Prove that the product of two regular spaces is regular.
Let be regular spaces.
We need to show is 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. 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 contains a and , we conclude that is contained in U.
Thus, X is regular.
A similar proof can be found here as well.