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 ofLetand letbe a compact subset ofsuch that

is closed becauseis T2. The setis a closed subset of the compact setso is compact.

We have

Sinceis T2 it is locally compact.