Exercise 6, § 9

Let $X, Y$ be two topological spaces, and let $A$ (resp. $B$) be a quasi-compact subset of $X$ (resp. $Y$). Show that if $U$ is any neighbourhood of $A \times B$ in $X \times Y$ then there is a neighbourhood $V$ of $A$ in $X$ and a neighbourhood $W$ of $B$ in $Y$ such that $V \times W \subset U$.