Proof That if a T0 or T1 Space is Homeomorphic to Another Space, That Space is T0 or T1 Respectively
Theorem
If
and
are homeomorphic then andis a T0 space or a T1 space then so is
Proof
Let
be a homeomorphism fromtoEvery singleton set
of is closed.
Let
be any point of
The set
is a singleton set in
and sinceis T1, every singleton set ofis closed so
is 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.
