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

Question

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.

Answer 1

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.