Introduction to GENERAL TOPOLOGY

 Chapter 1

Topological Spaces

Introduction

Tennis, football, baseball and hockey may all be exciting games but to play them

you must first learn (some of) the rules of the game. Mathematics is no different.

So we begin with the rules for topology.

This chapter opens with the definition of a topology and is then devoted to

some simple examples: finite topological spaces, discrete spaces, indiscrete spaces,

and spaces with the finite-closed topology.

Topology, like other branches of pure mathematics such as group theory, is an

axiomatic subject. We start with a set of axioms and we use these axioms to prove

propositions and theorems. It is extremely important to develop your skill at writing

proofs.

Why are proofs so important? Suppose our task were to construct a building.

We would start with the foundations. In our case these are the axioms or definitions

– everything else is built upon them. Each theorem or proposition represents a new

level of knowledge and must be firmly anchored to the previous level. We attach the

new level to the previous one using a proof. So the theorems and propositions are

the new heights of knowledge we achieve, while the proofs are essential as they are

the mortar which attaches them to the level below. Without proofs the structure

would collapse.

So what is a mathematical proof?

A mathematical proof is a watertight argument which begins with information

you are given, proceeds by logical argument, and ends with what you are asked to

prove.

You should begin a proof by writing down the information you are given and

then state what you are asked to prove. If the information you are given or what

you are required to prove contains technical terms, then you should write down the

definitions of those technical terms.

Every proof should consist of complete sentences. Each of these sentences

should be a consequence of (i) what has been stated previously or (ii) a theorem,

proposition or lemma that has already been proved.

In this book you will see many proofs, but note that mathematics is not a

spectator sport. It is a game for participants. The only way to learn to write proofs

is to try to write them yourself.

1.1 Topology

1.1.1 Definitions.

Let X be a non-empty set. A set τ of subsets of X is

said to be a topology on X if

(i) X and the empty set, Ø, belong to τ ,

(ii) the union of any (finite or infinite) number of sets in τ belongs to τ, and

(iii) the intersection of any two sets in τ belongs to τ.

The pair (X, τ) is called a topological space.

1.1.2 Example.

Let X = {a, b, c, d, e, f} and

τ 1 = {X, Ø, {a}, {c, d}, {a, c, d}, {b, c, d, e, f}}.

Then τ 1 is a topology on X as it satisfies conditions (i), (ii) and (iii) of Definitions

1.1.3 Example.

Let X = {a, b, c, d, e} and

τ 2 = {X, Ø, {a}, {c, d}, {a, c, e}, {b, c, d}}.

Then τ 2 is not a topology on X as the union

{c, d} ∪ {a, c, e} = {a, c, d, e}

of two members of τ 2 does not belong to τ 2 ; that is, τ 2 does not satisfy condition

(ii) of Definitions 1.1.1.

1.1.4 Example.

Let X = {a, b, c, d, e, f} and

τ 3 = {X, Ø, {a}, {f}, {a, f}, {a, c, f}, {b, c, d, e, f}} .

Then τ 3 is not a topology on X since the intersection

{a, c, f} ∩ {b, c, d, e, f} = {c, f}

of two sets in τ 3 does not belong to τ 3 ; that is, τ 3 does not have property (iii) of Definitions 1.1.1.

1.1.5 Example.

Let N be the set of all natural numbers (that is, the set of all

positive integers) and let τ 4 consist of N, Ø, and all finite subsets of N. Then τ 4

is not a topology on N, since the infinite union

{2} ∪ {3} ∪ · · · ∪ {n} ∪ · · · = {2, 3, . . . , n, . . . }

of members of τ 4 does not belong to τ 4 ; that is, τ 4 does not have property (ii)

of Definitions 1.1.1.

1.1.6 Definitions.

Let X be any non-empty set and let τ be the collection

of all subsets of X. Then τ is called the discrete topology on the set X. The

topological space (X, τ) is called a discrete space.

We note that τ in Definitions 1.1.6 does satisfy the conditions of Definitions

1.1.1 and so is indeed a topology.

Observe that the set X in Definitions 1.1.6 can be any non-empty set. So

there is an infinite number of discrete spaces – one for each set X.

1.1.7 Definitions.

Let X be any non-empty set and τ = {X, Ø}. Then τ

is called the indiscrete topology and (X, τ) is said to be an indiscrete space.

Once again we have to check that τ satisfies the conditions of 1.1.1 and so is

indeed a topology.

We observe again that the set X in Definitions 1.1.7 can be any non-empty

set. So there is an infinite number of indiscrete spaces – one for each set X.

In the introduction to this chapter we discussed the

importance of proofs and what is involved in writing

them. Our first experience with proofs is in Example

1.1.8 and Proposition 1.1.9. You should study these

proofs carefully.

You may like to watch the first of the YouTube videos

on proofs. It is called

“Topology Without Tears

1.1.8 Example.

If X = {a, b, c} and τ is a topology on X with {a} ∈ τ,

{b} ∈ τ, and {c} ∈ τ, prove that τ is the discrete topology.

Proof.

We are given that τ is a topology and that {a} ∈ τ, {b} ∈ τ, and {c} ∈ τ.

We are required to prove that τ is the discrete topology; that is, we are

required to prove (by Definitions 1.1.6) that τ contains all subsets of X.

Remember that τ is a topology and so satisfies conditions (i), (ii) and (iii)

of Definitions 1.1.1.

So we shall begin our proof by writing down all of the subsets of X.

The set X has 3 elements and so it has 23 distinct subsets. They are: S1 = Ø,

S2 = {a}, S3 = {b}, S4 = {c}, S5 = {a, b}, S6 = {a, c}, S7 = {b, c}, and

S8 = {a, b, c} = X.

We are required to prove that each of these subsets is in τ. As τ is a topology,

Definitions 1.1.1 (i) implies that X and Ø are in τ ; that is, S1 ∈ τ and S8 ∈ τ.

We are given that {a} ∈ τ, {b} ∈ τ and {c} ∈ τ ; that is, S2 ∈ τ, S3 ∈ τ and

S4 ∈ τ.

To complete the proof we need to show that S5 ∈ τ, S6 ∈ τ, and S7 ∈ τ. But

S5 = {a, b} = {a} ∪ {b}. As we are given that {a} and {b} are in τ, Definitions

1.1.1 (ii) implies that their union is also in τ ; that is, S5 = {a, b} ∈ τ.

Similarly

S6 = {a, c} = {a}∪{c} ∈ τ

and

S7 = {b, c} = {b}∪{c} ∈ τ.

In the introductory comments on this chapter we observed that mathematics

is not a spectator sport. You should be an active participant. Of course your

participation includes doing some of the exercises. But more than this is expected

of you. You have to think about the material presented to you.

One of your tasks is to look at the results that we prove and to ask pertinent

questions. For example, we have just shown that if each of the singleton sets

{a}, {b} and {c} is in τ and X = {a, b, c}, then τ is the discrete topology. You

should ask if this is but one example of a more general phenomenon; that is, if (X, τ)

is any topological space such that τ contains every singleton set, is τ necessarily

the discrete topology? The answer is “yes”, and this is proved in Proposition 1.1.9

1.1.9 Proposition.

If (X, τ) is a topological space such that, for every

x ∈ X, the singleton set {x} is in τ , then τ is the discrete topology.

Proof.

This result is a generalization of Example 1.1.8. Thus you might expect that

the proof would be similar. However, we cannot list all of the subsets of X

as we did in Example 1.1.8 because X may be an infinite set. Nevertheless

we must prove that every subset of X is in τ.

At this point you may be tempted to prove the result for some special

cases, for example taking X to consist of 4, 5 or even 100 elements. But this

approach is doomed to failure. Recall our opening comments in this chapter

where we described a mathematical proof as a watertight argument. We

cannot produce a watertight argument by considering a few special cases,

or even a very large number of special cases. The watertight argument

must cover all cases. So we must consider the general case of an arbitrary

non-empty set X. Somehow we must prove that every subset of X is in τ.

Looking again at the proof of Example 1.1.8 we see that the key is that

every subset of X is a union of singleton subsets of X and we already know

that all of the singleton subsets are in τ. This is also true in the general

case.

We begin the proof by recording the fact that every set is a union of its singleton

subsets. Let S be any subset of X. Then

S = [x∈S{x}.

Since we are given that each {x} is in τ, Definitions 1.1.1 (ii) and the above equation

imply that S ∈ τ. As S is an arbitrary subset of X, we have that τ is the discrete topology

That every set S is a union of its singleton subsets is a result which we shall

use from time to time throughout the book in many different contexts. Note that

it holds even when S = Ø as then we form what is called an empty union and get

Ø as the result.

Exercises 1.1

1.

Let X = {a, b, c, d, e, f}. Determine whether or not each of the following

collections of subsets of X is a topology on X:

(a) τ 1 = {X, Ø, {a}, {a, f}, {b, f}, {a, b, f}};

(b) τ 2 = {X, Ø, {a, b, f}, {a, b, d}, {a, b, d, f}};

(c) τ 3 = {X, Ø, {f}, {e, f}, {a, f}}.

2.

Let X = {a, b, c, d, e, f}. Which of the following collections of subsets of X is a

topology on X? (Justify your answers.)

(a) τ 1 = {X, Ø, {c}, {b, d, e}, {b, c, d, e}, {b}};

(b) τ 2 = {X, Ø, {a}, {b, d, e}, {a, b, d}, {a, b, d, e}};

(c) τ 3 = {X, Ø, {b}, {a, b, c}, {d, e, f}, {b, d, e, f}}.

3.

If X = {a, b, c, d, e, f} and τ is the discrete topology on X, which of the following

statements are true?

(a) X ∈ τ;

(b) {X} ∈ τ;

(c) {Ø} ∈ τ;

(d) Ø ∈ τ;

(e) Ø ∈ X;

(f) {Ø} ∈ X;

(g) {a} ∈ τ;

(h) a ∈ τ;

(i) Ø ⊆ X;

(j) {a} ∈ X;

(k) {Ø} ⊆ X;

(l) a ∈ X;

(m) X ⊆ τ;

(n) {a} ⊆ τ;

(o) {X} ⊆ τ;

(p) a ⊆ τ.

[Hint. Precisely six of the above are true.]

4.

Let (X, τ) be any topological space. Verify that the intersection of any finite

number of members of τ is a member of τ.

[Hint. To prove this result use “mathematical induction”.]

5.

Let R be the set of all real numbers. Prove that each of the following collections

of subsets of R is a topology.

(i) τ 1 consists of R, Ø, and every interval (−n, n), for n any positive integer;

(ii) τ 2 consists of R, Ø, and every interval [−n, n], for n any positive integer;

(iii) τ 3 consists of R, Ø, and every interval [n, ∞), for n any positive integer.

6.

Let N be the set of all positive integers. Prove that each of the following

collections of subsets of N is a topology.

(i) τ 1 consists of N, Ø, and every set {1, 2, . . . , n}, for n any positive integer.

(This is called the initial segment topology.)

(ii) τ 2 consists of N, Ø, and every set {n, n+ 1, . . . }, for n any positive integer.

(This is called the final segment topology.)

7.

List all possible topologies on the following sets:

(a) X = {a, b} ;

(b) Y = {a, b, c}.

8.

Let X be an infinite set and τ a topology on X. If every infinite subset of X

is in τ, prove that τ is the discrete topology

9.* Let R be the set of all real numbers. Precisely three of the following ten

collections of subsets of R are topologies? Identify these and justify your answer.

(i) τ 1 consists of R, Ø, and every interval (a, b), for a and b any real numbers

with a < b ;

(ii) τ 2 consists of R, Ø, and every interval (−r, r), for r any positive real number;

(iii) τ 3 consists of R, Ø, and every interval (−r, r), for r any positive rational

number;

(iv) τ 4 consists of R, Ø, and every interval [−r, r], for r any positive rational

number;

(v) τ 5 consists of R, Ø, and every interval (−r, r), for r any positive irrational

number;

(vi) τ 6 consists of R, Ø, and every interval [−r, r], for r any positive irrational

number;

(vii) τ 7 consists of R, Ø, and every interval [−r, r), for r any positive real number;

(viii) τ 8 consists of R, Ø, and every interval (−r, r], for r any positive real number;

(ix) τ 9 consists of R, Ø, every interval [−r, r], and every interval (−r, r), for r

any positive real number;

(x) τ 10 consists of R, Ø, every interval [−n, n], and every interval (−r, r), for n

any positive integer and r any positive real number.

Regards Ohm's-Jacobian.