Proof That the Alexandroff Compactification of a Space is T2 if and only if the Space is T2 and Locally Compact
The Alexandroff compactification of a spaceis T2 if and only if it is T2 andis locally compact.
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