Let $\{C_{n,k}: n\ge 1, k\ge 1\}$ be a collection of closed sets in a metric space $X$. Is it true that there exists a sequence $(F_n)$ of closed sets such that $$ C:=\bigcap_{k\ge 1}\bigcup_{n\ge 1}C_{n,k} = \{x \in X: x \in F_n \text{ for infinitely many }n\}? $$
(I know that the answer is affirmative in the case that $C=A \setminus B$, where $A$ is a closed real interval and $B$ is a countable set, but the method does not extend to the whole class of $F_{\sigma \delta}$ sets.)
I would add that, if the answer is affirmative in general, then it should be a known fact. However, I couldn't find it, e.g., in Kechris.