The Stacks project

Proof. The inclusion $\overline{AB} \subset \overline{A} \times _ k \overline{B}$ is immediate. We may replace $X$ and $Y$ by the reduced closed subschemes $\overline{A}$ and $\overline{B}$. Let $W \subset X \times _ k Y$ be a nonempty open subset. By Morphisms, Lemma 29.23.4 the subset $U = \text{pr}_ X(W)$ is nonempty open in $X$. Hence $A \cap U$ is nonempty. Pick $a \in A \cap U$. Denote $Y_{\kappa (a)} = \{ a\} \times _ k Y$ the fibre of $\text{pr}_ X : X \times _ k Y \to X$ over $a$. By Morphisms, Lemma 29.23.4 again the morphism $Y_ a \to Y$ is open as $\mathop{\mathrm{Spec}}(\kappa (a)) \to \mathop{\mathrm{Spec}}(k)$ is universally open. Hence the nonempty open subset $W_ a = W \times _{X \times _ k Y} Y_ a$ maps to a nonempty open subset of $Y$. We conclude there exists a $b \in B$ in the image. Hence $AB \cap W \not= \emptyset$ as desired. $\square$