Let N be an open set of YxZ. Is there exist an open set W in YxZ with compact boundry such that W is a subset of N?
Is the every open set in a product space contains an open set with compact boundry?
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general-topology
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1W should $b$e non-empty! – 2012-12-07
1 Answers
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Not in general, no. For example, let $Y$ be an infinite set with the particular point topology, and $Z$ be a one-point space. Then $N=\{(p,*)\}$ (where $p$ is the particular point of $Y$ and $*$ is the only point of $Z$ is a counterexample.
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0If you mean to as$k$ whether my example is Hausdorff, it isn't. If you mean to ask whether there's a Hausdorff example, I don't know. I expect so, though. – 2012-12-08