## Proof That Disjoint Compact Subsets of a Hausdorff Space are Contained in Disjoint Open Subsets

Theorem

Letandbe disjoint compact subsets of a Hausdorff spaceOpen setsand exist such thatand

Proof

LetSince

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).

Letand

Thenandand bothandare open.

For eachhence