Proof That any Element in the Complement of a Compact Subset of a Hausdorff Space is in a Open Subset of the Complement

Theorem

Letbe a compact subset of a Hausdorff spaceIfthen there is an open set such that

Open setsandexist such thatand

Thereforeand

To proveis open, letso that

Sinceis compact - hence closed - an open setexists such thathenceandis open.