# Product of hausdorff spaces

1. Sep 26, 2010

### 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....

2. Sep 27, 2010

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.