Countable compacta of continuity for projective functions
For a Polish space \(X\) let \(\mathcal{K}(X)\) denote the space of all compact subsets of \(X\) equipped with the Vietoris topology. In the paper under review the author investigates connections between complexity of a function \(f\) from a Polish space \(X\) to a Polish space \(Y\) and complexity of the set \(\widetilde C(f)=\{K\in \mathcal{K}(X)\); \(f|_K\) is continuous and \(K\) has exactly one limit point\}. In the main result he shows that for each positive integer \(n\) the following conditions are equivalent: (1)~\(\widetilde{C}(f)\in\Delta_n^1(\mathcal{K}(X))\); (2)~\(\widetilde{C}(f)\in \Sigma^1_n(\mathcal{K}(X))\); (3)~\(f\) is \(\Delta_n^1\)-measurable. In particular, \(f\) is Borel iff \(\widetilde{C}(f)\) is Borel.
scientific article; zbMATH DE number 5888211