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Topology: An Inquiry-Based Approach

Steven Schlicker

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Section Closed Sets in Metric Spaces

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Recall that Definition 10.1 defines a closed set in a metric space to be a set whose complement is open. We have seen that both βˆ… and X are open subsets of X. We now ask the same question, this time in terms of closed sets.
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Note that a subset of a metric space can be both open and closed. We call such sets clopen (for closed-open). When we discussed open sets, we saw that an arbitrary union of open sets is open, but that an arbitrary intersection of open sets may not be open (although a finite intersection of open sets is open). Since closed sets are complements of open sets, we should expect a similar result for closed sets.
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Activity 10.3.

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Let X=R with the Euclidean metric. Let An=[1n,1βˆ’1n] for each n∈Z+, nβ‰₯2.
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Activity 10.3 shows that an arbitrary union of closed sets is not necessarily closed. However, the following theorem tells us what we can say about unions and intersections of closed sets. The results should not be surprising given the relationship between open and closed sets.
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Proof.

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Let X be a metric space. To prove part 1, assume that {CΞ±} is a collection of closed sets in X for Ξ± in some indexing set I. DeMoivre’s Theorem shows that
Xβˆ–β‹‚Ξ±βˆˆICΞ±=β‹ƒΞ±βˆˆIXβˆ–CΞ±.
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The latter is an arbitrary union of open sets and so it an open set. By definition, then, β‹‚Ξ±βˆˆICΞ± is a closed set.
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For part 2, assume that C1, C2, …, Cn are closed sets in X for some n∈Z+. To show that C=⋃k=1nCk is a closed set, we will show that Xβˆ–C is an open set. Now
Xβˆ–β‹ƒi=1nCi=β‹‚i=1nXβˆ–Ci
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is a finite intersection of open sets, and so is an open set. Therefore, ⋃i=1nCi is a closed set.
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