= 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.
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