Unions and Intersections of Closed Sets

Section Unions and Intersections of Closed Sets

Now we have defined open and closed sets in topological spaces. In our preview activity we saw that a set can be both open and closed. As we did in metric spaces, we will call any set that is both open and closed a clopen (for closed-open) set.

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By definition, any union and any finite intersection of open sets in a topological space is open, so the fact that closed sets are complements of open sets implies the following theorem.

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Theorem 13.2.

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Let

X

be a topological space.

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Any intersection of closed sets in $X$ is a closed set in $X .$
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Any finite union of closed sets in $X$ is a closed set in $X .$

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Proof.

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Let

X

be a topological space. To prove part 1, assume that

C_{α}

is a collection of closed set in

X

for

α

in some indexing set

I .

Then

X ∖ ⋂_{α \in I} C_{α} = ⋃_{α \in I} X ∖ C_{α} .

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The latter is an arbitrary union of open sets and so it an open set. By definition, then,

⋂_{α \in I} C_{α}

is a closed set.

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For part 2, assume that

C_{1},

C_{2},

\dots,

C_{n}

are closed sets in

X

for some

n \in Z^{+} .

To show that

C = ⋂_{k = 1}^{n} C_{k}

is a closed set, we will show that

X ∖ C

is an open set. Now

X ∖ ⋃_{α \in I} C_{α} = ⋂_{α \in I} X ∖ C_{α}

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is a finite intersection of open sets, and so is an open set. Therefore,

⋃_{α \in I} C_{α}

is a closed set.

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Theorem 13.2 tells us that any intersection of closed sets is closed, but only finite unions of closed sets are closed. How do we know that non-finite unions of closed sets aren’t necessarily closed?

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