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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.

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