Proof That Disjoint Compact Subsets of a Hausdorff Space are Contained in Disjoint Open Subsets
Letandbe disjoint compact subsets of a Hausdorff spaceOpen setsand exist such thatand
is compact so open setsandexist such thatand(1)
The family of open setsforms an open cover of the compact setso a finite subcoverexists. For eachwe can find a correspondingsatisfying (1).
Thenandand bothandare open.