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Let $X$ be a complete metric space without isolated points and $A \subseteq X$ a numberable dense subspace. I want to probe that $A$ is not a $G_{\delta}$

I thought about constructing $A$ as de union of it's points, that are closed set with empty interior, but couldn't find a contradiction with putting $A$ as an intersection of open sets

Silkking
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Take countably many open sets whose intersection is $A$, and note that they're all dense because $A$ is. Also take the complements of all singletons in $A$, and note that they're all open and dense. Altogether, you have countably many open dense sets whose intersection is empty, contrary to Baire's theorem.

Andreas Blass
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