Proof That Every T3 Space is Also a T2 Space

Theorem

If a topological space is T3, it is also T2.

Proof

A regular T1 space is called a T3 space. A space is regular if given a closed setand an elementdisjoint open setsandexist withand

Letbe a T3 space and letbe distinct. Sinceis T1is closed. Sinceare distinct,Sinceis regular disjoint open setsexist such thatand

Hencebelong to disjoint open sets andis a Hausdorff T2 space.