Proof That Every Subspace of a Regular Space is Regular

Theorem

The property of being regular is inherited, so that a subset of aregular space is regular.

Proof

Letbea regular space and letbea subspace.

Letandletbeaclosedsubset ofsuchthatandwhereisthe closure ofin

Sinceisregular, open setsandexistsuch that

Butandareopensubsets of

becauseandSetsandaredisjoint because

Also, sinceandisalso regular.

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