Download the union of a locally finite collection of closed sets is

the union of a locally finite collection of
closed sets is closed∗
yark†
2013-03-21 21:11:01
The union of a collection of closed subsets of a topological space need not,
of course, be closed. However, we do have the following result:
Theorem. The union of a locally finite collection of closed subsets of a topological space is itself closed.
Proof. Let S be a locally finite collection of closed subsets of a topological
space X, and put Y = ∪S. Let x ∈ X \ Y . By local finiteness there is an
open neighbourhood U of x that
Sn meets only finitely many members of S, say
A1 , . . . , An . So U \ Y = U \ i=1 Ai , which is open. Thus U \ Y is an open
neighbourhood of x that does not meet Y . It follows that Y is closed.
One use for this result can be found in the entry on gluing together continuous functions.
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