Proof That a Locally Compact Hausdorff Space is Regular
A locally compact hausdorff - T2 - spaceis regular.
A spaceis regular if and only if, given anyand any neighbourhoodof there is a neighbourhoodofsuch that
Letbe a locally compact T2 space. Ifandis any neighbourhood ofthen there is a compact setsuch that
Sinceis compact,is closed, hence
Sinceis a neighbourhood ofis regular.