Theorem

Letbe a compact subset of a Hausdorff spaceIf a in X-A then open sets U and V exist such thatand

Proof

ChooseSinceand sinceis Hausdorff, open setsand exist such thatand

The family of setsforms an open cover ofso thatSince A is compact, a finite subcoverexists such that

Defineand

andare open, andand

But sinceforwe have