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


A subspace of a completely regular space is completely regular.


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.