Proof That a Closed Subspace of a Locally Compact T2 Space is Locally Compact
Theorem
A closed subspace of a locally compact T2 space is locally compact.
Proof
Letbe a closed subspace of
Let
and let
be a compact subset of
such that
is closed because
is T2. The set
is a closed subset of the compact set
so is compact.
We have
Sinceis T2 it is locally compact.