## Proof That a Compact Subset of a Hausdorff Space is Closed

Theorem

Any compact subsetof a Hausdorff spaceis closed.

Proof

Letbe a compact subset of a Hausdorff spaceand let

LetSinceis Hausdorff there are compact subsetsandofand respectively such thatandwith

The family of setsis an open cover ofSinceis compact there is a finite subcover

Letandthenand

Sinceandis open andis closed.