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 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