operator of X distributes over arbitrary intersections of subsets.• Closure operator. The closure operator of X distributes over arbitrary unions of subsets.

• Preorder characterizations:

• Specialization preorder. T is the finest topology consistent with the specialization preorder of X i.e.the finest topology giving the preorder ≤ satisfying x ≤ y if and only if x is in the closure of y in X.

• Open up-set. There is a preorder ≤ such that the open sets of X are precisely those that are upwardlyclosed i.e. if x is in the set and x ≤ y then y is in the set. (This preorder will be precisely the specializationpreorder.)

2

2.2. DUALITY WITH PREORDERED SETS 3

• Closed down-set. There is a preorder ≤ such that the closed sets of X are precisely those that aredownwardly closed i.e. if x is in the set and y ≤ x then y is in the set. (This preorder will be precisely thespecialization preorder.)

• Upward interior. A point x lies in the interior of a subset S of X if and only if there is a point y in Ssuch that y ≤ x where ≤ is the specialization preorder i.e. y lies in the closure of x.

• Downward closure. A point x lies in the closure of a subset S of X if and only if there is a point y in Ssuch that x ≤ y where ≤ is the specialization preorder i.e. x lies in the closure of y.

• Finite generation and category theoretic characterizations:

• Finite closure. A point x lies within the closure of a subset S of X if and only if there is a finite subsetF of S such that x lies in the closure of F.

• Finite subspace. T is coherent with the finite subspaces of X.• Finite inclusion map. The inclusion maps fi : Xi→ X of the finite subspaces of X form a final sink.• Finite generation. X is finitely generated i.e. it is in the final hull of the finite spaces. (This means thatthere is a final sink fi : Xi→ X where each Xi is a finite topological space.)

Topological spaces satisfying the above equivalent characterizations are called finitely generated spaces or Alexan-drov spaces and their topology T is called the Alexandrov topology, named after the Russian mathematician PavelAlexandrov who first investigated them.

2.2 Duality with preordered sets

2.2.1 The Alexandrov topology on a preordered set

Given a preordered set X = ⟨X,≤⟩ we can define an Alexandrov topology τ on X by choosing the open sets to bethe upper sets:

τ = G ⊆ X : ∀x, y ∈ X x ∈ G ∧ x ≤ y → y ∈ G,

We thus obtain a topological space T(X) = ⟨X, τ⟩ .The corresponding closed sets are the lower sets:

S ⊆ X : ∀x, y ∈ X x ∈ S ∧ y ≤ x → y ∈ S,

2.2.2 The specialization preorder on a topological space

Given a topological space X = <X, T> the specialization preorder on X is defined by:

x≤y if and only if x is in the closure of y.

We thus obtain a preordered setW(X) = <X, ≤>.

2.2.3 Equivalence between preorders and Alexandrov topologies

For every preordered set X = <X, ≤> we always have W(T(X)) = X, i.e. the preorder of X is recovered from thetopological space T(X) as the specialization preorder. Moreover for every Alexandrov space X, we have T(W(X)) =X, i.e. the Alexandrov topology of X is recovered as the topology induced by the specialization preorder.However for a topological space in general we do not have T(W(X)) = X. Rather T(W(X)) will be the set X with afiner topology than that of X (i.e. it will have more open sets).

4 CHAPTER 2. ALEXANDROV TOPOLOGY

2.2.4 Equivalence between monotony and continuity

Given a monotone function

f : X→Y

between two preordered sets (i.e. a function

f : X→Y

between the underlying sets such that x≤y in X implies f(x)≤f(y) in Y), let

T(f) : T(X)→T(Y)

be the same map as f considered as a map between the corresponding Alexandrov spaces. Then

T(f) : T(X)→T(Y)

is a continuous map.Conversely given a continuous map

f : X→Y

between two topological spaces, let

W(f) : W(X)→W(Y)

be the same map as f considered as a map between the corresponding preordered sets. Then

W(f) : W(X)→W(Y)

is a monotone function.Thus a map between two preordered sets is monotone if and only if it is a continuous map between the correspondingAlexandrov spaces. Conversely a map between two Alexandrov spaces is continuous if and only if it is a monotonefunction between the corresponding preordered sets.Notice however that in the case of topologies other than the Alexandrov topology, we can have a map between twotopological spaces that is not continuous but which is nevertheless still a monotone function between the correspondingpreordered sets. (To see this consider a non-Alexandrov space X and consider the identity map

i : X→T(W(X)).)

2.2.5 Category theoretic description of the duality

Let Set denote the category of sets and maps. Let Top denote the category of topological spaces and continuousmaps; and let Pro denote the category of preordered sets and monotone functions. Then

T : Pro→Top and

W : Top→Pro

are concrete functors over Set which are left and right adjoints respectively.Let Alx denote the full subcategory of Top consisting of the Alexandrov spaces. Then the restrictions

2.3. HISTORY 5

T : Pro→Alx and

W : Alx→Pro

are inverse concrete isomorphisms over Set.Alx is in fact a bico-reflective subcategory of Top with bico-reflector TW : Top→Alx. This means that given atopological space X, the identity map

i : T(W(X))→X

is continuous and for every continuous map

f : Y→X

where Y is an Alexandrov space, the composition

i −1f : Y→T(W(X))

is continuous.

2.2.6 Relationship to the construction of modal algebras from modal frames

Given a preordered set X, the interior operator and closure operator of T(X) are given by:

Int(S) = x ∈ X : for all y ∈ X, x≤y implies y ∈ S , for all S ⊆ X

Cl(S) = x ∈ X : there exists a y ∈ S with x≤y for all S ⊆ X

Considering the interior operator and closure operator to be modal operators on the power set Boolean algebra of X,this construction is a special case of the construction of a modal algebra from a modal frame i.e. a set with a singlebinary relation. (The latter construction is itself a special case of a more general construction of a complex algebrafrom a relational structure i.e. a set with relations defined on it.) The class of modal algebras that we obtain in thecase of a preordered set is the class of interior algebras—the algebraic abstractions of topological spaces.

2.3 History

Alexandrov spaces were first introduced in 1937 by P. S. Alexandrov under the name discrete spaces, where heprovided the characterizations in terms of sets and neighbourhoods.[1] The name discrete spaces later came to be usedfor topological spaces in which every subset is open and the original concept lay forgotten. With the advancement ofcategorical topology in the 1980s, Alexandrov spaces were rediscovered when the concept of finite generation wasapplied to general topology and the name finitely generated spaces was adopted for them. Alexandrov spaces werealso rediscovered around the same time in the context of topologies resulting from denotational semantics and domaintheory in computer science.In 1966 Michael C. McCord and A. K. Steiner each independently observed a duality between partially ordered setsand spaces which were precisely the T0 versions of the spaces that Alexandrov had introduced.[2][3] P. Johnstonereferred to such topologies as Alexandrov topologies.[4] F. G. Arenas independently proposed this name for thegeneral version of these topologies.[5] McCord also showed that these spaces are weak homotopy equivalent to theorder complex of the corresponding partially ordered set. Steiner demonstrated that the duality is a contravariantlattice isomorphism preserving arbitrary meets and joins as well as complementation.It was also a well known result in the field of modal logic that a duality exists between finite topological spaces andpreorders on finite sets (the finite modal frames for the modal logic S4). C. Naturman extended these results to aduality between Alexandrov spaces and preorders in general, providing the preorder characterizations as well as theinterior and closure algebraic characterizations.[6]

A systematic investigation of these spaces from the point of view of general topology which had been neglected sincethe original paper by Alexandrov, was taken up by F.G. Arenas.[5]

6 CHAPTER 2. ALEXANDROV TOPOLOGY

2.4 See also• P-space, a space satisfying the weaker condition that countable intersections of open sets are open

2.5 References[1] Alexandroff, P. (1937). “Diskrete Räume”. Mat. Sb. (N.S.) (in German) 2: 501–518.

[2] McCord, M. C. (1966). “Singular homology and homotopy groups of finite topological spaces”. DukeMathematical Journal33 (3): 465–474. doi:10.1215/S0012-7094-66-03352-7.

[3] Steiner, A. K. (1966). “The Lattice of Topologies: Structure and Complementation”. Transactions of the American Math-ematical Society 122 (2): 379–398. doi:10.2307/1994555. ISSN 0002-9947.

[4] Johnstone, P. T. (1986). Stone spaces (1st paperback ed.). New York: Cambridge University Press. ISBN 0-521-33779-8.

[5] Arenas, F. G. (1999). “Alexandroff spaces” (PDF). Acta Math. Univ. Comenianae 68 (1): 17–25.

[6] Naturman, C. A. (1991). Interior Algebras and Topology. Ph.D. thesis, University of Cape Town Department of Mathe-matics.

Chapter 3

Baire space

For the concept in set theory, see Baire space (set theory).

In mathematics, a Baire space is a topological space that has “enough” points that every intersection of a countablecollection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spacesare examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-LouisBaire who introduced the concept.

3.1 Motivation

In an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries ofdense open sets. These sets are, in a certain sense, “negligible”. Some examples are finite sets in ℝ, smooth curves inthe plane, and proper affine subspaces in a Euclidean space. If a topological space is a Baire space then it is “large”,meaning that it is not a countable union of negligible subsets. For example, the three-dimensional Euclidean space isnot a countable union of its affine planes.

3.2 Definition

The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailingneeds and viewpoints. First, we give the usual modern definition, and then we give a historical definition which iscloser to the definition originally given by Baire.

3.2.1 Modern definition

A Baire space is a topological space in which the union of every countable collection of closed sets with emptyinterior has empty interior.This definition is equivalent to each of the following conditions:

• Every intersection of countably many dense open sets is dense.

• The interior of every union of countably many closed nowhere dense sets is empty.

• Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsetsmust have an interior point.

3.2.2 Historical definition

Main article: Meagre set

7

8 CHAPTER 3. BAIRE SPACE

In his original definition, Baire defined a notion of category (unrelated to category theory) as follows.A subset of a topological space X is called

• nowhere dense in X if the interior of its closure is empty• of first category or meagre in X if it is a union of countably many nowhere dense subsets• of second category or nonmeagre in X if it is not of first category in X

The definition for a Baire space can then be stated as follows: a topological spaceX is a Baire space if every non-emptyopen set is of second category in X. This definition is equivalent to the modern definition.A subset A of X is comeagre if its complementX \A is meagre. A topological space X is a Baire space if and onlyif every comeager subset of X is dense.

3.3 Examples• The space R of real numbers with the usual topology, is a Baire space, and so is of second category in itself.The rational numbers are of first category and the irrational numbers are of second category in R .

• The Cantor set is a Baire space, and so is of second category in itself, but it is of first category in the interval[0, 1] with the usual topology.

• Here is an example of a set of second category in R with Lebesgue measure 0.

∞∩m=1

∞∪n=1

(rn − 1

2n+m, rn +

1

2n+m

)where rn∞n=1 is a sequence that enumerates the rational numbers.

• Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space,since it is the union of countably many closed sets without interior, the singletons.

3.4 Baire category theorem

Main article: Baire category theorem

The Baire category theorem gives sufficient conditions for a topological space to be a Baire space. It is an importanttool in topology and functional analysis.

• (BCT1) Every complete metric space is a Baire space. More generally, every topological space which ishomeomorphic to an open subset of a complete pseudometric space is a Baire space. In particular, everycompletely metrizable space is a Baire space.

• (BCT2) Every locally compact Hausdorff space (or more generally every locally compact sober space) is aBaire space.

BCT1 shows that each of the following is a Baire space:

• The space R of real numbers• The space of irrational numbers, which is homeomorphic to the Baire space ωω of set theory• The Cantor set• Indeed, every Polish space

BCT2 shows that every manifold is a Baire space, even if it is not paracompact, and hence not metrizable. Forexample, the long line is of second category.

3.5. PROPERTIES 9

3.5 Properties• Every non-empty Baire space is of second category in itself, and every intersection of countably many denseopen subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topologicaldisjoint sum of the rationals and the unit interval [0, 1].

• Every open subspace of a Baire space is a Baire space.

• Given a family of continuous functions fn:X→Y with pointwise limit f:X→Y. If X is a Baire space then thepoints where f is not continuous is a meagre set in X and the set of points where f is continuous is dense in X.A special case of this is the uniform boundedness principle.

• A closed subset of a Baire space is not necessarily Baire.

• The product of two Baire spaces is not necessarily Baire. However, there exist sufficient conditions that willguarantee that a product of arbitrarily many Baire spaces is again Baire.

3.6 See also• Banach–Mazur game

• Descriptive set theory

• Baire space (set theory)

3.7 References

3.8 Sources• Munkres, James, Topology, 2nd edition, Prentice Hall, 2000.

• Baire, René-Louis (1899), Sur les fonctions de variables réelles, Annali di Mat. Ser. 3 3, 1–123.

3.9 External links• Encyclopaedia of Mathematics article on Baire space

• Encyclopaedia of Mathematics article on Baire theorem

Chapter 4

Collectionwise Hausdorff space

In mathematics, in the field of topology, a topological space is said to be collectionwise Hausdorff if given any closeddiscrete collection of points in the topological space, there are pairwise disjoint open sets containing the points.[1] Aclosed discrete set S of a topology X is one where every point of X has a neighborhood that intersects at most onepoint from S. Every T1 space which is collectionwise Hausdorff is also Hausdorff.Metrizable spaces are collectionwise normal spaces and are hence, in particular, collectionwise Hausdorff.

4.1 References[1] FD Tall, The density topology, Pacific Journal of Mathematics, 1976

10

Chapter 5

Collectionwise normal space

In mathematics, a topological spaceX is called collectionwise normal if for every discrete family Fi (i ∈ I) of closedsubsets of X there exists a pairwise disjoint family of open sets Ui (i ∈ I), such that Fi ⊂ Ui. A family F of subsetsof X is called discrete when every point of X has a neighbourhood that intersects at most one of the sets from F .An equivalent definition demands that the above Ui (i ∈ I) are themselves a discrete family, which is stronger thanpairwise disjoint.Many authors assume that X is also a T1 space as part of the definition, i. e., for every pair of distinct points, eachhas an open neighborhood not containing the other. A collectionwise normal T1 space is a collectionwise Hausdorffspace.Every collectionwise normal space is normal (i. e., any two disjoint closed sets can be separated by neighbourhoods),and every paracompact space (i. e., every topological space in which every open cover admits a locally finite openrefinement) is collectionwise normal. The property is therefore intermediate in strength between paracompactnessand normality.Every metrizable space (i. e., every topological space that is homeomorphic to a metric space) is collectionwisenormal. The Moore metrisation theorem states that every collectionwise normal Moore space is metrizable.An Fσ-set in a collectionwise normal space is also collectionwise normal in the subspace topology. In particular, thisholds for closed subsets.

5.1 References• Engelking, Ryszard, General Topology, Heldermann Verlag Berlin, 1989. ISBN 3-88538-006-4

11

Chapter 6

Compact space

“Compactness” redirects here. For the concept in first-order logic, see Compactness theorem.In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of

The interval A = (-∞, −2] is not compact because it is not bounded. The interval C = (2, 4) is not compact because it is not closed.The interval B = [0, 1] is compact because it is both closed and bounded.

a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all itspoints lie within some fixed distance of each other). Examples include a closed interval, a rectangle, or a finite set ofpoints. This notion is defined for more general topological spaces than Euclidean space in various ways.One such generalization is that a space is sequentially compact if any infinite sequence of points sampled from thespace must frequently (infinitely often) get arbitrarily close to some point of the space. An equivalent definition isthat every sequence of points must have an infinite subsequence that converges to some point of the space. TheHeine-Borel theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it isclosed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval [0, 1] some of thosepoints must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3,5/6, 1/4, 6/7, … accumulate to 0 (others accumulate to 1). The same set of points would not accumulate to any pointof the open unit interval (0, 1); so the open unit interval is not compact. Euclidean space itself is not compact sinceit is not bounded. In particular, the sequence of points 0, 1, 2, 3, … has no subsequence that converges to any givenreal number.Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spacesconsisting not of geometrical points but of functions. The term compact was introduced into mathematics by MauriceFréchet in 1904 as a distillation of this concept. Compactness in this more general situation plays an extremelyimportant role in mathematical analysis, because many classical and important theorems of 19th century analysis,such as the extreme value theorem, are easily generalized to this situation. A typical application is furnished by the

12

6.1. HISTORICAL DEVELOPMENT 13

Arzelà–Ascoli theorem or the Peano existence theorem, in which one is able to conclude the existence of a functionwith some required properties as a limiting case of some more elementary construction.Various equivalent notions of compactness, including sequential compactness and limit point compactness, can bedeveloped in general metric spaces. In general topological spaces, however, different notions of compactness are notnecessarily equivalent. The most useful notion, which is the standard definition of the unqualified term compactness,is phrased in terms of the existence of finite families of open sets that "cover" the space in the sense that each pointof the space must lie in some set contained in the family. This more subtle notion, introduced by Pavel Alexandrovand Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. In spaces that are compact inthis sense, it is often possible to patch together information that holds locally—that is, in a neighborhood of eachpoint—into corresponding statements that hold throughout the space, and many theorems are of this character.The term compact set is sometimes a synonym for compact space, but usually refers to a compact subspace of atopological space.

6.1 Historical development

In the 19th century, several disparate mathematical properties were understood that would later be seen as conse-quences of compactness. On the one hand, Bernard Bolzano (1817) had been aware that any bounded sequence ofpoints (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some otherpoint, called a limit point. Bolzano’s proof relied on the method of bisection: the sequence was placed into an intervalthat was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected.The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts until itcloses down on the desired limit point. The full significance of Bolzano’s theorem, and its method of proof, wouldnot emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.[1]

In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for spacesof functions rather than just numbers or geometrical points. The idea of regarding functions as themselves pointsof a generalized space dates back to the investigations of Giulio Ascoli and Cesare Arzelà.[2] The culmination oftheir investigations, the Arzelà–Ascoli theorem, was a generalization of the Bolzano–Weierstrass theorem to familiesof continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergentsequence of functions from a suitable family of functions. The uniform limit of this sequence then played preciselythe same role as Bolzano’s “limit point”. Towards the beginning of the twentieth century, results similar to that ofArzelà and Ascoli began to accumulate in the area of integral equations, as investigated by David Hilbert and ErhardSchmidt. For a certain class of Green functions coming from solutions of integral equations, Schmidt had shown thata property analogous to the Arzelà–Ascoli theorem held in the sense of mean convergence—or convergence in whatwould later be dubbed a Hilbert space. This ultimately led to the notion of a compact operator as an offshoot of thegeneral notion of a compact space. It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property and coined the term compactness to refer to this general phenomenon (he used the term alreadyin his 1904 paper[3] which led to the famous 1906 thesis) .However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century fromthe study of the continuum, which was seen as fundamental for the rigorous formulation of analysis. In 1870, EduardHeine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous.In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller openintervals, it was possible to select a finite number of these that also covered it. The significance of this lemma wasrecognized by Émile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895)and Henri Lebesgue (1904). The Heine–Borel theorem, as the result is now known, is another special propertypossessed by closed and bounded sets of real numbers.This property was significant because it allowed for the passage from local information about a set (such as thecontinuity of a function) to global information about the set (such as the uniform continuity of a function). Thissentiment was expressed by Lebesgue (1904), who also exploited it in the development of the integral now bearinghis name. Ultimately the Russian school of point-set topology, under the direction of Pavel Alexandrov and PavelUrysohn, formulated Heine–Borel compactness in a way that could be applied to the modern notion of a topologicalspace. Alexandrov & Urysohn (1929) showed that the earlier version of compactness due to Fréchet, now called(relative) sequential compactness, under appropriate conditions followed from the version of compactness that wasformulated in terms of the existence of finite subcovers. It was this notion of compactness that became the dominantone, because it was not only a stronger property, but it could be formulated in a more general setting with a minimumof additional technical machinery, as it relied only on the structure of the open sets in a space.

14 CHAPTER 6. COMPACT SPACE

6.2 Basic examples

An example of a compact space is the (closed) unit interval [0,1] of real numbers. If one chooses an infinite numberof distinct points in the unit interval, then there must be some accumulation point in that interval. For instance,the odd-numbered terms of the sequence 1, 1/2, 1/3, 3/4, 1/5, 5/6, 1/7, 7/8, … get arbitrarily close to 0, while theeven-numbered ones get arbitrarily close to 1. The given example sequence shows the importance of including theboundary points of the interval, since the limit points must be in the space itself — an open (or half-open) interval ofthe real numbers is not compact. It is also crucial that the interval be bounded, since in the interval [0,∞) one couldchoose the sequence of points 0, 1, 2, 3, …, of which no sub-sequence ultimately gets arbitrarily close to any givenreal number.In two dimensions, closed disks are compact since for any infinite number of points sampled from a disk, some subsetof those points must get arbitrarily close either to a point within the disc, or to a point on the boundary. However, anopen disk is not compact, because a sequence of points can tend to the boundary without getting arbitrarily close toany point in the interior. Likewise, spheres are compact, but a sphere missing a point is not since a sequence of pointscan tend to the missing point, thereby not getting arbitrarily close to any point within the space. Lines and planes arenot compact, since one can take a set of equally-spaced points in any given direction without approaching any point.

6.3 Definitions

Various definitions of compactness may apply, depending on the level of generality. A subset of Euclidean space inparticular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that anyinfinite sequence from the set has a subsequence that converges to a point in the set. Various equivalent notions ofcompactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces.In general topological spaces, however, the different notions of compactness are not equivalent, and the most usefulnotion of compactness—originally called bicompactness—is defined using covers consisting of open sets (see Opencover definition below). That this form of compactness holds for closed and bounded subsets of Euclidean space isknown as the Heine–Borel theorem. Compactness, when defined in this manner, often allows one to take informationthat is known locally—in a neighbourhood of each point of the space—and to extend it to information that holdsglobally throughout the space. An example of this phenomenon is Dirichlet’s theorem, to which it was originallyapplied by Heine, that a continuous function on a compact interval is uniformly continuous; here, continuity is a localproperty of the function, and uniform continuity the corresponding global property.

6.3.1 Open cover definition

Formally, a topological space X is called compact if each of its open covers has a finite subcover. Otherwise, it iscalled non-compact. Explicitly, this means that for every arbitrary collection

Uαα∈A

of open subsets of X such that

X =∪α∈A

Uα,

there is a finite subset J of A such that

X =∪i∈J

Ui.

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki,use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are bothHausdorff and quasi-compact. A compact set is sometimes referred to as a compactum, plural compacta.

6.3. DEFINITIONS 15

6.3.2 Equivalent definitions

Assuming the axiom of choice, the following are equivalent:

1. A topological space X is compact.

2. Every open cover of X has a finite subcover.

3. X has a sub-base such that every cover of the space bymembers of the sub-base has a finite subcover (Alexander’ssub-base theorem)

4. Any collection of closed subsets of X with the finite intersection property has nonempty intersection.

5. Every net on X has a convergent subnet (see the article on nets for a proof).

6. Every filter on X has a convergent refinement.

7. Every ultrafilter on X converges to at least one point.

8. Every infinite subset of X has a complete accumulation point.[4]

Euclidean space

For any subset A of Euclidean space Rn, A is compact if and only if it is closed and bounded; this is the Heine–Boreltheorem.As a Euclidean space is a metric space, the conditions in the next subsection also apply to all of its subsets. Of allof the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for aclosed interval or closed n-ball.

Metric spaces

For any metric space (X,d), the following are equivalent:

1. (X,d) is compact.

2. (X,d) is complete and totally bounded (this is also equivalent to compactness for uniform spaces).[5]

3. (X,d) is sequentially compact; that is, every sequence in X has a convergent subsequence whose limit is in X(this is also equivalent to compactness for first-countable uniform spaces).

4. (X,d) is limit point compact; that is, every infinite subset of X has at least one limit point in X.

5. (X,d) is an image of a continuous function from the Cantor set.[6]

A compact metric space (X,d) also satisfies the following properties:

1. Lebesgue’s number lemma: For every open cover of X, there exists a number δ > 0 such that every subset ofX of diameter < δ is contained in some member of the cover.

2. (X,d) is second-countable, separable and Lindelöf – these three conditions are equivalent for metric spaces.The converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact.

3. X is closed and bounded (as a subset of any metric space whose restricted metric is d). The converse may failfor a non-Euclidean space; e.g. the real line equipped with the discrete topology is closed and bounded but notcompact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.It is complete but not totally bounded.

16 CHAPTER 6. COMPACT SPACE

Characterization by continuous functions

Let X be a topological space and C(X) the ring of real continuous functions on X. For each p∈X, the evaluation map

evp : C(X) → R

given by evp(f)=f(p) is a ring homomorphism. The kernel of evp is a maximal ideal, since the residue field C(X)/kerevp is the field of real numbers, by the first isomorphism theorem. A topological spaceX is pseudocompact if and onlyif every maximal ideal in C(X) has residue field the real numbers. For completely regular spaces, this is equivalent toevery maximal ideal being the kernel of an evaluation homomorphism.[7] There are pseudocompact spaces that arenot compact, though.In general, for non-pseudocompact spaces there are always maximal ideals m in C(X) such that the residue fieldC(X)/m is a (non-archimedean) hyperreal field. The framework of non-standard analysis allows for the followingalternative characterization of compactness:[8] a topological space X is compact if and only if every point x of thenatural extension *X is infinitely close to a point x0 of X (more precisely, x is contained in the monad of x0).

Hyperreal definition

A space X is compact if its natural extension *X (for example, an ultrapower) has the property that every point of *Xis infinitely close to a suitable point ofX ⊂ ∗X . For example, an open real interval X=(0,1) is not compact becauseits hyperreal extension *(0,1) contains infinitesimals, which are infinitely close to 0, which is not a point of X.

6.3.3 Compactness of subspaces

A subset K of a topological space X is called compact if it is compact as a subspace. Explicitly, this means that forevery arbitrary collection

Uαα∈A

of open subsets of X such that

K ⊆∪α∈A

Uα,

there is a finite subset J of A such that

K ⊆∪i∈J

Ui.

6.4 Properties of compact spaces

6.4.1 Functions and compact spaces

A continuous image of a compact space is compact.[9] This implies the extreme value theorem: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.[10] (Slightly more generally,this is true for an upper semicontinuous function.) As a sort of converse to the above statements, the pre-image of acompact space under a proper map is compact.

6.4.2 Compact spaces and set operations

A closed subset of a compact space is compact.,[11] and a finite union of compact sets is compact.

6.5. EXAMPLES 17

The product of any collection of compact spaces is compact. (Tychonoff’s theorem, which is equivalent to the axiomof choice)Every topological space X is an open dense subspace of a compact space having at most one point more than X, bythe Alexandroff one-point compactification. By the same construction, every locally compact Hausdorff space X isan open dense subspace of a compact Hausdorff space having at most one point more than X.

6.4.3 Ordered compact spaces

A nonempty compact subset of the real numbers has a greatest element and a least element.Let X be a simply ordered set endowed with the order topology. Then X is compact if and only if X is a completelattice (i.e. all subsets have suprema and infima).[12]

6.5 Examples• Any finite topological space, including the empty set, is compact. More generally, any space with a finitetopology (only finitely many open sets) is compact; this includes in particular the trivial topology.

• Any space carrying the cofinite topology is compact.

• Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, bymeans of Alexandroff one-point compactification. The one-point compactification of R is homeomorphic tothe circle S1; the one-point compactification of R2 is homeomorphic to the sphere S2. Using the one-pointcompactification, one can also easily construct compact spaces which are not Hausdorff, by starting with anon-Hausdorff space.

• The right order topology or left order topology on any bounded totally ordered set is compact. In particular,Sierpinski space is compact.

• R, carrying the lower limit topology, satisfies the property that no uncountable set is compact.

• In the cocountable topology on an uncountable set, no infinite set is compact. Like the previous example, thespace as a whole is not locally compact but is still Lindelöf.

• The closed unit interval [0,1] is compact. This follows from the Heine–Borel theorem. The open interval (0,1)is not compact: the open cover

(1

n, 1− 1

n

)for n = 3, 4, … does not have a finite subcover. Similarly, the set of rational numbers in the closedinterval [0,1] is not compact: the sets of rational numbers in the intervals[0,

1

π− 1

n

]and

[1

π+

1

n, 1

]cover all the rationals in [0, 1] for n = 4, 5, … but this cover does not have a finite subcover. (Note thatthe sets are open in the subspace topology even though they are not open as subsets of R.)

• The set R of all real numbers is not compact as there is a cover of open intervals that does not have a finitesubcover. For example, intervals (n−1, n+1) , where n takes all integer values in Z, cover R but there is nofinite subcover.

• For every natural number n, the n-sphere is compact. Again from the Heine–Borel theorem, the closed unitball of any finite-dimensional normed vector space is compact. This is not true for infinite dimensions; in fact,a normed vector space is finite-dimensional if and only if its closed unit ball is compact.

• On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology.(Alaoglu’s theorem)

18 CHAPTER 6. COMPACT SPACE

• The Cantor set is compact. In fact, every compact metric space is a continuous image of the Cantor set.

• Consider the set K of all functions f : R→ [0,1] from the real number line to the closed unit interval, and definea topology on K so that a sequence fn in K converges towards f ∈ K if and only if fn(x) convergestowards f(x) for all real numbers x. There is only one such topology; it is called the topology of pointwiseconvergence or the product topology. Then K is a compact topological space; this follows from the Tychonofftheorem.

• Consider the set K of all functions f : [0,1] → [0,1] satisfying the Lipschitz condition |f(x) − f(y)| ≤ |x − y| forall x, y ∈ [0,1]. Consider on K the metric induced by the uniform distance

d(f, g) = supx∈[0,1]

|f(x)− g(x)|.

Then by Arzelà–Ascoli theorem the space K is compact.

• The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complexnumbers C. Conversely, any compact subset of C arises in this manner, as the spectrum of some boundedlinear operator. For instance, a diagonal operator on the Hilbert space ℓ2 may have any compact nonemptysubset of C as spectrum.

6.5.1 Algebraic examples

• Compact groups such as an orthogonal group are compact, while groups such as a general linear group are not.

• Since the p-adic integers are homeomorphic to the Cantor set, they form a compact set.

• The spectrum of any commutative ring with the Zariski topology (that is, the set of all prime ideals) is compact,but never Hausdorff (except in trivial cases). In algebraic geometry, such topological spaces are examples ofquasi-compact schemes, “quasi” referring to the non-Hausdorff nature of the topology.

• The spectrum of a Boolean algebra is compact, a fact which is part of the Stone representation theorem. Stonespaces, compact totally disconnected Hausdorff spaces, form the abstract framework in which these spectraare studied. Such spaces are also useful in the study of profinite groups.

• The structure space of a commutative unital Banach algebra is a compact Hausdorff space.

• The Hilbert cube is compact, again a consequence of Tychonoff’s theorem.

• A profinite group (e.g., Galois group) is compact.

6.6 See also

• Compactly generated space

• Eberlein compactum

• Exhaustion by compact sets

• Lindelöf space

• Metacompact space

• Noetherian space

• Orthocompact space

• Paracompact space

6.7. NOTES 19

6.7 Notes[1] Kline 1972, pp. 952–953; Boyer & Merzbach 1991, p. 561

[2] Kline 1972, Chapter 46, §2

[3] Frechet, M. 1904. Generalisation d'un theorem de Weierstrass. Analyse Mathematique.

[4] (Kelley 1955, p. 163)

[5] Arkhangel’skii & Fedorchuk 1990, Theorem 5.3.7

[6] Willard 1970 Theorem 30.7.

[7] Gillman & Jerison 1976, §5.6

[8] Robinson, Theorem 4.1.13

[9] Arkhangel’skii &Fedorchuk 1990, Theorem 5.2.2; See also Compactness is preserved under a continuousmap at PlanetMath.org.

[10] Arkhangel’skii & Fedorchuk 1990, Corollary 5.2.1

[11] Arkhangel’skii & Fedorchuk 1990, Theorem 5.2.3; Closed set in a compact space is compact at PlanetMath.org. ; Closedsubsets of a compact set are compact at PlanetMath.org.

[12] (Steen & Seebach 1995, p. 67)

6.8 References• Alexandrov, Pavel; Urysohn, Pavel (1929), “Mémoire sur les espaces topologiques compacts”, KoninklijkeNederlandse Akademie van Wetenschappen te Amsterdam, Proceedings of the section of mathematical sciences14.

• Arkhangel’skii, A.V.; Fedorchuk, V.V. (1990), “The basic concepts and constructions of general topology”,in Arkhangel’skii, A.V.; Pontrjagin, L.S., General topology I, Encyclopedia of the Mathematical Sciences 17,Springer, ISBN 978-0-387-18178-3.

• Arkhangel’skii, A.V. (2001), “Compact space”, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4.

• Bolzano, Bernard (1817), Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein ent-gegengesetzes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege, Wilhelm Engelmann (Purelyanalytic proof of the theorem that between any two values which give results of opposite sign, there lies at leastone real root of the equation).

• Borel, Émile (1895), “Sur quelques points de la théorie des fonctions”, Annales Scientifiques de l'École NormaleSupérieure, 3 12: 9–55, JFM 26.0429.03

• Boyer, Carl B. (1959), The history of the calculus and its conceptual development, New York: Dover Publica-tions, MR 0124178.

• Arzelà, Cesare (1895), “Sulle funzioni di linee”, Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat. 5 (5):55–74.

• Arzelà, Cesare (1882–1883), “Un'osservazione intorno alle serie di funzioni”, Rend. Dell' Accad. R. Delle Sci.Dell'Istituto di Bologna: 142–159.

• Ascoli, G. (1883–1884), “Le curve limiti di una varietà data di curve”, Atti della R. Accad. Dei Lincei Memoriedella Cl. Sci. Fis. Mat. Nat. 18 (3): 521–586.

• Fréchet, Maurice (1906), “Sur quelques points du calcul fonctionnel”, Rendiconti del Circolo Matematico diPalermo 22 (1): 1–72, doi:10.1007/BF03018603.

• Gillman, Leonard; Jerison, Meyer (1976), Rings of continuous functions, Springer-Verlag.

• Kelley, John (1955), General topology, Graduate Texts in Mathematics 27, Springer-Verlag.

20 CHAPTER 6. COMPACT SPACE

• Kline, Morris (1972), Mathematical thought from ancient to modern times (3rd ed.), Oxford University Press(published 1990), ISBN 978-0-19-506136-9.

• Lebesgue, Henri (1904), Leçons sur l'intégration et la recherche des fonctions primitives, Gauthier-Villars.

• Robinson, Abraham (1996), Non-standard analysis, Princeton University Press, ISBN 978-0-691-04490-3,MR 0205854.

• Scarborough, C.T.; Stone, A.H. (1966), “Products of nearly compact spaces”, Transactions of the AmericanMathematical Society (Transactions of the American Mathematical Society, Vol. 124, No. 1) 124 (1): 131–147, doi:10.2307/1994440, JSTOR 1994440.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

• Willard, Stephen (1970), General Topology, Dover publications, ISBN 0-486-43479-6

6.9 External links• Countably compact at PlanetMath.org.

• Sundström, Manya Raman (2010). “A pedagogical history of compactness”. arXiv:1006.4131v1 [math.HO].

This article incorporates material from Examples of compact spaces on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

Chapter 7

Connected space

For other uses, see Connection (disambiguation).Connected and disconnected subspaces of R²

A

B

C

D

E4

E1

E2

E3

From top to bottom: red space A, pink space B, yellow space C and orange space D are all connected, whereas greenspace E (made of subsets E1, E2, E3, and E4) is not connected. Furthermore, A and B are also simply connected(genus 0), while C and D are not: C has genus 1 and D has genus 4.

In topology and related branches of mathematics, a connected space is a topological space that cannot be representedas the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topologicalproperties that is used to distinguish topological spaces. A stronger notion is that of a path-connected space, whichis a space where any two points can be joined by a path.

21

22 CHAPTER 7. CONNECTED SPACE

A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X.An example of a space that is not connected is a plane with an infinite line deleted from it. Other examples ofdisconnected spaces (that is, spaces which are not connected) include the plane with an annulus removed, as well asthe union of two disjoint closed disks, where all examples of this paragraph bear the subspace topology induced bytwo-dimensional Euclidean space.

7.1 Formal definition

A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, Xis said to be connected. A subset of a topological space is said to be connected if it is connected under its subspacetopology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article doesnot follow that practice.For a topological space X the following conditions are equivalent:

1. X is connected.

2. X cannot be divided into two disjoint nonempty closed sets.

3. The only subsets of X which are both open and closed (clopen sets) are X and the empty set.

4. The only subsets of X with empty boundary are X and the empty set.

5. X cannot be written as the union of two nonempty separated sets (sets for which each is disjoint from the other’sclosure).

6. All continuous functions from X to 0,1 are constant, where 0,1 is the two-point space endowed with thediscrete topology.

7.1.1 Connected components

The maximal connected subsets (ordered by inclusion) of a nonempty topological space are called the connectedcomponents of the space. The components of any topological space X form a partition of X: they are disjoint,nonempty, and their union is the whole space. Every component is a closed subset of the original space. It followsthat, in the case where their number is finite, each component is also an open subset. However, if their number isinfinite, this might not be the case; for instance, the connected components of the set of the rational numbers are theone-point sets, which are not open.Let Γx be the connected component of x in a topological space X, and Γ′

x be the intersection of all clopen setscontaining x (called quasi-component of x.) Then Γx ⊂ Γ′

x where the equality holds if X is compact Hausdorff orlocally connected.

7.1.2 Disconnected spaces

A space in which all components are one-point sets is called totally disconnected. Related to this property, a space Xis called totally separated if, for any two distinct elements x and y of X, there exist disjoint open neighborhoods Uof x and V of y such that X is the union of U and V. Clearly any totally separated space is totally disconnected, butthe converse does not hold. For example take two copies of the rational numbers Q, and identify them at every pointexcept zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering thetwo copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the conditionof being totally separated is strictly stronger than the condition of being Hausdorff.

7.2 Examples• The closed interval [0, 2] in the standard subspace topology is connected; although it can, for example, bewritten as the union of [0, 1) and [1, 2], the second set is not open in the chosen topology of [0, 2].

7.3. PATH CONNECTEDNESS 23

• The union of [0, 1) and (1, 2] is disconnected; both of these intervals are open in the standard topological space[0, 1) ∪ (1, 2].

• (0, 1) ∪ 3 is disconnected.

• A convex set is connected; it is actually simply connected.

• AEuclidean plane excluding the origin, (0, 0), is connected, but is not simply connected. The three-dimensionalEuclidean space without the origin is connected, and even simply connected. In contrast, the one-dimensionalEuclidean space without the origin is not connected.

• A Euclidean plane with a straight line removed is not connected since it consists of two half-planes.

• ℝ, The space of real numbers with the usual topology, is connected.

• If even a single point is removed from ℝ, the remainder is disconnected. However, if even a countable infinityof points are removed from ℝn, where n≥2, the remainder is connected.

• Any topological vector space over a connected field is connected.

• Every discrete topological space with at least two elements is disconnected, in fact such a space is totallydisconnected. The simplest example is the discrete two-point space.[1]

• On the other hand, a finite set might be connected. For example, the spectrum of a discrete valuation ringconsists of two points and is connected. It is an example of a Sierpiński space.

• The Cantor set is totally disconnected; since the set contains uncountably many points, it has uncountably manycomponents.

• If a space X is homotopy equivalent to a connected space, then X is itself connected.

• The topologist’s sine curve is an example of a set that is connected but is neither path connected nor locallyconnected.

• The general linear group GL(n,R) (that is, the group of n-by-n real, invertible matrices) consists of two con-nected components: the one with matrices of positive determinant and the other of negative determinant.In particular, it is not connected. In contrast, GL(n,C) is connected. More generally, the set of invertiblebounded operators on a (complex) Hilbert space is connected.

• The spectra of commutative local ring and integral domains are connected. More generally, the following areequivalent[2]

1. The spectrum of a commutative ring R is connected2. Every finitely generated projective module over R has constant rank.3. R has no idempotent = 0, 1 (i.e., R is not a product of two rings in a nontrivial way).

7.3 Path connectedness

A path from a point x to a point y in a topological space X is a continuous function f from the unit interval [0,1] toX with f(0) = x and f(1) = y. A path-component of X is an equivalence class of X under the equivalence relationwhich makes x equivalent to y if there is a path from x to y. The space X is said to be path-connected (or pathwiseconnected or 0-connected) if there is exactly one path-component, i.e. if there is a path joining any two points inX. Again, many authors exclude the empty space.Every path-connected space is connected. The converse is not always true: examples of connected spaces that arenot path-connected include the extended long line L* and the topologist’s sine curve.However, subsets of the real lineR are connected if and only if they are path-connected; these subsets are the intervalsofR. Also, open subsets ofRn orCn are connected if and only if they are path-connected. Additionally, connectednessand path-connectedness are the same for finite topological spaces.

24 CHAPTER 7. CONNECTED SPACE

This subspace of R² is path-connected, because a path can be drawn between any two points in the space.

7.4 Arc connectedness

A space X is said to be arc-connected or arcwise connected if any two distinct points can be joined by an arc, thatis a path f which is a homeomorphism between the unit interval [0, 1] and its image f([0, 1]). It can be shown anyHausdorff space which is path-connected is also arc-connected. An example of a space which is path-connected butnot arc-connected is provided by adding a second copy 0' of 0 to the nonnegative real numbers [0, ∞). One endowsthis set with a partial order by specifying that 0'<a for any positive number a, but leaving 0 and 0' incomparable.One then endows this set with the order topology, that is one takes the open intervals (a, b) = x | a < x < b and thehalf-open intervals [0, a) = x | 0 ≤ x < a, [0', a) = x | 0' ≤ x < a as a base for the topology. The resulting spaceis a T1 space but not a Hausdorff space. Clearly 0 and 0' can be connected by a path but not by an arc in this space.

7.5 Local connectedness

Main article: Locally connected space

A topological space is said to be locally connected at a point x if every neighbourhood of x contains a connectedopen neighbourhood. It is locally connected if it has a base of connected sets. It can be shown that a space X islocally connected if and only if every component of every open set of X is open. The topologist’s sine curve is anexample of a connected space that is not locally connected.Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets. An opensubset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlierstatement about Rn and Cn, each of which is locally path-connected. More generally, any topological manifold islocally path-connected.

7.6 Set operations

The intersection of connected sets is not necessarily connected.

7.6. SET OPERATIONS 25

AB

A B

A

B

A

Bconnexenon connexe

intersection intersection

connexe non connexe

union union

Examples of unions and intersections of connected sets

The union of connected sets is not necessarily connected. Consider a collection Xi of connected sets whose unionisX = ∪iXi . IfX is disconnected and U ∪ V is a separation ofX (with U, V disjoint and open inX ), then eachXi must be entirely contained in either U or V , since otherwise, Xi ∩ U and Xi ∩ V (which are disjoint and openin Xi ) would be a separation of Xi , contradicting the assumption that it is connected.This means that, if the union X is disconnected, then the collection Xi can be partitioned to two sub-collections,such that the unions of the sub-collections are disjoint and open inX (see picture). This implies that in several cases,a union of connected sets is necessarily connected. In particular:

1. If the common intersection of all sets is not empty ( ∩Xi = ∅ ), then obviously they cannot be partitioned tocollections with disjoint unions. Hence the union of connected sets with non-empty intersection is connected.

2. If the intersection of each pair of sets is not empty ( ∀i, j : Xi∩Xj = ∅ ) then again they cannot be partitionedto collections with disjoint unions, so their union must be connected.

3. If the sets can be ordered as a “linked chain”, i.e. indexed by integer indices and ∀i : Xi ∩Xi+1 = ∅ , thenagain their union must be connected.

4. If the sets are pairwise-disjoint and the quotient space X/Xi is connected, then X must be connected.Otherwise, if U ∪ V is a separation of X then q(U) ∪ q(V ) is a separation of the quotient space (sinceq(U), q(V ) are disjoint and open in the quotient space).[3]

26 CHAPTER 7. CONNECTED SPACE

Each ellipse is a connected set, but the union is not connected, since it can be partitioned to two disjoint open sets U and V.

Two connected sets whose difference is not connected

The set difference of connected sets is not necessarily connected. However, if X⊇Y and their difference X\Y isdisconnected (and thus can be written as a union of two open sets X1 and X2), then the union of Y with each suchcomponent is connected (i.e. Y∪Xi is connected for all i). Proof:[4] By contradiction, suppose Y∪X1 is not connected.So it can be written as the union of two disjoint open sets, e.g. Y∪X1 = Z1∪Z2. Because Y is connected, it must be

7.7. THEOREMS 27

entirely contained in one of these components, say Z1, and thus Z2 is contained in X1. Now we know that:

X = (Y∪X1)∪X2 = (Z1∪Z2)∪X2 = (Z1∪X2)∪(Z2∩X1)

The two sets in the last union are disjoint and open in X, so there is a separation of X, contradicting the fact that X isconnected.

7.7 Theorems

“Main theorem of connectedness” redirects to here.

• Main theorem: Let X and Y be topological spaces and let f : X → Y be a continuous function. If X is (path-)connected then the image f(X) is (path-)connected. This result can be considered a generalization of theintermediate value theorem.

• Every path-connected space is connected.

• Every locally path-connected space is locally connected.

• A locally path-connected space is path-connected if and only if it is connected.

• The closure of a connected subset is connected.

• The connected components are always closed (but in general not open)

• The connected components of a locally connected space are also open.

• The connected components of a space are disjoint unions of the path-connected components (which in generalare neither open nor closed).

• Every quotient of a connected (resp. locally connected, path-connected, locally path-connected) space is con-nected (resp. locally connected, path-connected, locally path-connected).

• Every product of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).

• Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locallypath-connected).

• Every manifold is locally path-connected.

7.8 Graphs

Graphs have path connected subsets, namely those subsets for which every pair of points has a path of edges joiningthem. But it is not always possible to find a topology on the set of points which induces the same connected sets. The5-cycle graph (and any n-cycle with n>3 odd) is one such example.As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit,there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivityaxioms; their morphisms are those functions which map connected sets to connected sets (Muscat & Buhagiar 2006).Topological spaces and graphs are special cases of connective spaces; indeed, the finite connective spaces are preciselythe finite graphs.However, every graph can be canonically made into a topological space, by treating vertices as points and edges ascopies of the unit interval (see topological graph theory#Graphs as topological spaces). Then one can show that thegraph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.

28 CHAPTER 7. CONNECTED SPACE

7.9 Stronger forms of connectedness

There are stronger forms of connectedness for topological spaces, for instance:

• If there exist no two disjoint non-empty open sets in a topological space, X, X must be connected, and thushyperconnected spaces are also connected.

• Since a simply connected space is, by definition, also required to be path connected, any simply connected spaceis also connected. Note however, that if the “path connectedness” requirement is dropped from the definitionof simple connectivity, a simply connected space does not need to be connected.

• Yet stronger versions of connectivity include the notion of a contractible space. Every contractible space ispath connected and thus also connected.

In general, note that any path connected space must be connected but there exist connected spaces that are not pathconnected. The deleted comb space furnishes such an example, as does the above-mentioned topologist’s sine curve.

7.10 See also• uniformly connected space

• locally connected space

• connected component (graph theory)

• n-connected

• Connectedness locus

• Extremally disconnected space

7.11 References

7.11.1 Notes[1] George F. Simmons (1968). Introduction to Topology and Modern Analysis. McGraw Hill Book Company. p. 144. ISBN

0-89874-551-9.

[2] Charles Weibel, The K-book: An introduction to algebraic K-theory

[3] Credit: Saaqib Mahmuud and Henno Brandsma at Math StackExchange.

[4] Credit: Marek at Math StackExchange

7.11.2 General references

• Munkres, James R. (2000). Topology, Second Edition. Prentice Hall. ISBN 0-13-181629-2.

• Weisstein, Eric W., “Connected Set”, MathWorld.

• V. I. Malykhin (2001), “Connected space”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Muscat, J; Buhagiar, D (2006). “Connective Spaces” (PDF). Mem. Fac. Sci. Eng. Shimane Univ., Series B:Math. Sc. 39: 1–13..

Chapter 8

Contractible space

In mathematics, a topological spaceX is contractible if the identity map onX is null-homotopic, i.e. if it is homotopicto some constant map.[1][2] Intuitively, a contractible space is one that can be continuously shrunk to a point.

8.1 Properties

A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of acontractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly,since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial.For a topological space X the following are all equivalent (here Y is an arbitrary topological space):

• X is contractible (i.e. the identity map is null-homotopic).

• X is homotopy equivalent to a one-point space.

• X deformation retracts onto a point. (However, there exist contractible spaces which do not strongly deformationretract to a point.)

• Any two maps f,g: Y → X are homotopic.

• Any map f: Y → X is null-homotopic.

The cone on a space X is always contractible. Therefore any space can be embedded in a contractible one (which alsoillustrates that subspaces of contractible spaces need not be contractible).Furthermore, X is contractible if and only if there exists a retraction from the cone of X to X.Every contractible space is path connected and simply connected. Moreover, since all the higher homotopy groupsvanish, every contractible space is n-connected for all n ≥ 0.

8.2 Locally contractible spaces

A topological space is locally contractible if every point has a local base of contractible neighborhoods. Contractiblespaces are not necessarily locally contractible nor vice versa. For example, the comb space is contractible but notlocally contractible (if it were, it would be locally connected which it is not). Locally contractible spaces are locally n-connected for all n≥ 0. In particular, they are locally simply connected, locally path connected, and locally connected.

8.3 Examples and counterexamples

• Any Euclidean space is contractible, as is any star domain on a Euclidean space.

29

30 CHAPTER 8. CONTRACTIBLE SPACE

• The Whitehead manifold is contractible.

• Spheres of any finite dimension are not contractible.

• The unit sphere in an infinite-dimensional Hilbert space is contractible.

• The house with two rooms is standard example of a space which is contractible, but not intuitively so.

• Dunce hat

• The cone on a Hawaiian earring is contractible (since it is a cone), but not locally contractible or even locallysimply connected.

• All manifolds and CW complexes are locally contractible, but in general not contractible.

8.4 References[1] Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

[2] Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0.

Chapter 9

Countably compact space

In mathematics a topological space is countably compact if every countable open cover has a finite subcover.

9.1 Examples• The first uncountable ordinal (with the order topology) is an example of a countably compact space that is notcompact.

9.2 Properties• A compact space is countably compact.

• A countably compact space is always limit point compact.

• For metrizable spaces, countable compactness, sequential compactness, limit point compactness and compact-ness are all equivalent.

• The example of the set of all real numbers with the standard topology shows that neither local compactnessnor σ-compactness nor paracompactness imply countable compactness.

• For T1 spaces, countable compactness and limit point compactness are equivalent.

9.3 See also• Sequentially compact space

• Compact space

• Limit point compact

9.4 References• James Munkres (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

31

Chapter 10

Door space

In mathematics, in the field of topology, a topological space is said to be a door space if every subset is either openor closed (or both).[1] The term comes from the introductory topology mnemonic that “a subset is not like a door: itcan be open, closed, both, or neither”.Here are some easy facts about door spaces:

• A Hausdorff door space has at most one accumulation point.

• In a Hausdorff door space if x is not an accumulation point then x is open.

To prove the first assertion, let X be a Hausdorff door space, and let x ≠ y be distinct points. Since X is Hausdorffthere are open neighborhoods U and V of x and y respectively such that U ∩ V = ∅. Suppose y is an accumulationpoint. Then U \ x ∪ y is closed, since if it were open, then we could say that y = (U \ x ∪ y) ∩ V is open,contradicting that y is an accumulation point. So we conclude that as U \ x ∪ y is closed, X \ (U \ x ∪ y) isopen and hence x = U ∩ [X \ (U \ x ∪ y)] is open, implying that x is not an accumulation point.

10.1 Notes[1] Kelley, ch.2, Exercise C, p. 76.

10.2 References• Kelley, John L. (1991). General Topology. Springer. ISBN 3540901256.

32

Chapter 11

Dowker space

In the mathematical field of general topology, a Dowker space is a topological space that is T4 but not countablyparacompact. They are named after Clifford Hugh Dowker.The non-trivial task of providing an example of a Dowker space (and therefore also proving their existence as math-ematical objects) helped mathematicians better understand the nature and variety of topological spaces. Topologicalspaces are sets together with some subsets (designated as “open sets”) satisfying certain properties. Topologicalspaces arose as generalization of the open sets of spaces studied in elementary mathematics, such as open disks inthe Euclidean plane, open balls in the Euclidean space, and open intervals of the real line.

11.1 Equivalences

Dowker showed, in 1951, the following:If X is a normal T1 space (that is, a T4 space), then the following are equivalent:

• X is a Dowker space

• The product of X with the unit interval is not normal.[1]

• X is not countably metacompact.

Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until M. E. Rudin con-structed one[2] in 1971. Rudin’s counterexample is a very large space (of cardinality ℵℵ0

ω ). Zoltán Balogh gave thefirst ZFC construction[3] of a small (cardinality continuum) example, which was more well-behaved than Rudin’s.Using PCF theory, M. Kojman and S. Shelah constructed[4] a subspace of Rudin’s Dowker space of cardinality ℵω+1

that is also Dowker.

11.2 References[1] Dowker, C. H. (1951). “On countably paracompact spaces” (PDF). Can. J. Math. 3: 219–224. doi:10.4153/CJM-1951-

026-2. Zbl 0042.41007. Retrieved March 29, 2015.

[2] Rudin, Mary Ellen (1971). “A normal space X for which X × I is not normal” (PDF). Fundam. Math. (Polish Academyof Sciences) 73 (2): 179–186. Zbl 0224.54019. Retrieved March 29, 2015.

[3] Balogh, Zoltan T. (August 1996). “A small Dowker space in ZFC” (PDF). Proc. Amer. Math. Soc. 124 (8): 2555–2560.Zbl 0876.54016. Retrieved March 29, 2015.

[4] Kojman, Menachem; Shelah, Saharon (1998). “A ZFC Dowker space in ℵω+1 : an application of PCF theory to topology”(PDF). Proc. Amer. Math. Soc. (American Mathematical Society) 126 (8): 2459–2465. Retrieved March 29, 2015.

33

Chapter 12

Dyadic space

See also Dyadic space (disambiguation)

In mathematics, a dyadic compactum is a Hausdorff topological space that is the image of a product of discrete two-point sets,[1] and a dyadic space is a topological space isomorphic to a subspace of a dyadic compactum.[2] Diadiccompacta and spaces satisfy the Suslin condition, and were introduced by Russian mathematician Pavel Alexandrov.[1]Polyadic spaces are generalisation of dyadic spaces.[3]

12.1 References[1] Efimov, B.A. (2001), “Dyadic compactum”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-

1-55608-010-4

[2] Efimov, B.A. (2001), “Dyadic space”, in Hazewinkel, Michiel, Encyclopedia ofMathematics, Springer, ISBN 978-1-55608-010-4

[3] Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2003). Encyclopedia of General Topology. Elsevier Science. p.193. ISBN 978-0444503558.

34

Chapter 13

End (topology)

In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected com-ponents of the “ideal boundary” of the space. That is, each end represents a topologically distinct way to move toinfinity within the space. Adding a point at each end yields a compactification of the original space, known as theend compactification.

13.1 Definition

Let X be a topological space, and suppose that

K1 ⊂ K2 ⊂ K3 ⊂ · · ·

is an ascending sequence of compact subsets of X whose interiors cover X. Then X has one end for every sequence

U1 ⊃ U2 ⊃ U3 ⊃ · · ·,

where each Un is a connected component of X \ Kn. The number of ends does not depend on the specific sequenceKi of compact sets; there is a natural bijection between the sets of ends associated with any two such sequences.Using this definition, a neighborhood of an end Ui is an open set V such that V ⊃ Un for some n. Such neigh-borhoods represent the neighborhoods of the corresponding point at infinity in the end compactification (this “com-pactification” isn’t always compact; the topological space X has to be connected and locally connected).The definition of ends given above applies only to spaces X that possess an exhaustion by compact sets (that is, Xmust be hemicompact). However, it can be generalized as follows: let X be any topological space, and consider thedirect system Kα of compact subsets of X and inclusion maps. There is a corresponding inverse system π0( X \Kα ) , where π0(Y) denotes the set of connected components of a space Y, and each inclusion map Y → Z inducesa function π0(Y) → π0(Z). Then set of ends of X is defined to be the inverse limit of this inverse system. Under thisdefinition, the set of ends is a functor from the category of topological spaces to the category of sets. The originaldefinition above represents the special case where the direct system of compact subsets has a cofinal sequence.

13.2 Examples

• The set of ends of any compact space is the empty set.

• The real line R has two ends. For example, if we let Kn be the closed interval [−n, n], then the two ends arethe sequences of open sets Un = (n, ∞) and Vn = (−∞, −n). These ends are usually referred to as “infinity”and “minus infinity”, respectively.

• If n > 1, then the Euclidean space Rn has only one end. This is because Rn\K has only one unbounded compo-nent for any compact set K.

35

36 CHAPTER 13. END (TOPOLOGY)

• More generally, if M is a compact manifold with boundary, then the number of ends of the interior of M isequal to the number of connected components of the boundary of M.

• The union of n distinct rays emanating from the origin in R2 has n ends.

• The infinite complete binary tree has uncountably many ends, corresponding to the uncountably many differentdescending paths starting at the root. (This can be seen by letting Kn be the complete binary tree of depth n.)These ends can be thought of as the “leaves” of the infinite tree. In the end compactification, the set of endshas the topology of a Cantor set.

13.3 History

The notion of an end of a topological space was introduced by Hans Freudenthal (1931).

13.4 Ends of graphs and groups

Main article: End (graph theory)

In infinite graph theory, an end is defined slightly differently, as an equivalence class of semi-infinite paths in the graph,or as a haven, a function mapping finite sets of vertices to connected components of their complements. However,for locally finite graphs (graphs in which each vertex has finite degree), the ends defined in this way correspondone-for-one with the ends of topological spaces defined from the graph (Diestel & Kühn 2003).The ends of a finitely generated group are defined to be the ends of the corresponding Cayley graph; this definition isinsensitive to the choice of generating set. Every finitely-generated infinite group has either 1, 2, or infinitely manyends, and Stallings theorem about ends of groups provides a decomposition for groups with more than one end.

13.5 Ends of a CW complex

For a path connected CW-complex, the ends can be characterized as homotopy classes of proper maps R+→X ,called rays in X: more precisely, if between the restriction -to the subset N - of any two of these maps exists a properhomotopy we say that they are equivalent and they define an equivalence class of proper rays. This set is called anend of X.

13.6 References• Diestel, Reinhard; Kühn, Daniela (2003), “Graph-theoretical versus topological ends of graphs”, Journal ofCombinatorial Theory, Series B 87 (1): 197–206, doi:10.1016/S0095-8956(02)00034-5, MR 1967888.

• Freudenthal, Hans (1931), "Über die Enden topologischer Räume und Gruppen”, Mathematische Zeitschrift(Springer Berlin / Heidelberg) 33: 692–713, doi:10.1007/BF01174375, ISSN 0025-5874, Zbl 0002.05603

• Ross Geoghegan, Topological methods in group theory, GTM-243 (2008), Springer ISBN 978-0-387-74611-1.

• Peter Scott, Terry Wall, Topological methods in group theory, London Math. Soc. Lecture Note Ser., 36,Cambridge Univ. Press (1979) 137-203.

Chapter 14

Extremally disconnected space

In mathematics, a topological space is termed extremally disconnected or extremely disconnected if the closureof every open set in it is open. (The term “extremally disconnected” is usual, even though the word “extremally” doesnot appear in most dictionaries.[1])An extremally disconnected space that is also compact and Hausdorff is sometimes called a Stonean space. (Notethat this is different from a Stone space, which is usually a totally disconnected compact Hausdorff space.) A theoremdue to Andrew Gleason says that the projective objects of the category of compact Hausdorff spaces are exactly theextremally disconnected compact Hausdorff spaces. In the duality between Stone spaces and Boolean algebras, theStonean spaces correspond to the complete Boolean algebras.An extremally disconnected first countable collectionwise Hausdorff space must be discrete. In particular, for metricspaces, the property of being extremally disconnected (the closure of every open set is open) is equivalent to theproperty of being discrete (every set is open).

14.1 Examples• Every discrete space is extremally disconnected.

• The Stone–Čech compactification of a discrete space is extremally disconnected.

• The spectrum of an abelian von Neumann algebra is extremally disconnected.

• Any set with the cofinite topology is extremally disconnected, but if the set is infinite this space is connected.

14.2 References[1] “extremally” in the O.E.D.

• A. V. Arkhangelskii (2001), “Extremally-disconnected space”, in Hazewinkel, Michiel, Encyclopedia of Math-ematics, Springer, ISBN 978-1-55608-010-4

• Johnstone, Peter T (1982). Stone spaces. CUP. ISBN 0-521-23893-5.

37

Chapter 15

Feebly compact space

In mathematics, a topological space is feebly compact if every locally finite cover by nonempty open sets is finite.Some facts:

• Every compact space is feebly compact.

• Every feebly compact paracompact space is compact.

• Every feebly compact space is pseudocompact but the converse is not necessarily true.

• For a completely regular Hausdorff space the properties of being feebly compact and pseudocompact are equiv-alent.

• Any maximal feebly compact space is submaximal.

38

Chapter 16

First-countable space

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the “first axiom ofcountability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis(local base). That is, for each point x in X there exists a sequence N1, N2, … of neighbourhoods of x such that forany neighbourhood N of x there exists an integer i with Ni contained in N. Since every neighborhood of any pointcontains an open neighborhood of that point the neighbourhood basis can be chosen without loss of generality toconsist of open neighborhoods.

16.1 Examples and counterexamples

The majority of 'everyday' spaces in mathematics are first-countable. In particular, every metric space is first-countable. To see this, note that the set of open balls centered at x with radius 1/n for integers n > 0 form a countablelocal base at x.An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the realline).Another counterexample is the ordinal space ω1+1 = [0,ω1] where ω1 is the first uncountable ordinal number. Theelement ω1 is a limit point of the subset [0,ω1) even though no sequence of elements in [0,ω1) has the element ω1

as its limit. In particular, the point ω1 in the space ω1+1 = [0,ω1] does not have a countable local base. Since ω1 isthe only such point, however, the subspace ω1 = [0,ω1) is first-countable.The quotient spaceR/Nwhere the natural numbers on the real line are identified as a single point is not first countable.However, this space has the property that for any subset A and every element x in the closure of A, there is a sequencein A converging to x. A space with this sequence property is sometimes called a Fréchet-Urysohn space.First-countability is strictly weaker than second-countability. Every second-countable space is first-countable, butany uncountable discrete space is first-countable but not second-countable.

16.2 Properties

One of the most important properties of first-countable spaces is that given a subset A, a point x lies in the closureof A if and only if there exists a sequence xn in A which converges to x. This has consequences for limits andcontinuity. In particular, if f is a function on a first-countable space, then f has a limit L at the point x if and onlyif for every sequence xn→ x, where xn ≠ x for all n, we have f(xn) → L. Also, if f is a function on a first-countablespace, then f is continuous if and only if whenever xn→ x, then f(xn) → f(x).In first-countable spaces, sequential compactness and countable compactness are equivalent properties. However,there exist examples of sequentially compact, first-countable spaces which aren't compact (these are necessarily non-metric spaces). One such space is the ordinal space [0,ω1). Every first-countable space is compactly generated.Every subspace of a first-countable space is first-countable. Any countable product of a first-countable space is first-countable, although uncountable products need not be.

39

40 CHAPTER 16. FIRST-COUNTABLE SPACE

16.3 See also• Second-countable space

• Separable space

16.4 References• Hazewinkel, Michiel, ed. (2001), “first axiom of countability”, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

Chapter 17

Glossary of topology

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolutedistinction between different areas of topology, the focus here is on general topology. The following definitions arealso fundamental to algebraic topology, differential topology and geometric topology.See the article on topological spaces for basic definitions and examples, and see the article on topology for a briefhistory and description of the subject area. See Naive set theory, Axiomatic set theory, and Function for definitionsconcerning sets and functions. The following articles may also be useful. These either contain specialised vocabularywithin general topology or provide more detailed expositions of the definitions given below. The list of generaltopology topics and the list of examples in general topology will also be very helpful.

• Compact space

• Connected space

• Continuity

• Metric space

• Separated sets

• Separation axiom

• Topological space

• Uniform space

See also: Glossary of Riemannian and metric geometry

All spaces in this glossary are assumed to be topological spaces unless stated otherwise.Contents :

• Top

• 0–9

• A

• B

• C

• D

• E

• F

41

42 CHAPTER 17. GLOSSARY OF TOPOLOGY

• G

• H

• I

• J

• K

• L

• M

• N

• O

• P

• Q

• R

• S

• T

• U

• V

• W

• X

• Y

• Z

17.1 A

Absolutely closed See H-closed

Accessible See T1 .

Accumulation point See limit point.

Alexandrov topology The topology of a space X is an Alexandrov topology (or is finitely generated) if arbitraryintersections of open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, againequivalently, if the open sets are the upper sets of a poset.[1]

Almost discrete A space is almost discrete if every open set is closed (hence clopen). The almost discrete spacesare precisely the finitely generated zero-dimensional spaces.

Approach space An approach space is a generalization of metric space based on point-to-set distances, instead ofpoint-to-point.

17.2. B 43

17.2 BBaire space This has two distinct common meanings:

1. A space is a Baire space if the intersection of any countable collection of dense open sets is dense; seeBaire space.

2. Baire space is the set of all functions from the natural numbers to the natural numbers, with the topologyof pointwise convergence; see Baire space (set theory).

Base A collection B of open sets is a base (or basis) for a topology τ if every open set in τ is a union of sets in B .The topology τ is the smallest topology on X containing B and is said to be generated by B .

Basis See Base.

Borel algebra The Borel algebra on a topological space (X, τ) is the smallest σ -algebra containing all the open sets.It is obtained by taking intersection of all σ -algebras on X containing τ .

Borel set A Borel set is an element of a Borel algebra.

Boundary The boundary (or frontier) of a set is the set’s closure minus its interior. Equivalently, the boundary of aset is the intersection of its closure with the closure of its complement. Boundary of a set A is denoted by ∂Aor bd A .

Bounded A set in a metric space is bounded if it has finite diameter. Equivalently, a set is bounded if it is containedin some open ball of finite radius. A function taking values in a metric space is bounded if its image is abounded set.

17.3 CCategory of topological spaces The categoryTop has topological spaces as objects and continuousmaps asmorphisms.

Cauchy sequence A sequence xn in a metric space (M, d) is a Cauchy sequence if, for every positive real numberr, there is an integer N such that for all integers m, n > N, we have d(xm, xn) < r.

Clopen set A set is clopen if it is both open and closed.

Closed ball If (M, d) is a metric space, a closed ball is a set of the form D(x; r) := y inM : d(x, y) ≤ r, where x isinM and r is a positive real number, the radius of the ball. A closed ball of radius r is a closed r-ball. Everyclosed ball is a closed set in the topology induced on M by d. Note that the closed ball D(x; r) might not beequal to the closure of the open ball B(x; r).

Closed set A set is closed if its complement is a member of the topology.

Closed function A function from one space to another is closed if the image of every closed set is closed.

Closure The closure of a set is the smallest closed set containing the original set. It is equal to the intersection of allclosed sets which contain it. An element of the closure of a set S is a point of closure of S.

Closure operator See Kuratowski closure axioms.

Coarser topology If X is a set, and if T1 and T2 are topologies on X, then T1 is coarser (or smaller, weaker) thanT2 if T1 is contained in T2. Beware, some authors, especially analysts, use the term stronger.

Comeagre A subset A of a spaceX is comeagre (comeager) if its complementX\A is meagre. Also called residual.

44 CHAPTER 17. GLOSSARY OF TOPOLOGY

Compact A space is compact if every open cover has a finite subcover. Every compact space is Lindelöf andparacompact. Therefore, every compact Hausdorff space is normal. See also quasicompact.

Compact-open topology The compact-open topology on the set C(X, Y) of all continuous maps between two spacesX and Y is defined as follows: given a compact subset K of X and an open subset U of Y, let V(K, U) denotethe set of all maps f in C(X, Y) such that f(K) is contained in U. Then the collection of all such V(K, U) is asubbase for the compact-open topology.

Complete A metric space is complete if every Cauchy sequence converges.

Completely metrizable/completely metrisable See complete space.

Completely normal A space is completely normal if any two separated sets have disjoint neighbourhoods.

Completely normal Hausdorff A completely normal Hausdorff space (or T5 space) is a completely normal T1

space. (A completely normal space is Hausdorff if and only if it is T1, so the terminology is consistent.) Everycompletely normal Hausdorff space is normal Hausdorff.

Completely regular A space is completely regular if, whenever C is a closed set and x is a point not in C, then Cand x are functionally separated.

Completely T3 See Tychonoff.

Component See Connected component/Path-connected component.

Connected A space is connected if it is not the union of a pair of disjoint nonempty open sets. Equivalently, a spaceis connected if the only clopen sets are the whole space and the empty set.

Connected component A connected component of a space is a maximal nonempty connected subspace. Each con-nected component is closed, and the set of connected components of a space is a partition of that space.

Continuous A function from one space to another is continuous if the preimage of every open set is open.

Continuum A space is called a continuum if it a compact, connected Hausdorff space.

Contractible A space X is contractible if the identity map on X is homotopic to a constant map. Every contractiblespace is simply connected.

Coproduct topology If Xi is a collection of spaces and X is the (set-theoretic) disjoint union of Xi, then thecoproduct topology (or disjoint union topology, topological sum of the Xi) on X is the finest topology forwhich all the injection maps are continuous.

Cosmic space A continuous image of some separable metric space.[2]

Countable chain condition A space X satisfies the countable chain condition if every family of non-empty, pairs-wise disjoint open sets is countable.

Countably compact A space is countably compact if every countable open cover has a finite subcover. Every count-ably compact space is pseudocompact and weakly countably compact.

Countably locally finite A collection of subsets of a space X is countably locally finite (or σ-locally finite) if it isthe union of a countable collection of locally finite collections of subsets of X.

Cover A collection of subsets of a space is a cover (or covering) of that space if the union of the collection is thewhole space.

Covering See Cover.

Cut point If X is a connected space with more than one point, then a point x of X is a cut point if the subspace X −x is disconnected.

17.4. D 45

17.4 D

Dense set A set is dense if it has nonempty intersection with every nonempty open set. Equivalently, a set is denseif its closure is the whole space.

Dense-in-itself set A set is dense-in-itself if it has no isolated point.

Density the minimal cardinality of a dense subset of a topological space. A set of density ℵ0 is a separable space.[3]

Derived set If X is a space and S is a subset of X, the derived set of S in X is the set of limit points of S in X.

Developable space A topological space with a development.[4]

Development A countable collection of open covers of a topological space, such that for any closed set C and anypoint p in its complement there exists a cover in the collection such that every neighbourhood of p in the coveris disjoint from C.[4]

Diameter If (M, d) is a metric space and S is a subset ofM, the diameter of S is the supremum of the distances d(x,y), where x and y range over S.

Discrete metric The discrete metric on a set X is the function d : X × X→ R such that for all x, y in X, d(x, x) = 0and d(x, y) = 1 if x ≠ y. The discrete metric induces the discrete topology on X.

Discrete space A space X is discrete if every subset of X is open. We say that X carries the discrete topology.[5]

Discrete topology See discrete space.

Disjoint union topology See Coproduct topology.

Dispersion point If X is a connected space with more than one point, then a point x of X is a dispersion point if thesubspace X − x is hereditarily disconnected (its only connected components are the one-point sets).

Distance See metric space.

Dunce hat (topology)

17.5 E

Entourage See Uniform space.

Exterior The exterior of a set is the interior of its complement.

17.6 F

Fσ set An Fσ set is a countable union of closed sets.[6]

Filter A filter on a space X is a nonempty family F of subsets of X such that the following conditions hold:

1. The empty set is not in F.2. The intersection of any finite number of elements of F is again in F.3. If A is in F and if B contains A, then B is in F.

46 CHAPTER 17. GLOSSARY OF TOPOLOGY

Final topology On a set X with respect to a family of functions into X , is the finest topology on X which makesthose functions continuous.[7]

Fine topology (potential theory) On Euclidean spaceRn , the coarsest topology making all subharmonic functions(equivalently all superharmonic functions) continuous.[8]

Finer topology If X is a set, and if T1 and T2 are topologies on X, then T2 is finer (or larger, stronger) than T1 ifT2 contains T1. Beware, some authors, especially analysts, use the term weaker.

Finitely generated See Alexandrov topology.

First category SeeMeagre.

First-countable A space is first-countable if every point has a countable local base.

Fréchet See T1.

Frontier See Boundary.

Full set A compact subset K of the complex plane is called full if its complement is connected. For example, theclosed unit disk is full, while the unit circle is not.

Functionally separated Two sets A and B in a space X are functionally separated if there is a continuous map f: X→ [0, 1] such that f(A) = 0 and f(B) = 1.

17.7 G

Gδ set A Gδ set or inner limiting set is a countable intersection of open sets.[6]

Gδ space A space in which every closed set is a Gδ set.[6]

Generic point A generic point for a closed set is a point for which the closed set is the closure of the singleton setcontaining that point.[9]

17.8 H

Hausdorff A Hausdorff space (or T2 space) is one in which every two distinct points have disjoint neighbourhoods.Every Hausdorff space is T1.

H-closed A space is H-closed, or Hausdorff closed or absolutely closed, if it is closed in every Hausdorff spacecontaining it.

Hereditarily P A space is hereditarily P for some property P if every subspace is also P.

Hereditary A property of spaces is said to be hereditary if whenever a space has that property, then so does everysubspace of it.[10] For example, second-countability is a hereditary property.

Homeomorphism If X and Y are spaces, a homeomorphism from X to Y is a bijective function f : X → Y suchthat f and f−1 are continuous. The spaces X and Y are then said to be homeomorphic. From the standpointof topology, homeomorphic spaces are identical.

Homogeneous A space X is homogeneous if, for every x and y in X, there is a homeomorphism f : X→ X such thatf(x) = y. Intuitively, the space looks the same at every point. Every topological group is homogeneous.

17.9. I 47

Homotopic maps Two continuous maps f, g : X→ Y are homotopic (in Y) if there is a continuous map H : X × [0,1] → Y such that H(x, 0) = f(x) and H(x, 1) = g(x) for all x in X. Here, X × [0, 1] is given the product topology.The function H is called a homotopy (in Y) between f and g.

Homotopy See Homotopic maps.

Hyper-connected A space is hyper-connected if no two non-empty open sets are disjoint[11] Every hyper-connectedspace is connected.[11]

17.9 I

Identification map See Quotient map.

Identification space See Quotient space.

Indiscrete space See Trivial topology.

Infinite-dimensional topology See Hilbert manifold and Q-manifolds, i.e. (generalized) manifolds modelled onthe Hilbert space and on the Hilbert cube respectively.

Inner limiting set A Gδ set.[6]

Interior The interior of a set is the largest open set contained in the original set. It is equal to the union of all opensets contained in it. An element of the interior of a set S is an interior point of S.

Interior point See Interior.

Isolated point A point x is an isolated point if the singleton x is open. More generally, if S is a subset of a spaceX, and if x is a point of S, then x is an isolated point of S if x is open in the subspace topology on S.

Isometric isomorphism If M1 and M2 are metric spaces, an isometric isomorphism from M1 to M2 is a bijectiveisometry f : M1 →M2. The metric spaces are then said to be isometrically isomorphic. From the standpointof metric space theory, isometrically isomorphic spaces are identical.

Isometry If (M1, d1) and (M2, d2) are metric spaces, an isometry from M1 to M2 is a function f : M1 → M2 suchthat d2(f(x), f(y)) = d1(x, y) for all x, y in M1. Every isometry is injective, although not every isometry issurjective.

17.10 K

Kolmogorov axiom See T0.

Kuratowski closure axioms The Kuratowski closure axioms is a set of axioms satisfied by the function which takeseach subset of X to its closure:

1. Isotonicity: Every set is contained in its closure.2. Idempotence: The closure of the closure of a set is equal to the closure of that set.3. Preservation of binary unions: The closure of the union of two sets is the union of their closures.4. Preservation of nullary unions: The closure of the empty set is empty.

If c is a function from the power set of X to itself, then c is a closure operator if it satisfies the Kuratowski closureaxioms. The Kuratowski closure axioms can then be used to define a topology on X by declaring the closedsets to be the fixed points of this operator, i.e. a set A is closed if and only if c(A) = A.

Kolmogorov topology TKol = R,∅ ∪(a,∞): a is real number; the pair (R,TKol) is named Kolmogorov Straight.

48 CHAPTER 17. GLOSSARY OF TOPOLOGY

17.11 LL-space An L-space is a hereditarily Lindelöf space which is not hereditarily separable. A Suslin line would be an

L-space.[12]

Larger topology See Finer topology.

Limit point A point x in a space X is a limit point of a subset S if every open set containing x also contains a pointof S other than x itself. This is equivalent to requiring that every neighbourhood of x contains a point of S otherthan x itself.

Limit point compact SeeWeakly countably compact.

Lindelöf A space is Lindelöf if every open cover has a countable subcover.

Local base A set B of neighbourhoods of a point x of a space X is a local base (or local basis, neighbourhoodbase, neighbourhood basis) at x if every neighbourhood of x contains some member of B.

Local basis See Local base.

Locally (P) space There are two definitions for a space to be “locally (P)" where (P) is a topological or set-theoreticproperty: that each point has a neighbourhood with property (P), or that every point has a neighourbood basefor which each member has property (P). The first definition is usually taken for locally compact, countablycompact, metrisable, separable, countable; the second for locally connected.[13]

Locally closed subset A subset of a topological space that is the intersection of an open and a closed subset. Equiv-alently, it is a relatively open subset of its closure.

Locally compact A space is locally compact if every point has a compact neighbourhood: the alternative definitionthat each point has a local base consisting of compact neighbourhoods is sometimes used: these are equivalentfor Hausdorff spaces.[13] Every locally compact Hausdorff space is Tychonoff.

Locally connected A space is locally connected if every point has a local base consisting of connected neighbourhoods.[13]

Locally finite A collection of subsets of a space is locally finite if every point has a neighbourhood which hasnonempty intersection with only finitely many of the subsets. See also countably locally finite, point finite.

Locally metrizable/Locally metrisable A space is locallymetrizable if every point has ametrizable neighbourhood.[13]

Locally path-connected A space is locally path-connected if every point has a local base consisting of path-connectedneighbourhoods.[13] A locally path-connected space is connected if and only if it is path-connected.

Locally simply connected A space is locally simply connected if every point has a local base consisting of simplyconnected neighbourhoods.

Loop If x is a point in a space X, a loop at x in X (or a loop in X with basepoint x) is a path f in X, such that f(0) =f(1) = x. Equivalently, a loop in X is a continuous map from the unit circle S1 into X.

17.12 MMeagre If X is a space and A is a subset of X, then A is meagre in X (or of first category in X) if it is the countable

union of nowhere dense sets. If A is not meagre in X, A is of second category in X.[14]

Metacompact A space is metacompact if every open cover has a point finite open refinement.

17.13. N 49

Metric SeeMetric space.

Metric invariant A metric invariant is a property which is preserved under isometric isomorphism.

Metric map If X and Y are metric spaces with metrics dX and dY respectively, then a metric map is a function ffrom X to Y, such that for any points x and y in X, dY(f(x), f(y)) ≤ dX(x, y). A metric map is strictly metricif the above inequality is strict for all x and y in X.

Metric space A metric space (M, d) is a set M equipped with a function d : M × M → R satisfying the followingaxioms for all x, y, and z in M:

1. d(x, y) ≥ 02. d(x, x) = 03. if d(x, y) = 0 then x = y (identity of indiscernibles)4. d(x, y) = d(y, x) (symmetry)5. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

The function d is ametric onM, and d(x, y) is the distance between x and y. The collection of all openballs of M is a base for a topology on M; this is the topology on M induced by d. Every metric space isHausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable.

Metrizable/Metrisable A space is metrizable if it is homeomorphic to a metric space. Every metrizable space isHausdorff and paracompact (and hence normal and Tychonoff). Every metrizable space is first-countable.

Monolith Every non-empty ultra-connected compact space X has a largest proper open subset; this subset is calledamonolith.

Moore space A Moore space is a developable regular Hausdorff space.[4]

17.13 NNeighbourhood/Neighborhood A neighbourhood of a point x is a set containing an open set which in turn contains

the point x. More generally, a neighbourhood of a set S is a set containing an open set which in turn containsthe set S. A neighbourhood of a point x is thus a neighbourhood of the singleton set x. (Note that under thisdefinition, the neighbourhood itself need not be open. Many authors require that neighbourhoods be open; becareful to note conventions.)

Neighbourhood base/basis See Local base.

Neighbourhood system for a point x A neighbourhood system at a point x in a space is the collection of all neigh-bourhoods of x.

Net A net in a space X is a map from a directed set A to X. A net from A to X is usually denoted (xα), where α is anindex variable ranging over A. Every sequence is a net, taking A to be the directed set of natural numbers withthe usual ordering.

Normal A space is normal if any two disjoint closed sets have disjoint neighbourhoods.[6] Every normal space admitsa partition of unity.

Normal Hausdorff A normal Hausdorff space (or T4 space) is a normal T1 space. (A normal space is Hausdorff ifand only if it is T1, so the terminology is consistent.) Every normal Hausdorff space is Tychonoff.

Nowhere dense A nowhere dense set is a set whose closure has empty interior.

50 CHAPTER 17. GLOSSARY OF TOPOLOGY

17.14 OOpen cover An open cover is a cover consisting of open sets.[4]

Open ball If (M, d) is a metric space, an open ball is a set of the form B(x; r) := y in M : d(x, y) < r, where x isin M and r is a positive real number, the radius of the ball. An open ball of radius r is an open r-ball. Everyopen ball is an open set in the topology on M induced by d.

Open condition See open property.

Open set An open set is a member of the topology.

Open function A function from one space to another is open if the image of every open set is open.

Open property A property of points in a topological space is said to be “open” if those points which possess it forman open set. Such conditions often take a common form, and that form can be said to be an open condition;for example, in metric spaces, one defines an open ball as above, and says that “strict inequality is an opencondition”.

17.15 PParacompact A space is paracompact if every open cover has a locally finite open refinement. Paracompact implies

metacompact.[15] Paracompact Hausdorff spaces are normal.[16]

Partition of unity A partition of unity of a space X is a set of continuous functions from X to [0, 1] such that anypoint has a neighbourhood where all but a finite number of the functions are identically zero, and the sum ofall the functions on the entire space is identically 1.

Path A path in a space X is a continuous map f from the closed unit interval [0, 1] into X. The point f(0) is theinitial point of f; the point f(1) is the terminal point of f.[11]

Path-connected A space X is path-connected if, for every two points x, y in X, there is a path f from x to y, i.e., apath with initial point f(0) = x and terminal point f(1) = y. Every path-connected space is connected.[11]

Path-connected component A path-connected component of a space is a maximal nonempty path-connected sub-space. The set of path-connected components of a space is a partition of that space, which is finer than thepartition into connected components.[11] The set of path-connected components of a space X is denoted π0(X).

Perfectly normal a normal space which is also a Gδ.[6]

π-base A collection B of nonempty open sets is a π-base for a topology τ if every nonempty open set in τ includesa set from B.[17]

Point A point is an element of a topological space. More generally, a point is an element of any set with an underlyingtopological structure; e.g. an element of a metric space or a topological group is also a “point”.

Point of closure See Closure.

Polish A space is Polish if it is separable and completely metrizable, i.e. if it is homeomorphic to a separable andcomplete metric space.

Polyadic A space is polyadic if it is the continuous image of the power of a one-point compactification of a locallycompact, non-compact Hausdorff space.

P-point A point of a topological space is a P-point if its filter of neighbourhoods is closed under countable intersec-tions.

17.16. Q 51

Pre-compact See Relatively compact.

Prodiscrete topology The prodiscrete topology on a product AG is the product topology when each factor A is giventhe discrete topology.[18]

Product topology If Xi is a collection of spaces and X is the (set-theoretic) product of Xi, then the producttopology on X is the coarsest topology for which all the projection maps are continuous.

Proper function/mapping A continuous function f from a space X to a space Y is proper if f−1(C) is a compactset in X for any compact subspace C of Y.

Proximity space Aproximity space (X, δ) is a setX equippedwith a binary relation δ between subsets ofX satisfyingthe following properties:

For all subsets A, B and C of X,

1. A δ B implies B δ A2. A δ B implies A is non-empty3. If A and B have non-empty intersection, then A δ B4. A δ (B ∪ C) iff (A δ B or A δ C)5. If, for all subsets E of X, we have (A δ E or B δ E), then we must have A δ (X − B)

Pseudocompact A space is pseudocompact if every real-valued continuous function on the space is bounded.

Pseudometric See Pseudometric space.

Pseudometric space A pseudometric space (M, d) is a setM equipped with a function d : M ×M →R satisfying allthe conditions of a metric space, except possibly the identity of indiscernibles. That is, points in a pseudometricspace may be “infinitely close” without being identical. The function d is a pseudometric onM. Every metricis a pseudometric.

Punctured neighbourhood/Punctured neighborhood Apunctured neighbourhood of a point x is a neighbourhoodof x, minus x. For instance, the interval (−1, 1) = y : −1 < y < 1 is a neighbourhood of x = 0 in the realline, so the set (−1, 0) ∪ (0, 1) = (−1, 1) − 0 is a punctured neighbourhood of 0.

17.16 Q

Quasicompact See compact. Some authors define “compact” to include the Hausdorff separation axiom, and theyuse the term quasicompact to mean what we call in this glossary simply “compact” (without the Hausdorffaxiom). This convention is most commonly found in French, and branches of mathematics heavily influencedby the French.

Quotient map If X and Y are spaces, and if f is a surjection from X to Y, then f is a quotient map (or identificationmap) if, for every subset U of Y, U is open in Y if and only if f −1(U) is open in X. In other words, Y hasthe f-strong topology. Equivalently, f is a quotient map if and only if it is the transfinite composition of mapsX → X/Z , where Z ⊂ X is a subset. Note that this doesn't imply that f is an open function.

Quotient space If X is a space, Y is a set, and f : X→ Y is any surjective function, then the quotient topology on Yinduced by f is the finest topology for which f is continuous. The space X is a quotient space or identificationspace. By definition, f is a quotient map. The most common example of this is to consider an equivalencerelation on X, with Y the set of equivalence classes and f the natural projection map. This construction is dualto the construction of the subspace topology.

52 CHAPTER 17. GLOSSARY OF TOPOLOGY

17.17 RRefinement A cover K is a refinement of a cover L if every member of K is a subset of some member of L.

Regular A space is regular if, whenever C is a closed set and x is a point not in C, then C and x have disjointneighbourhoods.

Regular Hausdorff A space is regular Hausdorff (or T3) if it is a regular T0 space. (A regular space is Hausdorffif and only if it is T0, so the terminology is consistent.)

Regular open A subset of a space X is regular open if it equals the interior of its closure; dually, a regular closed setis equal to the closure of its interior.[19] An example of a non-regular open set is the set U = (0,1) ∪ (1,2) in Rwith its normal topology, since 1 is in the interior of the closure of U, but not in U. The regular open subsetsof a space form a complete Boolean algebra.[19]

Relatively compact A subset Y of a space X is relatively compact in X if the closure of Y in X is compact.

Residual If X is a space and A is a subset of X, then A is residual in X if the complement of A is meagre in X. Alsocalled comeagre or comeager.

Resolvable A topological space is called resolvable if it is expressible as the union of two disjoint dense subsets.

Rim-compact A space is rim-compact if it has a base of open sets whose boundaries are compact.

17.18 SS-space An S-space is a hereditarily separable space which is not hereditarily Lindelöf.[12]

Scattered A space X is scattered if every nonempty subset A of X contains a point isolated in A.

Scott The Scott topology on a poset is that in which the open sets are those Upper sets inaccessible by directedjoins.[20]

Second category SeeMeagre.

Second-countable A space is second-countable or perfectly separable if it has a countable base for its topology.[6]Every second-countable space is first-countable, separable, and Lindelöf.

Semilocally simply connected A space X is semilocally simply connected if, for every point x in X, there is aneighbourhood U of x such that every loop at x in U is homotopic in X to the constant loop x. Every simplyconnected space and every locally simply connected space is semilocally simply connected. (Compare withlocally simply connected; here, the homotopy is allowed to live in X, whereas in the definition of locally simplyconnected, the homotopy must live in U.)

Semiregular A space is semiregular if the regular open sets form a base.

Separable A space is separable if it has a countable dense subset.[6][14]

Separated Two sets A and B are separated if each is disjoint from the other’s closure.

Sequentially compact A space is sequentially compact if every sequence has a convergent subsequence. Everysequentially compact space is countably compact, and every first-countable, countably compact space is se-quentially compact.

Short map See metric map

Simply connected A space is simply connected if it is path-connected and every loop is homotopic to a constantmap.

17.19. T 53

Smaller topology See Coarser topology.

Sober In a sober space, every irreducible closed subset is the closure of exactly one point: that is, has a uniquegeneric point.[21]

Star The star of a point in a given cover of a topological space is the union of all the sets in the cover that containthe point. See star refinement.

f -Strong topologyLet f : X → Y be a map of topological spaces. We say that Y has the f -strong topology if, for every subsetU ⊂ Y , one has that U is open in Y if and only if f−1(U) is open in X

Stronger topology See Finer topology. Beware, some authors, especially analysts, use the term weaker topology.

Subbase A collection of open sets is a subbase (or subbasis) for a topology if every non-empty proper open set inthe topology is a union of finite intersections of sets in the subbase. If B is any collection of subsets of a setX, the topology on X generated by B is the smallest topology containing B; this topology consists of the emptyset, X and all unions of finite intersections of elements of B.

Subbasis See Subbase.

Subcover A cover K is a subcover (or subcovering) of a cover L if every member of K is a member of L.

Subcovering See Subcover.

Submaximal space A topological space is said to be submaximal if every subset of it is locally closed, that is, everysubset is the intersection of an open set and a closed set.

Here are some facts about submaximality as a property of topological spaces:

• Every door space is submaximal.

• Every submaximal space is weakly submaximal viz every finite set is locally closed.

• Every submaximal space is irresolvable[22]

Subspace If T is a topology on a space X, and if A is a subset of X, then the subspace topology on A induced byT consists of all intersections of open sets in T with A. This construction is dual to the construction of thequotient topology.

17.19 T

T0 A space is T0 (orKolmogorov) if for every pair of distinct points x and y in the space, either there is an open setcontaining x but not y, or there is an open set containing y but not x.

T1 A space is T1 (or Fréchet or accessible) if for every pair of distinct points x and y in the space, there is an openset containing x but not y. (Compare with T0; here, we are allowed to specify which point will be contained inthe open set.) Equivalently, a space is T1 if all its singletons are closed. Every T1 space is T0.

T2 See Hausdorff space.

T3 See Regular Hausdorff.

T₃½ See Tychonoff space.

54 CHAPTER 17. GLOSSARY OF TOPOLOGY

T4 See Normal Hausdorff.

T5 See Completely normal Hausdorff.

Top See Category of topological spaces.

Topological invariant A topological invariant is a property which is preserved under homeomorphism. For exam-ple, compactness and connectedness are topological properties, whereas boundedness and completeness arenot. Algebraic topology is the study of topologically invariant abstract algebra constructions on topologicalspaces.

Topological space A topological space (X, T) is a set X equipped with a collection T of subsets of X satisfying thefollowing axioms:

1. The empty set and X are in T.2. The union of any collection of sets in T is also in T.3. The intersection of any pair of sets in T is also in T.

The collection T is a topology on X.

Topological sum See Coproduct topology.

Topologically complete Completely metrizable spaces (i. e. topological spaces homeomorphic to complete metricspaces) are often called topologically complete; sometimes the term is also used for Čech-complete spaces orcompletely uniformizable spaces.

Topology See Topological space.

Totally bounded A metric space M is totally bounded if, for every r > 0, there exist a finite cover of M by openballs of radius r. A metric space is compact if and only if it is complete and totally bounded.

Totally disconnected A space is totally disconnected if it has no connected subset with more than one point.

Trivial topology The trivial topology (or indiscrete topology) on a set X consists of precisely the empty set and theentire space X.

Tychonoff A Tychonoff space (or completely regular Hausdorff space, completely T3 space, T₃.₅ space) is acompletely regular T0 space. (A completely regular space is Hausdorff if and only if it is T0, so the terminologyis consistent.) Every Tychonoff space is regular Hausdorff.

17.20 UUltra-connected A space is ultra-connected if no two non-empty closed sets are disjoint.[11] Every ultra-connected

space is path-connected.

Ultrametric A metric is an ultrametric if it satisfies the following stronger version of the triangle inequality: for allx, y, z in M, d(x, z) ≤ max(d(x, y), d(y, z)).

Uniform isomorphism If X and Y are uniform spaces, a uniform isomorphism from X to Y is a bijective functionf : X→ Y such that f and f−1 are uniformly continuous. The spaces are then said to be uniformly isomorphicand share the same uniform properties.

Uniformizable/Uniformisable A space is uniformizable if it is homeomorphic to a uniform space.

17.21. W 55

Uniform space A uniform space is a set U equipped with a nonempty collection Φ of subsets of the Cartesianproduct X × X satisfying the following axioms:

1. if U is in Φ, then U contains (x, x) | x in X .2. if U is in Φ, then (y, x) | (x, y) in U is also in Φ3. if U is in Φ and V is a subset of X × X which contains U, then V is in Φ4. if U and V are in Φ, then U ∩ V is in Φ5. if U is in Φ, then there exists V in Φ such that, whenever (x, y) and (y, z) are in V, then (x, z) is in

U.

The elements of Φ are called entourages, and Φ itself is called a uniform structure on U.

Uniform structure See Uniform space.

17.21 WWeak topology The weak topology on a set, with respect to a collection of functions from that set into topological

spaces, is the coarsest topology on the set which makes all the functions continuous.

Weaker topology See Coarser topology. Beware, some authors, especially analysts, use the term stronger topol-ogy.

Weakly countably compact A space is weakly countably compact (or limit point compact) if every infinite subsethas a limit point.

Weakly hereditary A property of spaces is said to be weakly hereditary if whenever a space has that property, thenso does every closed subspace of it. For example, compactness and the Lindelöf property are both weaklyhereditary properties, although neither is hereditary.

Weight The weight of a space X is the smallest cardinal number κ such that X has a base of cardinal κ. (Note thatsuch a cardinal number exists, because the entire topology forms a base, and because the class of cardinalnumbers is well-ordered.)

Well-connected See Ultra-connected. (Some authors use this term strictly for ultra-connected compact spaces.)

17.22 ZZero-dimensional A space is zero-dimensional if it has a base of clopen sets.[23]

17.23 References[1] Vickers (1989) p.22

[2] Deza, Michel Marie; Deza, Elena (2012). Encyclopedia of Distances. Springer-Verlag. p. 64. ISBN 3642309585.

[3] Nagata (1985) p.104

[4] Steen & Seebach (1978) p.163

[5] Steen & Seebach (1978) p.41

[6] Steen & Seebach (1978) p.162

[7] Willard, Stephen (1970). General Topology. Addison-Wesley Series in Mathematics. Reading, MA: Addison-Wesley. Zbl0205.26601.

[8] Conway, John B. (1995). Functions of One Complex Variable II. Graduate Texts in Mathematics 159. Springer-Verlag. pp.367–376. ISBN 0-387-94460-5. Zbl 0887.30003.

56 CHAPTER 17. GLOSSARY OF TOPOLOGY

[9] Vickers (1989) p.65

[10] Steen & Seebach p.4

[11] Steen & Seebach (1978) p.29

[12] Gabbay, Dov M.; Kanamori, Akihiro; Woods, John Hayden, eds. (2012). Sets and Extensions in the Twentieth Century.Elsevier. p. 290. ISBN 0444516212.

[13] Hart et al (2004) p.65

[14] Steen & Seebach (1978) p.7

[15] Steen & Seebach (1978) p.23

[16] Steen & Seebach (1978) p.25

[17] Hart, Nagata, Vaughan Sect. d-22, page 227

[18] Ceccherini-Silberstein, Tullio; Coornaert, Michel (2010). Cellular automata and groups. Springer Monographs in Mathe-matics. Berlin: Springer-Verlag. p. 3. ISBN 978-3-642-14033-4. Zbl 1218.37004.

[19] Steen & Seebach (1978) p.6

[20] Vickers (1989) p.95

[21] Vickers (1989) p.66

[22] Miroslav Hušek; J. van Mill (2002), Recent progress in general topology, Recent Progress in General Topology 2, Elsevier,p. 21, ISBN 0-444-50980-1

[23] Steen & Seebach (1978) p.33

• Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004). Encyclopedia of general topology. Elsevier.ISBN 978-0-444-50355-8.

• Kunen, Kenneth; Vaughan, Jerry E. (editors). Handbook of Set-Theoretic Topology. North-Holland. ISBN0-444-86580-2.

• Nagata, Jun-iti (1985). Modern general topology. North-Holland Mathematical Library 33 (2nd revised ed.).Amsterdam-New York-Oxford: North-Holland. ISBN 0080933793. Zbl 0598.54001.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (Dover reprint of 1978 ed.).Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446.

• Vickers, Steven (1989). Topology via Logic. Cambridge Tracts in Theoretic Computer Science 5. ISBN0-521-36062-5. Zbl 0668.54001.

• Willard, Stephen (1970). General Topology. Addison-Wesley Series in Mathematics. Reading, MA: Addison-Wesley. ISBN 978-0-201-08707-9. Zbl 0205.26601. Also available as Dover reprint.

17.24 External links• A glossary of definitions in topology

Chapter 18

H-closed space

In mathematics, a topological space X is said to be H-closed, or Hausdorff closed, or absolutely closed if it isclosed in every Hausdorff space containing it as a subspace. This property is a generalization of compactness, sincea compact subset of a Hausdorff space is closed. Thus, every compact Hausdorff space is H-closed. The notion ofan H-closed space has been introduced in 1924 by P. Alexandroff and P. Urysohn.

18.1 Examples and equivalent formulations• The unit interval [0, 1] , endowed with the smallest topology which refines the euclidean topology, and contains

Q ∩ [0, 1] as an open set is H-closed but not compact.

• Every regular Hausdorff H-closed space is compact.

• A Hausdorff space is H-closed if and only if every open cover has a finite subfamily with dense union.

18.2 See also• Compact space

18.3 References• K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), Encyclopedia of General Topology, Chapter d20 (by JackPorter and Johannes Vermeer)

57

Chapter 19

Heine–Borel theorem

In the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:For a subset S of Euclidean space Rn, the following two statements are equivalent:

• S is closed and bounded

• S is compact (that is, every open cover of S has a finite subcover).

In the context of real analysis, the former property is sometimes used as the defining property of compactness.However, the two definitions cease to be equivalent when we consider subsets of more general metric spaces and inthis generality only the latter property is used to define compactness. In fact, the Heine–Borel theorem for arbitrarymetric spaces reads:

A subset of a metric space is compact if and only if it is complete and totally bounded.

19.1 History and motivation

The history of what today is called the Heine–Borel theorem starts in the 19th century, with the search for solidfoundations of real analysis. Central to the theory was the concept of uniform continuity and the theorem stating thatevery continuous function on a closed interval is uniformly continuous. Peter Gustav Lejeune Dirichlet was the firstto prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval inhis proof. He used this proof in his 1862 lectures, which were published only in 1904. Later Eduard Heine, KarlWeierstrass and Salvatore Pincherle used similar techniques. Émile Borel in 1895 was the first to state and provea form of what is now called the Heine–Borel theorem. His formulation was restricted to countable covers. PierreCousin (1895), Lebesgue (1898) and Schoenflies (1900) generalized it to arbitrary covers.[1]

19.2 Proof

If a set is compact, then it must be closed.Let S be a subset of Rn. Observe first the following: if a is a limit point of S, then any finite collection C of opensets, such that each open set U ∈ C is disjoint from some neighborhood VU of a, fails to be a cover of S. Indeed, theintersection of the finite family of sets VU is a neighborhood W of a in Rn. Since a is a limit point of S, W mustcontain a point x in S. This x ∈ S is not covered by the family C, because every U in C is disjoint from VU and hencedisjoint fromW, which contains x.If S is compact but not closed, then it has an accumulation point a not in S. Consider a collection C ′ consisting of anopen neighborhood N(x) for each x ∈ S, chosen small enough to not intersect some neighborhood Vx of a. Then C ′is an open cover of S, but any finite subcollection of C ′ has the form of C discussed previously, and thus cannot bean open subcover of S. This contradicts the compactness of S. Hence, every accumulation point of S is in S, so S isclosed.

58

19.3. GENERALIZATIONS 59

The proof above applies with almost no change to showing that any compact subset S of a Hausdorff topological spaceX is closed in X.If a set is compact, then it is bounded.Consider the open balls centered upon a common point, with any radius. This can cover any set, because all pointsin the set are some distance away from that point. Any finite subcover of this cover must be bounded, because allballs in the subcover are contained in the largest open ball within that subcover. Therefore, any set covered by thissubcover must also be bounded.A closed subset of a compact set is compact.Let K be a closed subset of a compact set T in Rn and let CK be an open cover of K. Then U = Rn \ K is an open setand

CT = CK ∪ U

is an open cover of T. Since T is compact, then CT has a finite subcover C ′T , that also covers the smaller set K. Since

U does not contain any point of K, the set K is already covered by C ′K = C ′

T \ U, that is a finite subcollection ofthe original collection CK. It is thus possible to extract from any open cover CK of K a finite subcover.If a set is closed and bounded, then it is compact.If a set S in Rn is bounded, then it can be enclosed within an n-box

T0 = [−a, a]n

where a > 0. By the property above, it is enough to show that T0 is compact.Assume, by way of contradiction, that T0 is not compact. Then there exists an infinite open cover C of T0 that doesnot admit any finite subcover. Through bisection of each of the sides of T0, the box T0 can be broken up into 2n subn-boxes, each of which has diameter equal to half the diameter of T0. Then at least one of the 2n sections of T0 mustrequire an infinite subcover of C, otherwise C itself would have a finite subcover, by uniting together the finite coversof the sections. Call this section T1.Likewise, the sides of T1 can be bisected, yielding 2n sections of T1, at least one of which must require an infinitesubcover of C. Continuing in like manner yields a decreasing sequence of nested n-boxes:

T0 ⊃ T1 ⊃ T2 ⊃ . . . ⊃ Tk ⊃ . . .

where the side length of Tk is (2 a) / 2k, which tends to 0 as k tends to infinity. Let us define a sequence (x ) suchthat each x is in T . This sequence is Cauchy, so it must converge to some limit L. Since each Tk is closed, and foreach k the sequence (x ) is eventually always inside T , we see that that L ∈ T for each k.Since C covers T0, then it has some member U ∈ C such that L ∈ U. Since U is open, there is an n-ball B(L) ⊆ U.For large enough k, one has Tk ⊆ B(L) ⊆ U, but then the infinite number of members of C needed to cover Tk canbe replaced by just one: U, a contradiction.Thus, T0 is compact. Since S is closed and a subset of the compact set T0, then S is also compact (see above).

19.3 Generalizations

The theorem does not hold as stated for general metric spaces. A metric space (or topological vector space) is said tohave the Heine–Borel property if every closed and bounded subset is compact. Many metric spaces fail to have theHeine–Borel property. For instance, the metric space of rational numbers (or indeed any incomplete metric space)fails to have the Heine–Borel property. Complete metric spaces may also fail to have the property. For instance,no infinite-dimensional Banach space has the Heine–Borel property. On the other hand, some infinite-dimensionalFréchet spaces do have the Heine–Borel property. For instance, the space C∞(K) of smooth functions on a compactset K ⊂ Rn , considered as a Fréchet space, has the Heine–Borel property, as can be shown by using the Arzelà–Ascoli theorem. More generally, any nuclear Fréchet space has the Heine–Borel property. For a topological space, aset S has the Heine-Borel property if every open covering of S contains a finite subcovering.

60 CHAPTER 19. HEINE–BOREL THEOREM

The Heine–Borel theorem can be generalized to arbitrary metric spaces by strengthening the conditions required forcompactness:

A subset of a metric space is compact if and only if it is complete and totally bounded.

This generalisation also applies to topological vector spaces and, more generally, to uniform spaces.Here is a sketch of the "⇒ "-part of the proof, in the context of a general metric space, according to Jean Dieudonné:

1. It is obvious that any compact set E is totally bounded.

2. Let (xn) be an arbitrary Cauchy sequence in E; let Fn be the closure of the set xk : k ≥ n in E and Un :=E − Fn. If the intersection of all Fn were empty, (Un) would be an open cover of E, hence there would be afinite subcover (Unk) of E, hence the intersection of the Fnk would be empty; this implies that Fn is emptyfor all n larger than any of the nk, which is a contradiction. Hence, the intersection of all Fn is not empty, andany point in this intersection is an accumulation point of the sequence (xn).

3. Any accumulation point of a Cauchy sequence is a limit point (xn); hence any Cauchy sequence in E convergesin E, in other words: E is complete.

A proof of the "⇐ "-part can be sketched as follows:

1. If E were not compact, there would exist a cover (U ) of E having no finite subcover of E. Use the totalboundedness of E to define inductively a sequence of balls (Bn) in E with

• the radius of Bn is 2−n;• there is no finite subcover (U ∩Bn) of Bn;• Bn₊₁ ∩ Bn is not empty.

2. Let xn be the center point of Bn and let yn be any point in Bn₊₁ ∩ Bn; hence we have d(xn₊₁, xn) ≤ d(xn₊₁, yn)+ d(yn, xn) ≤ 2−n−1 + 2−n ≤ 2−n+1. It follows for n ≤ p < q: d(xp, xq) ≤ d(xp, xp₊₁) + ... + d(xq₋₁, xq) ≤ 2−p+1 +... + 2−q+2 ≤ 2−n+2. Therefore, (xn) is a Cauchy sequence in E, converging to some limit point a in E, becauseE is complete.

3. Let I0 be an index such that UI0contains a; since (xn) converges to a and UI0

is open, there is a large n suchthat the ball Bn is a subset of UI0

–v in contradiction to the construction of Bn.

The proof of the "⇒ " part easily generalises to arbitrary uniform spaces, but the proof of the "⇐ " part (of a similarversion with “sequences” replaced with “filters”) is more complicated and is equivalent to the ultrafilter principle,[2]a weaker form of the Axiom of Choice. (Already, in general metric spaces, the "⇐ " direction requires the Axiomof dependent choice.)

19.4 See also• Bolzano–Weierstrass theorem

19.5 Notes[1] Sundström, Manya Raman (2010). “A pedagogical history of compactness”. arXiv:1006.4131v1 [math.HO].

[2] Eric Schechter (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8. UF24, p. 506.

19.6 References• P. Dugac (1989). “Sur la correspondance de Borel et le théorème de Dirichlet–Heine–Weierstrass–Borel–Schoenflies–Lebesgue”. Arch. Internat. Hist. Sci. 39: 69–110.

• proof of Heine-Borel theorem at PlanetMath.org.

19.7. EXTERNAL LINKS 61

19.7 External links• Ivan Kenig, Dr. Prof. Hans-Christian Graf v. Botthmer, Dmitrij Tiessen, Andreas Timm, Viktor Wittman(2004). The Heine–Borel Theorem (avi • mp4 • mov • swf • streamed video). Hannover: Leibniz Universität.

• Hazewinkel, Michiel, ed. (2001), “Borel-Lebesgue covering theorem”, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

Chapter 20

Hemicompact space

In mathematics, in the field of topology, a topological space is said to be hemicompact if it has a sequence of compactsubsets such that every compact subset of the space lies inside some compact set in the sequence. Clearly, this forcesthe union of the sequence to be the whole space, because every point is compact and hence must lie in one of thecompact sets.

20.1 Examples• Every compact space is hemicompact.

• The real line is hemicompact.

• Every locally compact Lindelöf space is hemicompact.

20.2 Properties

Every hemicompact space is σ-compact and if in addition it is first countable then it is locally compact.If X is a hemicompact space, then the space C(X,M) of all continuous functions f : X → M to a metric space(M, δ) with the compact-open topology is metrizable. To see this, take a sequence K1,K2, . . . of compact subsetsof X such that every compact subset of X lies inside some compact set in this sequence (the existence of such asequence follows from the hemicompactness ofX ). Denote

dn(f, g) = supx∈Kn

δ(f(x), g(x))

for f, g ∈ C(X,M) and n ∈ N . Then

d(f, g) =∞∑

n=1

1

2n· dn(f, g)

1 + dn(f, g)

defines a metric on C(X,M) which induces the compact-open topology.

20.3 See also• Compact space

• Locally compact space

• Lindelöf space

62

20.4. REFERENCES 63

20.4 References• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

Chapter 21

Hyperconnected space

For computer networking term, see hyperconnectivity.

In mathematics, a hyperconnected space is a topological space X that cannot be written as the union of two non-empty closed sets (whether disjoint or non-disjoint). The name irreducible space is preferred in algebraic geometry.For a topological space X the following conditions are equivalent:

• No two nonempty open sets are disjoint.

• X cannot be written as the union of two proper closed sets.

• Every nonempty open set is dense in X.

• The interior of every proper closed set is empty.

A space which satisfies any one of these conditions is called hyperconnected or irreducible.An irreducible set is a subset of a topological space for which the subspace topology is irreducible. Some authorsdo not consider the empty set to be irreducible (even though it vacuously satisfies the above conditions).

21.1 Examples

Examples of hyperconnected spaces include the cofinite topology on any infinite space and the Zariski topology onan algebraic variety.

21.2 Hyperconnectedness vs. connectedness

Every hyperconnected space is both connected and locally connected (though not necessarily path-connected or locallypath-connected).Note that in the definition of hyper-connectedness, the closed sets don't have to be disjoint. This is in contrast to thedefinition of connectedness, in which the open sets are disjoint.For example, the space of reals with the standard topology is connected but not hyperconnected. This is because itcannot be written as a union of two disjoint open sets, but it can be written as a union of two (non-disjoint) closedsets.

21.3 Properties

The (nonempty) open subsets of a hyperconnected space are “large” in the sense that each one is dense in X and anypair of them intersects. Thus, a hyperconnected space cannot be Hausdorff unless it contains only a single point.

64

21.4. IRREDUCIBLE COMPONENTS 65

Every hyperconnected space is both connected and locally connected (though not necessarily path-connected or locallypath-connected).The continuous image of a hyperconnected space is hyperconnected. In particular, any continuous function froma hyperconnected space to a Hausdorff space must be constant. It follows that every hyperconnected space ispseudocompact.Every open subspace of a hyperconnected space is hyperconnected. A closed subspace need not be hyperconnected,however, the closure of any hyperconnected subspace is always hyperconnected.

21.4 Irreducible components

An irreducible component in a topological space is a maximal irreducible subset (i.e. an irreducible set that is notcontained in any larger irreducible set). The irreducible components are always closed.Unlike the connected components of a space, the irreducible components need not be disjoint (i.e. they need notform a partition). In general, the irreducible components will overlap. Since every irreducible space is connected,the irreducible components will always lie in the connected components.The irreducible components of a Hausdorff space are just the singleton sets.Every subset of a Noetherian topological space is Noetherian, and hence has finitely many irreducible components.

21.5 See also• Ultraconnected space

• Sober space

21.6 References• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

• Hyperconnected space at PlanetMath.org.

Chapter 22

Kolmogorov space

In topology and related branches of mathematics, a topological space X is a T0 space orKolmogorov space (namedafter Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them has an open neighborhoodnot containing the other. In a T0 space all points are topologically distinguishable.This condition, called the T0 condition, is the weakest of the separation axioms. Nearly all topological spacesnormally studied in mathematics are T0 spaces. In particular, all T1 spaces, i.e., all spaces in which for every pair ofdistinct points each has a neighborhood not containing the other, are T0 spaces. This includes all T2 (or Hausdorff)spaces, i.e., all topological spaces in which distinct points have disjoint neighbourhoods. Given any topological spaceone can construct a T0 space by identifying topologically indistinguishable points.T0 spaces that are not T1 spaces are exactly those spaces for which the specialization preorder is a nontrivial partialorder. Such spaces naturally occur in computer science, specifically in denotational semantics.

22.1 Definition

A T0 space is a topological space in which every pair of distinct points is topologically distinguishable. That is, forany two different points x and y there is an open set which contains one of these points and not the other.Note that topologically distinguishable points are automatically distinct. On the other hand, if the singleton sets xand y are separated, then the points x and y must be topologically distinguishable. That is,

separated ⇒ topologically distinguishable⇒ distinct

The property of being topologically distinguishable is, in general, stronger than being distinct but weaker than beingseparated. In a T0 space, the second arrow above reverses; points are distinct if and only if they are distinguishable.This is how the T0 axiom fits in with the rest of the separation axioms.

22.2 Examples and nonexamples

Nearly all topological spaces normally studied in mathematics are T0. In particular, all Hausdorff (T2) spaces and T1

spaces are T0.

22.2.1 Spaces which are not T0

• A set with more than one element, with the trivial topology. No points are distinguishable.

• The set R2 where the open sets are the Cartesian product of an open set in R and R itself, i.e., the producttopology of R with the usual topology and R with the trivial topology; points (a,b) and (a,c) are not distin-guishable.

66

22.3. OPERATING WITH T0 SPACES 67

• The space of all measurable functions f from the real line R to the complex plane C such that the Lebesgueintegral of |f(x)|2 over the entire real line is finite. Two functions which are equal almost everywhere areindistinguishable. See also below.

22.2.2 Spaces which are T0 but not T1

• The Zariski topology on Spec(R), the prime spectrum of a commutative ring R is always T0 but generallynot T1. The non-closed points correspond to prime ideals which are not maximal. They are important to theunderstanding of schemes.

• The particular point topology on any set with at least two elements is T0 but not T1 since the particular pointis not closed (its closure is the whole space). An important special case is the Sierpiński space which is theparticular point topology on the set 0,1.

• The excluded point topology on any set with at least two elements is T0 but not T1. The only closed point isthe excluded point.

• The Alexandrov topology on a partially ordered set is T0 but will not be T1 unless the order is discrete (agreeswith equality). Every finite T0 space is of this type. This also includes the particular point and excluded pointtopologies as special cases.

• The right order topology on a totally ordered set is a related example.

• The overlapping interval topology is similar to the particular point topology since every open set includes 0.

• Quite generally, a topological space X will be T0 if and only if the specialization preorder on X is a partialorder. However, X will be T1 if and only if the order is discrete (i.e. agrees with equality). So a space will beT0 but not T1 if and only if the specialization preorder on X is a non-discrete partial order.

22.3 Operating with T0 spaces

Examples of topological space typically studied are T0. Indeed, whenmathematicians in many fields, notably analysis,naturally run across non-T0 spaces, they usually replace them with T0 spaces, in a manner to be described below.To motivate the ideas involved, consider a well-known example. The space L2(R) is meant to be the space of allmeasurable functions f from the real line R to the complex plane C such that the Lebesgue integral of |f(x)|2 over theentire real line is finite. This space should become a normed vector space by defining the norm ||f || to be the squareroot of that integral. The problem is that this is not really a norm, only a seminorm, because there are functionsother than the zero function whose (semi)norms are zero. The standard solution is to define L2(R) to be a set ofequivalence classes of functions instead of a set of functions directly. This constructs a quotient space of the originalseminormed vector space, and this quotient is a normed vector space. It inherits several convenient properties fromthe seminormed space; see below.In general, when dealing with a fixed topology T on a set X, it is helpful if that topology is T0. On the other hand,when X is fixed but T is allowed to vary within certain boundaries, to force T to be T0 may be inconvenient, sincenon-T0 topologies are often important special cases. Thus, it can be important to understand both T0 and non-T0

versions of the various conditions that can be placed on a topological space.

22.4 The Kolmogorov quotient

Topological indistinguishability of points is an equivalence relation. No matter what topological space X might beto begin with, the quotient space under this equivalence relation is always T0. This quotient space is called theKolmogorov quotient of X, which we will denote KQ(X). Of course, if X was T0 to begin with, then KQ(X) andX are naturally homeomorphic. Categorically, Kolmogorov spaces are a reflective subcategory of topological spaces,and the Kolmogorov quotient is the reflector.Topological spacesX and Y areKolmogorov equivalentwhen their Kolmogorov quotients are homeomorphic. Manyproperties of topological spaces are preserved by this equivalence; that is, if X and Y are Kolmogorov equivalent, thenX has such a property if and only if Y does. On the other hand, most of the other properties of topological spaces imply

68 CHAPTER 22. KOLMOGOROV SPACE

T0-ness; that is, if X has such a property, then Xmust be T0. Only a few properties, such as being an indiscrete space,are exceptions to this rule of thumb. Even better, many structures defined on topological spaces can be transferredbetween X and KQ(X). The result is that, if you have a non-T0 topological space with a certain structure or property,then you can usually form a T0 space with the same structures and properties by taking the Kolmogorov quotient.The example of L2(R) displays these features. From the point of view of topology, the seminormed vector space thatwe started with has a lot of extra structure; for example, it is a vector space, and it has a seminorm, and these definea pseudometric and a uniform structure that are compatible with the topology. Also, there are several properties ofthese structures; for example, the seminorm satisfies the parallelogram identity and the uniform structure is complete.The space is not T0 since any two functions in L2(R) which are equal almost everywhere are indistinguishable with thistopology. When we form the Kolmogorov quotient, the actual L2(R), these structures and properties are preserved.Thus, L2(R) is also a complete seminormed vector space satisfying the parallelogram identity. But we actually get abit more, since the space is now T0. A seminorm is a norm if and only if the underlying topology is T0, so L2(R) isactually a complete normed vector space satisfying the parallelogram identity — otherwise known as a Hilbert space.And it is a Hilbert space that mathematicians (and physicists, in quantum mechanics) generally want to study. Notethat the notation L2(R) usually denotes the Kolmogorov quotient, the set of equivalence classes of square integrablefunctions which differ on sets of measure zero, rather than simply the vector space of square integrable functionswhich the notation suggests.

22.5 Removing T0

Although norms were historically defined first, people came up with the definition of seminorm as well, which is asort of non-T0 version of a norm. In general, it is possible to define non-T0 versions of both properties and structuresof topological spaces. First, consider a property of topological spaces, such as being Hausdorff. One can then defineanother property of topological spaces by defining the space X to satisfy the property if and only if the Kolmogorovquotient KQ(X) is Hausdorff. This is a sensible, albeit less famous, property; in this case, such a space X is calledpreregular. (There even turns out to be a more direct definition of preregularity). Now consider a structure that canbe placed on topological spaces, such as a metric. We can define a new structure on topological spaces by letting anexample of the structure on X be simply a metric on KQ(X). This is a sensible structure on X; it is a pseudometric.(Again, there is a more direct definition of pseudometric.)In this way, there is a natural way to remove T0-ness from the requirements for a property or structure. It is generallyeasier to study spaces that are T0, but it may also be easier to allow structures that aren't T0 to get a fuller picture.The T0 requirement can be added or removed arbitrarily using the concept of Kolmogorov quotient.

22.6 External links• History of weak separation axioms (PDF file)

Chapter 23

Limit point compact

In mathematics, a topological space X is said to be limit point compact[1] or weakly countably compact if everyinfinite subset of X has a limit point in X. This property generalizes a property of compact spaces. In a metric space,limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces,however, these three notions of compactness are not equivalent.

23.1 Properties and Examples• Limit point compactness is equivalent to countable compactness ifX is a T1-space and is equivalent to compactnessif X is a metric space.

• An example of a space X that is not weakly countably compact is any countable (or larger) set with the discretetopology. A more interesting example is the countable complement topology.

• Even though a continuous function from a compact space X, to an ordered set Y in the order topology, mustbe bounded, the same thing does not hold if X is limit point compact. An example is given by the spaceX ×Z(where X = 1, 2 carries the indiscrete topology and Z is the set of all integers carrying the discrete topology)and the function f = πZ given by projection onto the second coordinate. Clearly, ƒ is continuous and X × Zis limit point compact (in fact, every nonempty subset ofX ×Z has a limit point) but ƒ is not bounded, and infact f(X × Z) = Z is not even limit point compact.

• Every countably compact space (and hence every compact space) is weakly countably compact, but the converseis not true.

• For metrizable spaces, compactness, limit point compactness, and sequential compactness are all equivalent.

• The set of all real numbers, R, is not limit point compact; the integers are an infinite set but do not have a limitpoint in R.

• If (X, T) and (X, T*) are topological spaces with T* finer than T and (X, T*) is limit point compact, then so is(X, T).

• A finite space is vacuously limit point compact.

23.2 See also• Compact space

• Sequential compactness

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70 CHAPTER 23. LIMIT POINT COMPACT

• Metric space

• Bolzano-Weierstrass theorem

• Countably compact space

23.3 Notes[1] The terminology “limit point compact” appears in a topology textbook by James Munkres, and is apparently due to him.

According to him, some call the property "Fréchet compactness”, while others call it the "Bolzano-Weierstrass property".Munkres, p. 178–179.

23.4 References• James Munkres (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

• This article incorporates material from Weakly countably compact on PlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

Chapter 24

Lindelöf space

In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. TheLindelöf property is a weakening of the more commonly used notion of compactness, which requires the existenceof a finite subcover.A strongly Lindelöf space is a topological space such that every open subspace is Lindelöf. Such spaces are alsoknown as hereditarily Lindelöf spaces, because all subspaces of such a space are Lindelöf.Lindelöf spaces are named for the Finnish mathematician Ernst Leonard Lindelöf.

24.1 Properties of Lindelöf spaces

In general, no implications hold (in either direction) between the Lindelöf property and other compactness properties,such as paracompactness. But by the Morita theorem, every regular Lindelöf space is paracompact.Any second-countable space is a Lindelöf space, but not conversely. However, the matter is simpler for metric spaces.A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable.An open subspace of a Lindelöf space is not necessarily Lindelöf. However, a closed subspace must be Lindelöf.Lindelöf is preserved by continuous maps. However, it is not necessarily preserved by products, not even by finiteproducts.A Lindelöf space is compact if and only if it is countably compact.Any σ-compact space is Lindelöf.

24.2 Properties of strongly Lindelöf spaces• Any second-countable space is a strongly Lindelöf space

• Any Suslin space is strongly Lindelöf.

• Strongly Lindelöf spaces are closed under taking countable unions, subspaces, and continuous images.

• Every Radon measure on a strongly Lindelöf space is moderated.

24.3 Product of Lindelöf spaces

The product of Lindelöf spaces is not necessarily Lindelöf. The usual example of this is the Sorgenfrey plane S ,which is the product of the real line R under the half-open interval topology with itself. Open sets in the Sorgenfreyplane are unions of half-open rectangles that include the south and west edges and omit the north and east edges,including the northwest, northeast, and southeast corners. The antidiagonal of S is the set of points (x, y) such thatx+ y = 0 .

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72 CHAPTER 24. LINDELÖF SPACE

Consider the open covering of S which consists of:

1. The set of all rectangles (−∞, x)× (−∞, y) , where (x, y) is on the antidiagonal.

2. The set of all rectangles [x,+∞)× [y,+∞) , where (x, y) is on the antidiagonal.

The thing to notice here is that each point on the antidiagonal is contained in exactly one set of the covering, so allthese sets are needed.Another way to see that S is not Lindelöf is to note that the antidiagonal defines a closed and uncountable discretesubspace of S . This subspace is not Lindelöf, and so the whole space cannot be Lindelöf as well (as closed subspacesof Lindelöf spaces are also Lindelöf).The product of a Lindelöf space and a compact space is Lindelöf.

24.4 Generalisation

The following definition generalises the definitions of compact and Lindelöf: a topological space is κ -compact (or κ-Lindelöf), where κ is any cardinal, if every open cover has a subcover of cardinality strictly less than κ . Compact isthen ℵ0 -compact and Lindelöf is then ℵ1 -compact.The Lindelöf degree, or Lindelöf number l(X) , is the smallest cardinal κ such that every open cover of the spaceXhas a subcover of size at most κ . In this notation, X is Lindelöf if l(X) = ℵ0 . The Lindelöf number as definedabove does not distinguish between compact spaces and Lindelöf non compact spaces. Some authors gave the nameLindelöf number to a different notion: the smallest cardinal κ such that every open cover of the spaceX has a subcoverof size strictly less than κ .[1] In this latter (and less used) sense the Lindelöf number is the smallest cardinal κ suchthat a topological space X is κ -compact. This notion is sometimes also called the compactness degree of the spaceX .[2]

24.5 See also• Axioms of countability

• Lindelöf’s lemma

24.6 Notes[1] Mary Ellen Rudin, Lectures on set theoretic topology, Conference Board of the Mathematical Sciences, American Math-

ematical Society, 1975, p. 4, retrievable on Google Books

[2] Hušek,Miroslav (1969), “The class of k-compact spaces is simple”,Mathematische Zeitschrift 110: 123–126, doi:10.1007/BF01124977,MR 0244947.

24.7 References• Michael Gemignani, Elementary Topology (ISBN 0-486-66522-4) (see especially section 7.2)

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446.

• I. Juhász (1980). Cardinal functions in topology - ten years later. Math. Centre Tracts, Amsterdam. ISBN90-6196-196-3.

• Munkres, James. Topology, 2nd ed.

• http://arxiv.org/abs/1301.5340 Generalized Lob’s Theorem.Strong Reflection Principles and Large CardinalAxioms.Consistency Results in Topology

Chapter 25

Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking,each small portion of the space looks like a small portion of a compact space.

25.1 Formal definition

Let X be a topological space. Most commonly X is called locally compact, if every point of X has a compactneighbourhood.There are other common definitions: They are all equivalent if X is a Hausdorff space (or preregular). But theyare not equivalent in general:

1. every point of X has a compact neighbourhood.2. every point of X has a closed compact neighbourhood.2′. every point of X has a relatively compact neighbourhood.2″. every point of X has a local base of relatively compact neighbourhoods.3. every point of X has a local base of compact neighbourhoods.3′. for every point x of X, every neighbourhood of x contains a compact neighbourhood of x.4. X is Hausdorff and satisfies any (all) of the previous conditions.

Logical relations among the conditions:

• Conditions (2), (2′), (2″) are equivalent.

• Conditions (3), (3′) are equivalent.

• Neither of conditions (2), (3) implies the other.

• Each condition implies (1).

• Compactness implies conditions (1) and (2), but not (3).

Condition (1) is probably the most commonly used definition, since it is the least restrictive and the others are equiv-alent to it when X is Hausdorff. This equivalence is a consequence of the facts that compact subsets of Hausdorffspaces are closed, and closed subsets of compact spaces are compact.Condition (4) is used, for example, in Bourbaki.[1] In almost all applications, locally compact spaces are indeed alsoHausdorff. These locally compact Hausdorff (LCH) spaces are thus the spaces that this article is primarily concernedwith.

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74 CHAPTER 25. LOCALLY COMPACT SPACE

25.2 Examples and counterexamples

25.2.1 Compact Hausdorff spaces

Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in thearticle compact space. Here we mention only:

• the unit interval [0,1];

• the Cantor set;

• the Hilbert cube.

25.2.2 Locally compact Hausdorff spaces that are not compact

• The Euclidean spacesRn (and in particular the real lineR) are locally compact as a consequence of the Heine–Borel theorem.

• Topological manifolds share the local properties of Euclidean spaces and are therefore also all locally compact.This even includes nonparacompact manifolds such as the long line.

• All discrete spaces are locally compact and Hausdorff (they are just the zero-dimensional manifolds). Theseare compact only if they are finite.

• All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology.This provides several examples of locally compact subsets of Euclidean spaces, such as the unit disc (either theopen or closed version).

• The space Qp of p-adic numbers is locally compact, because it is homeomorphic to the Cantor set minus onepoint. Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.

25.2.3 Hausdorff spaces that are not locally compact

Asmentioned in the following section, no Hausdorff space can possibly be locally compact if it is not also a Tychonoffspace; there are some examples of Hausdorff spaces that are not Tychonoff spaces in that article. But there are alsoexamples of Tychonoff spaces that fail to be locally compact, such as:

• the spaceQ of rational numbers (endowed with the topology fromR), since its compact subsets all have emptyinterior and therefore are not neighborhoods;

• the subspace (0,0) union (x,y) : x > 0 of R2, since the origin does not have a compact neighborhood;

• the lower limit topology or upper limit topology on the set R of real numbers (useful in the study of one-sidedlimits);

• any T0, hence Hausdorff, topological vector space that is infinite-dimensional, such as an infinite-dimensionalHilbert space.

The first two examples show that a subset of a locally compact space need not be locally compact, which contrastswith the open and closed subsets in the previous section. The last example contrasts with the Euclidean spaces inthe previous section; to be more specific, a Hausdorff topological vector space is locally compact if and only if it isfinite-dimensional (in which case it is a Euclidean space). This example also contrasts with the Hilbert cube as anexample of a compact space; there is no contradiction because the cube cannot be a neighbourhood of any point inHilbert space.

25.3. PROPERTIES 75

25.2.4 Non-Hausdorff examples

• The one-point compactification of the rational numbers Q is compact and therefore locally compact in senses(1) and (2) but it is not locally compact in sense (3).

• The particular point topology on any infinite set is locally compact in sense (3) but not in sense (2), because ithas no nonempty closed compact subspaces containing the particular point. The same holds for the real linewith the upper topology.

25.3 Properties

Every locally compact preregular space is, in fact, completely regular. It follows that every locally compact Hausdorffspace is a Tychonoff space. Since straight regularity is a more familiar condition than either preregularity (which isusually weaker) or complete regularity (which is usually stronger), locally compact preregular spaces are normallyreferred to in the mathematical literature as locally compact regular spaces. Similarly locally compact Tychonoffspaces are usually just referred to as locally compact Hausdorff spaces.Every locally compact Hausdorff space is a Baire space. That is, the conclusion of the Baire category theorem holds:the interior of every union of countably many nowhere dense subsets is empty.A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X can be written as the set-theoretic difference of two closed subsets of Y. As a corollary, a dense subspace X of a locally compact Hausdorffspace Y is locally compact if and only if X is an open subset of Y. Furthermore, if a subspace X of any Hausdorffspace Y is locally compact, then X still must be the difference of two closed subsets of Y, although the converseneedn't hold in this case.Quotient spaces of locally compact Hausdorff spaces are compactly generated. Conversely, every compactly gener-ated Hausdorff space is a quotient of some locally compact Hausdorff space.For locally compact spaces local uniform convergence is the same as compact convergence.

25.3.1 The point at infinity

Since every locally compact Hausdorff space X is Tychonoff, it can be embedded in a compact Hausdorff space b(X)using the Stone–Čech compactification. But in fact, there is a simpler method available in the locally compact case;the one-point compactification will embed X in a compact Hausdorff space a(X) with just one extra point. (The one-point compactification can be applied to other spaces, but a(X) will be Hausdorff if and only if X is locally compactand Hausdorff.) The locally compact Hausdorff spaces can thus be characterised as the open subsets of compactHausdorff spaces.Intuitively, the extra point in a(X) can be thought of as a point at infinity. The point at infinity should be thought ofas lying outside every compact subset of X. Many intuitive notions about tendency towards infinity can be formulatedin locally compact Hausdorff spaces using this idea. For example, a continuous real or complex valued function fwith domain X is said to vanish at infinity if, given any positive number e, there is a compact subset K of X suchthat |f(x)| < e whenever the point x lies outside of K. This definition makes sense for any topological space X. IfX is locally compact and Hausdorff, such functions are precisely those extendable to a continuous function g on itsone-point compactification a(X) = X ∪ ∞ where g(∞) = 0.The set C0(X) of all continuous complex-valued functions that vanish at infinity is a C* algebra. In fact, everycommutative C* algebra is isomorphic to C0(X) for some unique (up to homeomorphism) locally compact Hausdorffspace X. More precisely, the categories of locally compact Hausdorff spaces and of commutative C* algebras aredual; this is shown using the Gelfand representation. Forming the one-point compactification a(X) of X correspondsunder this duality to adjoining an identity element to C0(X).

25.3.2 Locally compact groups

The notion of local compactness is important in the study of topological groups mainly because every Hausdorfflocally compact group G carries natural measures called the Haar measures which allow one to integrate measurablefunctions defined on G. The Lebesgue measure on the real line R is a special case of this.

76 CHAPTER 25. LOCALLY COMPACT SPACE

The Pontryagin dual of a topological abelian group A is locally compact if and only if A is locally compact. Moreprecisely, Pontryagin duality defines a self-duality of the category of locally compact abelian groups. The study oflocally compact abelian groups is the foundation of harmonic analysis, a field that has since spread to non-abelianlocally compact groups.

25.4 Notes[1] Bourbaki, Nicolas (1989). General Topology, Part I (reprint of the 1966 ed.). Berlin: Springer-Verlag. ISBN 3-540-

19374-X.

25.5 References• Kelley, John (1975). General Topology. Springer. ISBN 978-0387901251.

• Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 978-0131816299.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446.

• Willard, Stephen (1970). General Topology. Addison-Wesley. ISBN 978-0486434797.

Chapter 26

Locally connected space

In this topological space, V is a neighbourhood of p and it contains a connected neighbourhood (the dark green disk) that containsp.

In topology and other branches of mathematics, a topological space X is locally connected if every point admits aneighbourhood basis consisting entirely of open, connected sets.

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78 CHAPTER 26. LOCALLY CONNECTED SPACE

26.1 Background

Throughout the history of topology, connectedness and compactness have been two of the most widely studied topo-logical properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognitionof their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion ofa topological property and thus a topological space. However, whereas the structure of compact subsets of Euclideanspace was understood quite early on via the Heine–Borel theorem, connected subsets of Rn (for n > 1) proved to bemuch more complicated. Indeed, while any compact Hausdorff space is locally compact, a connected space – andeven a connected subset of the Euclidean plane – need not be locally connected (see below).This led to a rich vein of research in the first half of the twentieth century, in which topologists studied the implicationsbetween increasingly subtle and complex variations on the notion of a locally connected space. As an example, thenotion of weak local connectedness at a point and its relation to local connectedness will be considered later on in thearticle.In the latter part of the twentieth century, research trends shifted to more intense study of spaces like manifoldswhich are locally well understood (being locally homeomorphic to Euclidean space) but have complicated globalbehavior. By this it is meant that although the basic point-set topology of manifolds is relatively simple (as manifoldsare essentially metrizable according to most definitions of the concept), their algebraic topology is far more complex.From this modern perspective, the stronger property of local path connectedness turns out to be more important: forinstance, in order for a space to admit a universal cover it must be connected and locally path connected. Local pathconnectedness will be discussed as well.A space is locally connected if and only if for every open set U, the connected components of U (in the subspacetopology) are open. It follows, for instance, that a continuous function from a locally connected space to a totallydisconnected space must be locally constant. In fact the openness of components is so natural that one must be sureto keep in mind that it is not true in general: for instance Cantor space is totally disconnected but not discrete.

26.2 Definitions and first examples

Let X be a topological space, and let x be a point of X.We say that X is locally connected at x if for every open set V containing x there exists a connected, open set Uwith x ∈ U ⊂ V . The space X is said to be locally connected if it is locally connected at x for all x in X.[1] Notethat local connectedness and connectedness are not related to one another; a space may possess one or both of theseproperties, or neither.By contrast, we say that X is weakly locally connected at x (or connected im kleinen at x) if for every open setV containing x there exists a connected subset N of V such that x lies in the interior of N. An equivalent definitionis: each open set V containing x contains an open neighborhood U of x such that any two points in U lie in someconnected subset of V.[2] The space X is said to be weakly locally connected if it is weakly locally connected at xfor all x in X.In other words, the only difference between the two definitions is that for local connectedness at x we require aneighborhood base of open connected sets containing x, whereas for weak local connectedness at x we require only aneighborhood base of connected sets containing x.Evidently a space which is locally connected at x is weakly locally connected at x. The converse does not hold (acounterexample, the broom space, is given below). On the other hand, it is equally clear that a locally connectedspace is weakly locally connected, and here it turns out that the converse does hold: a space which is weakly locallyconnected at all of its points is necessarily locally connected at all of its points.[3] A proof is given below.We say that X is locally path connected at x if for every open set V containing x there exists a path connected, openset U with x ∈ U ⊂ V . The space X is said to be locally path connected if it is locally path connected at x for all xin X.Since path connected spaces are connected, locally path connected spaces are locally connected. This time the con-verse does not hold (see example 6 below).

26.3. PROPERTIES 79

26.2.1 First examples

1. For any positive integer n, the Euclidean space Rn is connected and locally connected.

2. The subspace [0, 1] ∪ [2, 3] of the real line R1 is locally connected but not connected.

3. The topologist’s sine curve is a subspace of the Euclidean plane which is connected, but not locally connected.[4]

4. The space Q of rational numbers endowed with the standard Euclidean topology, is neither connected norlocally connected.

5. The comb space is path connected but not locally path connected.

6. A countably infinite set endowed with the cofinite topology is locally connected (indeed, hyperconnected) butnot locally path connected.[5]

Further examples are given later on in the article.

26.3 Properties1. Local connectedness is, by definition, a local property of topological spaces, i.e., a topological property P such

that a space X possesses property P if and only if each point x in X admits a neighborhood base of sets whichhave property P. Accordingly, all the “metaproperties” held by a local property hold for local connectedness.In particular:

2. A space is locally connected if and only if it admits a base of connected subsets.

3. The disjoint union⨿

i Xi of a family Xi of spaces is locally connected if and only if each Xi is locallyconnected. In particular, since a single point is certainly locally connected, it follows that any discrete space islocally connected. On the other hand, a discrete space is totally disconnected, so is connected only if it has atmost one point.

4. Conversely, a totally disconnected space is locally connected if and only if it is discrete. This can be used toexplain the aforementioned fact that the rational numbers are not locally connected.

26.4 Components and path components

The following result follows almost immediately from the definitions but will be quite useful:Lemma: Let X be a space, and Yi a family of subsets of X. Suppose that

∩i Yi is nonempty. Then, if each Yi is

connected (respectively, path connected) then the union∪

i Yi is connected (respectively, path connected).[6]

Now consider two relations on a topological space X: for x, y ∈ X , write:

x ≡c y if there is a connected subset of X containing both x and y; andx ≡pc y if there is a path connected subset of X containing both x and y.

Evidently both relations are reflexive and symmetric. Moreover, if x and y are contained in a connected (respectively,path connected) subset A and y and z are connected in a connected (respectively, path connected) subset B, thenthe Lemma implies that A ∪ B is a connected (respectively, path connected) subset containing x, y and z. Thuseach relation is an equivalence relation, and defines a partition of X into equivalence classes. We consider these twopartitions in turn.For x in X, the set Cx of all points y such that y ≡c x is called the connected component of x.[7] The Lemma impliesthat Cx is the unique maximal connected subset of X containing x.[8] Since the closure of Cx is also a connectedsubset containing x,[9] it follows that Cx is closed.[10]

If X has only finitely many connected components, then each component is the complement of a finite union ofclosed sets and therefore open. In general, the connected components need not be open, since, e.g., there existtotally disconnected spaces (i.e., Cx = x for all points x) which are not discrete, like Cantor space. However,

80 CHAPTER 26. LOCALLY CONNECTED SPACE

the connected components of a locally connected space are also open, and thus are clopen sets.[11] It follows that alocally connected space X is a topological disjoint union

⨿Cx of its distinct connected components. Conversely, if

for every open subset U of X, the connected components of U are open, then X admits a base of connected sets andis therefore locally connected.[12]

Similarly x in X, the set PCx of all points y such that y ≡pc x is called the path component of x.[13] As above, PCx isalso the union of all path connected subsets of X which contain x, so by the Lemma is itself path connected. Becausepath connected sets are connected, we have PCx ⊂ Cx for all x in X.However the closure of a path connected set need not be path connected: for instance, the topologist’s sine curve isthe closure of the open subset U consisting of all points (x,y) with x > 0, and U, being homeomorphic to an intervalon the real line, is certainly path connected. Moreover, the path components of the topologist’s sine curve C are U,which is open but not closed, and C \ U , which is closed but not open.A space is locally path connected if and only if for all open subsets U, the path components of U are open.[13]Therefore the path components of a locally path connected space give a partition of X into pairwise disjoint opensets. It follows that an open connected subspace of a locally path connected space is necessarily path connected.[14]Moreover, if a space is locally path connected, then it is also locally connected, so for all x in X, Cx is connectedand locally path connected, hence path connected, i.e., Cx = PCx . That is, for a locally path connected space thecomponents and path components coincide.

26.4.1 Examples

1. The set I × I (where I = [0,1]) in the dictionary order topology has exactly one component (because it isconnected) but has uncountablymany path components. Indeed, any set of the form a × I is a path componentfor each a belonging to I.

2. Let f be a continuous map from R to Rℓ (R in the lower limit topology). Since R is connected, and the imageof a connected space under a continuous map must be connected, the image of R under f must be connected.Therefore, the image of R under f must be a subset of a component of Rℓ. Since this image is nonempty, theonly continuous maps from R to Rℓ, are the constant maps. In fact, any continuous map from a connectedspace to a totally disconnected space must be constant.

26.5 Quasicomponents

Let X be a topological space. We define a third relation on X: x ≡qc y if there is no separation of X into open sets Aand B such that x is an element of A and y is an element of B. This is an equivalence relation on X and the equivalenceclass QCx containing x is called the quasicomponent of x.[8]

QCx can also be characterized as the intersection of all clopen subsets of X which contain x.[8] Accordingly QCx isclosed; in general it need not be open.Evidently Cx ⊆ QCx for all x in X.[8] Overall we have the following containments among path components, compo-nents and quasicomponents at x:

PCx ⊆ Cx ⊆ QCx.

If X is locally connected, then, as above, Cx is a clopen set containing x, soQCx ⊆ Cx and thusQCx = Cx . Sincelocal path connectedness implies local connectedness, it follows that at all points x of a locally path connected spacewe have

PCx = Cx = QCx.

26.5.1 Examples

1. An example of a space whose quasicomponents are not equal to its components is a countable set, X, with thediscrete topology along with two points a and b such that any neighbourhood of a either contains b or all but

26.6. MORE ON LOCAL CONNECTEDNESS VERSUS WEAK LOCAL CONNECTEDNESS 81

finitely many points of X, and any neighbourhood of b either contains a or all but finitely many points of X.The point a lies in the same quasicomponent of b but not in the same component as b.

2. The Arens–Fort space is not locally connected, but nevertheless the components and

the quasicomponents coincide: indeed QCx = Cx = x for all points x.[4]

26.6 More on local connectedness versus weak local connectedness

TheoremLet X be a weakly locally connected space. Then X is locally connected.ProofIt is sufficient to show that the components of open sets are open. Let U be open in X and let C be a component of U.Let x be an element of C. Then x is an element of U so that there is a connected subspace A of X contained in U andcontaining a neighbourhood V of x. Since A is connected and A contains x, A must be a subset of C (the componentcontaining x). Therefore, the neighbourhood V of x is a subset of C. Since x was arbitrary, we have shown that each xin C has a neighbourhood V contained in C. This shows that C is open relative toU. Therefore, X is locally connected.A certain infinite union of decreasing broom spaces is an example of a space which is weakly locally connected at aparticular point, but not locally connected at that point.[15]

26.7 Notes[1] Willard, Definition 27.4, p. 199

[2] Willard, Definition 27.14, p. 201

[3] Willard, Theorem 27.16, p. 201

[4] Steen & Seebach, pp. 137–138

[5] Steen & Seebach, pp. 49–50

[6] Willard, Theorem 26.7a, p. 192

[7] Willard, Definition 26.11, p.194

[8] Willard, Problem 26B, pp. 195–196

[9] Kelley, Theorem 20, p. 54; Willard, Theorem 26.8, p.193

[10] Willard, Theorem 26.12, p. 194

[11] Willard, Corollary 27.10, p. 200

[12] Willard, Theorem 27.9, p. 200

[13] Willard, Problem 27D, p. 202

[14] Willard, Theorem 27.5, p. 199

[15] Steen & Seebach, example 119.4, p. 139

26.8 See also• Comb space

• Connected space

• Equivalence relation

82 CHAPTER 26. LOCALLY CONNECTED SPACE

• Sorgenfrey line

• Topologist’s sine curve

• Totally disconnected space

• Locally simply connected space

• Semi-locally simply connected

26.9 References• John L. Kelley; General Topology; ISBN 0-387-90125-6

• Munkres, James (1999), Topology (2nd ed.), Prentice Hall, ISBN 0-13-181629-2.

• Stephen Willard; General Topology; Dover Publications, 2004.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Mineola, NY: Dover Publications, Inc., ISBN 978-0-486-68735-3, MR 1382863

26.10 Further reading• Coppin, C. A. (1972), “Continuous Functions from a Connected Locally Connected Space into a ConnectedSpace with a Dispersion Point”, Proceedings of the American Mathematical Society (American Mathemati-cal Society) 32 (2): 625–626, doi:10.1090/S0002-9939-1972-0296913-7, JSTOR 2037874. For Hausdorffspaces, it is shown that any continuous function from a connected locally connected space into a connectedspace with a dispersion point is constant

• Davis, H. S. (1968), “A Note on Connectedness Im Kleinen”, Proceedings of the American Mathematical Soci-ety (American Mathematical Society) 19 (5): 1237–1241, doi:10.1090/s0002-9939-1968-0254814-3, JSTOR2036067.

Chapter 27

Locally finite collection

In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space.It is fundamental in the study of paracompactness and topological dimension.A collection of subsets of a topological spaceX is said to be locally finite, if each point in the space has a neighbourhoodthat intersects only finitely many of the sets in the collection.Note that the term locally finite has different meanings in other mathematical fields.

27.1 Examples and properties

A finite collection of subsets of a topological space is locally finite. Infinite collections can also be locally finite: forexample, the collection of all subsets of R of the form (n, n + 2) with integer n. A countable collection of subsetsneed not be locally finite, as shown by the collection of all subsets of R of the form (−n, n) with integer n.If a collection of sets is locally finite, the collection of all closures of these sets is also locally finite. The reasonfor this is that if an open set containing a point intersects the closure of a set, it necessarily intersects the set itself,hence a neighborhood can intersect at most the same number of closures (it may intersect fewer, since two distinct,indeed disjoint, sets can have the same closure). The converse, however, can fail if the closures of the sets are notdistinct. For example, in the finite complement topology on R the collection of all open sets is not locally finite, butthe collection of all closures of these sets is locally finite (since the only closures are R and the empty set).

27.1.1 Compact spaces

No infinite collection of a compact space can be locally finite. Indeed, let Ga be an infinite family of subsets ofa space and suppose this collection is locally finite. For each point x of this space, choose a neighbourhood Ux thatintersects the collection Ga at only finitely many values of a. Clearly:

Ux for each x in X (the union over all x) is an open covering in X

and hence has a finite subcover, Ua1 ∪ ...... ∪ Uan. Since each Uai intersects Ga for only finitely many values ofa, the union of all such Uai intersects the collection Ga for only finitely many values of a. It follows that X (thewhole space!) intersects the collection Ga at only finitely many values of a, contradicting the infinite cardinality ofthe collection Ga.A topological space in which every open cover admits a locally finite open refinement is called paracompact. Everylocally finite collection of subsets of a topological space X is also point-finite. A topological space in which everyopen cover admits a point-finite open refinement is called metacompact.

27.1.2 Second countable spaces

No uncountable cover of a Lindelöf space can be locally finite, by essentially the same argument as in the case ofcompact spaces. In particular, no uncountable cover of a second-countable space is locally finite.

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84 CHAPTER 27. LOCALLY FINITE COLLECTION

27.2 Closed sets

It is clear from the definition of a topology that a finite union of closed sets is closed. One can readily give an exampleof an infinite union of closed sets that is not closed. However, if we consider a locally finite collection of closed sets,the union is closed. To see this we note that if x is a point outside the union of this locally finite collection of closedsets, we merely choose a neighbourhoodV of x that intersects this collection at only finitely many of these sets. Definea bijective map from the collection of sets that V intersects to 1, ..., k thus giving an index to each of these sets.Then for each set, choose an open set Ui containing x that doesn't intersect it. The intersection of all such Ui for 1 ≤i ≤ k intersected with V, is a neighbourhood of x that does not intersect the union of this collection of closed sets.

27.3 Countably locally finite collections

A collection in a space is countably locally finite (or σ-locally finite) if it is the union of a countable family of locallyfinite collections of subsets of X. Countable local finiteness is a key hypothesis in the Nagata–Smirnov metrizationtheorem, which states that a topological space is metrizable if and only if it is regular and has a countably locally finitebasis.

27.4 References• James R. Munkres (2000), Topology (2nd ed.), Prentice Hall, ISBN 0-13-181629-2

Chapter 28

Locally finite space

In the mathematical field of topology, a locally finite space is a topological space in which every point has a finiteneighborhood.A locally finite space is Alexandrov.A T1 space is locally finite if and only if it is discrete.

28.1 References• Nakaoka, Fumie; Oda, Nobuyuki (2001), “Some applications of minimal open sets”, International Journal ofMathematics and Mathematical Sciences 29 (8): 471–476, doi:10.1155/S0161171201006482

85

Chapter 29

Locally Hausdorff space

In mathematics, in the field of topology, a topological space is said to be locally Hausdorff if every point has anopen neighbourhood that is a Hausdorff space under the subspace topology.[1]

Here are some facts:

• Every Hausdorff space is locally Hausdorff.

• Every locally Hausdorff space is T1.[2]

• There are locally Hausdorff spaces where a sequence has more than one limit. This can never happen for aHausdorff space.

• The bug-eyed line is locally Hausdorff (it is in fact locally metrizable) but not Hausdorff.

• The etale space for the sheaf of differentiable functions on a differential manifold is not Hausdorff, but it islocally Hausdorff.

• A T1 space need not be locally Hausdorff; an example of this is an infinite set given the cofinite topology.

• Let X be a set given the particular point topology. Then X is locally Hausdorff at precisely one point. Fromthe last example, it will follow that a set (with more than one point) given the particular point topology isnot a topological group. Note that if x is the 'particular point' of X, and y is distinct from x, then any setcontaining y that doesn't also contain x inherits the discrete topology and is therefore Hausdorff. However, noneighbourhood of y is actually Hausdorff so that the space cannot be locally Hausdorff at y.

• IfG is a topological group that is locally Hausdorff at x for some point x ofG, thenG is Hausdorff. This followsfrom the fact that if y is a point of G, there exists a homeomorphism from G to itself carrying x to y, so G islocally Hausdorff at every point, and is therefore T1 (and T1 topological groups are Hausdorff).

29.1 References[1] Niefield, Susan B. (1991), “Weak products over a locally Hausdorff locale”, Category theory (Como, 1990), Lecture Notes

in Math. 1488, Springer, Berlin, pp. 298–305, doi:10.1007/BFb0084228, MR 1173020.

[2] Clark, Lisa Orloff; an Huef, Astrid; Raeburn, Iain (2013), “The equivalence relations of local homeomorphisms and Fellalgebras”, New York Journal of Mathematics 19: 367–394, MR 3084709. See remarks prior to Lemma 3.2.

86

Chapter 30

Locally normal space

In mathematics, particularly topology, a topological space X is locally normal if intuitively it looks locally like anormal space. More precisely, a locally normal space satisfies the property that each point of the space belongs to aneighbourhood of the space that is normal under the subspace topology.

30.1 Formal definition

A topological space X is said to be locally normal if and only if each point, x, of X has a neighbourhood that isnormal under the subspace topology.Note that not every neighbourhood of x has to be normal, but at least one neighbourhood of x has to be normal (underthe subspace topology).Note however, that if a space were called locally normal if and only if each point of the space belonged to a subset ofthe space that was normal under the subspace topology, then every topological space would be locally normal. Thisis because, the singleton x is vacuously normal and contains x. Therefore, the definition is more restrictive.

30.2 Examples and properties• Every locally normal T1 space is locally regular and locally Hausdorff.

• A locally compact Hausdorff space is always locally normal.

• A normal space is always locally normal.

• A T1 space need not be locally normal as the set of all real numbers endowed with the cofinite topology shows.

30.3 Theorems

Theorem 1If X is homeomorphic to Y and X is locally normal, then so is Y.ProofThis follows from the fact that the image of a normal space under a homeomorphism is always normal.

30.4 See also• Locally Hausdorff space

• Locally compact space

87

88 CHAPTER 30. LOCALLY NORMAL SPACE

• Locally metrizable space

• Normal space

• Homeomorphism

• Locally regular space

30.5 References

Chapter 31

Locally regular space

In mathematics, particularly topology, a topological space X is locally regular if intuitively it looks locally like aregular space. More precisely, a locally regular space satisfies the property that each point of the space belongs to anopen subset of the space that is regular under the subspace topology.

31.1 Formal definition

A topological space X is said to be locally regular if and only if each point, x, of X has a neighbourhood that isregular under the subspace topology. Equivalently, a space X is locally regular if and only if the collection of all opensets that are regular under the subspace topology forms a base for the topology on X.

31.2 Examples and properties• Every locally regular T0 space is locally Hausdorff.

• A regular space is always locally regular.

• A locally compact Hausdorff space is regular, hence locally regular.

• A T1 space need not be locally regular as the set of all real numbers endowed with the cofinite topology shows.

31.3 See also• Locally Hausdorff space

• Locally compact space

• Locally metrizable space

• Normal space

• Homeomorphism

• Locally normal space

31.4 References

89

Chapter 32

Locally simply connected space

In mathematics, a locally simply connected space is a topological space that admits a basis of simply connectedsets.[1][2] Every locally simply connected space is also locally path-connected and locally connected.

Hawaiian earring

The circle is an example of a locally simply connected space which is not simply connected. The Hawaiian earring

90

32.1. REFERENCES 91

is a space which is neither locally simply connected nor simply connected. The cone on the Hawaiian earring iscontractible and therefore simply connected, but still not locally simply connected.All topological manifolds and CW complexes are locally simply connected. In fact, these satisfy the much strongerproperty of being locally contractible.A strictly weaker condition is that of being semi-locally simply connected. Both locally simply connected spaces andsimply connected spaces are semi-locally simply connected, but neither converse holds.

32.1 References[1] Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

[2] Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0.

Chapter 33

Luzin space

For continuous images of separable complete metric spaces, known as Lusin spaces, see Polish space #Lusin spaces.

In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, is an uncountable topological T1 spacewithout isolated points in which every nowhere-dense subset is countable. There are many minor variations of thisdefinition in use: the T1 condition can be replaced by T2 or T3, and some authors allow a countable or even arbitrarynumber of isolated points.The existence of a Luzin space is independent of the axioms of ZFC. Luzin (1914) showed that the continuumhypothesis implies that a Luzin space exists. Kunen (1977) showed that assuming Martin’s Axiom and the negationof the continuum hypothesis, there are no Hausdorff Luzin spaces.

33.1 In real analysis

In real analysis and descriptive set theory, a Luzin set (or Lusin set), is defined as an uncountable subset A of thereals such that every uncountable subset of A is nonmeager; that is, of second Baire category. Equivalently, A is anuncountable set of reals which meets every first category set in only countably many points. Luzin proved that, ifthe continuum hypothesis holds, then every nonmeager set has a Luzin subset. Obvious properties of a Luzin set arethat it must be nonmeager (otherwise the set itself is an uncountable meager subset) and of measure zero, becauseevery set of positive measure contains a meager set which also has positive measure, and is therefore uncountable.A weakly Luzin set is an uncountable subset of a real vector space such that for any uncountable subset the set ofdirections between different elements of the subset is dense in the sphere of directions.The measure-category duality provides a measure analogue of Luzin sets – sets of positive outer measure, everyuncountable subset of which has positive outer measure. These sets are called Sierpiński sets, afterWacław Sierpiński.Sierpiński sets are weakly Luzin sets but are not Luzin sets.

33.2 Example of a Luzin set

Choose a collection of 2ℵ0 meager subsets of R such that every meager subset is contained in one of them. By thecontinuum hypothesis, it is possible to enumerate them as Sα for countable ordinals α. For each countable ordinal βchoose a real number xᵦ that is not in any of the sets Sα for α<β, which is possible as the union of these sets is meagerso is not the whole of R. Then the uncountable set X of all these real numbers xᵦ has only a countable number ofelements in each set Sα, so is a Luzin set.More complicated variations of this construction produce examples of Luzin sets that are subgroups, subfields orreal-closed subfields of the real numbers.

92

33.3. REFERENCES 93

33.3 References• Arkhangelskii, A V (1978), “STRUCTURE AND CLASSIFICATION OF TOPOLOGICAL SPACES ANDCARDINAL INVARIANTS”,RussianMathematical Surveys 33 (6): 33–96, doi:10.1070/RM1978v033n06ABEH003884Paper mentioning Luzin spaces

• Efimov, B.A. (2001), “Luzin space”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Kunen, Kenneth (1977), “Luzin spaces”, Topology Proceedings, Vol. I (Conf., Auburn Univ., Auburn, Ala.,1976), pp. 191–199, MR 0450063

• Lusin, N.N. (1914), “Sur un problème de M. Baire”, C.R. Acad. Sci. Paris 158: 1258–1261

• Oxtoby, John C. (1980), Measure and category: a survey of the analogies between topological and measurespaces, Berlin: Springer-Verlag, ISBN 0-387-90508-1

Chapter 34

Mesocompact space

In mathematics, in the field of general topology, a topological space is said to bemesocompact if every open cover hasa compact-finite open refinement.[1] That is, given any open cover, we can find an open refinement with the propertythat every compact set meets only finitely many members of the refinement.[2]

The following facts are true about mesocompactness:

• Every compact space, and more generally every paracompact space is mesocompact. This follows from thefact that any locally finite cover is automatically compact-finite.

• Every mesocompact space is metacompact, and hence also orthocompact. This follows from the fact that pointsare compact, and hence any compact-finite cover is automatically point finite.

34.1 Notes[1] Hart, Nagata & Vaughan, p200

[2] Pearl, p23

34.2 References• K.P. Hart; J. Nagata; J.E. Vaughan, eds. (2004), Encyclopedia of General Topology, Elsevier, ISBN 0-444-50355-2

• Pearl, Elliott, ed. (2007), Open Problems in Topology II, Elsevier, ISBN 0-444-52208-5

94

Chapter 35

Metacompact space

In mathematics, in the field of general topology, a topological space is said to be metacompact if every open coverhas a point finite open refinement. That is, given any open cover of the topological space, there is a refinement whichis again an open cover with the property that every point is contained only in finitely many sets of the refining cover.A space is countably metacompact if every countable open cover has a point finite open refinement.

35.1 Properties

The following can be said about metacompactness in relation to other properties of topological spaces:

• Every paracompact space is metacompact. This implies that every compact space is metacompact, and everymetric space is metacompact. The converse does not hold: a counter-example is the Dieudonné plank.

• Every metacompact space is orthocompact.

• Every metacompact normal space is a shrinking space

• The product of a compact space and a metacompact space is metacompact. This follows from the tube lemma.

• An easy example of a non-metacompact space (but a countably metacompact space) is the Moore plane.

• In order for a Tychonoff space X to be compact it is necessary and sufficient that X be metacompact andpseudocompact (see Watson).

35.2 Covering dimension

A topological space X is said to be of covering dimension n if every open cover of X has a point finite open refinementsuch that no point of X is included in more than n + 1 sets in the refinement and if n is the minimum value for whichthis is true. If no such minimal n exists, the space is said to be of infinite covering dimension.

35.3 See also

• Compact space

• Paracompact space

• Normal space

• Realcompact space

• Pseudocompact space

95

96 CHAPTER 35. METACOMPACT SPACE

• Mesocompact space

• Tychonoff space

• Glossary of topology

35.4 References• Watson, W. Stephen (1981). “Pseudocompact metacompact spaces are compact”. Proc. Amer. Math. Soc.81: 151–152. doi:10.1090/s0002-9939-1981-0589159-1.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446. P.23.

Chapter 36

Michael selection theorem

In functional analysis, a branch of mathematics, the most popular version of theMichael selection theorem, namedafter Ernest Michael, states the following:

Let E be a Banach space, X a paracompact space and φ : X → E a lower hemicontinuous multivaluedmap with nonempty convex closed values. Then there exists a continuous selection f : X→ E of φ.

Conversely, if any lower semicontinuous multimap from topological space X to a Banach space, withnonempty convex closed values admits continuous selection, then X is paracompact. This provides an-other characterization for paracompactness.

36.1 Other selection theorems

• Zero-dimensional Michael selection theorem

• Aumann measurable selection theorem

• Bressan-Colombo directionally continuous selection theorem

• Castaing representation theorem

• Fryszkowski decomposable map selection

• Helly’s selection theorem

• Kuratowski, Ryll-Nardzewski measurable selection theorem

36.2 References

• Michael, Ernest (1956), “Continuous selections. I”, Annals of Mathematics. Second Series (Annals of Mathe-matics) 63 (2): 361–382, doi:10.2307/1969615, JSTOR 1969615, MR 0077107

• Dušan Repovš; Pavel V. Semenov (2014). “Continuous Selections of Multivalued Mappings”. In Hart, K.P.; van Mill, J.; Simon, P. Recent progress in general topology III. Berlin: Springer. pp. 711–749. ISBN978-94-6239-023-2.

• Jean-Pierre Aubin, Arrigo Cellina Differential Inclusions, Set-Valued Maps And Viability Theory, Grundl. derMath. Wiss., vol. 264, Springer - Verlag, Berlin, 1984

• J.-P. Aubin and H. Frankowska Set-Valued Analysis, Birkh¨auser, Basel, 1990

• Klaus Deimling Multivalued Differential Equations, Walter de Gruyter, 1992

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98 CHAPTER 36. MICHAEL SELECTION THEOREM

• D.Repovs and P. V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Academic Publishers,Dordrecht 1998.

• D.Repovs and P. V. Semenov, Ernest Michael and theory of continuous selections, Topol. Appl. 155:8 (2008),755-763.

• Aliprantis, Kim C. Border Infinite dimensional analysis. Hitchhiker’s guide Springer

• S.Hu, N.Papageorgiou Handbook of multivalued analysis. Vol. I Kluwer

Chapter 37

Monotonically normal space

In mathematics, amonotonically normal space is a particular kind of normal space, with some special characteris-tics, and is such that it is hereditarily normal, and any two separated subsets are strongly separated. They are definedin terms of a monotone normality operator.A T1 topological space (X, T ) is said to be monotonically normal if the following condition holds:For every x ∈ G , where G is open, there is an open set µ(x,G) such that

1. x ∈ µ(x,G) ⊆ G

2. if µ(x,G) ∩ µ(y,H) = ∅ then either x ∈ H or y ∈ G .

There are some equivalent criteria of monotone normality.

37.1 Equivalent definitions

37.1.1 Definition 2

A space X is called monotonically normal if it is T1 and for each pair of disjoint closed subsets A,B there is an openset G(A,B) with the properties

1. A ⊆ G(A,B) ⊆ G(A,B)− ⊆ X\B and2. G(A,B) ⊆ G(A′, B′) , whenever A ⊆ A′ and B′ ⊆ B .

This operator G is called monotone normality operator.Note that if G is a monotone normality operator, then G defined by G(A,B) = G(A,B)\G(B,A)− is also amonotone normality operator; and G satisfies

G(A,B) ∩ G(B,A) = ∅

For this reason we some time take the monotone normality operator so as to satisfy the above requirement; and thatfacilitates the proof of some theorems and of the equivalence of the definitions as well.

37.1.2 Definition 3

A space X is called monotonically normal if it is T1 ,and to each pair (A, B) of subsets of X, with A ∩ B− = ∅ =B ∩A− , one can assign an open subset G(A, B) of X such that

1. A ⊆ G(A,B) ⊆ G(A,B)− ⊆ X\B,

2. G(A,B) ⊆ G(A′, B′) whenever A ⊆ A′ and B′ ⊆ B .

99

100 CHAPTER 37. MONOTONICALLY NORMAL SPACE

37.1.3 Definition 4

A space X is called monotonically normal if it is T1 and there is a function H that assigns to each ordered pair (p,C)where C is closed and p is without C, an open set H(p,C) satisfying:

1. p ∈ H(p, C) ⊆ X\C

2. if D is closed and p ∈ C ⊇ D thenH(p, C) ⊆ H(p,D)

3. if p = q are points in X, then H(p, q) ∩H(q, p) = ∅ .

37.2 Properties

An important example of these spaces would be, assuming Axiom of Choice, the linearly ordered spaces; however,it really needs axiom of choice for an arbitrary linear order to be normal (see van Douwen’s paper). Any generalisedmetric is monotonically normal even without choice. An important property of monotonically normal spaces is thatany two separated subsets are strongly separated there. Monotone normality is hereditary property and a monotoni-cally normal space is always normal by the first condition of the second equivalent definition.We list up some of the properties :

1. A closed map preserves monotone normality.

2. A monotonically normal space is hereditarily collectionwise normal.

3. Elastic spaces are monotonically normal.

37.3 Some discussion links• Heath, R. W.; Lutzer, D. J.; Zenor, P. L. (April 1973). “Monotonically Normal Spaces” (PDF). Transactionsof the American Mathematical Society 178: 481–493.

• Borges, Carlos R. (March 1973). “A Study of Monotonically Normal Spaces” (PDF). Proceedings of theAmerican Mathematical Society 38 (1): 211–214.

• van Douwen, Eric K. (September 1985). “Horrors of Topology Without AC: A Nonnormal Orderable Space”(PDF). Proceedings of the American Mathematical Society 95 (1): 101–105.

• Gartside, P. M. (1997). “Cardinal Invariants of Monotonically Normal Spaces” (PDF). Topology and its Ap-plications 77 (3): 303–314.

• Henno Brandsma’s discussion about Monotone Normality in Topology Atlas can be viewed here

Chapter 38

n-connected

This article is about the concept in algebraic topology. For the concept in graph theory, see Connectivity (graphtheory).

In the mathematical branch of algebraic topology, specifically homotopy theory, n-connectedness is a way to say thata space vanishes or that a map is an isomorphism “up to dimension n, in homotopy".

38.1 n-connected space

A topological space X is said to be n-connected when it is non-empty, path-connected, and its first n homotopygroups vanish identically, that is

πi(X) ≡ 0 , 1 ≤ i ≤ n,

where the left-hand side denotes the i-th homotopy group.The requirements of being non-empty and path-connected can be interpreted as (−1)-connected and 0-connected,respectively, which is useful in defining 0-connected and 1-connected maps, as below. The 0-th homotopy set can bedefined as:

π0(X, ∗) := [(S0, ∗), (X, ∗)].

This is only a pointed set, not a group, unless X is itself a topological group; the distinguished point is the class of thetrivial map, sending S0 to the base point of X. Using this set, a space is 0-connected if and only if the 0th homotopyset is the one-point set. The definition of homotopy groups and this homotopy set require that X be pointed (have achosen base point), which cannot be done if X is empty.A topological spaceX is path-connected if and only if its 0-th homotopy group vanishes identically, as path-connectednessimplies that any two points x1 and x2 in X can be connected with a continuous path which starts in x1 and ends in x2,which is equivalent to the assertion that every mapping from S0 (a discrete set of two points) to X can be deformedcontinuously to a constant map. With this definition, we can define X to be n-connected if and only if

πi(X) ≡ 0, 0 ≤ i ≤ n.

38.1.1 Examples

• A space X is (−1)-connected if and only if it is non-empty.

• A space X is 0-connected if and only if it is non-empty and path-connected.

101

102 CHAPTER 38. N-CONNECTED

• A space is 1-connected if and only if it is simply connected.

• An n-sphere is (n-1)-connected.

38.2 n-connected map

The corresponding relative notion to the absolute notion of an n-connected space is an n-connected map, which isdefined as a map whose homotopy fiber Ff is an (n − 1)-connected space. In terms of homotopy groups, it meansthat a map f : X → Y is n-connected if and only if:

• πi(f) : πi(X)∼→πi(Y ) is an isomorphism for i < n , and

• πn(f) : πn(X) ↠ πn(Y ) is a surjection.

The last condition is frequently confusing; it is because the vanishing of the (n − 1)-st homotopy group of the homotopyfiber Ff corresponds to a surjection on the nth homotopy groups, in the exact sequence:

πn(X)πn(f)→ πn(Y ) → πn−1(Ff).

If the group on the right πn−1(Ff) vanishes, then the map on the left is a surjection.Low-dimensional examples:

• A connected map (0-connected map) is one that is onto path components (0th homotopy group); this corre-sponds to the homotopy fiber being non-empty.

• A simply connected map (1-connected map) is one that is an isomorphism on path components (0th homotopygroup) and onto the fundamental group (1st homotopy group).

n-connectivity for spaces can in turn be defined in terms of n-connectivity of maps: a space X with basepoint x0 isan n-connected space if and only if the inclusion of the basepoint x0 → X is an n-connected map. The single pointset is contractible, so all its homotopy groups vanish, and thus “isomorphism below n and onto at n" corresponds tothe first n homotopy groups of X vanishing.

38.2.1 Interpretation

This is instructive for a subset: an n-connected inclusion A → X is one such that, up to dimension n−1, homotopiesin the larger space X can be homotoped into homotopies in the subset A.For example, for an inclusion map A → X to be 1-connected, it must be:

• onto π0(X),

• one-to-one on π0(A) → π0(X), and

• onto π1(X).

One-to-one on π0(A) → π0(X) means that if there is a path connecting two points a, b ∈ A by passing through X,there is a path in A connecting them, while onto π1(X) means that in fact a path in X is homotopic to a path in A.In other words, a function which is an isomorphism on πn−1(A) → πn−1(X) only implies that any element ofπn−1(A) that are homotopic in X are abstractly homotopic in A – the homotopy in A may be unrelated to thehomotopy in X – while being n-connected (so also onto πn(X) ) means that (up to dimension n−1) homotopies in Xcan be pushed into homotopies in A.This gives a more concrete explanation for the utility of the definition of n-connectedness: for example, a spacesuch that the inclusion of the k-skeleton in n-connected (for n>k) – such as the inclusion of a point in the n-sphere –means that any cells in dimension between k and n are not affecting the homotopy type from the point of view of lowdimensions.

38.3. APPLICATIONS 103

38.3 Applications

The concept of n-connectedness is used in the Hurewicz theorem which describes the relation between singularhomology and the higher homotopy groups.In geometric topology, cases when the inclusion of a geometrically-defined space, such as the space of immersionsM → N, into a more general topological space, such as the space of all continuous maps between two associatedspaces X(M) → X(N), are n-connected are said to satisfy a homotopy principle or “h-principle”. There are anumber of powerful general techniques for proving h-principles.

38.4 See also• connected space

• simply connected

• path-connected

• connective spectrum

Chapter 39

Negative-dimensional space

In topology, a discipline within mathematics, a negative-dimensional space is an extension of the usual notion ofspace allowing for negative dimensions.[1]

39.1 Definition

Suppose that Mt0 is a compact space of Hausdorff dimension t0, which is an element of a scale of compact spacesembedded in each other and parametrized by t (0 < t < ∞). Such scales are considered equivalent with respect toMt0if the compact spaces constituting them coincide for t ≥ t0. It is said that the compact space Mt0 is the hole in thisequivalent set of scales, and −t0 is the negative dimension of the corresponding equivalence class.[2]

39.2 History

By the 1940s, the science of topology had developed and studied a thorough basic theory of topological spaces ofpositive dimension. Motivated by computations, and to some extent aesthetics, topologists searched for mathematicalframeworks that extended our notion of space to allow for negative dimensions. Such dimensions, as well as thefourth and higher dimensions, are hard to imagine since we are not able to directly observe them. It wasn’t until the1960s that a special topological framework was constructed—the category of spectra. A spectrum is a generalizationof space that allows for negative dimensions. The concept of negative-dimensional spaces is applied, for example, toanalyze linguistic statistics.[3]

39.3 See also

• Cone (topology)

• Equidimensionality

• Join (topology)

• Suspension/desuspension

• Spectrum (topology)

39.4 References[1] Wolcott, Luke; McTernan, Elizabeth (2012). “ImaginingNegative-Dimensional Space” (PDF). In Bosch, Robert; McKenna,

Douglas; Sarhangi, Reza. Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture. Phoenix, Arizona,USA: Tessellations Publishing. pp. 637–642. ISBN 978-1-938664-00-7. ISSN 1099-6702. Retrieved 25 June 2015.

104

39.5. EXTERNAL LINKS 105

[2] Maslov, V.P. “General Notion of a Topological Space ofNegativeDimension andQuantization of Its Density”. springer.com.Retrieved 2015-06-23.

[3] Maslov, V.P. “Negative Dimension in General and Asymptotic Topology”. arxiv.org. Retrieved 2015-06-25.

39.5 External links• Отрицательная асимптотическая топологическая размерность, новый конденсат и их связь с квантованнымзаконом Ципфа. For a translation into English, see Maslov, V.P. (November 2006). “Negative asymptotictopological dimension, a new condensate, and their relation to the quantized Zipf law”. Mathematical Notes 80(5-6): 806–813. Retrieved 30 June 2015.

Chapter 40

Noetherian topological space

In mathematics, aNoetherian topological space is a topological space in which closed subsets satisfy the descendingchain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they arethe complements of the closed subsets. It can also be shown to be equivalent that every open subset of such a spaceis compact, and in fact the seemingly stronger statement that every subset is compact.

40.1 Definition

A topological space X is called Noetherian if it satisfies the descending chain condition for closed subsets: for anysequence

Y1 ⊇ Y2 ⊇ · · ·

of closed subsets Yi of X , there is an integerm such that Ym = Ym+1 = · · · .

40.2 Relation to compactness

The Noetherian condition can be seen as a strong compactness condition:

• Every Noetherian topological space is compact.

• A topological space X is Noetherian if and only if every subspace of X is compact. (i.e. X is hereditarilycompact).

40.3 Noetherian topological spaces from algebraic geometry

Many examples of Noetherian topological spaces come from algebraic geometry, where for the Zariski topologyan irreducible set has the intuitive property that any closed proper subset has smaller dimension. Since dimensioncan only 'jump down' a finite number of times, and algebraic sets are made up of finite unions of irreducible sets,descending chains of Zariski closed sets must eventually be constant.A more algebraic way to see this is that the associated ideals defining algebraic sets must satisfy the ascending chaincondition. That follows because the rings of algebraic geometry, in the classical sense, are Noetherian rings. Thisclass of examples therefore also explains the name.If R is a commutative Noetherian ring, then Spec(R), the prime spectrum of R, is a Noetherian topological space.More generally, a Noetherian scheme is a Noetherian topological space. The converse does not hold, since Spec(R)of a one-dimensional valuation domain R consists of exactly two points and therefore is Noetherian, but there areexamples of such rings which are not Noetherian.

106

40.4. EXAMPLE 107

40.4 Example

The space Ank (affine n -space over a field k ) under the Zariski topology is an example of a Noetherian topological

space. By properties of the ideal of a subset of Ank , we know that if

Y1 ⊇ Y2 ⊇ Y3 ⊇ · · ·

is a descending chain of Zariski-closed subsets, then

I(Y1) ⊆ I(Y2) ⊆ I(Y3) ⊆ · · ·

is an ascending chain of ideals of k[x1, . . . , xn]. Since k[x1, . . . , xn] is a Noetherian ring, there exists an integermsuch that

I(Ym) = I(Ym+1) = I(Ym+2) = · · · .

Since V (I(Y )) is the closure of Y for all Y, V (I(Yi)) = Yi for all i. Hence

Ym = Ym+1 = Ym+2 = · · ·

40.5 References• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

This article incorporates material fromNoetherian topological space on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

Chapter 41

Normal space

For normal vector space, see normal (geometry).

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4:every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called aT4 space. These conditions are examples of separation axioms and their further strengthenings define completelynormal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces.

41.1 Definitions

A topological space X is a normal space if, given any disjoint closed sets E and F, there are open neighbourhoodsU of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated byneighbourhoods.

U

E

V

F

The closed sets E and F, here represented by closed disks on opposite sides of the picture, are separated by their respective neigh-bourhoods U and V, here represented by larger, but still disjoint, open disks.

A T4 space is a T1 space X that is normal; this is equivalent to X being normal and Hausdorff.A completely normal space or a hereditarily normal space is a topological space X such that every subspace of Xwith subspace topology is a normal space. It turns out that X is completely normal if and only if every two separatedsets can be separated by neighbourhoods.A completely T4 space, or T5 space is a completely normal T1 space topological space X, which implies that X is

108

41.2. EXAMPLES OF NORMAL SPACES 109

Hausdorff; equivalently, every subspace of X must be a T4 space.A perfectly normal space is a topological space X in which every two disjoint closed sets E and F can be preciselyseparated by a continuous function f fromX to the real lineR: the preimages of 0 and 1 under f are, respectively,E and F. (In this definition, the real line can be replaced with the unit interval [0,1].)It turns out that X is perfectly normal if and only if X is normal and every closed set is a Gδ set. Equivalently, X isperfectly normal if and only if every closed set is a zero set. Every perfectly normal space is automatically completelynormal.[1]

A Hausdorff perfectly normal space X is a T6 space, or perfectly T4 space.Note that the terms “normal space” and “T4" and derived concepts occasionally have a different meaning. (Nonethe-less, “T5" always means the same as “completely T4", whatever that may be.) The definitions given here are the onesusually used today. For more on this issue, see History of the separation axioms.Terms like “normal regular space" and “normal Hausdorff space” also turn up in the literature – they simply meanthat the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space isthe same thing as a T4 space. Given the historical confusion of the meaning of the terms, verbal descriptions whenapplicable are helpful, that is, “normal Hausdorff” instead of “T4", or “completely normal Hausdorff” instead of “T5".Fully normal spaces and fully T4 spaces are discussed elsewhere; they are related to paracompactness.A locally normal space is a topological space where every point has an open neighbourhood that is normal. Everynormal space is locally normal, but the converse is not true. A classical example of a completely regular locallynormal space that is not normal is the Nemytskii plane.

41.2 Examples of normal spaces

Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:

• All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff;

• All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not ingeneral Hausdorff;

• All compact Hausdorff spaces are normal;

• In particular, the Stone–Čech compactification of a Tychonoff space is normal Hausdorff;

• Generalizing the above examples, all paracompact Hausdorff spaces are normal, and all paracompact regularspaces are normal;

• All paracompact topological manifolds are perfectly normal Hausdorff. However, there exist non-paracompactmanifolds which are not even normal.

• All order topologies on totally ordered sets are hereditarily normal and Hausdorff.

• Every regular second-countable space is completely normal, and every regular Lindelöf space is normal.

Also, all fully normal spaces are normal (even if not regular). Sierpinski space is an example of a normal space thatis not regular.

41.3 Examples of non-normal spaces

An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on thespectrum of a ring, which is used in algebraic geometry.A non-normal space of some relevance to analysis is the topological vector space of all functions from the real lineR to itself, with the topology of pointwise convergence. More generally, a theorem of A. H. Stone states that theproduct of uncountably many non-compact metric spaces is never normal.

110 CHAPTER 41. NORMAL SPACE

41.4 Properties

Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.[2]

The main significance of normal spaces lies in the fact that they admit “enough” continuous real-valued functions, asexpressed by the following theorems valid for any normal space X.Urysohn’s lemma: If A and B are two disjoint closed subsets of X, then there exists a continuous function f from Xto the real line R such that f(x) = 0 for all x in A and f(x) = 1 for all x in B. In fact, we can take the values of f to beentirely within the unit interval [0,1]. (In fancier terms, disjoint closed sets are not only separated by neighbourhoods,but also separated by a function.)More generally, the Tietze extension theorem: If A is a closed subset of X and f is a continuous function from A toR, then there exists a continuous function F: X→ R which extends f in the sense that F(x) = f(x) for all x in A.If U is a locally finite open cover of a normal space X, then there is a partition of unity precisely subordinate to U.(This shows the relationship of normal spaces to paracompactness.)In fact, any space that satisfies any one of these conditions must be normal.A product of normal spaces is not necessarily normal. This fact was first proved by Robert Sorgenfrey. An example ofthis phenomenon is the Sorgenfrey plane. Also, a subset of a normal space need not be normal (i.e. not every normalHausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čechcompactification (which is normal Hausdorff). A more explicit example is the Tychonoff plank.

41.5 Relationships to other separation axioms

If a normal space is R0, then it is in fact completely regular. Thus, anything from “normal R0" to “normal completelyregular” is the same as what we normally call normal regular. Taking Kolmogorov quotients, we see that all normalT1 spaces are Tychonoff. These are what we normally call normal Hausdorff spaces.A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, thereare disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa.Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpinski spaceis normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.

41.6 Citations[1] Munkres 2000, p. 213

[2] Willard, Stephen (1970). General topology. Reading, Mass.: Addison-Wesley Pub. Co. pp. 100–101. ISBN 0486434796.

41.7 References• Kemoto, Nobuyuki (2004). “Higher Separation Axioms”. In K.P. Hart, J. Nagata, and J.E. Vaughan. Ency-clopedia of General Topology. Amsterdam: Elsevier Science. ISBN 0-444-50355-2.

• Munkres, James R. (2000). Topology (2nd ed.). Prentice-Hall. ISBN 0-13-181629-2.

• Sorgenfrey, R.H. (1947). “On the topological product of paracompact spaces”. Bull. Amer. Math. Soc. 53:631–632. doi:10.1090/S0002-9904-1947-08858-3.

• Stone, A. H. (1948). “Paracompactness and product spaces”. Bull. Amer. Math. Soc. 54: 977–982.doi:10.1090/S0002-9904-1948-09118-2.

• Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6.

Chapter 42

Orthocompact space

In mathematics, in the field of general topology, a topological space is said to be orthocompact if every open coverhas an interior preserving open refinement. That is, given an open cover of the topological space, there is a refinementwhich is also an open cover, with the further property that at any point, the intersection of all open sets in the refinementcontaining that point, is also open.If the number of open sets containing the point is finite, then their intersection is clearly open. That is, every pointfinite open cover is interior preserving. Hence, we have the following: every metacompact space, and in particular,every paracompact space, is orthocompact.Useful theorems:

• Orthocompactness is a topological invariant; that is, it is preserved by homeomorphisms.

• Every closed subspace of an orthocompact space is orthocompact.

• A topological space X is orthocompact if and only if every open cover of X by basic open subsets of X has aninterior-preserving refinement that is an open cover of X.

• The product X × [0,1] of the closed unit interval with an orthocompact space X is orthocompact if and only ifX is countably metacompact. (B.M. Scott) [1]

• Every orthocompact space is countably orthocompact.

• Every countably orthocompact Lindelöf space is orthocompact.

42.1 References[1] B.M. Scott, Towards a product theory for orthocompactness, “Studies in Topology”, N.M. Stavrakas and K.R. Allen, eds

(1975), 517–537.

• P. Fletcher, W.F. Lindgren, Quasi-uniform Spaces, Marcel Dekker, 1982, ISBN 0-8247-1839-9. Chap.V.

111

Chapter 43

P-space

For the complexity class, see PSPACE.

In the mathematical field of topology, there are various notions of a P-space and of a p-space.

43.1 Generic use

The expression P-space might be used generically to denote a topological space satisfying some given and previouslyintroduced topological invariant P.[1] This might apply also to spaces of a different kind, e. g., non topological, orwith additional structure.

43.2 P-spaces in the sense of Gillman–Henriksen

A P-space in the sense of Gillman–Henriksen is a topological space in which every countable intersection of opensets is open. An equivalent condition is that countable unions of closed sets are closed. In other words, Gδ sets areopen and Fσ sets are closed. The letter P stands for both pseudo-discrete and prime.Different authors restrict their attention to topological spaces that satisfy various separation axioms. With the rightaxioms, one may characterize P-spaces in terms of their rings of continuous real-valued functions.Special kinds of P-spaces include Alexandrov spaces, in which arbitrary intersections of open sets are open. Thesein turn include locally finite spaces, which include finite spaces and discrete spaces.

43.3 P-spaces in the sense of Morita

A different notion of a P-space has been introduced by Kiiti Morita in 1964, in connection with his (now solved)conjectures (see the relative entry for more information). Spaces satisfying the covering property introduced byMorita are sometimes also called Morita P-spaces or normal P-spaces.

43.4 p-spaces

A notion of a p-space has been introduced by Alexander Arhangelskii.[2]

43.5 References[1] Aisling E. McCluskey, Comparison of Topologies (Minimal and Maximal Topologies), Chapter a7 in Encyclopedia of Gen-

eral Topology, Edited by Klaas Pieter Hart, Jun-iti Nagata and Jerry E. Vaughan, 2003 Elsevier B.V.

112

43.6. FURTHER READING 113

[2] Encyclopedia of General Topology, p. 278.

43.6 Further reading• Gillman, Leonard; Henriksen, Melvin (September 1954), “Concerning Rings of Continuous Functions”, Trans-actions of the American Mathematical Society 77 (2): 340–362, doi:10.2307/1990875, retrieved January 28,2014

• Misra, Arvind K. (December 1972), “A topological view of P-spaces”, General Topology and its Applications2 (4): 349–362, doi:10.1016/0016-660X(72)90026-8

43.7 External links• Hart, K.P. (2001), “P-space”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• P-space at PlanetMath.org.

Chapter 44

Paracompact space

In mathematics, a paracompact space is a topological space in which every open cover has an open refinement thatis locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Everyparacompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions ofunity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces arealways closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact spaceis called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact.Tychonoff’s theorem (which states that the product of any collection of compact topological spaces is compact) doesnot generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact. However,the product of a paracompact space and a compact space is always paracompact.Every metric space is paracompact. A topological space is metrizable if and only if it is a paracompact and locallymetrizable Hausdorff space.

44.1 Paracompactness

A cover of a set X is a collection of subsets of X whose union contains X. In symbols, if U = Uα : α in A is anindexed family of subsets of X, then U is a cover of X if

X ⊆∪α∈A

Uα.

A cover of a topological space X is open if all its members are open sets. A refinement of a cover of a space X is anew cover of the same space such that every set in the new cover is a subset of some set in the old cover. In symbols,the cover V = Vᵦ : β in B is a refinement of the cover U = Uα : α in A if and only if, for any Vᵦ in V, thereexists some Uα in U such that Vᵦ⊆Uα.An open cover of a space X is locally finite if every point of the space has a neighborhood that intersects only finitelymany sets in the cover. In symbols, U = Uα : α in A is locally finite if and only if, for any x in X, there exists someneighbourhood V(x) of x such that the set

α ∈ A : Uα ∩ V (x) = ∅

is finite. A topological spaceX is now said to be paracompact if every open cover has a locally finite open refinement.

44.2 Examples• Every compact space is paracompact.

114

44.3. PROPERTIES 115

• Every regular Lindelöf space is paracompact. In particular, every locally compact Hausdorff second-countablespace is paracompact.

• The Sorgenfrey line is paracompact, even though it is neither compact, locally compact, second countable, normetrizable.

• Every CW complex is paracompact [1]

• (Theorem of A. H. Stone) Every metric space is paracompact.[2] Early proofs were somewhat involved, butan elementary one was found by M. E. Rudin.[3] Existing proofs of this require the axiom of choice for thenon-separable case. It has been shown that neither ZF theory nor ZF theory with the axiom of dependentchoice is sufficient.[4]

Some examples of spaces that are not paracompact include:

• The most famous counterexample is the long line, which is a nonparacompact topological manifold. (The longline is locally compact, but not second countable.)

• Another counterexample is a product of uncountably many copies of an infinite discrete space. Any infiniteset carrying the particular point topology is not paracompact; in fact it is not even metacompact.

• The Prüfer manifold is a non-paracompact surface.

• The bagpipe theorem shows that there are 2ℵ1 isomorphism classes of non-paracompact surfaces.

44.3 Properties

Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact. This canbe extended to F-sigma subspaces as well.

• A regular space is paracompact if every open cover admits a locally finite refinement. (Here, the refinement isnot required to be open.) In particular, every regular Lindelof space is paracompact.

• (Smirnov metrization theorem) A topological space is metrizable if and only if it is paracompact, Hausdorff,and locally metrizable.

• Michael selection theorem states that lower semicontinuousmultifunctions fromX into nonempty closed convexsubsets of Banach spaces admit continuous selection iff X is paracompact.

Although a product of paracompact spaces need not be paracompact, the following are true:

• The product of a paracompact space and a compact space is paracompact.

• The product of a metacompact space and a compact space is metacompact.

Both these results can be proved by the tube lemma which is used in the proof that a product of finitely many compactspaces is compact.

44.4 Paracompact Hausdorff Spaces

Paracompact spaces are sometimes required to also be Hausdorff to extend their properties.

• (Theorem of Jean Dieudonné) Every paracompact Hausdorff space is normal.

• Every paracompact Hausdorff space is a shrinking space, that is, every open cover of a paracompact Hausdorffspace has a shrinking: another open cover indexed by the same set such that the closure of every set in the newcover lies inside the corresponding set in the old cover.

• On paracompact Hausdorff spaces, sheaf cohomology and Čech cohomology are equal.[5]

116 CHAPTER 44. PARACOMPACT SPACE

44.4.1 Partitions of unity

The most important feature of paracompact Hausdorff spaces is that they are normal and admit partitions of unitysubordinate to any open cover. This means the following: if X is a paracompact Hausdorff space with a given opencover, then there exists a collection of continuous functions on X with values in the unit interval [0, 1] such that:

• for every function f: X → R from the collection, there is an open set U from the cover such that the supportof f is contained in U;

• for every point x in X, there is a neighborhood V of x such that all but finitely many of the functions in thecollection are identically 0 in V and the sum of the nonzero functions is identically 1 in V.

In fact, a T1 space is Hausdorff and paracompact if and only if it admits partitions of unity subordinate to any opencover (see below). This property is sometimes used to define paracompact spaces (at least in the Hausdorff case).Partitions of unity are useful because they often allow one to extend local constructions to the whole space. Forinstance, the integral of differential forms on paracompact manifolds is first defined locally (where the manifold lookslike Euclidean space and the integral is well known), and this definition is then extended to the whole space via apartition of unity.

Proof that paracompact Hausdorff spaces admit partitions of unity

A Hausdorff space X is paracompact if and only if it every open cover admits a subordinate partition of unity. Theif direction is straightforward. Now for the only if direction, we do this in a few stages.

Lemma 1: If O is a locally finite open cover, then there exists open sets WU for each U ∈ O , suchthat each WU ⊆ U and WU : U ∈ O is a locally finite refinement.

Lemma 2: IfO is a locally finite open cover, then there are continuous functions fU : X → [0, 1] suchthat supp fU ⊆ U and such that f :=

∑U∈O fU is a continuous function which is always non-zero

and finite.

Theorem: In a paracompact Hausdorff spaceX , ifO is an open cover, then there exists a partition ofunity subordinate to it.

Proof (Lemma 1): Let V be the collection of open sets meeting only finitely many sets in O , andwhose closure is contained in a set in O . One can check as an exercise that this provides an openrefinement, since paracompact Hausdorff spaces are regular, and since O is locally finite. Now replaceV by a locally finite open refinement. One can easily check that each set in this refinement has the sameproperty as that which characterised the original cover.

Now we define WU =∪A ∈ V : A ⊆ U . We have that each WU ⊆ U ; for otherwise: suppose

there is x ∈ WU \ U . We will show that there is closed set C ⊃ WU such that x /∈ C (this meanssimply x /∈ WU by definition of closure). Since we chose V to be locally finite there is neighbourhoodV [x] of x such that only finitely many sets U1, ..., Un ∈ A ∈ V : A ⊆ U have non-empty intersectionwith V [x] . We take their closures U1, ..., Un and then V := V [x] \ ∪Ui is an open set (since sum isfinite) such that V ∩WU = ∅ . Moreover x ∈ V , because ∀i = 1, ..., n we have Ui ⊆ U and weknow that x /∈ U . Then C := X \ V is closed set without x which conatins WU . So x /∈ WU andwe've reached contradiction. And it easy to see that WU : U ∈ O is an open refinement of O .

Finally, to verify that this cover is locally finite, fix x ∈ X and letN be a neighbourhood of x . We knowthat for each U we haveWU ⊆ U . Since O is locally finite there are only finitely many sets U1, ..., Uk

having non-empty intersection withN . Then only setsWU1 , ...,WUkhave non-empty intersection with

N , because for every other U ′ we have N ∩WU ′ ⊆ N ∩ U ′ = ∅

44.5. RELATIONSHIP WITH COMPACTNESS 117

Proof (Lemma 2): Applying Lemma 1, let fU : X → [0, 1] be coninuous maps with fU WU = 1and supp fU ⊆ U (by Urysohn’s lemma for disjoint closed sets in normal spaces, which a paracompactHausdorff space is). Note by the support of a function, we here mean the points not mapping to zero(and not the closure of this set). To show that f =

∑U∈O fU is always finite and non-zero, take x ∈ X

, and let N a neighbourhood of x meeting only finitely many sets in O ; thus x belongs to only finitelymany sets in O ; thus fU (x) = 0 for all but finitely many U ; moreover x ∈ WU for some U ,thus fU (x) = 1 ; so f(x) is finite and ≥ 1 . To establish continuity, take x,N as before, and letS = U ∈ O : N meets U , which is finite; then f N =

∑U∈S fU N , which is a continuous

function; hence the preimage under f of a neighbourhood of f(x) will be a neighbourhood of x .

Proof (Theorem): TakeO∗ a locally finite subcover of the refinement cover: V open : (∃U ∈ O)V ⊆U . Applying Lemma 2, we obtain continuous functions fW : X → [0, 1] with supp fW ⊆ W (thusthe usual closed version of the support is contained in some U ∈ O , for eachW ∈ O∗ ; for which theirsum constitutes a continuous function which is always finite non-zero (hence 1/f is continuous positive,finite-valued). So replacing each fW by fW /f , we have now — all things remaining the same — thattheir sum is everywhere 1 . Finally for x ∈ X , lettingN be a neighbourhood of x meeting only finitelymany sets in O∗ , we have fW N = 0 for all but finitely manyW ∈ O∗ since each supp fW ⊆ W. Thus we have a partition of unity subordinate to the original open cover.

44.5 Relationship with compactness

There is a similarity between the definitions of compactness and paracompactness: For paracompactness, “subcover”is replaced by “open refinement” and “finite” by is replaced by “locally finite”. Both of these changes are significant:if we take the definition of paracompact and change “open refinement” back to “subcover”, or “locally finite” back to“finite”, we end up with the compact spaces in both cases.Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topologicalspace entities into manageable pieces.

44.5.1 Comparison of properties with compactness

Paracompactness is similar to compactness in the following respects:

• Every closed subset of a paracompact space is paracompact.

• Every paracompact Hausdorff space is normal.

It is different in these respects:

• A paracompact subset of a Hausdorff space need not be closed. In fact, for metric spaces, all subsets areparacompact.

• A product of paracompact spaces need not be paracompact. The square of the real line R in the lower limittopology is a classical example for this.

44.6 Variations

There are several variations of the notion of paracompactness. To define them, we first need to extend the list ofterms above:A topological space is:

• metacompact if every open cover has an open pointwise finite refinement.

• orthocompact if every open cover has an open refinement such that the intersection of all the open sets aboutany point in this refinement is open.

118 CHAPTER 44. PARACOMPACT SPACE

• fully normal if every open cover has an open star refinement, and fully T4 if it is fully normal and T1 (seeseparation axioms).

The adverb "countably" can be added to any of the adjectives “paracompact”, “metacompact”, and “fully normal” tomake the requirement apply only to countable open covers.Every paracompact space is metacompact, and every metacompact space is orthocompact.

44.6.1 Definition of relevant terms for the variations

• Given a cover and a point, the star of the point in the cover is the union of all the sets in the cover that containthe point. In symbols, the star of x in U = Uα : α in A is

U∗(x) :=∪

Uα∋x

Uα.

The notation for the star is not standardised in the literature, and this is just one possibility.

• A star refinement of a cover of a space X is a new cover of the same space such that, given any point in thespace, the star of the point in the new cover is a subset of some set in the old cover. In symbols, V is a starrefinement of U = Uα : α in A if and only if, for any x in X, there exists a Uα in U, such that V*(x) iscontained in Uα.

• A cover of a space X is pointwise finite if every point of the space belongs to only finitely many sets in thecover. In symbols, U is pointwise finite if and only if, for any x in X, the set

α ∈ A : x ∈ Uα

is finite.

As the name implies, a fully normal space is normal. Every fully T4 space is paracompact. In fact, for Hausdorffspaces, paracompactness and full normality are equivalent. Thus, a fully T4 space is the same thing as a paracompactHausdorff space.As an historical note: fully normal spaces were defined before paracompact spaces. The proof that all metrizablespaces are fully normal is easy. When it was proved by A.H. Stone that for Hausdorff spaces fully normal andparacompact are equivalent, he implicitly proved that all metrizable spaces are paracompact. Later M.E. Rudin gavea direct proof of the latter fact.

44.7 See also• a-paracompact space

• Paranormal space

44.8 Notes[1] Hatcher, Allen, Vector bundles and K-theory, preliminary version available on the author’s homepage

[2] Stone, A. H. Paracompactness and product spaces. Bull. Amer. Math. Soc. 54 (1948), 977-982

[3] Rudin, Mary Ellen. A new proof that metric spaces are paracompact. Proceedings of the American Mathematical Society,Vol. 20, No. 2. (Feb., 1969), p. 603.

[4] C. Good, I. J. Tree, and W. S. Watson. On Stone’s Theorem and the Axiom of Choice. Proceedings of the AmericanMathematical Society, Vol. 126, No. 4. (April, 1998), pp. 1211–1218.

[5] Brylinski, Jean-Luc (2007), Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics 107,Springer, p. 32, ISBN 9780817647308.

44.9. REFERENCES 119

44.9 References• Dieudonné, Jean (1944), “Une généralisation des espaces compacts”, Journal de Mathématiques Pures et Ap-pliquées, Neuvième Série 23: 65–76, ISSN 0021-7824, MR 0013297

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology (2 ed), Springer Verlag, 1978,ISBN 3-540-90312-7. P.23.

• Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6.

• Mathew, Akhil. “Topology/Paracompactness”.

44.10 External links• Hazewinkel, Michiel, ed. (2001), “Paracompact space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 45

Paranormal space

Not to be confused with Paranormal phenomena outside the range of normal experience.

In mathematics, in the realm of topology, a paranormal space (Nyikos 1984) is a topological space in which everycountable discrete collection of closed sets has a locally finite open expansion.

45.1 See also• Normal space – a topological space in which every two disjoint closed sets have disjoint open neighborhoods

• Paracompact space – a topological space in which every open cover admits an open locally finite refinement

45.2 References• Nyikos (1984), “Problem Section: Problem B. 25,”, Top. Proc. 9

• Smith, Kerry D.; Szeptycki, Paul J. (2000), “Paranormal spaces under ◊*", Proceedings of the American Math-ematical Society 128 (3): 903–908, doi:10.1090/S0002-9939-99-05032-7, ISSN 0002-9939, MR 1622981

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Chapter 46

Perfect set

In mathematics, in the field of topology, a perfect set is a closed set with no isolated points, and a perfect spaceis any topological space with no isolated points. In such spaces, every point can be approximated arbitrarily wellby other points – given any point and any topological neighborhood of the point, there is another point within theneighborhood.The term perfect space is also used, incompatibly, to refer to other properties of a topological space, such as being aGδ space. Context is required to determine which meaning is intended.In this article, a space which is not perfect will be referred to as imperfect.

46.1 Examples and nonexamples

The real line R is a connected perfect space, while the Cantor space 2ω and Baire space ωω are perfect, totallydisconnected zero-dimensional spaces.Any nonempty set admits an imperfect topology: the discrete topology. Any set with more than one point admits aperfect topology: the indiscrete topology.

46.2 Imperfection of a space

Define the imperfection of a topological space to be the number of isolated points. This is a cardinal invariant – i.e.,a mapping which assigns to each topological space a cardinal number such that homeomorphic spaces get assignedthe same number.A space is perfect if and only if it has imperfection zero.

46.3 Closure properties

Every nonempty perfect space has subsets which are imperfect in the subspace topology, namely the singleton sets.However, any open subspace of a perfect space is perfect.Perfection is a local property of a topological space: a space is perfect if and only if every point in the space admitsa basis of neighborhoods each of which is perfect in the subspace topology.Let Xii∈I be a family of topological spaces. As for any local property, the disjoint union

⨿i Xi is perfect if and

only if every Xi is perfect.The Cartesian product of a family Xii∈I is perfect in the product topology if and only if at least one of the followingholds:(i) At least oneXi is perfect.(ii) I = ∅ .

121

122 CHAPTER 46. PERFECT SET

(iii) The set of indices i ∈ I such that Xi has at least two points is infinite.A continuous image, and even a quotient, of a perfect space need not be perfect. For example, let X = R − 0, letY = 1, 2 given the discrete topology and let f be a function defined such that f(x) = 2 if x > 0 and f(x) = 1 if x <0. However, every image of a perfect space under an injective continuous map is perfect.

46.4 Connection with other topological properties

It is natural to compare the concept of a perfect space – in which no singleton set is open – to that of a T1 space – inwhich every singleton set is closed.A T1 space is perfect if and only if every point of the space is an ω -accumulation point. In particular a nonemptyperfect T1 space is infinite.Any connected T1 space with more than one point is perfect. (More interesting therefore are disconnected perfectspaces, especially totally disconnected perfect spaces like Cantor space and Baire space.)On the other hand, the set X = , • endowed with the topology ∅, , X is connected, T0 (and even sober)but not perfect (this space is called Sierpinski space).Suppose X is a homogeneous topological space, i.e., the group Aut(X) of self-homeomorphisms acts transitively onX. Then X is either perfect or discrete. This holds in particular for all topological groups.A space which is of the first category is necessarily perfect (so, similar to compactifiying a space, we can 'make' aspace to be of the second category by taking the disjoint union with a one-point space).

46.5 Perfect spaces in descriptive set theory

Classical results in descriptive set theory establish limits on the cardinality of non-empty, perfect spaces with addi-tional completeness properties. These results show that:

• IfX is a complete metric space with no isolated points, then the Cantor space 2ω can be continuously embeddedinto X. Thus X has cardinality at least 2ℵ0 . If X is a separable, complete metric space with no isolated points,the cardinality of X is exactly 2ℵ0 .

• If X is a locally compact Hausdorff space with no isolated points, there is an injective function (not necessarilycontinuous) from Cantor space to X, and so X has cardinality at least 2ℵ0 .

46.6 See also• Dense-in-itself

• Finite intersection property

• Derived set (mathematics)

• Subspace topology

46.7 References• Kechris, A. S. (1995), Classical Descriptive Set Theory, Berlin, NewYork: Springer-Verlag, ISBN 3540943749

• Levy, A. (1979), Basic Set Theory, Berlin, New York: Springer-Verlag

• edited by Elliott Pearl. (2007), Pearl, Elliott, ed., Open problems in topology. II, Elsevier, ISBN 978-0-444-52208-5, MR 2367385

Chapter 47

Polyadic space

In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topologicalpower of an Alexandroff one-point compactification of a discrete topological space.

47.1 History

Polyadic spaces were first studied by S. Mrówka in 1970 as a generalisation of dyadic spaces.[1] The theory wasdeveloped further by R. H. Marty, János Gerlits and Murray G. Bell,[2] the latter of whom introduced the concept ofthe more general centred spaces.[1]

47.2 Background

A subset K of a topological space X is said to be compact if every open cover of K contains a finite subcover. It issaid to be locally compact at a point x ∈ X if x lies in the interior of some compact subset of X. X is a locally compactspace if it is locally compact at every point in the space.[3]

A proper subset A ⊂ X is said to be dense if the closure Ā = X. A space whose set has a countable, dense subset iscalled a separable space.For a non-compact, locally compact Hausdorff topological space (X, τX) , we define the Alexandroff one-pointcompactification as the topological space with the set ω∪X , denoted ωX , where ω /∈ X , with the topology τωX

defined as follows:[4][2]

• τX ⊆ τωX

• X \ C ∪ ω ∈ τωX , for every compact subset C ⊆ X .

47.3 Definition

LetX be a discrete topological space, and let ωX be an Alexandroff one-point compactification ofX . A Hausdorffspace P is polyadic if for some cardinal number λ , there exists a continuous surjective function f : ωXλ → P ,where ωXλ is the product space obtained by multiplying ωX with itself λ times.[5]

47.4 Examples

Take the set of natural numbers Z+ with the discrete topology. Its Alexandroff one-point compactification is ωZ+. Choose λ = 1 and define the homeomorphism h : ωZ+ → [0, 1] with the mapping

123