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
Letbe a compact subset of a Hausdorff space
If a in X-A then open sets U and V exist such that
and
Proof
ChooseSince
and since
is Hausdorff, open sets
and
exist such that
and



Since A is compact, a finite subcoverexists such that
Defineand
and
are open, and
and
But sincefor
we have