Study Guide - Topology I

Curtis Toupin

Fall 2015

CONTENTS

1 Metric Spaces 1

2 Topological Spaces 3

3 Subspaces 10

4 Continuous Functions 11

5 Product Spaces 13

6 Quotient Spaces 15

7 Sequences and Nets 17

8 Separation Axioms 21

9 Regularity and Complete Regularity 25

10 Normal Spaces 27

11 Countability 29

12 Compact Spaces 31

13 Filters and Ultranets and Shit 34

14 Locally Compact Spaces 36

15 Connectedness 37

16 The Homotopy Relation 42

17 The Fundamental Group 45

i

SECTION 1

METRIC SPACES

1.1 Definition

A metric on a set M is a function ρ : M ×M → R such that

a) ρ(x, y) = 0 ⇐⇒ x = y

b) ρ(x, y) = ρ(y, x) ∀x, y ∈M

c) ρ(x, z) ≤ ρ(x, y) + ρ(y, z) ∀x, y, z ∈M

Note that some include a fourth condition ρ(x, y) ≥ 0 for all x, y ∈M , but this is made unnecessaryby the other three, as

0 = ρ(x, x) ≤ ρ(x, y) + ρ(y, x) = 2ρ(x, y) ⇒ ρ(x, y) ≥ 0 ∀x, y ∈M

1.2 Definition

If (M,ρ) and (N, σ) are metrix spaces, a function f : M → N is continuous at x if M if and onlyif for each ε > 0, ∃δ > 0 uch that σ(f(x), f(y)) < ε whenever ρ(x, y) < δ.

1.3 Definition

Let (M,ρ) be a metric space, x ∈M . For ε > 0, we define

Uρ(x, ε) = y ∈M | ρ(x, y) < ε

called the ε-disk about x.

1.4 Definition

A set E in a metric space M is open if and only if for each x ∈ E, ∃ε > 0 such that U(x, ε) ⊆ E.A set is closed if and only if it is the complement of an open set.

1

1.5. THEOREM SECTION 1. METRIC SPACES

1.5 Theorem

If (M,ρ) and (N, σ) are metric spaces, a function f : M → N is continuous at x0 ∈M if and onlyif for each open set V in N containing f(x0), there is an open set U in M containing x0 such thatf(U) ⊆ V .

2

SECTION 2

TOPOLOGICAL SPACES

2.1 Definition

A topology on a set X is a collection τ of subsets of X called the open sets, satisfying

1) Ø, X ∈ τ

2) τ is closed under arbitrary unions

3) τ is closed under finite intersection

2.2 Definition

If X is a topological space and E ⊆ X, we say E is closed if and only if Ec = E \X is open (thatis, Ec ∈ τ).

2.3 Theorem

If F is the collection of closed sets in a topological space X, then

F-a) Ø, X ∈ F

F-b) F is closed under arbitrary intersections

F-c) F is closed under finite unions

Moreover, any collection of sets satisfying the above 3 axioms determines a unique topology on X.

2.4 Definition

If X is a topological space and E ⊆ X, the closure of E in X is the smallest closed set containingE, constructed as

E = cl(E) =⋂K ⊆ X | K is closed and E ⊆ K

3

2.5. LEMMA SECTION 2. TOPOLOGICAL SPACES

2.5 Lemma

If A ⊆ B then A ⊆ B.

Proof:Since B ⊆ B and A ⊆ B, A ⊆ B. Since B is a closed set containing A, it must also contain thesmallest closed set containing A, whence A ⊆ B.

2.6 Theorem

The operation A 7→ A in a topological space has the following properties

K-a) E ⊆ E

K-b) (E) = E

K-c) A ∪B = A ∪B

K-d) Ø = Ø

K-e) E is closed in X if and only if E = E

Moreover, a mapping A 7→ A satisfying K − a through K − d defines a unique topology where theclosed sets are exactly those described in K− e. The closure operation of this topology will be theoperation A 7→ A which we began with.

Proof:Suppose X is a topological space. A ∪ B is closed and contains A ∪ B. Thus it must containA ∪B. On the other hand, A ⊆ A ∪ B ⊆ A ∪B, and so A ⊆ A ∪B. A similar argument showsB ⊆ A ∪B as well, and so A ∪B ⊆ A ∪B, whence A ∪B = A ∪B, verifying K − c. K − a holds

by definition. K − b holds since by K − a, we have E ⊆ E ⇒ E ⊆ (E), and Moreover (E) is

contained in all closed sets containing E. Since E is a closed set containing E, we have (E) ⊆ E,

and so (E) = E. K − d and K − e follow trivially from the definition.

Now, suppose A 7→ A is a mapping satisfying K − a through K − d, and let

F = A ⊆ X | A = A

First note that if A ⊆ B, then B = A ∪B \ A = A ∪B \ A ⇒ A ⊆ B.

Now suppose Fλ is some collection of sets in F . Then⋂Fλ ⊆

⋂Fλ by K − a. And since⋂

Fλ ⊆ Fλ for all λ,⋂Fλ ⊆ Fλ for all λ. Thus

⋂Fλ ⊆

⋂Fλ =

⋂Fλ, so

⋂Fλ ∈ F .

If A,B ∈ F , then A = A and B = B, so A ∪B = A ∪B = A ∪B, so A ∪B ∈ F .

By K − d, Ø = Ø, and by K − a, X ⊆ X ⇒ X = X, so Ø, X ∈ F .

Thus F does determine a topology, it remains to show that the closure operation of this topology

4

2.7. DEFINITION SECTION 2. TOPOLOGICAL SPACES

is A 7→ A. By K−b, E ∈ F for all E, so if A ⊆ X and K ∈ F such that A ⊆ K, then A ⊆ K = K,thus A is the smallest closed set containing A, as required.

Note that an operation satisfying K − a through K − d is called a Kuratowski closure opera-tion.

2.7 Definition

If X is a topological space and E ⊆ X, the interior of E in X is thet set

E = int(E) =⋃G ⊆ X | G is open and G ⊆ E

Note thatX \ E = X \ E and X \ E = (X \ E)

2.8 Lemma

If A ⊆ B then A ⊆ B.

2.9 Theorem

The interior operation A 7→ A in a topological space has the following properties:

I -a) A ⊆ A

I -b) (A) = A

I -c) (A ∩B) = A ∩B

I -d) X = X

I -e) G is open if and only if G = G

Conversely, given any map A 7→ A satisfying I − a through I − d determines a unique topology,where the open sets are exactly those described by I−e. Moreover the resulting interior operationis A 7→ A.

Proof:Similar to proof for closure operation.

2.10 Definition

If X is a topological space and E ⊆ X, the boundary or frontier of E is

∂E = E ∩X \ E

5

2.11. THEOREM SECTION 2. TOPOLOGICAL SPACES

2.11 Theorem

For any subset E of a topological space X,

a) E = E ∪ ∂E

b) E = E \ ∂E

c) X = E ∪ ∂E ∪ (X \ E)

Proof:

a)

E ∪ ∂E = E ∪(E ∩X \ E

)= (E ∪ E) ∩ (E ∪X \ E) = E ∩X = E

b)E − ∂E = E − (E ∩X \ E) = (E \ E) ∪ (E \X \ E) = E \X \ E = E

c) Since ∂E ∪ (X \ E) = X \ E, and since X \ E = X \ E,

X = E ∪X \ E = E ∪ ∂E ∪ (X \ E)

2.12 Definition

If X is a topological space and x ∈ X, a neighbourhood of x is a set U which contains an openset V containing x. Thus U is a neighbourhood of x if and only if x ∈ U. The collection Ux of allneighbourhoods of x is the neighbourhood system at x.

2.13 Theorem

The neighbourhood system Ux at x ∈ X has the following properties:

N-a) If U ∈ Ux, then x ∈ U .

N-b) If U, V ∈ Ux, then U ∩ V ∈ Ux.

N-c) If U ∈ Ux, then ∃V ∈ Ux such that U ∈ Uy for each y ∈ V .

N-d) If U ∈ Ux and U ⊆ V , then V ∈ Ux.

N-e) G ⊆ X is open if and only if G contains a neighbourhood for each of its points.

Moreover, if Ux is a nonempty collection of subsets of a set X for each x ∈ X satisfying N − athrough N − d, then these define a unique topology on X where the open sets are exactly thosedescribed by N − e, and Ux is the neighbourhood system at x for each x ∈ X.

Proof:

a) Trivial

6

2.14. DEFINITION SECTION 2. TOPOLOGICAL SPACES

b) If U, V ∈ Ux, then x ∈ U and V ∈ U, so x ∈ U ∩ V = (U ∩ V ) ⊆ U ∩ V , so U ∩ V ∈ Ux.

c) If U ∈ Ux, then for each y ∈ U, U ∈ Uy.

d) U ∈ Ux, so x ∈ U ⊆ U ⊆ V , so V ∈ Ux.

e) If G is open, then G = G is a neighbourhood of each of its points. On the other hand, if Gcontains a neighbourhood Vx of each x ∈ G, then G =

⋃x∈G

V x is a union of open sets, and hence

open.

2.14 Definition

A neighbourhood base at x in the topological space X is a subcollection Bx taken from theneighbourhood system Ux having the property that each U ∈ Ux contains some V ∈ Bx. That is,Ux must be determined by Bx as follows:

Ux = U ⊆ X | V ⊆ U for some V ∈ Bx

The elements of a neighbourhood base are called basic neighbourhoods.

2.15 Theorem

Let X be a topological space and for each x ∈ X, let Bx be a neighbourhood base at x. Then

V-a) If V ∈ Bx, then x ∈ V

V-b) If V1, V2 ∈ Bx, then there is some V3 ∈ Bx such that V3 ⊆ V1 ∩ V2

V-c) If V ∈ Bx, there is some V0 ∈ Bx such that if y ∈ V0, then there is some W ∈ By withW ⊆ V .

V-d) G ⊆ X is open if and only if G contains a basic neighbourhood at each of its points.

Conversely, in a set X, if a collection of Bx of subsets of X is assigned to each x ∈ X so as tosatisfy V − a, V − b, and V − c, and if we define ”open” by V − d, the result is a topology on Xin which Bx is a neighbourhood base a x for each x ∈ X.

Proof:Define a set for each x ∈ X

Ux = U ⊆ X | B ⊆ U for some B ∈ Bx

and show that Ux is a neighbourhood system at x.

7

2.16. THEOREM SECTION 2. TOPOLOGICAL SPACES

2.16 Theorem

Let X be a topological space and suppose a neighbourhood base has been fixed at each x ∈ X.Then

a) G ⊆ X is open if and only if G contains a basic neighbourhood at each of its points

b) F ⊆ X is closed if and only if each point x 6∈ F has a basic neighbourhood disjoint from F .

c) E = x ∈ X | each basic neighbourhood of x meets E

d) E = x ∈ X | some basice neighbourhood of x is contained in E

e) ∂E = x ∈ X | each basic neighbourhood of x meets both E and X \ E

Proof:

a) Trivial by definition

b) Trivial by a)

c) Recall that E = ∩K ⊆ X | K is closed and E ⊆ K. If some neighbourhood U of x does notmeet E, then x ∈ U and E ⊆ X \ U, which is closed. Thus E ⊆ X \ U, hence x 6∈ E. Onthe other hand, if x 6∈ E, then X \ E is an open set containing x, and hence contains a basicneighbourhood of x which does not meet E.

d) Follows from c) by deMorgan’s Law.

e) Follows from c) and the definition ∂E = E ∩X \ E.

2.17 Theorem - Haussdorf Criterion

For each x ∈ X, let B1x be a neighbourhood base at x for a topology τ1 on X, and let B2

x be aneighbourhood base at x for a topology τ2 on X. Then τ1 ⊆ τ2 if and only if at each x ∈ X, givenB1 ∈ B1

x, there is some B2 ∈ B2x such that B2 ⊆ B1.

Proof:Suppose τ1 ⊆ τ2, and let B1 ∈ B1

x. Then since B1 is a neighbourhood of x in (X, τ1), x is con-tained in some element B of τ1 which is contained in B1. But if B ∈ τ1, then B ∈ τ2, so B is aneighbourhood of x in (X, τ2). It follows that B2 ⊆ B for some B2 ∈ B2

x, so B2 ⊆ B1.

Conversely, if B ∈ τ1, then B contains some B1 ∈ B1x for each x ∈ B, hence B contains a

corresponding element B2 ∈ B2x for each x ∈ B, thus B ∈ τ2.

2.18 Definition

An accumulation point (cluster point) of a set A in a topological space X is a point x ∈ X suchthat each neighbourhood (basic neighbourhood if you prefer) of x contains some point of A otherthan x. The set A′ of all cluster points of A is called the derived set of A.

8

2.19. THEOREM SECTION 2. TOPOLOGICAL SPACES

2.19 Theorem

A = A ∪ A′

Proof:A′ ⊆ A by an earlier theorem, so A∪A′ ⊆ A. Further, if x ∈ A, then every neighbourhood ofx meets A, and so either x ∈ A, or every neighbourhood of x meets A \ x, in which case x ∈ A′,so A ⊆ A ∪ A′.

2.20 Definition

If (X, τ) is a topological space, a base of τ is a collection B ⊆ τ such that

τ = ⋃B∈C

B∣∣∣ C ⊆ B

2.21 Theorem

B is a base for a topology on X if and only if

a) X =⋃B∈B

B

b) whenever B1, B2 ∈ B with p ∈ B1 ∩B2, there is some B3 ∈ B with p ∈ B3 ⊆ B1 ∩B2.

2.22 Theorem

If B is a collection of open sets in X, B is a base for X if and only if for each x ∈ X, the collectionBx = B ∈ B | x ∈ B is a neighbourhood base at x.

2.23 Definition

If (X, τ) is a topological space, a subbase for τ is a coolection C ⊆ τ such that the collection of allfinite intersections of elements from C forms a base for τ .

2.24 Theorem

Any collection of subsets of a set X is a subbase for some topology on X.

9

SECTION 3

SUBSPACES

3.1 Definition

If (X, τ) is a topological space and A ⊆ X, the collection τ ′ = G ∩ A | G ∈ τ is a topology forA, called the relative topology for A. A is called a subspace of X.

3.2 Theorem

If A is a subspace of a topological space X, then

a) H ⊆ A is open in A if and only if H = G ∩ A where G is open in X

b) F ⊆ A is closed in A if and only if F = K ∩ A where K is closed in X

c) If E ⊆ A, then clA(E) = A ∩ clX(E).

d) If x ∈ A, then V is a neighbourhood of x in A if and only if V = U∩A for some neighbourhood Uof x in X.

e) If x ∈ A and if Bx is a neighbourhood base at x ∈ X, then B ∩ A | B ∈ Bx is aneighbourhood base at x ∈ A.

f) If B is a base for X, then B ∩ A | B ∈ B is a base for A.

10

SECTION 4

CONTINUOUS FUNCTIONS

4.1 Definition

Let X and H be topological spaces, and let f : X → Y . Then f is continuous at x0 ∈ X if andonly if for each neighbourhood V of f(x0) in Y , there is a neighbourhood U of x0 in X such thatf(U) ⊆ V . We say f is continuous on X if and only if f is continuous at each x0 ∈ X.

4.2 Theorem

If X and Y are topological spaces and f : X → Y , then the following are equivalent.

a) f is continuous

b) For each open set H in Y , f−1(H) is open in X.

c) For each closed set K in Y , f−1(K) is closed in X.

d) For each E ⊆ X, f(clX(E)) ⊆ clY (E).

4.3 Theorem

If X, Y, Z are topological spaces and f : X → Y and g : Y → Z are continuous, then gf : X → Zis continuous.

4.4 Theorem

If A ⊆ X and f : X → Y is continuous, then f |A : A→ Y is continuous.

4.5 Theorem

If X = A ∪ B where A and B are both open or both closed in X, and if f : X → Y is a functionsuch that both f |A and f |B are continuous, then f is continuous.

11

4.6. THEOREM SECTION 4. CONTINUOUS FUNCTIONS

4.6 Theorem

Suppose Y ⊆ Z and f : X → Y . Then f is continuous as a map from X to Y if and only if it iscontinuous as a map from X to Z.

4.7 Definition

If X and Y are topological spaces, a function f from X to Y is a homeomorphism if and only if itis injective, surjective, continuous, and has continuous inverse. In this case, we say X and Y arehomeomorphic. If f is everything but surjective, we say f is an embedding of X into Y .

4.8 Theorem

If X and Y are topological spaces, and f : X → Y is injective and surjective, then the followingare equivalent.

a) f is a homeomorphism

b) If G ⊆ X, then f(G) is open in Y if and only if G is open in X

c) If F ⊆ X, then f(F ) is closed in Y if and only if F is closed in X.

d) If E ⊆ X, then f(clX(E)) = clY (f(E)).

12

SECTION 5

PRODUCT SPACES

5.1 Definition

Let Xα be a set for each α ∈ A. The Cartesian product of the sets Xα is the set∏α∈A

Xα = x : A→⋃α∈A

Xα | x(α) ∈ Xα for each α ∈ A

The space Xα is called the αth factor space.

5.2 Definition

The map πβ :∏Xα → Xβ is defined by πβ(x) = xβ is called the projection map of

∏Xα on Xβ,

or more simply, the βth projection map.

5.3 Definition

The Tychonoff topology (or product topology) on∏Xα is obtained by taking as a base for the

open sets, sets of the form∏Uα where

P-a) Uα is open in Xα for each α ∈ A

P-b) For all but finitely many α, Uα = Xα.

Alternatively, P − a)′, Uα ∈ Bα where for each α, Bα is a fixed base for the topology on Xα.The product topology is precisely that topology which has for a subbase the collection

π−1α (Uα) | α ∈ A,Uα is open in Xα

In the case where the indexing set A is finite, the product topology coincides with the box topology.

13

5.4. DEFINITION SECTION 5. PRODUCT SPACES

5.4 Definition

If X and Y are both topological spaces and f : X → Y , we call f an open (closed) map if andonly if for each open (closed) set A in X, f(A) is an open (closed) set in Y . If f is injective andsurjective, then f is open ⇐⇒ f is closed ⇐⇒ f−1 is continuous. Thus a bijection f is ahomeomorphism if and only if it is continuous and open ⇐⇒ it is continuous and closed.

5.5 Theorem

For each β, πβ :∏Xα → Xβ is continuous and open, but need not be closed.

5.6 Theorem

The Tychonoff topology on∏Xα is the weakest topology which makes each πβ continuous.

Proof:If τ is any topology on the product in which each projection is continuous, then for each β if Uβ isopen in Xβ, π−1

β (Uβ) ∈ τ . Consequently, the members of a subbase for the Tychonoff topology allbelong to τ , and hence the Tychonoff topology is contained in τ .

5.7 Theorem

A map f : X →∏Xα is continuous if and only if πα f is continuous for all α ∈ A.

Proof:If f is continuous then πα f is continuous being the composition of continuous functions.

If each πα f is continuous, then let Uα be open in Xα and consider π−1(Uα), which is openin∏Xα. Then f−1(π−1

α (Uα)) = (πa f)−1(Uα) which is open by the continuity of πα f . As thesets of the form π−1

α (Uα) form a subbase for∏Xα, we have that f is continuous.

5.8 Definition

Let X be a set and Xα a topological space with fα : X → Xα, for each α ∈ A. The weaktopology induced on X by the collection fα | α ∈ A of functions is the smallest topology on Xmaking each fα continuous. It is that topology for which the sets f−1

α (Uα) for α ∈ A and Uα openin Xα form a subbase. The Tychonoff topology is the weak topology induced by the projections.

5.9 Theorem

If X has the weak topology induced by a collection of functions fα, fα : X → Xα, then f : Y →X is continuous if and only if each fα f : Y → Xα is continuous.

14

SECTION 6

QUOTIENT SPACES

6.1 Definition

If X is a topological space, Y is a set, and g : X → Y is a surjection, then the collection of subsetsof Y defined by

τg = G ⊆ Y | f−1(G) is open in Xis a topology on Y , called the quotient topology induced on Y by g. Y is called the quotient spaceand g is called the quotient map. The quotient topology on Y is the largest topology making gcontinuous.

6.2 Theorem

If X and Y are topological spaces and f : X → Y is continuous and either open or closed, thenthe topology τ on Y is the quotient topology τf .

Proof:If f is continuous and open, then τ ⊆ τf , as τf is the largest topology making f continuous.

Now if U ∈ τf , then f−1(U) ⊆ X is open, as f is continuous with respect to τf by definition.But then f(f−1(U)) = U ∈ τ as f is open with respect to τ . Thus τf ⊆ τ as well.

6.3 Theorem

Let Y have the quotient topology induced by a map f of X onto Y . Then an arbitrary mapg : Y → Z is continuous if and only if g f : X → Z is continuous.

6.4 Definition

let X be a topological space. A decomposition D of X is a collection of disjoint subsets of X whoseunion is X. If a decomposition D is endowed with the topology in which F ⊆ D is open if andonly if

⋃F | F ∈ F is open in X, then D is referred to as a decomposition space of X.

15

6.5. DEFINITION SECTION 6. QUOTIENT SPACES

6.5 Definition

If ∼ is an equivalence relation on the topological space X, then the identification space X/ ∼ isdefined to be the decomposition space D whose elements are the equivalence classes for ∼.

16

SECTION 7

SEQUENCES AND NETS

7.1 Definition

A sequence (xn) in a topological space X is said to converge to x ∈ X and we write xn → x if andonly if for each neighbourhood U of x, ∃N ∈ N such that ∀n ≥ N , xn ∈ U . In this case we say(xn) is eventually in U .

7.2 Definition

A topological spaceX is first countable if and only if each x ∈ X has a countable neighbourhood base.

7.3 Theorem

If X is a first countable space and E ⊆ X, then x ∈ E if and only if there is a sequence (xn)contained in E which converges to X.

Proof:If x ∈ E, pick a countable neighbourhood base Un | n = 1, 2, 3, . . . at x ∈ X. Replacing Un byn⋂k=1

Uk where necessary, we have without loss of generality that

U1 ⊇ U2 ⊇ U3 ⊇ · · ·

Since x ∈ E, Ui∩E 6= Ø for each i, so we can pick xn ∈ Un∩E. The result is the desired sequence.The converse is obvious.

7.4 Corollary

Let X and Y be first countable spaces. Then

a) U ⊆ X is open if and only if whenever xn → x ∈ U , then (xn) is eventually in U .

17

7.5. DEFINITION SECTION 7. SEQUENCES AND NETS

b) F ⊆ X is closed if and only if whenever (xn) ⊆ F and xn → x, then x ∈ F .

c) f : X → Y is continuous if and only if whenever xn → x in X, then f(xn)→ f(x) in Y .

7.5 Definition

A set Λ is a directed set if there is a relation ≤ on Λ satisfying

Λ-a) λ ≤ λ for each λ ∈ Λ.

Λ-b) If λ1 ≤ λ2, and λ2 ≤ λ3, then λ1 ≤ λ3

Λ-c) If λ1, λ2 ∈ Λ, then ∃λ3 ∈ Λ with λ1 ≤ λ3 and λ2 ≤ λ3.

7.6 Example

Reverse inclusion of neighbourhoods of a point x ∈ X, U1 ≤ U2 ⇐⇒ U2 ⊆ U1 is a directed set.

7.7 Definition

A net in a set X is a function P : Λ→ X where Λ is some directed set. The point P (λ) is usuallydenoted xλ, and we often speak of (xλ)λ∈Λ.

A subnet of a net P : Λ → X is the composition P ϕ where ϕ : M → Λ is an increasingcofinal function from a directed set M to Λ. That is,

a) ϕ(µ1) ≤ ϕ(µ2) whenever µ1 ≤ µ2 (ϕ is increasing)

b) for each λ ∈ Λ, ∃µ ∈M such that λ ≤ ϕ(µ) (ϕ is cofinal in Λ)

For µ ∈M the point P ϕ(µ) is often written xλµ

7.8 Definition

Let (xλ) be a net in a space X. Then (xλ) converges to x ∈ X (written xλ → x) provided foreach neighbourhood U of x, there is some λ0 ∈ Λ such that λ ≤ λ0 implies xλ ∈ U . Thus xλ → xif and only if each neighbourhood of x contains a tail of xλ, sometimes said (xλ) is residually oreventually in each neighbourhood of x.

7.9 Example

Let X be a topological space, x ∈ X, and Λ any fixed neighbourhood base at x ∈ X. Then theorder relation U1 ≤ U2 if and only if U2 ⊆ U1 (i.e. reverse inclusion) directs Λ. hence if we pickxU ∈ U for each U ∈ Λ, the result is a net (xU) in X. Moreover, xU → x, as for any givenneighbourhood V of x, we have U0 ⊆ V for some U0 ∈ Λ. Then U ≥ U0 implies U ⊆ U0, so thatxU ∈ U ⊆ V .

18

7.10. THEOREM SECTION 7. SEQUENCES AND NETS

If xλ → x, then every subnet of (xλ) converges to x.

If xλ = x for each λ ∈ Λ, then xλ → x.

7.10 Theorem

A net has y as a cluster point if and only if it has a subnet which converges to y.

Proof:Let y be a cluster point of (xλ). Define

M = (λ, U) | λ ∈ Λ, U a neighbourhood of y such that xλ ∈ U

and order M by (λ1, U1) ≤ (λ2, U2) if and only if λ1 ≤ λ2 and U2 ⊆ U1. This is a direction on M .Define ϕ : M → Λ by ϕ(λ, U) = λ. Then ϕ is increasing and cofinal in Λ, so ϕ defines a subnet of(xλ). Let U0 be any neighbourhood of y and find λ0 ∈ Λ such that xλ0 ∈ U0. Then (λ0, U0) ∈ M ,and moreover (λ, U) ≥ (λ0, U0)⇒ U ⊆ U0, so that xλ ∈ U ⊆ U0. It follows that the subnet definedby ϕ converges to y.

Now suppose that ϕ : M → Λ defines a subnet of (xλ) which converges to y. Then for eachneighbourhood U of y, there is some uU in M such that u ≥ uU implies xϕ(u) ∈ U . Suppose aneighbourhood U of y and a point λ0 ∈ Λ are given. Since ϕ(M) is cofinal in Λ, there is someu0 ∈M such that ϕ(u0) ≥ λ0. But there is also some uU ∈M such that u ≥ uU implies xϕ(u) ∈ U .Pick u∗ ∈ M such that u∗ ≥ u0 and u∗ ≥ uU . Then ϕ(u∗) = λ∗ ≥ λ0, since ϕ(u∗) ≥ ϕ(u0), andxλ∗ = xϕ(u∗) ∈ U , sence u∗ ≥ uU . Thus for any neighbourhood U of y and any λ0 ∈ Λ, there issome λ∗ ≥ λ0 with xλ∗ ∈ U . It follows that y is a cluster point of (xλ).

7.11 Corollary

If a subnet of (xλ) has y as a cluster point, so does (xl).

Proof:A subnet of a subnet of (xλ) is a subnet of (xλ).

7.12 Theorem

If E ⊆ X, then x ∈ E if and only if there is a net (xλ) in E with xλ → x.

Proof:If x ∈ E, then each neighbourhood U of x meets E in at least one point xU . Then (xU) is a netcontained in E which converges to x.

Conversely, if (xλ) converges to x ∈ X, and xλ ∈ E for all λ ∈ Λ, then for each neighbourhood Uof x, (xλ) is eventually in U , so U ∩ E 6= Ø, so x ∈ E.

19

7.13. THEOREM SECTION 7. SEQUENCES AND NETS

7.13 Theorem

Let f : X → Y . Then f is continuous at x0 ∈ X if and only if whenever xλ → x0 in X, thenf(xλ)→ f(x0) in Y .

Proof:Suppose f is continuous at x0 ∈ X and that xλ → x0. Then for each neighbourhood V of f(x0)in Y , U = f−1(V o) ⊆ f−1(V ) is open and contains x0, and so it is a neighbourhood of x0. Thusfor some λ0 ∈ Λ, xλ ∈ U for all λ ≥ λ0, and so f(xλ) ∈ f(U) = V o ⊆ V , so f(xλ) is residually inV , whence f(xλ)→ f(x).

If f is not continuous at x0, then for some neighbourhood V of f(x0), f(U) 6⊆ V for anyneighbourhood U of x0. Thus for each neighbourhood U of x0, we can pick xU ∈ U such thatf(xU) 6∈ V . But then (xU) is a net in X and xU → x0, while f(xU) 6→ f(x0).

7.14 Theorem

A net (xλ) in a product space X =∏Xa converges to x if and only if for each α ∈ A,

πα(xλ)→ πα(x) in Xα.

Proof:If xλ → x in

∏Xα, then since πα is continuous, πα(xλ)→ πα(x) by the previous theorem for each α.

Suppose on the other hand that πα(xλ)→ πα(x) for each α ∈ A. Let

π−1α1

(Uα1) ∩ · · · ∩ π−1αn (Uαn)

be a basic neighbourhood of x in the product space. Then for each i ∈ 1, . . . , n, there is a λi suchthat whenever λ ≥ λi, παi(xλi) ∈ Uαi . Thus if λ0 is picked greater than all of λ1, . . . , λn, we haveπαi(xλ) ∈ Uαi for each i, for all λ ≥ λ0. It follows that for λ ≥ λ0, xλ ∈ ∩π−1

αi(Uαi) and hence that

xλ → x in the product.

7.15 Definition

A net (xλ) in a set X is an ultranet (universal net) if an only if for each subset E of X, (xλ) iseither residually in E or residually in X \ E. It follows from this definition that if an ultranet isfrequently in E, then it is residually in E. In particular, an ultranet in a topological space mustconverge to each of its cluster points.

7.16 Theorem

If (xλ) is an ultranet in X and f : X → Y , then (f(xλ)) is an ultranet in Y .

20

SECTION 8

SEPARATION AXIOMS

8.1 Definitions

1. A topological space is a T0 space if and only if whenever x, y ∈ X such that x 6= y, there isan open set in X containing one and not the other.

2. A topological space is a T1 space if and only if whenever x, y ∈ X such that x 6= y, there areopen sets U, V in X with x ∈ U and y 6∈ U , and y ∈ V and x 6∈ V .

3. A topological space is T2 or Hausdorff if and only if whenever x, y ∈ X such that x 6= y,there are open sets U, V in X such that x ∈ U , y ∈ V , and U ∩ V = Ø.

8.2 Examples

• The trivial topology on a non-singleton set X is not T0.

• The Sierpinksi topology on the set X = a, b (i.e. τ = Ø, a, X) is T0 but not T1.

• The cofinite topology on an infinite set x is T1, but not T2.

• Any metric space is T2

8.3 Theorem

A pseudometric ρ on X is a metric if and only if the topology it generates is T0.

Proof:If the topology generated by ρ is T0, then whenever x 6= y ∈ X, there is some open set and hencesome ε-ball about one, not containing the other. Then ρ(x, y) ≥ ε > 0, showing that ρ is a metric.Conversely, if ρ is a metric, then it is trivial to show that the topology it generates is T0.

21

8.4. THEOREM SECTION 8. SEPARATION AXIOMS

8.4 Theorem

For a topological space X, the following are equivalent:

a) X is T1

b) each one point set in X is closed

c) each subset of X is the intersection of the open sets containing it.

Proof:If X is T1 and x ∈ X, then each y 6= x has a neighbourhood disjoint from x, so X \ x is anopen set, and thus x is closed.

If A ⊆ X and each singleton in X is closed, then A =⋂x 6∈A

X \ x, and each X \ x is open.

If c) holds, then x is the intersection of its open neighbourhoods, and hence for any y 6= x,there is an open set containing x and not y.

8.5 Corollary

Subspaces and products of T1 spaces are T1.

8.6 Theorem

T2 ⇒ T1 ⇒ T0

8.7 Theorem

For a topological space X, the following are equivalent:

a) X is Hausdorff

b) limits in X are unique

c) the diagonal ∆ = (x, x) | x ∈ X is closed in X ×X.

Proof:Suppose X is T2, and let (xλ) be a net converging to x ∈ X. If y ∈ X and x 6= y, then there areneighbourhoods U of x and V of y such that U ∩ V = Ø. Thus, as (xλ) is residually in U , wecannot have (xλ) be residually in V , so (xλ) does not converge to y.

If limits in X are unique, and (zλ) = ((xλ, xλ)) is a net in ∆ which converges to z = (x, y) ∈ X×X,then we have that xλ → x and xλ → y. As limits are unique, x = y, and so z ∈ ∆, whence ∆ isclosed.

If ∆ is closed in X × X, and let x 6= y in X. Then (x, y) 6∈ ∆, and hence there is a basicneighbourhood U × V of (x, y) in X ×X which does not meet ∆. Hence, for all (x, y) ∈ U × V ,x 6= y, and so U ∩ V = Ø, whence X is Hausdorff.

22

8.8. THEOREM SECTION 8. SEPARATION AXIOMS

8.8 Theorem

a) Every subspace of a T2 space is T2

b) A nonempty product space is T2 if and only if each factor is T2

c) Quotients of T2 spaces need not be T2.

Proof:

a) If X is T2, and A is a subspace of X, distinct points a and b in A have disjoint neighbourhoods Uand V in X, and so U ∩ A and U ∩B are disjoint neighbourhoods of a and b in A.

b) If Xα is a T2 space for each α ∈ A and x 6= y ∈∏Xα, then for some α, xα 6= yα So disjoint

neighbourhoods Uα of xα and Vα of yα exist in Xα. Thus π−1α (Uα) and π−1

α (Vα) are disjointneighbourhoods of x and y in

∏Xα.

Conversely if∏Xα is a nonempty T2 space, pick a fixed point bα ∈ Xα for each α ∈ A.

Then the subspace

Bα = x ∈∏

Xα | xβ = bβ unless β = α

is T2 by part a), and is homeomorphic to Xα under the restriction to Bα of the projection map.Thus Xα is T2 for each α.

c) For example, the continuous open image of a Hausdorff space need not be Hausdorff. Let Xbe the union of the lines y = 0 and y = 1 in R2, and let Y be the quotient of X obtainedby identifying (x, 0) with (x, 1) for x 6= 0. Then the resulting projection map p : X → Y iscontinuous and open, but p((0, 0)) and p((0, 1)) are distinct points of Y which do not havedisjoint neighbourhoods .

8.9 Theorem

If f : X → Y is continuous and Y is Hausdorff, then (x1, x2) | f(x1) = f(x1) is a closed subsetof X ×X.Proof:Let A be the set defined above. If (x1, x2 6∈ A, then f(x1) and f(x02) are distinct and hencehave disjoint neighbourhoods U and V in Y . Then since f is continuous, f−1(U) and f−1(V ) areneighbourhoods of x1 and x2 respectively, and so f−1(U)× f−1(V ) is a neighbourhood of (x1, x2).Obviously, this neighbourhood cannot meet A, so A is closed.

8.10 Theorem

If f is an open map of X onto Y and the set (x1, x2) | f(x1) = f(x2) is closed in X ×X, thenY is Hausdorff.

Proof:Let A be the above set and suppose f(x1) and f(x2) are distinct points in Y . Then (x1, x2) 6∈ A,so there are open neighbourhoods U of x1 and V of x2 such that (U ×V )∩A = Ø. Then since f isopen, f(U) and f(V ) are neighbourhoods of f(x1) and f(x2), respectively, and f(U)∩ f(V ) = Ø.

23

8.11. THEOREM SECTION 8. SEPARATION AXIOMS

8.11 Theorem

If f is a continuous open map of X onto Y , then Y is Hausdorff if and only if (x1, x2) | f(x1) =f(x2) is a closed subset of X ×X.

8.12 Theorem

If f, g : X → Y are continuous and Y is Hausdorff, then x | f(x) = g(x) is closed in X.

Proof:Let A be the above set. If (xλ) is a net in A and xλ → x, then by continuity, we have bothf(xλ)→ f(x) and g(xλ)→ g(x). Since f(xλ) = g(xλ) for all λ and limits are unique in Y as it isT2, we must have f(x) = g(x). Thus x ∈ A and A is closed.

24

SECTION 9

REGULARITY AND COMPLETE REGULARITY

9.1 Definition

A topological space is regular if for any closed set A ⊆ X and x 6∈ A, ∃U, V ⊆ X open such thatx ∈ U , A ⊆ V , and U ∩ V = Ø.

9.2 Definition

A T3 space is a space which is regular and T1.

9.3 Theorem

The following are equivalent:

• X is regular

• If U is open in X and x ∈ U , then there is an open set V such that x ∈ V ⊆ V ⊆ U .

• Each x ∈ X has a neighbourhood base consisting of closed sets.

9.4 Theorem

• Every subspace of a regular space (T3 space) is regular (T3).

• A nonempty product space is regular (T3) if and only if each factor space is regular (T3).

• Quotients of T3 spaces need not be regular.

9.5 Theorem

If X is T3 and f is continuous, open and closed map of X onto Y , then Y is T2.

25

9.6. THEOREM SECTION 9. REGULARITY AND COMPLETE REGULARITY

9.6 Theorem

If X is T3 and Y is obtained from X by identifying a single closed set A in X with a point, thenY is T2.

9.7 Definition

A topological space X is completely regular if whenever A is a closed set in X and x 6∈ A, there isa continuous function f : X → I such that f(x) = 0 and f(A) = 1. It is enough to fine a functionf : X → R such that f(x) = b, f(A) = a, and b 6= a. Such a function is said to separate x and A.

9.8 Definition

A Tychonoff space is a completely regular T1 space.

9.9 Theorem

• Subspaces preserve complete regularity and Tychonoff.

• A nonempty product space is completely regular (Tychonoff) if and only if each factor spaceis.

• Quotients of Tychonoff spaces need not be completely regular or T2.

26

SECTION 10

NORMAL SPACES

10.1 Definition

A space is said to be normal if for any closed A,B ⊆ X such that A ∩ B = Ø, there are openU, V ⊆ X such that A ⊆ U , B ⊆ V , and U ∩ V = Ø

10.2 Definition

A T4 space is a normal T1 space.

10.3 Lemma

If X contains a dense set D and a closed, relatively discrete subspace S with |S| ≥ 2|D|, then X isnot normal.

10.4 Theorem

• Closed subspaces of normal (T4) spaces are normal (T4)

• Products of normal spaces need not be normal.

• The closed continuous image of a normal (or T4) space is normal (T4).

10.5 Urysohn’s Lemma

A space X is normal if and only if whenever A and B are disjoint closed sets in X, there is acontinuous function f : X → [0, 1] with f(A) = 0 and f(B) = 1.

10.6 Corollary

Every T4 space is Tychonoff.

27

10.7. TIETZE’S EXTENSION THEOREM SECTION 10. NORMAL SPACES

10.7 Tietze’s Extension Theorem

X is normal if and only if whenever A is a closed subset of x and f : A→ R is continuous, thereis an extension of f to all of X. That is, there is a continuous map F : X → R such that F |A = f .

10.8 Definition

A cover of a space X is a collection C of subsets of X whose union is all of X. A subcover of acover C is a subcollection C ′ of C which is also a cover. An open cover of X is a cover consistingof open sets.

28

SECTION 11

COUNTABILITY

11.1 Definition

X is second countable if its topology has a countable base.

11.2 Theorem

• The continuous open image of a second countable space is second countable.

• Subspaces of second countable spaces are second countable.

• A product of Hausdorrf spaces is second countable if and only if each factor is second count-able and all but countably many factors have the trivial topology.

11.3 Definition

A space X is separable if X has a countable dense subset. Note a discrete space is separable ifand only if it is countable.

11.4 Theorem

• The continuous image of a separable space is separable.

• Subspaces of separable spaces need not be separable. However an open subspace of a sepa-rable space is separable.

• A product of nontrivial Hausdorff spaces is separable if and only if each factor is separableand there are ≤ ℵ1 = |R| factors.

11.5 Definition

X is Lindelof if every open cover of X has a countable subcover.

29

11.6. THEOREM SECTION 11. COUNTABILITY

11.6 Theorem

• The continuous image of a Lindelof space is Lindelof .

• Closed subspaces of Lindelof spaces are Lindelof ; arbitrary subspaces of Lindelof spaces neednot be.

• Products of (even two) Lindelof spaces need not be Lindelof

11.7 Theorem

A regular, Lindelof space is normal.

11.8 Theorem

If X is second countable, then X is

• Lindelof

• Separable

11.9 Theorem

For a (pseudo)metric space X, The following are equivalent

• X is second countable

• X is Lindelof

• X is separable

30

SECTION 12

COMPACT SPACES

12.1 Definition

A space is compact if every open cover of X has a finite subcover.

12.2 Definition

A space is countably compact if every countable open cover has a finite subcover.

12.3 Definition

A family E of subsets of X has the finite intersection property if the intersection of any finitesubcollection of E is nonempty.

12.4 Theorem

The following are equivalent:

• X is compact

• each family E of closed subsets of X with the finite intersection property has nonemptyintersection

• each net in X has a cluster point

• each ultranet in X converges.

12.5 Theorem

• Every closed subset of a compact space is compact.

• A compact subset of a Hausdorff space is closed.

31

12.6. THEOREM SECTION 12. COMPACT SPACES

12.6 Theorem

• Disjoint compact subsets of a Hausdorff space can be separated by disjoint open sets.

• A compact set and a disjoint closed set in a regular space can be separated by disjoint opensets.

• If A×B is a compact subset of a product space X×Y contained in an open set W in X×Y ,then there are open sets U ⊆ X and V ⊆ Y such that A×B ⊆ U × V ⊆ W .

12.7 Theorem

The continuous image of a compact space is compact.

12.8 Remark

Let f : X → Y be continuous where X is compact and Y is Hausdorff. Then f is a closed map.

12.9 Corollary

A continuous bijection of a compact space X onto a Hausdorff space Y is a homeomorphism

12.10 Theorem

Let f : X → R be continuous where X is compact. Then ∃x, x ∈ X such that ∀x ∈ X, f(x) ≤f(x) ≤ f(x). That is, f attains a maximum and a minimum value on X.

12.11 Theorem

If A1, . . . , An are compact subsets of a topological space X, thenn⋃i=1

Ai is compact.

12.12 Definition

A topological space X is called sequentially compact if every sequence in X has a convergentsubsequence.

12.13 Remark

A first countable compact space is sequentially compact.

32

12.14. LEMMA SECTION 12. COMPACT SPACES

12.14 Lemma

Let (M,ρ) be a sequentially compact pseudometric space. Then ∀ε > 0 ∃x1, . . . , xn ∈M such that

M =n⋃i=1

Uρ(xi, ε).

12.15 Corollary

Every sequentially compact pseudometric space is separable.

12.16 Theorem

A pseudometric space (M,ρ) is compact if and only if it is sequentially compact.

12.17 Tychonoff’s Theorem

A nonempty product space is compact if and only if each factor space is compact.

12.18 Theorem

A compact Hausdorff space X is a T4 space.

12.19 Remark

A compact subset of a metric space is closed and bounded.

33

SECTION 13

FILTERS AND ULTRANETS AND SHIT

13.1 Definition

A net (xλ)λ∈Λ in a set X is called an ultranet if ∀E ⊆ X, either (xλ) is residually in E or Ec.

13.2 Definition

A filter in a set X is a nonempty collection F of nonempty subsets of x such that

• ∀F1, F2 ∈ F , F1 ∩ F2 ∈ F

• ∀F, F ′ ⊆ X, if F ∈ F and F ⊆ F ′, then F ′ ∈ F .

For example, for x ∈ X the neighbourhood system Ux is a filter.

13.3 Definition

A filter F in X is called an ultrafilter if there does not exist a filter F ′ in X such that F ⊆ F ′ andF 6= F ′. For example, for x ∈ X, take F = A ⊆ X | x ∈ A. Then F is an ultrafilter.

13.4 Proposition

A filter F is an ultrafilter of X if and only if ∀E ⊆ X, either E ∈ F or Ec ∈ F .

13.5 Definition

Let (A,≤) be a partially ordered set.

• A subset B of A is called a chain if ∀x, y ∈ B, either x ≤ y or y ≤ x.

• An uppter bound for a subset B ⊆ A is an element a ∈ A such that b ≤ a ∀b ∈ B.

• A maximal element in A is an element m ∈ A such that ∀x ∈ A, if m ≤ x, then m = x.

34

13.6. ZORN’S LEMMA SECTION 13. FILTERS AND ULTRANETS AND SHIT

13.6 Zorn’s Lemma

Let (A,≤) be a partially ordered set with an upper bound. Then A has a maximal element.

13.7 Lemma

Every filter is contained in an ultrafilter.

13.8 Proposition

Every net (xλ)λ∈Λ has a subnet which is an ultranet.

35

SECTION 14

LOCALLY COMPACT SPACES

14.1 Definition

A space X is locally compact if each point in X has a neighbourhood base consisting of compactsets.

14.2 Theorem

A Hausdorff spaceX is locally compact if and only if each point inX has a compact neighbourhood .

14.3 Theorem

In a locally compact Hausdorff space, the intersection of an open set with a closed set is locallycompact. Conversely, a locally compact subset of a Hausdorff space is the intersection of an openset and a closed set.

36

SECTION 15

CONNECTEDNESS

15.1 Definition

A topological space X is called disconnected if there are disjoint, open, nonempty sets H,K ⊆ Xsuch that X = H ∪K. X is called connected if it is not disconnected.

15.2 Theorem

The following are equivalent:

• X is connected.

• if A ⊆ X is both closed and open, then A = Ø or A = X.

• If A ⊆ X and ∂A = Ø, then A = Ø or A = X.

15.3 Definition

Subsets H,K of a topological space X are called mutually separated in X if H ∩K = H ∩K = Ø.

15.4 Theorem

A subspace E of X is disconnected if and only if there exist nonempty mutually separated setsH,K in X with E = H ∪K.

15.5 Corollary

If H,K are mutually separated in X and E is connected subset of H ∪K then E ⊆ H or E ⊆ K.

37

15.6. THEOREM SECTION 15. CONNECTEDNESS

15.6 Theorem

If E is a connected subset of X and E ⊆ A ⊆ E, then A is connected (in particular, the closureof a connected set is connected).

15.7 Theorem

• If X =⋃α∈A

Xα where each Xα is a (path) connected subspace of X and ∃α0 ∈ A such that

∀α ∈ A, Xα ∩Xα0 6= Ø, then X is (path) connected.

• If X =N⋃n=1

Xn where N ≤ ∞ and each Xn is a (path) connected subspace of X and ∀n < N ,

Xn ∩Xn+1 6= Ø, then X is (path) connected.

• If ∃x0 ∈ X such that ∀x ∈ X, ∃ a (path) connected set Ex with x0, x ∈ E, then X is (path)connected.

15.8 Theorem

A subset E ⊆ R is (path) connected if and only if E is an interval.

15.9 Theorem

The continuous image of a (path) connected space is (path) connected.

15.10 Definition

A function f : X → R is said to have the intermediate value property if ∀a, b ∈ X, ∀y ∈ R, iff(a) ≤ y ≤ f(b), then y = f(c) for some c ∈ X.

15.11 Theorem

X is connected if and only if every continuous function f : X → R has the intermediate valueproperty.

15.12 Theorem

A nonempty product space X =∏Xα is (path) connected if and only if each Xα is (path)

connected.

38

15.13. COMPONENTS SECTION 15. CONNECTEDNESS

15.13 Components

15.14 Definition

When x ∈ X where X is a topological space, the component Cx of x is

Cx =⋃C ⊆ X | x ∈ C and C is connected

15.15 Remark

• Cx is connected.

• Cx is the largest connected subset of X containing x.

• ∀x, y ∈ X, either Cx = Cy or Cx ∩ Cy = Ø.

•⋃x∈X

Cx = X

• Cx is closed ∀x ∈ X

• The relation ≈ on X defined by x ≈ y ⇐⇒ ∃ a connected set C with x, y ∈ C is anequivalence relation and the equivalence class of x is Cx.

15.16 Definition

A topological space X is called totally disconnected if ∀x ∈ X, Cx = x.

15.17 Remark

Every subspace of a totally disconnected space is totally disconnected.

15.18 Definition

A continuous function f : [0, 1]→ X such that f(0) = x and f(1) = y is called a path in X fromx to y.

15.19 Definition

A topological space X is called path connected if ∀x, y ∈ X, there is a path in X from x to y.

15.20 Theorem

Every path connected space is connected.

39

15.21. REMARK SECTION 15. CONNECTEDNESS

15.21 Remark

• Connected 6⇒ path connected

• The closure of a path connected set need not be path connected.

15.22 Remark

Now define a relation ∼ on a topological space X by a ∼ b ⇐⇒ ∃ a path from a to b. Equivalenceclasses of ∼ are called path components. The path component Px of x ∈ X is the largest pathconnected set containing x. Path components are contained in components (i.e. Px ⊆ Cx). Pathcomponents need not be closed.

15.23 Definition

A topological space X is called locally path connected if each point has a neighbourhood baseconsisting of path connected sets.

15.24 Theorem

Every path component of a locally path connected space X is both closed and open.

15.25 Theorem

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

15.26 Corollary

If X is locally path connected, then ∀x ∈ X, Cx = Px.

15.27 Definition

A topological space is locally connected if each point x ∈ X has a neighbourhood base consistingof connected sets.

15.28 Remark

Locally path connected ⇒ locally connected, but locally connected 6⇒ locally path connected.

15.29 Theorem

X is locally compact if and only if each (path) component of each open set is open.

40

15.30. COROLLARY SECTION 15. CONNECTEDNESS

15.30 Corollary

Every component of a locally connected space is clopen.

15.31 Theorem

Every quotient space of a locally (path) connected space is locally (path) connected.

15.32 Theorem

A nonempty product space X =∏Xα is locally (path) connected if and only if each Xα is locally

(path) connected and all but finitely many factors Xα are (path) connected.

41

SECTION 16

THE HOMOTOPY RELATION

16.1 Definition

Denote C(X, Y ) = f : X → Y | f is continuous

16.2 Definition

Given f, g ∈ C(X, Y ) we say f is homotopic to g and write f ' g if ∃H ∈ C(X × I, Y ) such thatH(x, 0) = f(x) and H(x, 1) = g(x) for all x ∈ X. H is called a homotopy between f and g andwe write H : f ' g.

16.3 Theorem

' is an equivalent relation on C(X, Y ). Proof:

• f ' f

• f ' g ⇒ ∃H : f ' g. Let H ′(x, t) = H(x, 1− t), then H ′ : g ' f , so g ' f

• If f ' g and g ' h, we have H1 : f ' g and H2 : g ' h. Define H(x, t) by

H(x, t) =

H1(x, 2t), t ∈ [0, 1

2], x ∈ X

H2(x, 2t− 1), t ∈ (12, 1], x ∈ X

Then H : f ' h, so f ' h.

16.4 Theorem

Let f1, g1 ∈ C(X, Y ) and f2, g2 ∈ C(Y, Z). If f1 ' g1 and f2 ' g2, then f2 f1 ' g2 g1.

42

16.5. DEFINITION SECTION 16. THE HOMOTOPY RELATION

16.5 Definition

A topological space is called contractible if the identity map idX : X → X is homotopic to someconstant function c : X → X.

16.6 Theorem

X is contractible if and only if for any topological space T , any two maps f, g ∈ C(T,X) arehomotopic .

16.7 Definition

Two topological spacesX, Y are said to be homotopically equivalent if there exists a continuous functionf ∈ C(X, Y ) and g ∈ C(X, Y ) such that f g ' idY and g f ' idX . The maps f and g are calledhomotopy equivalences and g is called a homotopy inverse of f (and vice versa).

16.8 Remark

Homotopy equivalence is an equivalence relation in the class of topological spaces.

16.9 Theorem

X is contractible if and only if X is homotopy equivalent to a singleton.

16.10 Definition

Let f, g ∈ C(X, Y ) and let A ⊆ X. We say that f ' g rel A or f ' g[A] if ther eis a homotopy H :f ' g such that H(x, t) = f(x) = g(x) ∀x ∈ A, ∀t ∈ I. Then H is called a homotopy between fand g relative to A. We write H : f ' g [A].

16.11 Remarks

• f ' g [Ø] ⇐⇒ f ' g

• f ' g [X] ⇐⇒ f = g

• f ' g [A] ⇒ f ' g and f |A = g|A (note the converse does not always hold)

16.12 Theorem

' [A] is an equivalence class on C(X, Y ).

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16.13. THEOREM SECTION 16. THE HOMOTOPY RELATION

16.13 Theorem

Let f1, g1 ∈ C(X, Y ), f2, g2 ∈ C(Y, Z), A ⊆ X ,B ⊆ Y . If f1 ' g1 [A], f2 ' g2 [B] and f1(A) ⊆ B,then f2 f1 ' g2 g1 [A].

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SECTION 17

THE FUNDAMENTAL GROUP

17.1 Definition

Given a topological space X and x1, x2 ∈ X, let Π(X, x1, x2) denote the set of paths from x1 to x2

in X. Given α, β ∈ Π(X, x1, x2), we write a 'p β for α ' β [0, 1] and when this holds, we willsay that α and β are path homotopic .

17.2 Remark

If α ∈ Π(X, x1, x2), β ∈ Π(X, x2, x3), define γ = α ∗ β by

(α ∗ β)(t) =

α(2t), t ∈ [0, 1

2]

β(2t− 1), t ∈ (12, 1]

Then γ = α ∗ β ∈ Π(X, x1, x3).

17.3 Remark

If α ∈ Π(X, x1, x2), then the function α(t) = α(1− t), t ∈ I is in Π(X, x2, x1).

17.4 Definition

Given x0 ∈ X, let εx0 denote the constant path εx0(t) = x0 ∀t ∈ I.

17.5 Proposition

• If α1, α2 ∈ Π(X, x1, x2), and β1, β2 ∈ Π(X, x2, x3) such that α1 'p α2 and β1 'p β2, thenα1 ∗ β1 'p α2 ∗ β2.

• If α1, α2 ∈ Π(X, x1, x2) and α1 'p α2 then α1 'p α2.

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17.6. DEFINITION SECTION 17. THE FUNDAMENTAL GROUP

• If α ∈ Π(X, x1, x2), β ∈ Π(X, x2, x3), γ ∈ Π(x3, x4), then (α ∗ β) ∗ γ 'p α ∗ (β ∗ γ) and

(α ∗ β) = β ∗ α.

• If α ∈ Π(X, x1, x2), then α ∗ α 'p εx1 , α ∗ α 'p εx2 , and α ∗ εx2 'p α 'p εx1 ∗ α

Proof:

•

•

• α ∗ (β ∗ γ)(s) =

a(2s), s ∈ [0, 1

2]

β ∗ γ(2s− 1), s ∈ (12, 1]

=

α(2s), s ∈ [0, 1

2]

β(4s− 2), s ∈ (12, 3

4]

γ(4s− 3), s ∈ (34, 1]

(α ∗ β) ∗ γ(s) =

α ∗ β(2s), s ∈ [0, 1

2]

γ(2s− 1) s ∈ (12, 1]

=

α(4s), s ∈ [0, 1

4]

β(4s− 1), s ∈ (14, 1

2]

γ(2s− 1), s ∈ (12, 1]

Take H(s, t) =

α(

4st+1

), 0 ≤ s ≤ t+1

4, t ∈ I

β(4s− t− 1), t+14≤ s ≤ t+2

4, t ∈ I

γ(

4s−t−22−t

), t+2

4≤ s ≤ 1

Then H : (α ∗ β) ∗ γ 'p α ∗ (β ∗ γ)

And

(α ∗ β)(t) = α ∗ β(1− t) =

α(2− 2t), 1− t ∈ [0, 1

2]

β(2− 2t− 1), 1− t ∈ [12, 1]

=

β(1− 2t), t ∈ [0, 1

2]

α(2− 2t), t ∈ [12, 1]

while

β ∗ α(t) =

β(2t), t ∈ [0, 1

2]

α(2t− 1), t ∈ [12, 1]

=

β(1− 2t), t ∈ [0, 1

2]

α(1− (2t− 1)), t ∈ [12, 1]

= (α ∗ β)(t)

• If H(s, t) =

α(2s(1− t)), 0 ≤ s ≤ 1

2, t ∈ I

α((2− 2s)(1− t)), 12≤ s ≤ 1, t ∈ I

17.6 Definition

Given x0 ∈ X, let Ω(X, x0) = Π(X, x0, x0) and call elements of Ω(X, x0) loops based at x0. Letπ1(X, x0) denote the set of equivalence classes of 'p in Ω(X, x0). Let [α] denote the equivalenceclass of α in Ω(X, x0).

17.7 Remark

There is a binary operation ∗ on π1(X, x0) such that [α] ∗ [β] = [α ∗ β] ∀α, β ∈ Ω(X, x0) sinceα ' α′ and β ' β′ ⇒ α ∗ β ' α′ ∗ β′.

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17.8. THEOREM SECTION 17. THE FUNDAMENTAL GROUP

17.8 Theorem(π1(X, x0), ∗

)is a group with identity [εx0 ]. Moreover, [α]−1 = [α] ∀α ∈ Ω(X, x0).

17.9 Definition

π1(X, x0) is called the fundamental group of X relative to the base point x0 or based at x0.

17.10 Theorem

Let x0, x1 ∈ X and let α ∈ Π(X, x0, x1). Then ∃ a group isomorphism hα : π1(X, x0)→ π1(X, x1)such that hα([β]) = [α ∗ (β ∗ α)] ∀β ∈ Ω(X, x0).

17.11 Corollary

If X is path connected, then ∀x, y ∈ X, π1(X, x) ∼= π1(X, y). That is, the fundamental group isunique up to isomorphism.

17.12 Theorem

Let f ∈ C(X, Y ), x0 ∈ X. Then ∃ a homomorphism f# : π1(X, x0) → π1(Y, f(x0)) such thatf#([β]) = [f β], ∀β ∈ Ω(X, x0).

17.13 Theorem

Let f ∈ C(X, y), g ∈ C(Y, Z), and x0 ∈ X. Then (g f)#x0

= g#f(x0) f#

x0.

17.14 Corollary

If f is a homemorphism of X onto Y and x0 ∈ X then f#x0

is an isomorphism of π1(X, x0) ontoπ1(Y, f(x0)).

17.15 Theorem

Let f, g ∈ C(X, Y ), x0 ∈ X. If f ' g [x0], then f#x0' g#

x0.

17.16 Theorem

Suppose that f, g ∈ C(X, Y ) are homotopic maps and let H : f ' g. Let x0 ∈ X and defineα(s) = H(x0, s), s ∈ I. Then α ∈ Π(Y, f(x0), g(x0)) and g#

x0= hα f#

x0where hα : π1(Y, f(x0))→

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17.17. THEOREM SECTION 17. THE FUNDAMENTAL GROUP

π1(Y, g(x0)) is given by hα([β]) = [α ∗ (β ∗ α)]. That is, the following diagram commutes

π1(X, x0)g#x0

((f#x0

π1(Y, f(x0))hα// π1(Y, g(x0))

17.17 Theorem

If f is a homotopy equivalence from X to Y , then ∀x0 ∈ X, f#x0

is an isomorphism of π1(X, x0)onto π1(Y, f(x0)).

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