Lemma 1. A -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.

Let be regular spaces.

We need to show is regular.

Pick an arbitrary pointain 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. Let be 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

Since a V containsaand , we conclude that is contained in U.

Thus, X is regular.

A similar proof can be found here as well.