Chapter 1 (Introduction to Topology) Open Sets, Closed Set and Clopen Sets

1.2 Open Sets, Closed Sets, and Clopen Sets

Rather than continually refer to “members of τ ", we find it more convenient to give

such sets a name. We call them “open sets”. We shall also name the complements

of open sets. They will be called “closed sets”. This nomenclature is not ideal, but

derives from the so-called “open intervals” and “closed intervals” on the real number

line. We shall have more to say about this in Chapter 2.

1.2.1 Definition.

Let (X, τ) be any topological space. Then the members

of τ are said to be open sets.

1.2.2 Proposition.

If (X, τ) is any topological space, then

(i) X and Ø are open sets,

(ii) the union of any (finite or infinite) number of open sets is an open set, and

(iii) the intersection of any finite number of open sets is an open set.

Proof.

Clearly (i) and (ii) are trivial consequences of Definition 1.2.1 and

Definitions 1.1.1 (i) and (ii). The condition (iii) follows from Definition 1.2.1 and

Exercises 1.1 #4.

On reading Proposition 1.2.2, a question should have popped into your mind:

while any finite or infinite union of open sets is open, we state only that finite

intersections of open sets are open. Are infinite intersections of open sets always

open? The next example shows that the answer is “no”.

1.2. OPEN SETS

1.2.3 Example.

Let N be the set of all positive integers and let τ consist of

Ø and each subset S of N such that the complement of S in N, N\S, is a finite set.

It is easily verified that τ satisfies Definitions 1.1.1 and so is a topology on N. (In

the next section we shall discuss this topology further. It is called the finite-closed

topology.) For each natural number n, define the set Sn as follows:

Sn = {1} ∪ {n + 1} ∪ {n + 2} ∪ {n + 3} ∪ · · · = {1} ∪

∞

[

m=n+1

{m}.

Clearly each Sn is an open set in the topology τ, since its complement is a finite

set. However,

∞\n=1

Sn = {1}.

(1)

As the complement of {1} is neither N nor a finite set, {1} is not open. So (1)

shows that the intersection of the open sets Sn is not open.

You might well ask: how did you find the example presented in Example 1.2.3?

The answer is unglamorous! It was by trial and error.

If we tried, for example, a discrete topology, we would find that each intersection

of open sets is indeed open. The same is true of the indiscrete topology. So what

you need to do is some intelligent guesswork.

Remember that to prove that the intersection of open sets is not necessarily

open, you need to find just one counterexample!

1.2.4 Definition.

Let (X, τ) be a topological space. A subset S of X is

said to be a closed set in (X, τ) if its complement in X, namely X \S, is open

in (X, τ).

In Example 1.1.2, the closed sets are

Ø, X, {b, c, d, e, f}, {a, b, e, f}, {b, e, f} and {a}.

If (X, τ) is a discrete space, then it is obvious that every subset of X is a closed

set. However in an indiscrete space, (X, τ), the only closed sets are X and Ø.

1.2.5 Proposition.

If (X, τ) is any topological space, then

(i) Ø and X are closed sets,

(ii) the intersection of any (finite or infinite) number of closed sets is a closed

set and

(iii) the union of any finite number of closed sets is a closed set.

Proof.

(i) follows immediately from Proposition 1.2.2 (i) and Definition 1.2.4, as

the complement of X is Ø and the complement of Ø is X.

To prove that (iii) is true, let S1, S2, . . . , Sn be closed sets. We are required to

prove that S1 ∪ S2 ∪ · · · ∪ Sn is a closed set. It suffices to show, by Definition 1.2.4,

that X \ (S1 ∪ S2 ∪ · · · ∪ Sn) is an open set.

As S1, S2, . . . , Sn are closed sets, their complements X \S1, X \S2, . . . , X \Sn

are open sets. But

X \ (S1 ∪ S2 ∪ · · · ∪ Sn) = (X \ S1) ∩ (X \ S2) ∩ · · · ∩ (X \ Sn).

(1)

As the right hand side of (1) is a finite intersection of open sets, it is an open

set. So the left hand side of (1) is an open set. Hence S1 ∪ S2 ∪ · · · ∪ Sn is a closed

set, as required. So (iii) is true

Warning.

The names “open” and “closed” often lead newcomers to the world

of topology into error. Despite the names, some open sets are also closed sets!

Moreover, some sets are neither open sets nor closed sets! Indeed, if we consider

Example 1.1.2 we see that

(i) the set {a} is both open and closed;

(ii) the set {b, c} is neither open nor closed;

(iii) the set {c, d} is open but not closed;

(iv) the set {a, b, e, f} is closed but not open.

In a discrete space every set is both open and closed, while in an indiscrete space

(X, τ), all subsets of X except X and Ø are neither open nor closed.

To remind you that sets can be both open and closed we introduce the following

definition.

1.2.6 Definition.

A subset S of a topological space (X, τ) is said to be

clopen if it is both open and closed in (X, τ).

In every topological space (X, τ) both X and Ø are clopen1.

In a discrete space all subsets of X are clopen.

In an indiscrete space the only clopen subsets are X and Ø.

Exercises 1.2

1.

List all 64 subsets of the set X in Example 1.1.2. Write down, next to each set,

whether it is (i) clopen; (ii) neither open nor closed; (iii) open but not closed;

(iv) closed but not open.

2.

Let (X, τ) be a topological space with the property that every subset is closed.

Prove that it is a discrete space.

3.

Observe that if (X, τ) is a discrete space or an indiscrete space, then every open

set is a clopen set. Find a topology τ on the set X = {a, b, c, d} which is not

discrete and is not indiscrete but has the property that every open set is clopen.

4.

Let X be an infinite set. If τ is a topology on X such that every infinite subset

of X is closed, prove that τ is the discrete topology.

5.

Let X be an infinite set and τ a topology on X with the property that the only

infinite subset of X which is open is X itself. Is (X, τ) necessarily an indiscrete

space?

6.

(i) Let τ be a topology on a set X such that τ consists of precisely four

sets; that is, τ = {X, Ø, A, B}, where A and B are non-empty distinct

proper subsets of X. [A is a proper subset of X means that A ⊆ X and

A 6= X. This is denoted by A ⊂ X.] Prove that A and B must satisfy

exactly one of the following conditions:

(a) B = X \ A; (b) A ⊂ B; (c) B ⊂ A.

[Hint. Firstly show that A and B must satisfy at least one of the conditions

and then show that they cannot satisfy more than one of the conditions.]

(ii) Using (i) list all topologies on X = {1, 2, 3, 4} which consist of exactly four

sets.

Distinct Topologies on Finite and Infinite Sets

7.

(i) As recorded in, the number of distinct topologies on a set with n ∈ N points can be very large even for small n; namely when n = 2, there are 4 topologies; when n = 3,

there are 29 topologies: when n = 4, there are 355 topologies; when n = 5,

there are 6942 topologies etc. Using mathematical induction, prove that as

n increases, the number of topologies increases.

[Hint. It suffices to show that if a set with n points has M distinct topologies,

then a set with n + 1 points has at least M + 1 topologies.]

(ii) Using mathematical induction prove that if the finite set X has n ∈ N points,

then it has at least (n − 1)! distinct topologies.

[Hint. Let X = {x1, . . . , xn} and Y = {x1, . . . , xn, xn+1}. if τ is any

topology on X, fix an i ∈ {1, 2, . . . , n}. Define a topology τi on Y as

follows: For each open set U ∈ τ, define Ui by replacing any occurrence of

xi

in U by xn+1; then τi consists of all such Ui plus the set Y . Verify that

τi

is indeed a topology on Y . Deduce that for each topology on X, there

are at least n distinct topologies on Y .]

(iii) If X is any infinite set of cardinality ℵ, prove that there are at least 2ℵ

distinct topologies on X. Deduce that every infinite set has an uncountable

number of distinct topologies on it.

[Hint. Prove that there at least 2ℵ distinct topologies with precisely 3 open

sets. For an introduction to cardinal numbers,

By Allen Oluwaseun Babatunde (B.Sc Ed.)