If each space (a A) is a Hausdorff space, then is a Hausdorff space in the box topologies.
Can anyone help to start the proof?
Suppose that and are distinct points in X. Then there must be at least one coordinate at which they differ. So choose such that . Apply the Hausdorff condition in the space , and use that to construct boxes separating and
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.