Proof That a Subset of a T4 Space Gives Rise to a T4 Space
Theorem
A subset
of a T4 space
gives rise to a T4 space.
Since every subspace of a T4 space is T1 andis T1
is also T1.is closed so a subset
ofis closed inif and only ifis also closed in
Hence if
and 
are disjoint subsets of
they are also disjoint closed subsets of
Hence open sets
exist such that
and
\Then
and
are disjoint subsets ofopen in
Since
is T1 and normal, it is T4.