A space is $F\_\sigma$-discrete space if it is the countable union of closed discrete subspaces. Is it true that every subset of an $F\_\sigma$-discrete space is of the type $G\_\delta$?



on $F_\sigma$-discrete  space
The answer is yes. Suppose that $X$ is an $F\_\sigma$ space as you have defined it, so that $X=\bigcup\_n X\_n$, where each $X\_n$ is a closed discrete space. Suppose that $Y\subset X$ is any subset. For each $n$, note that $X\_n-Y$ is closed in the discrete space $X\_n$ and hence also closed in $X$, and so the complement $U\_n=X-(X\_n-Y)$ is open in $X$. Furthermore, $\bigcap\_n U\_n=Y$, since $Y\subset U\_n$ and anything not in $Y$ is in some $X\_n$ and hence thrown out of that $U\_n$. So $Y$ is a countable intersection of open sets and hence $G\_\delta$. 


