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Math Help - Product of Hausdorff spaces

  1. #1
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    Product of Hausdorff spaces

    I have to prove that if X_\alpha is Hausdorff for each \alpha \in A, then X= \Pi_{\alpha \in A}X_\alpha is also Hausdorff.

    I have proved this for the two dimensional case, and I know if A is finite then it would be true for all of them by induction, but it doesn't say anything about A being finite or infinite. Is this still true by induction if A is infinite?
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  2. #2
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    No. Induction on n shows that "P(n)" is true for all finite n- it does not extend to infinite values.
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  3. #3
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    Take a pair of distinct points x, y \in X. Then for some index \beta \in A, x_\beta \neq y_\beta, where x_\beta, y_\beta \in X_\beta. Since X_\beta is Hausdorff, we can find disjoint open subsets U and V of X_\beta containing x_\beta and y_\beta respectively.

    Now let U_\alpha = X_\alpha for \alpha \neq \beta, and U_\beta = U, and similarly take V_\alpha = X_\alpha for \alpha \neq \beta, and V_\beta = V. Then \Pi_{\alpha \in A}U_\alpha and \Pi_{\alpha \in A}V_\alpha are disjoint open subsets of X containing x and y respectively, showing that X is Hausdorff.
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