 Hausdorff property of a nonempty product

For nonempty topological spaces $X,Y$, their product topology is Hausdorff exactly when both factors are Hausdorff spaces. Distinct pairs differ in some coordinate, and disjoint open neighbourhoods in that factor give disjoint open rectangles in the Cartesian product. Conversely, fixing $y_0\in Y$ gives a homeomorphism from $X$ onto the topological subspace $X\times \{y_0\}$; similarly for $Y$. The Hausdorff condition passes to topological subspaces. Nonemptiness matters: an empty set product is Hausdorff regardless of the other factor.