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We know that Baire Category Theorem implies that in a complete metric space, the countable intersection of open and dense sets is nonempty and actually dense itself.

But it is clear that a countable intersection of open sets need not be open.

So, can you find an example of a non-open set that is a countable intersection of open and dense sets in a complete metric space?

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    $\begingroup$ For each rational $r$, let $U_r=\Bbb R\setminus\{r\}$. Then the irrationals are the intersection of the sets $U_r$. $\endgroup$ Jun 3, 2014 at 17:29

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Sure:

For each rational number $ r$, let $U_r=\Bbb R\setminus\{\,r\,\}$. Then the set of irrational numbers is the intersection of the sets $U_r$.

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