Theorem
Any compact subset
of a Hausdorff space
is closed.
Proof
Letbe a compact subset of a Hausdorff spaceand let
Let
Sinceis Hausdorff there are compact subsets
and
of
and
respectively such that
and
with
The family of sets
is an open cover of
Sinceis compact there is a finite subcover
Let
and
then
and
Since
and
is open andis closed.