Typical properties of continuous functions via the Vietoris topology (Q1308855)

Let \(X\) be a compact metrizable space and let \(C(X)\) be the Banach space of all continuous real functions. Let \(K(X)\) be the topological space (with the Vietoris topology) of all nonempty closed sets in \(X\). The main result is the following: Let \(X\) be a closed subset of \(I = [0,1]\) and let \(A\) be the family of all analytic functions on \(I\). If \(E \subset K(X)\) is a \(G^*_\delta\) set containing all finite sets, then for any set \(D \subset A\) such that \(A \smallsetminus D\) is dense in \(C(I)\), the set of all \(f \in D \smallsetminus X\) such that for all \(h \in D\) the set \(\{x; f = h\} \in E\) is dense of type \(G_\delta\).