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Math Help - regular spaces

  1. #1
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    regular spaces

    Prove that the product of two regular spaces is regular.
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    Prove that the product of two regular spaces is regular.
    Lemma 1. A 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 X_1, X_2 be regular spaces.
    We need to show 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 p_{i} be a canonical projection mapping p_i:X \rightarrow X_i, i=1,2;let \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  V = \bigcap _{i=1,2} p_{i}^{-1}(V_{i}), where p_{i}(a) \in V_{i}, \bar{V_{i}} \subset U_{i}.

    Since a V contains a and  \bigcap _{i=1,2} p_{i}^{-1}(\overline{V_{i}}) \subset \bigcap _{i=1,2} p_{i}^{-1}(U_{i}), we conclude that  \bar{V} is contained in U.

    Thus, X is regular.

    A similar proof can be found here as well.
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