Let $X\subseteq \mathbb R$ such that

• $X$ is an $F_{\sigma\delta}$-set (in $\mathbb R$); and
• $X$ is a $G_{\delta\sigma}$-set.

It is not necessarily true that $X$ must be $F_\sigma$ or $G_\delta$. A counterexample is $[\mathbb Q \cap (-\infty,0)]\cup \mathbb P\cap (0,\infty)]$, where $\mathbb Q$ and $\mathbb P$ are the rationals and irrationals, respectively.

Question. Is there necessarily an open subset of $X$ which is $F_\sigma$ or $G_\delta$ in $\mathbb R$?

Has there been a study of zero-dimensional spaces which are both $F_{\sigma\delta}$ and $G_{\delta\sigma}$?