Theorem
Every topological space
which is T4 is also T3.
Proof
A normal space (so that any two disjoint closed sets are contained in disjoint open sets) which is also T1 is called a T4 space.
Letbe a T4 space thenis normal and T1. Suppose
and
is a closed subset of
Sinceis T1 the singleton set
is closed. Setsandare closed and disjoint. Sinceis normal, open sets
and
exist such that
Thereforeis regular and T3.