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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?

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Yes, take $W$ to be empty. –  Chris Eagle Dec 7 '12 at 18:58
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W should be non-empty! –  aliakbar Dec 7 '12 at 18:59

1 Answer 1

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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Is the your answer satisfies in a Hausdorff space? –  aliakbar Dec 8 '12 at 4:36
    
If you mean to ask 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. –  Chris Eagle Dec 8 '12 at 10:26

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