## Proof That a Subset of a Completely Regular Space is Completely Regular

Theorem

A subspace of a completely regular space is completely regular.

Proof

Letbe a completely regular space and letbe a subspace.

Letbe closed inand let

Thenfor someclosed in

Sinceandwe have

Sinceis completely regular a continuous functionexists such thatandfor

Thenis continuous withand for

Thenis completely regular.