# Existence of an open set with compact boundary

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?

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13 questions asked, no answer accepted. Way to go. – gnometorule Dec 15 '12 at 7:59
What have you tried so far? What does an open set in $X\times Y$ look like, expressed via open setd oin $X,Y$? – Hagen von Eitzen Dec 15 '12 at 8:52
13 questions asked and 11 of them are answered, but no answer accepted. – Paul Dec 15 '12 at 9:19