# Math Help - X_{\alpha} is Hausdorff

1. ## X_{\alpha} is Hausdorff

If each $X_{\alpha}$ is a Hausdorff space, then $\prod X_{\alpha}$ is a Hausdorff space in both the box and product top.

How do I go about proving this?

I know since $X_{\alpha}$ is Hausdorff, if for all $x_1,x_2\in X_{\alpha}$ where $x_1\neq x_2$ there exists neighborhoods $U_1 \ \text{and} \ U_2$ of $x_1 \ \text{and} \ x_2$ that are disjoint.

2. ## Re: X_{\alpha} is Hausdorff

If $(x_\alpha)$ and $(y_\alpha)$ are distinct points in the product space, then they must differ in at least one coordinate. So there exists $\alpha_0$ say, such that $x_{\alpha_0} \ne y_{\alpha_0}$. Use the Hausdorff property in the $\alpha_0$-coordinate space to separate $x_{\alpha_0}$ and $y_{\alpha_0}$ by disjoint neighbourhoods. Then use those neighbourhoods to construct neighbourhoods in the product space that separate $(x_\alpha)$ and $(y_\alpha)$. The same construction will work for both the product and the box topologies.