Let $N$ be an open set of the product space $Y\times Z$ such that the boundary of $N$ is compact. Is there an open subset $U$ of $Y$ and open subset $V$ of $Z$ such that the boundary of $U\times V$ is compact and $U\times V$ a subset of $N$?
The boundary set of product space
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general-topology
1 Answers
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Not necessarily: let $Y=Z=\Bbb Q$, and let $N=\Bbb Q^2\setminus\{\langle 0,0\rangle\}$. The boundary of $N$ is $\{\langle 0,0\rangle\}$, but no non-empty open set in $\Bbb Q^2$ of the form $U\times V$ has compact boundary.