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.