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Thread: Box Topology

  1. #1
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    Box Topology

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

    Can anyone help to start the proof?
    Suppose that $\displaystyle x=(x_a)$ and $\displaystyle y=(y_a)$ are distinct points in X. Then there must be at least one coordinate at which they differ. So choose $\displaystyle a\in A$ such that $\displaystyle x_a \ne y_a$. Apply the Hausdorff condition in the space $\displaystyle X_a$, and use that to construct boxes separating $\displaystyle x$ and $\displaystyle y.$
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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
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    Let your open sets be the entire space in each other coordinate.
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