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 $G\_{\delta\sigma}$?



Subsets of reals which are both $F_{\sigma\delta}$ and $G_{\delta\sigma}$
No, let $X$ be the set of those irrationals in $x\in (0,1)$ with binary expansion
$$x=0.x\_1x\_2\dots$$
such that if we define $x^{\text{even}}, x^{\text{odd}}$ by
$$x^{\text{even}}=0.x\_2x\_4x\_6\dots$$
$$x^{\text{odd}}=0.x\_1x\_3x\_5\dots$$
then
exactly one of $x^{\text{even}}$, $x^{\text{odd}}$ is irrational.


