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


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


ChooseSinceand sinceis Hausdorff, open setsand exist such thatand

The family of setsforms an open cover ofso that

Since A is compact, a finite subcoverexists such that


andare open, andand

But sinceforwe have

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