Prove that the product of two regular spaces is regular.
Lemma 1. A $\displaystyle T_1$-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 $\displaystyle X_1, X_2$ be regular spaces.
We need to show $\displaystyle X = X_1 \times X_2$ 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 $\displaystyle p_{i}$ be a canonical projection mapping $\displaystyle p_i:X \rightarrow X_i, i=1,2$;let $\displaystyle \bigcap _{i=1,2} p_{i}^{-1}(U_{i})$ be a basic open set in X which contains a and is contained in U.
Using a regularity and choose a V as $\displaystyle V = \bigcap _{i=1,2} p_{i}^{-1}(V_{i})$, where $\displaystyle p_{i}(a) \in V_{i}, \bar{V_{i}} \subset U_{i}. $
Since a V contains a and $\displaystyle \bigcap _{i=1,2} p_{i}^{-1}(\overline{V_{i}}) \subset \bigcap _{i=1,2} p_{i}^{-1}(U_{i})$, we conclude that $\displaystyle \bar{V}$ is contained in U.
Thus, X is regular.
A similar proof can be found here as well.