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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.

Paolo Leonetti
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    Exercise 23.5 on page 180 of Kechris' book claims (if I understood well) that if $X$ is separable metrizable space then the family of all sets of the form ${x:\text{$x\in A_n$ for infinitely many $n$}}$ where $A_n$ run through the class $\Pi^0_\xi(X)$, is exactly the class $\Pi^0_{\xi+2}(X)$. So for $A_n$ closed we should obtain all $F_{\sigma\delta}$ sets. – Peter Elias Aug 20 '17 at 01:46
  • It works, indeed :) – Paolo Leonetti Aug 20 '17 at 10:52

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