1. ## 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?

2. Originally Posted by problem
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 $\displaystyle y.$

3. 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.

4. Let your open sets be the entire space in each other coordinate.