Proof That Every T4 Space is Also a T3 Space
Every topological spacewhich is T4 is also T3.
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. Supposeandis a closed subset ofSinceis T1 the singleton setis closed. Setsandare closed and disjoint. Sinceis normal, open setsandexist such that
Thereforeis regular and T3.