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Theorem) If each space Xa(a∈A) is a Hausdorff space, then X=∏Xa is a Hausdorff space in both the box and product topologies.

I understand if a box topology, the theorem holds.

but if a product toplogy, I do not understand clearly.

I think if there are distinct points c,d in X, then Uc, Ud (arbitrary open sets in X contain c, d respectively) are equals Xa except for finitely many values of a, so Uc and Ud are not disjoint.

If I have a mistake, please point out it....

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# Product of hausdorff spaces

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