Theorem
A locally compact hausdorff - T2 - space
is regular.
Proof
A spaceis regular if and only if, given any
and any neighbourhood
of
there is a neighbourhood
of
such that
Letbe a locally compact T2 space. Ifandis any neighbourhood ofthen there is a compact set
such that
Sinceis compact,is closed, hence
Define
to obtain
Sinceis a neighbourhood of
is regular.