Given a Polish space $(X,\tau)$ can we characterize in some meaningful way the sequences $(A_n)_{n\in\omega}$ of $\boldsymbol{\Delta}_2^0(X)$ sets (we can assume them to be pairwise disjoint) such that their union $\bigcup_{n\in\omega} A_n$ is still a $\boldsymbol{\Delta}_2^0(X)$ set?
I was thinking about using the characterization of $\boldsymbol{\Delta}_2^0$ sets given by the Difference Hierarchy (Hausdorff-Kuratowski theorem) in some way, but I don't get much further.
Do you have any idea?
We could rephrase the same question using $\boldsymbol{\Delta}_\alpha^0$ instead of $\boldsymbol{\Delta}_2^0$
Thanks!