# Does their exist a countable $G_\delta$ set in $\Bbb{R}$ with usual metric.

Basically I have tried to examine the fact. Suppose , $$C=\{x_n\}$$ be countable $$G_\delta$$ set.
Hence, $$C=\cap V_n$$ where each $$V_n$$ is open.
Thus, $$\Bbb{R}=C\cup( \Bbb{R}\setminus C)=\cup \{x_n\} \cup\\ (\cup \Bbb{R}\setminus V_n)$$
If I can show int$$($$cl $$\Bbb{R}\setminus V_n)=\emptyset$$, then it will prove that $$\Bbb{R}$$ is first category set contradicting the Baire Category Theorem. But to do this $$V_n$$ should be dense.
I can't proceed the proof even I am not getting any counter example of it.
Can anybody give such example or can prove it? Thanks for assistance in advance.

• $\Bbb R$ is a "perfectly normal" space, that is, every closed set is $G_{\delta}.$ So any countable closed set (e.g. $\Bbb Z$ or $\{0\}\cup \{1/n: n\in \Bbb N\}$) will do Aug 24, 2019 at 1:59

Let $$V_n=\cup_{k=1}^{\infty} (k-\frac{1}{n},k+\frac{1}{n}).$$
Then, each $$V_n$$ is open, and $$\cap_{n\in \mathbb{N}} V_n=\mathbb{N},$$ which is countable.