Proof That Every Compact T2 Space is Locally Compact

Theorem

Every compact T2 spaceis locally compact.

Proof

Letbe a T2 space. Thenis locally compact if and only if, giventhere is a compact setsuch that

For a T2 space the existence of a compact subsetofsuch thatguarantees that for any neighbourhoodofthere is a compact setsuch that

Letbe a compact T2 space. The above condition is satisfied sinceis a compact subset of itself, henceis locally compact.