4 The topology of Rd

In this chapter we look at some properties of open sets,

and introduce the closed sets.

4.1 Properties of open sets

We begin with an elementary lemma.

Lemma 4.1.1 Let A and B be subsets of Rd. Suppose

that A ⊆ B. Then every interior point of A is also an

interior point of B, i.e., intA ⊆ intB.

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Theorem 4.1.2 Let U, V ⊆ Rd be open. Then U ∪ V and

U ∩ V are also open.

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It follows (by an easy induction) that finite unions and

intersections of open sets are open.

In fact, for unions, more is true.

Infinite unions of open sets are still open sets.

Let us first say what we mean by infinite unions.

Recall that if A1, A2, . . . , An are sets then

A1 ∪A2 ∪ · · · ∪An can be written as

A1 ∪A2 ∪ · · · ∪An =n⋃

i=1

Ai

= {x | x ∈ Ai for at least one i ∈ {1, . . . , n}}.

Suppose we have a sequence A1, A2, A3, . . . (i.e. an

ordered list) of sets.

We simply take the above reformulation of a finite union

and define⋃n∈N

An =∞⋃

n=1

An = {x | x ∈ An for at least one n ∈ N }.

Such unions are called countable unions.

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Theorem 4.1.3 Let (Un) be a sequence of open sets in R.

Then⋃∞

n=1 Un is open.

In other words: countable unions of open sets are again

open.

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One can also define countable intersections by⋂n∈N

An =∞⋂

n=1

An = {x | x lies in all of the sets An}

= {x | x ∈ An for all n ∈ N}.

In general it is not true that countable intersections of

open sets are again open, even though finite intersections

are!

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4.2 Closed sets

Complements of open sets are usually not open (though

there are some exceptions).

In fact these complements form an important class of

subsets as well.

Definition 4.2.1 A subset E of Rd is said to be closed

(or closed in Rd, or a closed subset of Rd) if its

complement Ec = Rd\E is open.

Example 4.2.2 1. Closed intervals in R and (more

generally) closed d-cells in Rd are closed subsets, and so

are closed balls in Rd.

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2. Finite subsets of Rd are closed.

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3. Both ∅ and Rd are closed in Rd.

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Warnings!

1. Most sets are neither open nor closed.

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2. The sets ∅ and Rd are both open and closed in Rd.

Sets which are both open and closed are often called

clopen sets.

As a tricky exercise (or see books), you can prove that

∅ and Rd are the only clopen subsets of Rd.

You may find it useful occasionally to quote this result

as a standard fact.

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3. It is very important that you remember the

distinction between closed and not open.

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Since closed sets are complements of open sets many of

their properties follow from properties of open sets. If

A,B ⊆ Rd then

(A ∪B)c = Ac ∩Bc.

(de Morgan’s law, as you checked on the first question

sheet).

Similarly

(A ∩B)c = Ac ∪Bc.

We even have ( ∞⋃n=1

An

)c

=∞⋂

n=1

Acn

for any sequence (An) of subsets of Rd, and also( ∞⋂n=1

An

)c

=∞⋃

n=1

Acn .

We leave the verification as a coursework exercise.

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The statement and applications of the next result are

examinable, but the proof is NEB (not examinable as

bookwork).

Theorem 4.2.3 (i) Finite or countable intersections of

closed subsets of Rd are closed.

(ii) Unions of finitely many closed subsets of Rd are

closed.

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