Yes, since closed discrete sets are hereditarily closed. Let $X$ be the space and $E_n$ the closed discrete sets, so $X = \bigcup\limits_{n=1}^\infty E_n$. Then given $S \subseteq X$, each $E_n \setminus S$ is still closed, so $S = \bigcap\limits_{n=1}^\infty (E_n \setminus S)^\complement$ shows that $S$ is a $G_\delta$.
Nathan
- 269
- 3
- 8