Proof That Any Element Not in a Compact Subset of a Hausdorff Space is in an Open Set That Has No Intersection With an Open Set Containing the Compact Subset
Theorem
Let
be a compact subset of a Hausdorff space
If a in X-A then open sets U and V exist such that
and
Proof
Choose
Since
and since
is Hausdorff, open sets
and
exist such that
and
forms an open cover of
so that
Since A is compact, a finite subcover
exists such that
Define
and
and
are open, and
and
But since
for
we have
