1 Metric Spaces Definition 1.1. A metric on a set X is a function d : X × X → R such that for any x, y and z in X , (i) d(x, y) ≥ 0, (ii) d(x, y)= d(y,x), (iii) d(x, y) = 0 if and only if x = y, (iv) d(x, y) ≤ d(y,z )+ d(z,x). “triangle inequality” A metric space is a nonempty set X together with a metric d on it, usually denoted by (X, d). Definition 1.2. Let (X, d) be a metric space, p ∈ X and r a positive real number. The d-open ball with center p and radius r is the set B d (p; r)= {x ∈ X | d(x, p) <r}. Definition 1.3. Let (X, d) be a metric space X . A set V ⊆ X is said to be a neighborhood of x if there is an ε> 0 such that B d (x; ε) ⊆ V . Denote by N(x) the collection of all neighborhoods of x. Definition 1.4. Let G be a subset of a metric space (X, d). G is said to be d-open if it is a neighborhood of each of its points. In other words, G is d-open if for any x in G, there is an ε> 0 such that B d (x; ε) ⊆ G. When the metric d is understood, we may simply say that G is an open set. Theorem 1.5. Any d-open ball is d-open. Theorem 1.6. For any metric space (X, d), (i) ∅ and X are d-open. (ii) any union of d-open sets is d-open. (iii) any finite intersection of d-open sets is d-open. Definition 1.7. Let (X, d) be a metric space. The family of d-open sets in X is called the topology for X generated by d. Definition 1.8. Two metrics d and ρ on X are said to be equivalent if they generate the same topology, i.e. τ d = τ ρ . Definition 1.9. A subset F of X is said to be d-closed if its complement X - F is d-open. 1
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1 Metric Spaces
Definition 1.1. A metric on a set X is a function d : X ×X → R such that
A metric space is a nonempty set X together with a metric d on it, usually
denoted by (X, d).
Definition 1.2. Let (X, d) be a metric space, p ∈ X and r a positive real
number. The d-open ball with center p and radius r is the set
Bd(p; r) = {x ∈ X | d(x, p) < r}.
Definition 1.3. Let (X, d) be a metric space X. A set V ⊆ X is said to be a
neighborhood of x if there is an ε > 0 such that Bd(x; ε) ⊆ V .
Denote by N(x) the collection of all neighborhoods of x.
Definition 1.4. Let G be a subset of a metric space (X, d). G is said to be
d-open if it is a neighborhood of each of its points. In other words, G is d-open
if for any x in G, there is an ε > 0 such that Bd(x; ε) ⊆ G. When the metric
d is understood, we may simply say that G is an open set.
Theorem 1.5. Any d-open ball is d-open.
Theorem 1.6. For any metric space (X, d),
(i) ∅ and X are d-open.
(ii) any union of d-open sets is d-open.
(iii) any finite intersection of d-open sets is d-open.
Definition 1.7. Let (X, d) be a metric space. The family of d-open sets in X
is called the topology for X generated by d.
Definition 1.8. Two metrics d and ρ on X are said to be equivalent if they
generate the same topology, i.e. τd = τρ.
Definition 1.9. A subset F of X is said to be d-closed if its complement
X − F is d-open.
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Corollary 1.10. For any metric space (X, d),
(i) ∅ and X are d-closed.
(ii) any intersection of d-closed sets is d-closed.
(iii) any finite union of d-closed sets is d-closed.
Definition 1.11. Let A and B be nonempty subsets of a metric space (X, d).
The distance between A and B, denoted by d(A, B), is defined to be
d(A, B) = inf{ d(a, b) | a ∈ A and b ∈ B }.
If B = {x}, then the distance between A and B is called the distance between
x and A and is denoted by d(x, A).
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2 Topological Spaces
Definition 2.1. A topology τ on a set X is a collection of subsets of X such
that
(i) ∅ and X belong to τ .
(ii) any union of elements of τ belongs to τ .
(iii) any finite intersection of elements of τ belongs to τ .
A topological space is a nonempty set X together with a topology τ on it,
usually denoted by (X, τ). The elements of τ are called the open sets (of X).
Examples.
1. On any nonempty set X, the collection of all subsets of X is a topology
on X, called the discrete topology on X. Also, the collection {∅,X} is also a
topology on X, called the indiscrete topology on X.
2. Let (X, d) be a metric space. The set of all d-open sets is a topology on X,
called the metric topology on X induced by d, denoted by τd.
Definition 2.2. Let (X, τ) be a topological space. If there exists a metric d
on X such that τ = τd, we say that (X, τ) is metrizable.
Definition 2.3. Let A be a subset of a topological space X. A set W ⊆ X is
said to be a neighborhood of A if there is an open set G such that A ⊆ G ⊆ W .
If A = {x}, a neighborhood of A is usually called a neighborhood of x.
Denote by N(x) the collection of all neighborhoods of x.
Theorem 2.4. Let x be an element of a topological space X.
(i) If W ∈ N(x), then x ∈ W .
(ii) If V , W ∈ N(x), then V ∩W ∈ N(x).
(iii) If W ∈ N(x) and W ⊆ V , then V ∈ N(x).
Theorem 2.5. Let G be a subset of a topological space X. G is open if and
only if it is a neighborhood of each of its points.
Definition 2.6. Let A be a subset of a topological space X. The closure
of A (in X), denoted by A, is the set of all points x in X such that every
neighborhood of x meets A.
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Theorem 2.7. Let A and B be arbitrary subsets of a topological space X.
(i) If A ⊆ B, then A ⊆ B.
(ii) A ⊆ A.
(iii) A is the smallest closed set containing A.
(iv) A is closed if and only if A = A.
(v) A = A.
(vi) A ∪B = A ∪B.
Definition 2.8. Let A be a subset of a topological space X. A point x in X is
said to be an accumulation point or a limit point of A if every neighborhood of
x meets A−{x}. The set of all accumulation points of A is called the derived
set of A, denoted by A′.
Theorem 2.9. A = A∪A′. In particular, A is closed if and only if it contains
all its accumulation points.
Definition 2.10. A subset A of a topological space X is said to be dense (in
X) if A = X.
Theorem 2.11. Let A be a subset of a topological space X. A is dense if and
only if every nonempty open subset of X meets A.
Definition 2.12. Let A be a subset of a topological space X. A point x ∈ X
is said to be an interior point of A if A is a neighborhood of x. The set of all
interior points of A is called the interior of A, denoted by Int A.
A point x ∈ X is said to be an exterior point of A if x is an interior point
of X−A. The exterior of A, denoted by Ext A, is the set of all exterior points
of A.
The frontier or boundary of A is the set Fr A = A ∩ X − A.
Theorem 2.13. Let A be a subset of a topological space X. Then Int A,
Ext A and Fr A form a partition of X.
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3 Continuous Functions
Definition 3.1. Let f be a function from a topological space X into a topo-
logical space Y . Let xo ∈ X. f is said to be continuous at xo if, whenever W
is a neighborhood of f(xo) in Y , f−1[W ] is a neighborhood of xo in X.
f is said to be continuous (on X) if f is continuous at every point in X.
f is said to be a homeomorphism if f is 1-1, onto and both f and f−1 are
continuous. In this case, X and Y are said to be homeomorphic.
Theorem 3.2. Let f : X → Y and xo ∈ X. Then f is continuous at xo if
and only if for every neighborhood W of f(xo), there is a neighborhood V of
xo such that f [V ] ⊆ W .
Theorem 3.3. Let (X, d) and (Y, ρ) be metric spaces, f : X → Y and xo ∈ X.
Then f is continuous at xo if and only if for every ε > 0, there is a δ > 0 such
that for every x ∈ X, d(x, xo) < δ implies ρ(f(x), f(xo)) < ε.
Theorem 3.4. Let f : X → Y and g : Y → Z. If f is continuous at xo and g
is continuous at f(xo), then g ◦ f is continuous at xo.
Theorem 3.5. Let F denote either the set of real numbers R or the set of
complex numbers C. Let f and g be continuous functions from a topological
space X into F. Prove that the functions f + g, f − g, f · g, |f | are continuous
and if g(x) 6= 0, for all x ∈ X, then f/g is also continuous.
Theorem 3.6. Let f : X → Y . Then the following statements are equivalent:
(a) f is continuous;
(b) If B is open in Y , then f−1[B] is open in X;
(c) If B is closed in Y , then f−1[B] is closed in X.
Definition 3.7. A function f : X → Y is said to be open (closed) if for any
open (closed) subset A of X, f [A] is open (closed) in Y .
Theorem 3.8. Let f : X → Y be 1-1 and onto. Then the following statements
are equivalent :
(a) f is a homeomorphism;
(b) f is continuous and open;
(c) f is continuous and closed.
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4 Subspaces
Theorem 4.1. Let (X, τ) be a topological space and S ⊆ X. Define
τS = {G ∩ S | G ∈ τ }.
Then τS is a topology on S. Moreover, τS is the smallest topology on S which
makes i : S → X continuous, where i(x) = x for all x ∈ S is the inclusion
map.
Definition 4.2. Let (X, τ) be a topological space and S ⊆ X. Then the
topology τS is called the relative topology on S and (S, τS) is called a subspace
of X.
Theorem 4.3. Let S be a subset of a metric space (X, d). The restriction of
d to S × S is a metric on S, and the metric topology induced by d |S×S is the
relative topology on S.
Theorem 4.4. Let S be a subspace of a topological space X and A ⊆ S.
Then A is closed in S if and only if there is a closed set F in X such that
A = F ∩ S.
Theorem 4.5. Let S be a subspace of a topological space X, and x ∈ S. A
subset W ⊆ S is a neighborhood of x in S if and only if there is a neighborhood
V of x in X such that W = V ∩ S.
Theorem 4.6. Let S be a subspace of a topological space X and A ⊆ S.
Then the closure of A in S is A ∩ S where A is the closure of A in X.
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5 Sequences
Definition 5.1. A sequence on a nonempty set X is a function from N into
X. The function { (n, f(n)) | n ∈ N } will be denoted by (xn).
Definition 5.2. A sequence (xn) in a topological space X is said to converge
to xo ∈ X if for any neighborhood V of xo there is an N ∈ N such that xn ∈ V
for any n ≥ N .
Theorem 5.3. Any sequence in a metric space converges to at most one limit.
Theorem 5.4. Let A be a subset of a topological space X and x ∈ X. If
there is a sequence (xn) of points in A which converges to x, then x ∈ A.
The converse holds if X is a metric space (first countable space).
Theorem 5.5. Let f : X → Y . If f is continuous at x ∈ X, then for any
sequence (xn) in X which converges to x, the sequence (f(xn)) converges to
f(x) in Y .
The converse holds if X is a metric space (first countable space).
Definition 5.6. Let (xn) be a sequence in X. A subsequence (xnk) of (xn) is
a sequence k 7→ xnkwhere (nk) is a strictly increasing sequence in N.
Theorem 5.7. In a topological space X, if a sequence (xn) converges to x,
then every subsequence of (xn) also converges to x.
Definition 5.8. Let (xn) be a sequence in a topological space X. A point
x ∈ X is called a cluster point or a limit point of a sequence (xn) if every
neighborhood of x contains xn for infinitely many n’s.
Theorem 5.9. Let (xn) be a sequence in a topological space. If (xn) has a
convergent subsequence, then (xn) has a cluster point.
The converse holds if X is a metric space (first countable space).
Definition 5.10. Let (X, d) be a metric space. A sequence (xn) in X is called
a Cauchy sequence if for any ε > 0, there is an N ∈ N such that d(xm, xn) < ε
for any m ≥ N , n ≥ N .
Theorem 5.11. Any convergent sequence in a metric space is a Cauchy se-
quence.
Theorem 5.12. If a Cauchy sequence in a metric space has a convergent
subsequence, then that sequence converges.
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6 Complete Metric Spaces
Definition 6.1. A metric space (X, d) is said to be complete if every Cauchy
sequence in X converges (to a point in X).
Theorem 6.2. A closed subset of a complete metric space is a complete
subspace.
Theorem 6.3. A complete subspace of a metric space is a closed subset.
Definition 6.4. Let A be a nonempty subset of a metric space (X, d). The
diameter of A is defined to be
diam(A) = sup{d(x, y) | x, y ∈ A}.
We say that A is bounded if diam(A) is finite.
Theorem 6.5. Let (X, d) be a complete metric space. If (Fn) is a sequence
of nonempty closed subsets of X such that Fn+1 ⊆ Fn for all n ∈ N and
(diam(Fn)) converges to 0. Then⋂∞
n=1 Fn is a singleton.
Definition 6.6. Let f be a function from a metric space (X, d) into a metric
space (Y, ρ). We say that f is uniformly continuous if given any ε > 0, there
exists a δ > 0 such that for any x, y ∈ X, d(x, y) < δ implies ρ(f(x), f(y)) < ε.
Theorem 6.7. A uniformly continuous function maps Cauchy sequences into
Cauchy sequences.
Definition 6.8. Let f be a function from a metric space (X, d) into a metric
space (Y, ρ). We say that f is an isometry if d(a, b) = ρ(f(a), f(b)) for any a,
b ∈ X.
Theorem 6.9. An isometry is uniformly continuous and is a homeomorphism
from X onto f [X].
Theorem 6.10. Let A be a dense subset of a metric space (X, d). Let f be a
uniformly continuous function (isometry) from A into a complete metric space
(Y, ρ). Then there is a unique uniformly continuous function (isometry) g from
X into Y which extends f .
Definition 6.11. A completion of a metric space (X, d) is a pair consisting of
a complete metric space (X∗, d∗) and an isometry ϕ of X into X∗ such that
ϕ[X] is dense in X∗.
Theorem 6.12. Every metric space has a completion.
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Theorem 6.13. A completion of a metric space is unique up to isometry. More
precisely, if {ϕ1, (X∗1 , d
∗1)} and {ϕ2, (X
∗2 , d
∗2)} are two completions of (X, d),
then there is a unique isometry f from X∗1 onto X∗
2 such that f ◦ ϕ1 = ϕ2.
Definition 6.14. A function f : (X, d) → (X, d) is said to be a contraction
map if there is a real number k < 1 such that d(f(x), f(y)) ≤ k d(x, y) for all
x, y ∈ X.
Theorem 6.15. Let f be a contraction map of a complete metric space (X, d)
into itself. Then f has a unique fixed point.
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7 Compactness I
Definition 7.1. A cover or a covering of a topological space X is a family Cof subsets of X whose union is X. A subcover of a cover C is a subfamily of Cwhich is a cover of X. An open cover of X is a cover consisting of open sets.
Definition 7.2. A topological space X is said to be compact if every open
cover of X has a finite subcover. A subset S of X is said to be compact if S
is compact with respect to the subspace topology.
Theorem 7.3. A subset S of a topological space X is compact if and only if
every open cover of S by open sets in X has a finite subcover.
Theorem 7.4. A closed subset of a compact space is compact.
Definition 7.5. A topological space X is said to be Hausdorff if any two
distinct points in X have disjoint neighborhoods.
Theorem 7.6. If A is a compact subset of a Hausdorff space X and x /∈ A,
then x and A have disjoint neighborhoods.
Theorem 7.7. Any compact subset of a Hausdorff space is closed.
Theorem 7.8. A continuous image of a compact space is compact.
Corollary 7.9. Let f : X → Y is a bijective continuous function. If X is
compact and Y is Hausdorff, then f is a homeomorphism.
Theorem 7.10. A continuous function of a compact metric space into a metric
space is uniformly continuous.
Definition 7.11. A metric space (X, d) is said to be totally bounded or pre-
compact if for any ε > 0, there is a finite cover of X by sets of diameter less
than ε.
Theorem 7.12. A subspace of a totally bounded metric space is totally
bounded.
Theorem 7.13. Every totally bounded subset of a metric space is bounded.
A bounded subset of Rn is totally bounded.
Theorem 7.14. A metric space is totally bounded if and only if every sequence
in it has a Cauchy subsequence.
Definition 7.15. A space X is said to be sequentially compact if every se-
quence in X has a convergent subsequence.
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Theorem 7.16. A metric space X is sequentially compact if and only if it is
complete and totally bounded.
Definition 7.17. Let C be a cover of a metric space X. A Lebesgue number
for C is a positive number λ such that any subset of X of diameter less than
or equal to λ is contained in some member of C.
Theorem 7.18. Every open cover of a sequentially compact metric space has
a Lebesgue number.
Definition 7.19. A space X is said to satisfy the Bolzano-Weierstrass prop-
erty if every infinite subset has an accumulation point in X.
Theorem 7.20. In a metric space X, the following statements are equivalent:
(a) X is compact;
(b) X has the Bolzano-Weierstrass property;
(c) X is sequentially compact;
(d) X is complete and totally bounded.
Theorem 7.21 (Heine-Borel). A subset of Rn is compact if and only if it
is closed and bounded.
Corollary 7.22 (Extreme Value Theorem). A real-valued continuous
function on a compact space has a maximum and a minimum.
Theorem 7.23 (Bolzano-Weierstrass). Every bounded infinite subset of
Rn has at least one accumulation point (in Rn).
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8 Bases and Subbases
Definition 8.1. Let (X, τ) be a topological space. A subset B of τ is called a
base for τ if every element of τ is a union of elements of B.
Theorem 8.2. Let (X, τ) be a topological space and B ⊆ τ . B is a base for
τ if and only if for each G ∈ τ and each x ∈ G, there is a B ∈ B such that
x ∈ B ⊆ G.
Corollary 8.3. Let B be a base for τ . A subset G of X is open if and only if
for each x ∈ G, there is a B ∈ B such that x ∈ B ⊆ G.
Corollary 8.4. If τ and τ ′ are topologies for a set X which have a common
base B, then τ = τ ′.
Theorem 8.5. Let X be a nonempty set. A family B of subsets of X is
a base for some topology τ on X if and only if X = ∪B and for every two
members U and V of B and each point x in U ∩ V , there is a W ∈ B such
that x ∈ W ⊆ U ∩ V .
Definition 8.6. Let C be a collection of subsets of X. The topology generated
by C is the smallest topology on X in which every element of C is open. C is
called a subbase for that topology.
Theorem 8.7. Every base for a topology is also a subbase.
Theorem 8.8. Let (X, τ) be a topological space and C ⊆ P(X). C is a
subbase for τ if and only if the set of finite intersections of elements of C is a
base for τ .
Definition 8.9. Let x be a point in a space X. A neighborhood base or a local
base at x is a set Bx of neighborhoods of x such that for each neighborhood V
of x, there is an element B in Bx such that B ⊆ V .
Theorem 8.10. Let (X, τ) be a topological space and B ⊆ τ . Then B is a
base for τ if and only if for each x ∈ X, the set Bx = {B ∈ B | x ∈ B } is a
neighborhood base at x.
Definition 8.11. Let (X,≤) be a linearly ordered set. The order topology
on X is the topology generated by all sets of the form {x ∈ X | x < a } or
{x ∈ X | x > a } for some a ∈ X.
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9 Product Topology
Definition 9.1. The Cartesian product of the family {Xα | α ∈ Λ} is defined
to be ∏α∈Λ
Xα ={
x : Λ →⋃α∈Λ
Xα
∣∣∣ x(α) ∈ Xα for each α ∈ Λ}
.
We usually denote x(α) by xα and call it the α-th coordinate of x.
For each β ∈ Λ, the function Pβ :∏
α∈Λ Xα → Xβ defined by Pβ(x) = xβ
is called the projection of∏
α∈Λ Xα on Xβ or the β-th projection map.
Definition 9.2. Let Xα be a topological space for each α ∈ Λ. The product
topology on∏
α∈Λ Xα is the topology on∏
α∈Λ Xα generated by
{P−1α [Oα] | α ∈ Λ and Oα is open in Xα}.
Theorem 9.3. The product topology on∏
Xα is the smallest topology in
which every projection Pβ :∏
α∈Λ Xα → Xβ is continuous.
Theorem 9.4. A base of the product topology on∏
Xα is the collection of
all subsets of∏
Xα of the form∏
α∈Λ Uα where Uα is open in Xα for each α
and Uα = Xα for all but finitely many α’s.
Theorem 9.5. A function f from a topological space X into a product space∏α∈Λ Xα is continuous if and only if Pα ◦ f is continuous for each α ∈ Λ.
Theorem 9.6. Let∏
α∈Λ Xα be a product space. For each α ∈ Λ, the projec-
tion map Pα is open.
Theorem 9.7. The set {∏
α∈Λ Gα | Gα is open in Xα for each α ∈ Λ } is a
base for a topology on∏
Xα. This topology is called the box topology on∏Xα.
Theorem 9.8. The box topology is in general larger than the product topol-
ogy. For a finite product space, both topologies are the same.
Theorem 9.9. Let (X1, d1), (X2, d2), . . . , (Xn, dn) be metric spaces. Define