Proof That any Element in the Complement of a Compact Subset of a Hausdorff Space is in a Open Subset of the Complement
Letbe a compact subset of a Hausdorff spaceIfthen there is an open set such that
Open setsandexist such thatand
To proveis open, letso that
Sinceis compact - hence closed - an open setexists such thathenceandis open.