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

Defineto obtain

Sinceis a neighbourhood ofis regular.