Proof That a Compact Subset of a Hausdorff Space is Closed


Any compact subsetof a Hausdorff spaceis closed.


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


Sinceandis open andis closed.

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