Properties of Topological Spaces_2

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 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2.1 Modern denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.2 Historical denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Hausdor 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 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.3.1 Open cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3.2 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Grammatical aspect 5718.1 Basic concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

18.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.1.2 Modern usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

18.2 Common aspectual distinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.3 Aspect vs. tense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.4 Lexical vs. grammatical aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.5 Indicating aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.6 Aspect by language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

18.6.1 English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6018.6.2 German vernacular and colloquial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.6.3 Slavic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6218.6.4 Romance languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6218.6.5 Finnic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.6.6 Austronesian languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.6.7 Creole languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.6.8 American Sign Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

18.7 Terms for various aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6518.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6618.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6618.10Other references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

19 Grammatical mood 68

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19.1 Realis moods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6819.1.1 Indicative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

19.2 Irrealis moods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6919.2.1 Subjunctive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6919.2.2 Conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7019.2.3 Optative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7019.2.4 Imperative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7019.2.5 Jussive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7119.2.6 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7119.2.7 Inferential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

19.3 Other moods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7119.3.1 Interrogative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7119.3.2 Deity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

19.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7219.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7219.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

20 H-closed space 7320.1 Examples and equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7320.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7320.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

21 HeineBorel theorem 7421.1 History and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7421.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7421.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7621.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7621.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7621.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

22 Hemicompact space 7822.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

23 Hyperconnected space 8023.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8023.2 Hyperconnectedness vs. connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8023.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8023.4 Irreducible components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8123.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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23.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

24 Kolmogorov space 8224.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8224.2 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

24.2.1 Spaces which are not T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8224.2.2 Spaces which are T0 but not T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

24.3 Operating with T0 spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.4 The Kolmogorov quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.5 Removing T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8424.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

25 Limit point compact 8525.1 Properties and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8525.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8525.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8625.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

26 Lindelf space 8726.1 Properties of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.2 Properties of strongly Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.3 Product of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

27 Locally compact space 8927.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8927.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

27.2.1 Compact Hausdor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.2.2 Locally compact Hausdor spaces that are not compact . . . . . . . . . . . . . . . . . . . 9027.2.3 Hausdor spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 9027.2.4 Non-Hausdor examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

27.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.3.1 The point at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

27.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

28 Locally connected space 9328.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.2 Denitions and rst examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

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28.2.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.4 Components and path components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

28.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.5 Quasicomponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

28.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.6 More on local connectedness versus weak local connectedness . . . . . . . . . . . . . . . . . . . . 9728.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9828.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

29 Locally nite collection 9929.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

29.1.1 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9929.1.2 Second countable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

29.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10029.3 Countably locally nite collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10029.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

30 Locally nite space 10130.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

31 Locally Hausdor space 10231.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

32 Locally normal space 10332.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10332.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10332.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10332.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10332.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

33 Locally regular space 10533.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

34 Locally simply connected space 10634.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

35 Luzin space 10835.1 In real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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35.2 Example of a Luzin set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10835.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

36 Mesocompact space 11036.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11036.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

37 Metacompact space 11137.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11137.2 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11137.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11137.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

38 Michael selection theorem 11338.1 Other selection theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11338.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

39 Monotonically normal space 11539.1 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

39.1.1 Denition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11539.1.2 Denition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11539.1.3 Denition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

39.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11639.3 Some discussion links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

40 n-connected 11740.1 n-connected space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

40.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11740.2 n-connected map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

40.2.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11840.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11940.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

41 Noetherian topological space 12041.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12041.2 Relation to compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12041.3 Noetherian topological spaces from algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . 12041.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12141.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

42 Normal space 12242.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12242.2 Examples of normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12342.3 Examples of non-normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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42.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12442.5 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12442.6 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12442.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

43 Orthocompact space 12543.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

44 P-space 12644.1 Generic use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12644.2 P-spaces in the sense of GillmanHenriksen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12644.3 P-spaces in the sense of Morita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12644.4 p-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12644.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12644.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12744.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

45 Paracompact space 12845.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12845.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12845.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12945.4 Paracompact Hausdor Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

45.4.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13045.5 Relationship with compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

45.5.1 Comparison of properties with compactness . . . . . . . . . . . . . . . . . . . . . . . . . 13145.6 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

45.6.1 Denition of relevant terms for the variations . . . . . . . . . . . . . . . . . . . . . . . . . 13245.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13245.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13245.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13345.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

46 Paranormal space 13446.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13446.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

47 Perfect set 13547.1 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13547.2 Imperfection of a space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13547.3 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13547.4 Connection with other topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13647.5 Perfect spaces in descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13647.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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47.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

48 Pluperfect 13748.1 Meaning of the pluperfect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13748.2 Examples from various languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

48.2.1 Greek and Latin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13848.2.2 English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13848.2.3 Other Germanic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13848.2.4 German . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13848.2.5 Dutch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13948.2.6 Romance languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13948.2.7 Slavic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14048.2.8 Other languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

48.3 Table of forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14148.4 Dierent perfect construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14148.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14148.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14248.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

49 Polyadic space 14349.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14349.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14349.3 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14349.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14349.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

49.5.1 Ramseys theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14449.5.2 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

49.6 Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14549.6.1 Centred space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14549.6.2 AD-compact space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14549.6.3 -adic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14549.6.4 Hyadic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

49.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14649.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

50 Pseudocompact space 14850.1 Properties related to pseudocompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14850.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14850.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

51 Pseudometric space 15051.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15051.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

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51.3 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15151.4 Metric identication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15151.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15151.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

52 Pseudonormal space 15352.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

53 Realcompact space 15453.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15453.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15453.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

54 Regular space 15654.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15654.2 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15754.3 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15754.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15854.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

55 Relatively compact subspace 15955.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15955.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

56 Resolvable space 16056.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16056.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16056.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

57 Rickart space 16157.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

58 Second-countable space 16258.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

58.1.1 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16258.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16358.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

59 Semi-locally simply connected 16459.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16459.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16459.3 Topology of fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16559.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16559.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

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60 Separable space 16660.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16660.2 Separability versus second countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16660.3 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16760.4 Constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16760.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

60.5.1 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16760.5.2 Non-separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

60.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16860.6.1 Embedding separable metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

60.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

61 Sequential space 17061.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17061.2 Sequential closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17061.3 FrchetUrysohn space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17161.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17161.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17161.6 Equivalent conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17261.7 Categorical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17261.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17261.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

62 Shrinking space 17462.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

63 Simply connected at innity 17563.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

64 Simply connected space 17664.1 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17664.2 Formal denition and equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17764.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17764.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17964.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17964.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

65 Sub-Stonean space 18065.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18065.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18065.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

66 Supercompact space 18166.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

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66.2 Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18166.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

67 T1 space 18367.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18367.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18367.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18467.4 Generalisations to other kinds of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18567.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

68 Tenseaspectmood 18668.1 Creoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

68.1.1 Hawaiian Creole English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18668.2 Modern Greek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18768.3 Slavic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

68.3.1 Russian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18868.4 Romance languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

68.4.1 French . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18868.4.2 Italian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18868.4.3 Portuguese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18968.4.4 Spanish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

68.5 Germanic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19068.5.1 German . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19068.5.2 Danish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19068.5.3 Dutch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19068.5.4 Icelandic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19168.5.5 English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

68.6 Basque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19568.7 Hawaiian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19568.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19568.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

69 Topological manifold 19769.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19769.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19769.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

69.3.1 The Hausdor axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19869.3.2 Compactness and countability axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19869.3.3 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

69.4 Coordinate charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19969.5 Classication of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19969.6 Manifolds with boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

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69.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20069.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20069.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

70 Topological property 20170.1 Common topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

70.1.1 Cardinal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20170.1.2 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20170.1.3 Countability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20270.1.4 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20270.1.5 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20370.1.6 Metrizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20370.1.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

70.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20470.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20470.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

71 Toronto space 20571.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

72 Totally disconnected space 20672.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20672.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20672.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20772.4 Constructing a disconnected space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20772.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20772.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

73 Ultraconnected space 20873.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20873.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20873.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

74 Uniformizable space 20974.1 Induced uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20974.2 Fine uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20974.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

75 Uses of English verb forms 21175.1 Inected forms of verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21175.2 Verbs in combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21175.3 Tenses, aspects and moods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

75.3.1 Tenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21275.3.2 Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

CONTENTS xv

75.3.3 Moods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21475.4 Active and passive voice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21575.5 Negation and questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21675.6 Modal verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21675.7 Uses of verb combination types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

75.7.1 Simple present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21675.7.2 Present progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21875.7.3 Present perfect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21875.7.4 Present perfect progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21975.7.5 Simple past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22075.7.6 Past progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22075.7.7 Past perfect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22175.7.8 Past perfect progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22275.7.9 Simple future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22275.7.10 Future progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22375.7.11 Future perfect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22375.7.12 Future perfect progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22375.7.13 Simple conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22475.7.14 Conditional progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22475.7.15 Conditional perfect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22575.7.16 Conditional perfect progressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

75.8 Have got and can see . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22575.9 Been and gone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22675.10Conditional sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22675.11Expressions of wish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22775.12Indirect speech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22875.13Dependent clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22975.14Uses of nonnite verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

75.14.1 Bare innitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22975.14.2 To-innitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23075.14.3 Present participle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23275.14.4 Past participle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23375.14.5 Gerund . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23475.14.6 Perfect and progressive nonnite constructions . . . . . . . . . . . . . . . . . . . . . . . . 235

75.15Deverbal uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23575.16Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23675.17References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

76 Volterra space 23876.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

77 Weak Hausdor space 239

xvi CONTENTS

77.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

78 Zero-dimensional space 24078.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24078.2 Properties of spaces with covering dimension zero . . . . . . . . . . . . . . . . . . . . . . . . . . 24078.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24078.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

79 -compact space 24279.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24279.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24279.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24379.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

80 -bounded space 24480.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

81 Topological space 24581.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

81.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24581.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24681.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24781.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

81.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24781.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24781.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24881.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24981.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24981.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24981.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24981.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24981.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25081.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25081.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25081.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25181.14Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 252

81.14.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25281.14.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25781.14.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

Chapter 1

a-paracompact space

In mathematics, in the eld of topology, a topological space is said to be a-paracompact if every open cover of thespace has a locally nite renement. In contrast to the denition of paracompactness, the renement 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 nite family of open sets isopen. In an Alexandrov space the nite 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 nitely generated spaces since their topology is uniquely determined by the familyof all nite subspaces. Alexandrov spaces can be viewed as a generalization of nite topological spaces.

2.1 Characterizations of Alexandrov topologiesAlexandrov topologies have numerous characterizations. Let X = 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 lter. The neighbourhood lter 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 nest topology consistent with the specialization preorder of X i.e.the nest 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 nite subset

F of S such that x lies in the closure of F. Finite subspace. T is coherent with the nite subspaces of X. Finite inclusion map. The inclusion maps fi : Xi X of the nite subspaces of X form a nal sink. Finite generation. X is nitely generated i.e. it is in the nal hull of the nite spaces. (This means thatthere is a nal sink fi : Xi X where each Xi is a nite topological space.)

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

2.2 Duality with preordered sets

2.2.1 The Alexandrov topology on a preordered setGiven a preordered set X = hX;i we can dene an Alexandrov topology on X by choosing the open sets to bethe upper sets:

= fG X : 8x; y 2 X x 2 G ^ x y ! y 2 G; gWe thus obtain a topological space T(X) = hX; i .The corresponding closed sets are the lower sets:

fS X : 8x; y 2 X x 2 S ^ y x ! y 2 S; g

2.2.2 The specialization preorder on a topological spaceGiven a topological space X = the specialization preorder on X is dened by:

xy if and only if x is in the closure of {y}.

We thus obtain a preordered set W(X) = .

2.2.3 Equivalence between preorders and Alexandrov topologiesFor every preordered set 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 aner 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 continuityGiven a monotone function

f : XY

between two preordered sets (i.e. a function

f : XY

between the underlying sets such that xy 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 : XY

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 : XT(W(X)).)

2.2.5 Category theoretic description of the dualityLet 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 : ProTop and

W : TopPro

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 : ProAlx and

W : AlxPro

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

i : T(W(X))X

is continuous and for every continuous map

f : YX

where Y is an Alexandrov space, the composition

i 1f : YT(W(X))

is continuous.

2.2.6 Relationship to the construction of modal algebras from modal framesGiven a preordered set X, the interior operator and closure operator of T(X) are given by:

Int(S) = { x X : for all y X, xy implies y S }, for all S X

Cl(S) = { x X : there exists a y S with xy } 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 dened on it.) The class of modal algebras that we obtain in thecase of a preordered set is the class of interior algebrasthe algebraic abstractions of topological spaces.

2.3 HistoryAlexandrov spaces were rst 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 nite generation wasapplied to general topology and the name nitely 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 eld of modal logic that a duality exists between nite topological spaces andpreorders on nite sets (the nite 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] Alexandro, P. (1937). Diskrete Rume. Mat. Sb. (N.S.) (in German) 2: 501518.

[2] McCord, M. C. (1966). Singular homology and homotopy groups of nite topological spaces. Duke Mathematical Journal33 (3): 465474. 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): 379398. 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). Alexandro spaces (PDF). Acta Math. Univ. Comenianae 68 (1): 1725.

[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 Hausdor spacesare examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of Ren-LouisBaire who introduced the concept.

3.1 MotivationIn 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 nite sets in , smooth curves inthe plane, and proper ane 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 ane planes.

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

3.2.1 Modern denitionA Baire space is a topological space in which the union of every countable collection of closed sets with emptyinterior has empty interior.This denition 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 denitionMain article: Meagre set

7

8 CHAPTER 3. BAIRE SPACE

In his original denition, Baire dened 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 rst 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 rst category in X

The denition 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 denition is equivalent to the modern denition.A subset A of X is comeagre if its complementX nA 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 rst 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 rst 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.

1\m=1

1[n=1

rn 1

2n+m; rn +

1

2n+m

where frng1n=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 theoremMain article: Baire category theorem

The Baire category theorem gives sucient 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 Hausdor 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 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:XY with pointwise limit f:XY. 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 sucient conditions that willguarantee that a product of arbitrarily many Baire spaces is again Baire.

3.6 See also BanachMazur game Descriptive set theory Baire space (set theory)

3.7 References

3.8 Sources Munkres, James, Topology, 2nd edition, Prentice Hall, 2000. Baire, Ren-Louis (1899), Sur les fonctions de variables relles, Annali di Mat. Ser. 3 3, 1123.

3.9 External links Encyclopaedia of Mathematics article on Baire space Encyclopaedia of Mathematics article on Baire theorem

Chapter 4

Collectionwise Hausdor space

In mathematics, in the eld of topology, a topological space is said to be collectionwise Hausdor 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 Hausdor is also Hausdor.Metrizable spaces are collectionwise normal spaces and are hence, in particular, collectionwise Hausdor.

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

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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 denition 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 denition, i. e., for every pair of distinct points, eachhas an open neighborhood not containing the other. A collectionwise normal T1 space is a collectionwise Hausdorspace.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 nite openrenement) 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 F-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

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Chapter 6

Compact space

Compactness redirects here. For the concept in rst-order logic, see Compactness theorem.In mathematics, and more specically 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 xed distance of each other). Examples include a closed interval, a rectangle, or a nite set ofpoints. This notion is dened for more general topological spaces than Euclidean space in various ways.One such generalization is that a space is sequentially compact if any innite sequence of points sampled from thespace must frequently (innitely often) get arbitrarily close to some point of the space. An equivalent denition isthat every sequence of points must have an innite 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 innite 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 MauriceFrchet 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

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6.1. HISTORICAL DEVELOPMENT 13

ArzelAscoli 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, dierent notions of compactness are notnecessarily equivalent. The most useful notion, which is the standard denition of the unqualied term compactness,is phrased in terms of the existence of nite 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 nite sets. In spaces that are compact inthis sense, it is often possible to patch together information that holds locallythat is, in a neighborhood of eachpointinto 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 developmentIn 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. Bolzanos 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 innitely 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 signicance of Bolzanos 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 BolzanoWeierstrass 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 Arzel.[2] The culmination oftheir investigations, the ArzelAscoli theorem, was a generalization of the BolzanoWeierstrass 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 Bolzanos limit point. Towards the beginning of the twentieth century, results similar to that ofArzel 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 ArzelAscoli theorem held in the sense of mean convergenceor convergence in whatwould later be dubbed a Hilbert space. This ultimately led to the notion of a compact operator as an oshoot of thegeneral notion of a compact space. It was Maurice Frchet who, in 1906, had distilled the essence of the BolzanoWeierstrass 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 dierent 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 dened 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 nite number of these that also covered it. The signicance of this lemma wasrecognized by mile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895)and Henri Lebesgue (1904). The HeineBorel theorem, as the result is now known, is another special propertypossessed by closed and bounded sets of real numbers.This property was signicant 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 HeineBorel 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 Frchet, now called(relative) sequential compactness, under appropriate conditions followed from the version of compactness that wasformulated in terms of the existence of nite 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 examplesAn example of a compact space is the (closed) unit interval [0,1] of real numbers. If one chooses an innite 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 innite 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 DenitionsVarious denitions 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 BolzanoWeierstrass theorem, that anyinnite sequence from the set has a subsequence that converges to a point in the set. Variou