Is the Product of Hausdorff Spaces Always Hausdorff?

[emptyboat]

Hello, everyone.

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...

 

[arkajad]

Start with this: if c and d are different points of X then there is an index a\in A for which the projections of c and d differ. Exploit this value of the index.

 

[emptyboat]

Thanks a lot, arkajad. I understand it.
if only one coordinate is different, they are disjoint.