Proposition 7

Every product of Hausdorff spaces is Hausdorff. Conversely, if a product of non-empty spaces is Hausdorff, then each factor is a Hausdorff space.

Let $X = \prod_{i \in I} X_i$ be a product of topological spaces. Then if $x, y$ are two distinct points of $X$, we have $\operatorname{pr}_i x \neq \operatorname{pr}_i y$ for some index $i$, and Proposition 5 of no. 1 shows that $X$ is Hausdorff if the $X_i$ are. Conversely, if $X$ is Hausdorff and the $X_i$ are non-empty, then each $X_i$ is homeomorphic to a subspace of $X$ (§ 4, no. 2, Proposition 4) and is therefore Hausdorff.