Ideals of compact sets associated with Borel functions.
For a Polish space \(X\), let \(J (X)\) be the hyperspace of compact subsets of \(X\) with the Hausdorff metric. Given a function \(f: X\to Y\) between Polish spaces \(X\) and \(Y\), put \({\mathcal C} (f) = \{ C \in J(X) : f \upharpoonright C\) is continuous\(\}\). The author establishes a number of results about the connection between the complexity of \(f\) and the complexity of \({\mathcal C} (f)\). For example, \(f\) is continuous iff \({\mathcal C} (f)\) is a \(G_\delta\)-subset of \(J(X)\). Also, if \({\mathcal C} (f)\) is a \(\Pi^0_3\) subset of \(J(X)\), then \(f\) is a Baire class one function which in turn implies that \({\mathcal C} (f)\) is a \(\Pi^0_4\) subset of \(J(X)\), with none of the two implications being reversible. Furthermore, for a Borel function \(f\), \(f\) has a \(G_\delta\)-graph iff \({\mathcal C} (f)\) is a Borel subset of \(J(X)\). The proofs use arguments from classical descriptive set theory.
scientific article; zbMATH DE number 2084016