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### Baire's Theorem and Irrationals [duplicate]

I am asked to show that the irrational numbers are not a countable union of closed subsets of $\mathbb{R}$ given that if a complete metric space is the countable union of of closed subsets then at ...
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### Why is $\Bbb Q$ not $G_{\delta}$? [duplicate]

I know of proofs that the rationals are not $G_{\delta}$, so I was just wondering what the following set is equal to: $$\bigcap_m \bigcup_{r_n\in \Bbb Q}(r_n-2^{-n}/m,r_n+2^{-n}/m)$$ I know it ...
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### Is it possible to have $D=\Bbb P$

Let $f:\Bbb R\to \Bbb R$ and $D=\{x\in \Bbb R: f$ is discontinuous at $x\}$. My problem is : Is it possible to have $D=\Bbb P$ where $\Bbb P$ is the set of irrationals in $\Bbb R$. I know the ...
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### Problem on Baire's category theorem.

I'd like to show that the set of irrational numbers in $[0,1]$ cannot be represented as a countable union of closed sets. The hint says to use Baire's category theorem. I know two versions of such ...