Theorem
Any compact subsetof a Hausdorff space
is closed.
Proof
Letbe a compact subset of a Hausdorff space
and let
LetSince
is Hausdorff there are compact subsets
and
of
and
respectively such that
and
with
The family of setsis an open cover of
Since
is compact there is a finite subcover
Letand
then
and
Sinceand
is open and
is closed.