show that in first countable hausdorff space every open set is G δ set?

Question

This is question from my assignment i have tried to do some proof on it Let $\{x\}$ be one point set in $X$ since $X$ is countable then there exists a countable basis for $x$ since $X$ is hausdorff for any $x,y\in X$ with $x$ not equal to $y$ there is a open set $U$ containing $x$ and $V$ containing $y$ im not able to do the proof further can u help me with this proof?

Answer 1

Borrowing my own answer from here:

Suppose $X = \{x_n: n \in \mathbb{N}\}$ to make it explicitly countable. As $X$ is $T_1$ in particular, finite sets are closed, so all sets $U_n = X\setminus\{x_n\}$ are open. Now $$\{x_m\} =\cap \{U_n: n \neq m\}$$ for every $m$, making all singletons a $G_\delta$.

Note that Hausdorff is slightly overkill, only $T_1$ is used.