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About LinkBacksProve that the product of two regular spaces is regular.
Lemma 1. AOriginally Posted by Andreamet
-space is regular iff for each a in X and each open set U containing a ,there is an open set W containing a whose closure is contained in U.
Letbe regular spaces.
We need to showis regular.
Pick an arbitrary point a in X and an open set U containing a. By lemma 1, it is sufficient to show that there is an open set V in X containing a whose closure is contained in U. Letbe a canonical projection mapping
;let
be a basic open set in X which contains a and is contained in U.
Using a regularity and choose a V as, where
 \in V_{i}, \bar{V_{i}} \subset U_{i}. )
Since a V contains a and, we conclude that
is contained in U.
Thus, X is regular.
A similar proof can be found here as well.
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