Proof That the Alexandroff Compactification of a Space is T2 if and only if the Space is T2 and Locally Compact

Theorem

The Alexandroff compactification of a spaceis T2 if and only if it is T2 andis locally compact.

Proof

Supposeis compact, then the Alexandroff compactification ofis

Ifis T2, it is also locally compact.

Supposeis not compact andis the Alexandroff compactification ofIfis T2 then so issinceis a subspace of

Sinceis T2 and compact,is also locally compact. LetwithIf thenandexist such that becauseis T2.

Supposeis an ideal point thenand a compact subset B ofexists such that

Henceis a neighbourhood ofis a neighbourhood ofand

Henceis T2.