Prove that the product of two regular spaces is regular.

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- Mar 10th 2009, 11:22 PMAndreametregular spaces
Prove that the product of two regular spaces is regular.

- Mar 11th 2009, 02:19 AMaliceinwonderland
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 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.