Proof That Being a T4 Space is a Topological Property
Theorem
If a T4 space
is homeomorphic to a space
thenis T4.
Proof
If
is a T1 space homeomorphic to a space
thenis T1. Supposeis a normal space and
is a homeomorphism. Let
and
be disjoint closed subsets ofthen 
and
are disjoint open sets of
Sinceis normal there are disjoint open sets
and
insuch that
and
The setsandare open inand
and
henceis normal and T4.
