Existence of an open set with compact boundary

Question

Let $X$ and $Y$ be two Hausdorff spaces such that for every open set $U$ in $X$ and $V$ in $Y$ , there exist open sets $W\subseteq U$ and $W'\subseteq V$ , such that $\operatorname{Bd}(W)$ and $\operatorname{Bd}(W')$ are compact. Is for every open set $N$ in $X\times Y$, there exist an open set $M$ such that $M\subseteq N$ and $\operatorname{Bd}(M)$ compact?