Proof That a Subset of a T4 Space Gives Rise to a T4 Space

Theorem

A subsetof a T4 spacegives rise to a T4 space.

Since every subspace of a T4 space is T1 andis T1is also T1.is closed so a subsetofis closed inif and only ifis also closed inHence ifand are disjoint subsets ofthey are also disjoint closed subsets of

Hence open setsexist such thatand

\Thenandare disjoint subsets ofopen inSinceis T1 and normal, it is T4.