An intersection of F-sigma-delta set and open set

Question

Let $X$ be a Polish space and let $U$ be an open subset of $X$. I know that $A\cap U$ is of $F_{\sigma\delta}$ subset of $X$. Is it true then that $A$ is of $F_{\sigma\delta}$ subset of $X$?

Answer 1

No. Let $X=\mathbb R, U=(0,1), A=U \cup B$ where $B$ is a Borel subset of $(1,2)$ which is not an $F_{\sigma \delta}$ set.