Theorem
If a T4 spaceis homeomorphic to a space
then
is T4.
Proof
Ifis a T1 space homeomorphic to a space
then
is T1. Suppose
is a normal space and
is a homeomorphism. Let
and
be disjoint closed subsets of
then
and
are disjoint open sets of
Sinceis normal there are disjoint open sets
and
in
such that
and
The setsand
are open in
and
and
hence
is normal and T4.