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Math Help - Box Topology

  1. #1
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    Box Topology

    If each space X_a (a \in A) is a Hausdorff space, then X=\prod X_a is a Hausdorff space in the box topologies.

    Can anyone help to start the proof?
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  2. #2
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    Quote Originally Posted by problem View Post
    If each space X_a (a \in A) is a Hausdorff space, then X=\prod X_a is a Hausdorff space in the box topologies.

    Can anyone help to start the proof?
    Suppose that x=(x_a) and y=(y_a) are distinct points in X. Then there must be at least one coordinate at which they differ. So choose a\in A such that x_a \ne y_a. Apply the Hausdorff condition in the space X_a, and use that to construct boxes separating x and y.
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  3. #3
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    I am stucked in constructing boxes to seperate x and y. I understand that for i-th coordinate, we can say x_i and y_i can be in open sets that are disjoint. But I do not know how to consider two open sets containing different x and y to be disjoint.
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  4. #4
    Senior Member Tinyboss's Avatar
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    Let your open sets be the entire space in each other coordinate.
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