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Transcript

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. ∗ hTheUnionOfALocallyFiniteCollectionOfClosedSetsIsClosedi created: h2013-03-21i by: hyarki version: h38349i Privacy setting: h1i hTheoremi h54A99i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1