1 Metric Spaces - Chulalongkorn University: Faculties and...

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

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

dp((x1, . . . , xn), (y1, . . . , yn)) = d1(x1, y1) + · · ·+ dn(xn, yn),

where (x1, . . . , xn), (y1, . . . , yn) ∈∏n

i=1 Xi. Then dp is a metric on∏n

i=1 Xi and

the metric topology induced by dp is the product topology for∏n

i=1 Xi.

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10 Countability Axioms

Definition 10.1. The space (X, τ) is said to be first countable or to satisfy the

first axiom of countability if for each x ∈ X, there is a countable neighborhood

base at x.

Theorem 10.2. Let A be a subset of a first countable space space X and

x ∈ X. Then x ∈ A if and only if there is a sequence (xn) of points in A

which converges to x.

Theorem 10.3. Let f : X → Y be a function, where X is first countable.

Then f is continuous at x ∈ X if and only if for any sequence (xn) in X which

converges to x, the sequence (f(xn)) converges to f(x) in Y .

Definition 10.4. The space (X, τ) is said to be second countable or to satisfy

the second axiom of countability if there is a countable base for τ .

Theorem 10.5. If (X, τ) is second countable, then it is first countable.

Definition 10.6. A space X is said to be separable if X contains a countable

dense subset.

Theorem 10.7. If (X, τ) is second countable, then it is separable.

Definition 10.8. A space X is said to be Lindelof if every open cover of X

has a countable subcover.

Theorem 10.9. If (X, τ) is second countable, then it is Lindelof.

Theorem 10.10. Let X be a metric space. Then the followings are equivalent:

(a) X is second countable.

(b) X is separable.

(c) X is Lindelof.

Theorem 10.11. Any subspace of a first (second) countable space is also first

(second) countable.

Remark. A subspace of a separable space may not be separable. A subspace

of a Lindelof space may not be Lindelof.

Theorem 10.12. An open subspace of a separable space is separable. A

closed subspace of a Lindelof space is Lindelof.

Theorem 10.13. A subspace of a separable metric space is separable.

14

Theorem 10.14. A continuous image of a separable space is separable. A

continuous image of a Lindelof space is Lindelof.

Remark. A continuous image of a first countable space may not be first

countable. A continuous image of a second countable space may not be second

countable.

Theorem 10.15. Let f : X → Y be an open continuous function, and X is

first (second) countable. Then f [X] is also first (second) countable.

Corollary 10.16. Separability, first countability and second countability are

preserved under a homeomorphism.

Theorem 10.17. A nonempty countable product space is first (second) count-

able if and only if each factor is first (second) countable.

Theorem 10.18. A nonempty countable product space is separable if and

only if each factor is separable.

Remark. The product of two Lindelof spaces need not be Lindelof.

15

11 Separation Axioms

Definition 11.1. A topological space X is said to be a T0–space if for any

two distinct points in X, there is a neighborhood of one not containing the

other.

Definition 11.2. A topological space X is said to be a T1–space if for any two

distinct points in X, each point has a neighborhood which does not contain

the other.

Theorem 11.3. A space X is T1 if and only if each singleton is closed. Hence,

a space X is T1 if and only if each finite subset is closed.

Theorem 11.4. If x is an accumulation point of a subset A of a T1–space,

then every neighborhood of x contains infinitely many points of A.

Definition 11.5. A space X is said to be a T2–space or a Hausdorff space if

any two distinct points have disjoint neighborhoods.

Theorem 11.6. Any sequence in a Hausdorff space has at most one limit.

Theorem 11.7. A space X is Hausdorff if and only if the diagonal ∆ =

{(x, x) | x ∈ X} is closed in X ×X.

Theorem 11.8. Let f and g be functions from a space X into a space Y . If f

and g are continuous and Y is Hausdorff, then the set {x ∈ X | f(x) = g(x)}is closed in X.

Corollary 11.9. Let f and g be functions from a space X into a Hausdorff

space Y . If f and g are continuous and f = g on a dense subset of X, then

f = g on X.

Corollary 11.10. If f is a continuous function from a dense subset of a space

X into a Hausdorff space Y , then there exists at most one continuous extension

of f .

Definition 11.11. A T1–space is said to be a T3–space or a regular space if any

point and any closed set not containing that point have disjoint neighborhoods.

Theorem 11.12. For any space X, the following statements are equivalent:

(a) For any closed set A and any point x /∈ A, x and A have disjoint neigh-

borhoods.

(b) For each x ∈ X and each neighborhood V of x, there exists an open set

G such that x ∈ G ⊆ G ⊆ V .

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(c) For each x ∈ X and a closed set A not containing x, there is a neighbor-

hood V of x such that V ∩ A = ∅.

Definition 11.13. A T1–space is said to be a T3 12–space or a completely regular

space or a Tychonoff space if for any closed set A and any x /∈ A, there exists

a continuous function f : X → [0, 1] such that f(x) = 0 and f [A] = 1.

Definition 11.14. A T1–space is said to be a T4–space or a normal space if

any two disjoint closed sets have disjoint neighborhoods.

Theorem 11.15. A space is normal if and only if for each closed set A and

each neighborhood V of A, there is an open set G such that A ⊆ G ⊆ G ⊆ V.

Theorem 11.16. A compact Hausdorff space is normal.

Theorem 11.17. A regular Lindelof space is normal.

Theorem 11.18 (Urysohn’s Lemma). For any space X, the following are

equivalent:

(a) Each pair of closed sets have disjoint neighborhoods.

(b) If A and B are disjoint closed sets, then there is a continuous function

f : X → [0, 1] such that f [A] = {0} and f [B] = {1}.

(c) If A and B are disjoint closed sets, then there is a continuous function

f : X → [a, b] such that f [A] = {a} and f [B] = {b}.

Theorem 11.19 (Tietze Extension Theorem). For any space X, the fol-

lowing are equivalent:

(a) Each pair of closed sets have disjoint neighborhoods.

(b) For any closed subset A of X, every continuous function f : A → R has

a continuous extension g : X → R. Furthermore, if f [A] ⊆ [a, b], then g

can be chosen so that g[X] ⊆ [a, b].

Theorem 11.20. If X is a Ti–space and i ≥ j, then X is a Tj–space.

Any metric space is a Ti–space for all i.

Theorem 11.21. Subspaces of Ti–spaces are Ti–spaces for i = 0, 1, 2, 3, 312.

Closed subspaces of a T4–space are T4-spaces.

Theorem 11.22. Let f : X → Y be a homeomorphism and X is a Ti–space,

then Y is a Ti–space for i = 0, 1, 2, 3, 312, 4.

Theorem 11.23. A nonempty product space is a Ti–space if and only if each

factor is a Ti–space, i = 0, 1, 2, 3, 312. (A product of two normal spaces may

not be normal.)

17

12 Connectedness

Definition 12.1. A topological space is said to be disconnected if it is the

union of two nonempty disjoint open sets. A space is connected if it is not

disconnected.

A subspace of a topological space is said to be connected if it is a connected

space with respect to the subspace topology.

Definition 12.2. Two nonempty subsets A and B of a space X are called

separated sets if A ∩B = ∅ and B ∩ A = ∅.

Theorem 12.3. The following statements are equivalent for a topological

space X:

1. X is disconnected.

2. X is the union of two disjoint, nonempty closed sets.

3. X is the union of two separated sets.

4. X has a proper subset A which is both open and closed.

5. X has a proper subset A such that Fr(A) = ∅.

6. There is a continuous function from X onto a discrete two-point space

{a, b}.

Corollary 12.4. The following statements are equivalent for a topological

space X:

1. X is connected.

2. X is not the union of two disjoint, nonempty closed sets.

3. X is not the union of two separated sets.

4. The only subsets of X which are both open and closed are X and ∅.

5. X has no proper subset A such that Fr(A) = ∅.

6. Every continuous function from X into a discrete space is constant.

Definition 12.5. A subset I of R is said to be an interval if for any real

numbers x, y, if x, y belong to I, then any real number between x and y also

belongs to I.

18

Theorem 12.6. Let I be a subset of R. Then I is an interval if and only

if I is one of the following sets: ∅, [a, b], [a, b), (a, b], (a, b), [a,∞), (a,∞),

(−∞, b], (−∞, b), R.

Theorem 12.7. A subset of R is connected if and only if it is an interval.

Theorem 12.8. Let A and B be subsets of a topological space X. If A

is connected and A ⊆ B ⊆ A, then B is connected. In particular, if A is

connected, then A is connected.

Theorem 12.9. A continuous image of a connected space is connected.

Theorem 12.10 (Intermediate Value Theorem). Let f be a continuous

real-valued function defined on a connected space X. If a and b are in the

range of f , then for any c between a and b, there is an x ∈ X such that

f(x) = c.

Theorem 12.11. Let {Aα | α ∈ Λ} be a family of connected subsets of a

topological space X such that⋂

α∈Λ Aα 6= ∅. Then⋃

α∈Λ Aα is connected.

Corollary 12.12. Let An be connected subsets of a topological space X for

each n ∈ N. If An ∩ An+1 6= ∅ for each n ∈ N, then⋃∞

n=1 An is connected.

Corollary 12.13. A topological space X is connected if and only if every two

points of X lie in a connected subset of X.

Definition 12.14. A component of a space X is a maximal connected subset

of X.

Theorem 12.15. A component of a space is a closed set (but may not be an

open set).

Theorem 12.16. The components of a topological space form a partition of

it.

Theorem 12.17. A nonempty connected subspace of X which is both closed

and open in X is a component.

Definition 12.18. A space X is said to be totally disconnected if every com-

ponent of X is a singleton.

Theorem 12.19. A nonempty product space is connected if and only if each

factor is connected.

Corollary 12.20. The components of a nonempty product spaces are the

products of the components of the factors.

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Corollary 12.21. A nonempty product space is totally disconnected if each

factor is totally disconnected.

Definition 12.22. Let X be a topological space and a, b ∈ X. A path from a

to b is a continuous function f : [0, 1] → X such that f(0) = a and f(1) = b.

A space X is said to be path–connected if every pair of points of X can be

joined by a path in X.

A subspace A of X is path–connected if it is a path–connected space with

respect to the subspace topology.

Theorem 12.23. A path–connected space is connected.

Theorem 12.24. A continuous image of a path–connected space is path–

connected.

Theorem 12.25. A nonempty product of path–connected space is path–

connected if and only if each factor is path–connected.

Definition 12.26. A space is said to be locally connected if every neighbor-

hood of a point x ∈ X contains a connected neighborhood of x. Equivalently,

A space is locally connected if every point has a neighborhood base consisting

of connected sets.

Theorem 12.27. A space is locally connected if and only if the components

of open subsets are open.

Corollary 12.28. The components of a locally connected space is open.

Corollary 12.29. A nonempty open subset of R is the union of a countable

family of open disjoint intervals.

Theorem 12.30. A topological space (X, τ) is locally connected if and only

if τ has a base consisting of connected subsets.

Theorem 12.31. An open subspace of a locally connected space is locally

connected.

Theorem 12.32. A continuous open (or closed) function maps a locally con-

nected space onto a locally connected space.

Corollary 12.33. Local connectedness is preserved under a homeomorphism.

Theorem 12.34. A nonempty product space is locally connected if and only if

each factor is locally connected and all but finitely many factors are connected.

Corollary 12.35. A nonempty finite product space is locally connected if and

only if each factor is locally connected.

20

13 Quotient Spaces

Definition 13.1. Let f be a continuous function of a space X onto a set Y .

The quotient topology on Y induced by f is the largest topology on Y such

that f is continuous. The set Y with this topology is called a quotient space

of X, and f is called a quotient map.

Theorem 13.2. The quotient topology on Y induced by f is given by

τf = {G ⊆ Y | f−1[G] ∈ τX}.

Theorem 13.3. Let f be a continuous function from X onto Y . If f is either

open or closed, then f is a quotient map.

Corollary 13.4. Suppose f : X → Y is a continuous function from X onto

Y , X is compact and Y is Hausdorff. Then f is a quotient map.

Theorem 13.5. Suppose that Y has the quotient topology induced by a map

f : X → Y . Let g be a function from Y into a space Z. Then g is continuous

if and only if g ◦ f is continuous.

Definition 13.6. Let X be a topological space and r an equivalence relation

on X. The quotient (identification) space of X determined by r is the set of

equivalence classes X/r with the topology induced by the canonical projection

π : X → X/r, where π(x) = [x] = { a ∈ X | a r x}.

Theorem 13.7. Let f be a continuous function from X onto Y . Let r be

an equivalence relation on X such that f is constant on each equivalence

class. Then there exists a unique continuous function g : X/r → Y such that

f = g ◦ π.

Theorem 13.8. Let f be a continuous function from X onto Y . Define

an equivalence relation rf on X by a rf b if and only if f(a) = f(b). Let

g : X/rf → Y be the unique continuous function such that f = g ◦ π. Then g

is a homeomorphism if and only if f is a quotient map.

21

14 Nets

Definition 14.1. A directed set is a pair {D,�} where D is a set and � is a

relation on D such that

(i) λ � λ for every λ ∈ D.

(ii) For any α, β, γ ∈ D, if α � β and β � γ, then α � γ.

(iii) For any α, β ∈ D, there is some γ ∈ D such that γ � α and γ � β.

Definition 14.2. Let X be a set. A net in X is a function of a directed set

{D,�} into X. The net { (λ, xλ) | λ ∈ D} will be denoted by (xλ)λ∈D.

Definition 14.3. A net (xλ)λ∈D in a topological space X is said to converge

to a point x if for any neighborhood V of x, there is an element λV ∈ D such

that for any λ ∈ D, λ � λV implies xλ ∈ V . We call x a limit of the net

(xλ)λ∈D.

Theorem 14.4. Let A be a subset of a topological space X and x ∈ X. Then

x ∈ A if and only if there is a net in A which converges to x.

Corollary 14.5. Let A be a subset of a space X. A point x ∈ X is an

accumulation point of A if and only if there is a net in A−{x} which converges

to x.

Theorem 14.6. Let f : X → Y . Then f is continuous at x ∈ X if and only if

for any net (xλ)λ∈D in X, (xλ)λ∈D converges to x implies (f(xλ))λ∈D converges

to f(x).

Theorem 14.7. A net (xλ)λ∈D converges in a product space∏

α∈Λ Xα to a

point b if and only if (Pα(xλ))λ∈D converges to bα for each α ∈ Λ.

Theorem 14.8. A space X is Hausdorff if and only if every net in X converges

to at most one limit.

Definition 14.9. Let f : D → X be a net; let f(λ) = xλ. If M is a directed

set and g : M → D is a function such that

(i) µ1 � µ2 implies g(µ1) � g(µ2),

(ii) for each λ ∈ D, there is some µ ∈ M such that g(µ) � λ,

then the composite function f ◦ g : M → X is called a subnet of (xλ)λ∈D. For

µ ∈ M , the net f ◦ g(µ) is denoted by (xλµ)µ∈M .

Theorem 14.10. If the net converges to a point x, then so does any subnet.

22

Definition 14.11. We say that x ∈ X is a cluster poin of a net (xλ)λ∈D if for

each neighborhood U of x and for each λo ∈ D there is some λ � λo such that

xλ ∈ U .

Theorem 14.12. A net has a cluster point if and only if it has a convergent

subnet.

23

15 Compactness II

Theorem 15.1. Let X be a topological space. The following statements are

equivalent:

(a) X is compact.

(b) Every net in X has a cluster point.

(c) Every net in X has a convergent subnet.

Definition 15.2. A space X is said to be countably compact if every countable

open cover of X has a finite subcover.

Theorem 15.3. Let X be a topological space. The following statements are

equivalent:

(a) X is countably compact.

(b) Every countable family of closed subsets of X with the finite intersection

property has a nonempty intersection.

(c) Every sequence in X has a cluster point.

Theorem 15.4. A continuous image of a countably compact space is count-

ably compact. A closed subspace of a countably compact space is countably

compact.

Theorem 15.5. If X is compact, then X is countably compact. The converse

holds if X is Lindelof.

Theorem 15.6. If X is sequentially compact, then X is countably compact.

The converse holds if X is first countable.

Corollary 15.7. For a second countable space, compactness, countable com-

pactness and sequential compactness are equivalent.

Corollary 15.8. For a metric space, compactness, countable compactness and

sequential compactness are equivalent.

Theorem 15.9 (Tychonoff’s theorem). A nonempty product space is com-

pact if and only if each factor is compact.

Definition 15.10. A space X is said to be locally compact if every point has

a compact neighborhood.

Theorem 15.11. Let X be a locally compact Hausdorff space. Then for any

x ∈ X, every neighborhood of x contains a compact neighborhood of x.

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Theorem 15.12. A locally compact Hausdorff space is completely regular.

Theorem 15.13. A closed subset of a locally compact space is locally com-

pact. An open subset of a locally compact Hausdorff space is locally compact.

Theorem 15.14. If f is a continuous open map from X onto Y and X is

locally compact, then so is Y .

Theorem 15.15. A nonempty product space is locally compact if and only if

each factor is locally compact and all but finitely many factors are compact.

Corollary 15.16. A nonempty finite product is locally compact if and only

if each factor is locally compact.

Definition 15.17. A compactification of a space X is a pair {X, h} consisting

of a compact space X and a homeomorphism h of X into X such that h[X] is

dense in X.

Theorem 15.18. Let X be a space and ω /∈ X. Let X = X ∪ {ω}. Define

τ = τ ∪ {G ∪ {ω} | G ∈ τ and X −G is compact }

Then τ is a topology on X with the following properties:

(i) X is a subspace of X.

(ii) (X, τ) is a compact space.

(iii) X is dense in X if and only if X is not compact.

(iv) X is Hausdorff if and only if X is locally compact Hausdorff.

Definition 15.19. If X is not compact, then {X, i} is called a one-point

compactification or an Alexandroff compactification of X.

Theorem 15.20. A one-point compactification is unique up to homeomor-

phism.

Theorem 15.21 (Urysohn’s Lemma, LCH version). Let X be a locally

compact Hausdorff space. Let K be compact and G open subsets of X such

that K ⊆ G. Then there exists a continuous function f : X → [0, 1] such that

f [K] = {1} and f = 0 outside a compact subset of G.

25