Outline · 2018-01-30 · Outline 1 Prologue 2 Introduction The Denition of a Topological Space ... The topology T generated by a basis B is a topology. We have already proved this

Outline

1 Prologue

2 IntroductionThe Definition of a Topological SpaceMetric TopologiesBases

Algebraic Topology Lecture 1 6 October, 2015

Prologue

Spheres

bb

bc

bc

S0

bc

bc

S1

bc

bc

S2

How are they different from one another?S0 is in two pieces.If you remove two points from S1, it would be in two pieces.Removing any finite number of points from S2 would not break itinto two pieces.


Prologue

S1

I

The Cylinder = S1 × I.

I I0

1 0

1

The Möbius strip= I × I/{(0, t) ∼ (1,1− t)}.


Prologue

b

b

a a

The Torus

= S1 × S1 = I × I/({(x ,0) ∼ (x ,1)} ∪ {(0, y) ∼ (1, y)}).


Prologue

S1

S1

b

b

a a

The Torus = S1 × S1

= I × I/({(x ,0) ∼ (x ,1)} ∪ {(0, y) ∼ (1, y)}).


Prologue

ba b

b

a a

The Torus = S1 × S1 = I × I/({(x ,0) ∼ (x ,1)} ∪ {(0, y) ∼ (1, y)}).


Prologue

What happens if we reverse some of the arrows?

a a

b

b

?

The Torus

?

a a

b

b

T

?

T

a a

b

b

T

?

T


Introduction The Definition of a Topological Space

Definition (Topological Space)A topological space is a set X together with a collection T of subsetsof X satisfying the following properties.

1 ∅ and X are in T .2 The union of the elements of any subcollection of T is in T .3 The intersection of the elements of any finite subcollection of T is

in T .The set T is called a topology.The subsets that appear in T are called open sets.

Technically a topological space is a pair (X , T ). However, we will oftenrefer to X as a topological space if the topology T is understood.



Example (The Real Numbers)The set of real numbers R is a topological space with the topology Tconsisting of all sets U that satisfy the following property.

For all x ∈ U there is some ε > 0 such that Bε(x) :={y ∈ R | |x − y | < ε} (the ball of radius ε around x) iscontained in U.

(∗)

Let’s show that this is indeed a topology on R.





(∗)

Proof.1 ∅ and X are in T .

∅ contains no points, so it satisfies (∗) vacuously.Here X = R. Pick x ∈ R. Then every ball Bε(x) is contained in R,so R satisfies (∗).





(∗)

Proof.2 The union of the elements of any subcollection of T is in T .

Suppose S ⊂ T and pick x ∈⋃

U∈S U.Then x ∈ U for some U ∈ S, so, since U satisfies (∗), there issome Bε(x) contained in U.But then Bε(x) ⊂

⋃U∈S U, so

⋃U∈S U satisfies (∗).





(∗)

Proof.3 The intersection of the elements of any finite subset of T is in T .

Pick U1, . . . ,Un ∈ T , and pick x ∈⋂n

i=1 Ui .Each Ui satisfies (∗), so for each i there is εi > 0 such thatBεi (x) ⊂ Ui .Take ε = min{εi}ni=1.Then Bε(x) ⊂ Bεi (x) ⊂ Ui for all i , so Bε(x) ⊂

⋂ni=1 Ui .



Example (An Open Set in R)The interval (0,1) is open in the topology T on R defined earlier.

Pick x ∈ (0,1). Then 0 < x < 1.Pick ε < min{1− x , x − 0}.x − ε > x − (x − 0) = 0.x + ε < x + (1− x) = 1.Therefore 0 < x − ε < x < x + ε < 1, so Bε(x) ⊂ (0,1) and (0,1)satisfies (∗).



Example (An Non-open Set in R)The interval [0,1) is not open in the topology T on R defined earlier.

For any x ∈ [0,1) with x 6= 0 we can find an open ball satisfyingthe requirements of (∗).But if x = 0, then any open ball around x has the form (−ε, ε) forsome ε > 0.This contains numbers not in [0,1)—for example, −ε/2.



Exercise.1 Construct an example of an open subset of R.2 Construct an example of an open subset of R that consists of

multiple disjoint intervals.3 Construct an example of an open subset of R that consists of

infinitely many disjoint intervals.4 Construct an example of an open subset of R that consists of a

single interval that is not bounded.

Definition (Bounded (in R))A subset X ⊂ R is called bounded if there exists some M > 0 such that|x − y | < M for all x , y ∈ X .

Exercise.5 Construct an example of an open subset of R that consists of

infinitely many disjoint intervals, and that is bounded.


Introduction Metric Topologies

Example (Rn)We can define a topology on Rn in a similar way. T consists of all setsU ⊂ Rn satisfying the following property.

For all x ∈ U there is some ε > 0 such that Bε(x) :={y ∈ Rn | ‖x − y‖ < ε} is contained in U.

These are all examples of metric topologies.



Definition (Metric)A metric on a set X is a function

d : X × X → Rhaving the following properties.

1 d(x , y) ≥ 0 for all x , y ∈ X , and d(x , y) = 0 if and only if x = y .2 d(x , y) = d(y , x) for all x , y ∈ X (symmetry).3 d(x , y) + d(y , z) ≥ d(x , z) for all x , y , z ∈ X (triangle inequality).

d(x , y) is called the distance between x and y .Given x ∈ X , a metric d on X , and ε > 0, the ball of radius ε centred atx is Bε(x) := {y ∈ X | d(x , y) < ε}.

Definition (Metric Topology)Given a set X with a metric d , the metric topology induced by dconsists of all sets U satisfying the following condition.

For all x ∈ U there is some ε > 0 such that Bε(x) ⊂ U.



Definition (Euclidean metric)The Euclidean metric on Rn is defined by d(x , y) = ‖x − y‖, where ‖ · ‖is the Euclidean norm:

‖(z1, z2, . . . , zn)‖ :=√

z21 + z2

2 + · · ·+ z2n .


Introduction Bases

The collection of open balls in a metric topology is an instance of amore general concept, called a basis.

Definition (Basis)A basis for a topology on a set X is a collection B of subsets of X(called basis elements) such that

1 For each x ∈ X there is at least one basis element containing x .2 If x belongs to the intersection of two basis elements B1 and B2,

then there exists a basis element B3 such that x ∈ B3 ⊂ B1 ∩ B2.

Definition (Topology Generated by a Basis)If B satisfies these two conditions, then we define the topology Tgenerated by B as follows. A subset U of X is in T if for each x ∈ Uthere exists a basis element B ∈ B such that x ∈ B ⊂ U.

Note that by this definition each basis element B is an element of T .


Introduction Bases

In order to justify the use of the term “topology generated by a basis,”we need the following theorem.

TheoremThe topology T generated by a basis B is a topology.

We have already proved this for the metric topology on R. The prooffor the general case is vey similar, and is left to you to do.The main difference is that we don’t have a metric, so, when showingthat T is closed under taking finite intersections, we can’t take theminimum of the radii εi . Instead we have to use condition 2 in thedefinition of a basis.


Introduction Bases

TheoremThe collection of open balls in a metric space is a basis.

b

b

d(x , y)

δ

d(x , z)

d(y , z)bzb

y

ǫ

bx

Proof.Condition 1 is easy.To prove condition 2, first show that,given y ∈ Bε(x), there is δ > 0 suchthat Bδ(y) ⊂ Bε(x).Let δ = ε− d(x , y).Now pick z ∈ Bδ(y).

d(x , z) ≤ d(x , y) + d(y , z)< d(x , y) + δ

= d(x , y) + ε− d(x , y) = ε.

Therefore z ∈ Bε(x).


Introduction Bases

TheoremThe collection of open balls in a metric space is a basis.

b

b

δ

by

ǫ

bx

x ′

b

ǫ′

δ′

Proof.Now suppose that y ∈ Bε(x) ∩ Bε′(x ′).Bδ(y) ⊂ Bε(x) for some δ andBδ′(y) ⊂ Bε′(x ′) for some δ′.Let δ′′ = min{δ, δ′}.Then Bδ′′(y) ⊂ Bε(x) ∩ Bε′(x ′).Therefore the collection of open ballsin a metric space satisfies condition 2of a basis, so it is a basis.


Introduction Bases

Example (The Lower Limit Topology on R)The lower limit topology on R is the topology generated by the basis Bconsisting of all intervals of the form [a,b), where a < b.

Exercise.1 Construct an example of an open set in this topology.2 Construct an example of an open set in this topology consisting of

multiple disjoint intervals.3 Is the interval (0,1) open in this topology? Yes:

(0,1) =⋃

n≥2[1/n,1).


Documents

Outline · 2018-01-30 · Outline 1 Prologue 2 Introduction The Denition of a Topological Space ... The topology T generated by a basis B is a topology. We have already proved this