Exercise 6, § 9

Let $X$, $Y$ be two topological spaces, $A$ (resp. $B$) a quasi-compact subset of $X$ (resp. $Y$). Now prove that, for every neighbourhood $U$ of $A × B$ in $X × Y$, there exists a neighbourhood $V$ of $A$ in $X$ and a neighbourhood $W$ of $B$ in $Y$ such that $V × W \subset U$.