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$\mathbb{R}$ is a baire space, so a countable union of closed no-where dense sets: $\bigcup C_n$ has empty interior. That is, $$\bigcup_{A \text{ open }, \; A\subset \bigcup C_n}A = \emptyset $$

So any open set $\mathcal{O}$ that has nonempty intersection with $\bigcup C_n$ will not intersect in a set that's open, otherwise the interior of $\bigcup C_n$ would be nonempty. My question is: is there some open subset of $\mathcal{O}$ that is disjoint from $\bigcup C_n$?

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The sets $\{q\}$ for $q\in\Bbb Q$ are nowhere dense, but their union over $\Bbb Q$, which is $\Bbb Q$, is dense.

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