Proof That if a T0 or T1 Space is Homeomorphic to Another Space, That Space is T0 or T1 Respectively

Theorem

Ifandare homeomorphic then andis a T0 space or a T1 space then so is

Proof

Letbe a homeomorphism fromtoEvery singleton setof is closed.

Letbe any point ofThe setis a singleton set inand sinceis T1, every singleton set ofis closed sois closed in

Sinceis a homeomorphism it maps closed sets to closed sets so

Henceis a T1 space

The proof forbeing a T0 space is similar.