The Baire property of certain hypo-graph spaces (Q498626)

Let \(X\) be a compact metrizable space and \(Y\) be a non-degenerate dendrite (i.e., a connected, locally connected, compact metrizable space, containing no closed simple curves) with an end point \(\mathbf{0}\), and let \(C(X,Y)\) be the set of all continuous functions from \(X\) to \(Y\). For \(f\in C(X,Y)\), let \( \downarrow f=\cup \{\{x\}\times [\mathbf{0},f(x)]:x\in X\}\subset X\times Y\) be the hypo-graph of \(f\). Let \(\downarrow C(X,Y)=\{\downarrow f: f\in C(X,Y)\}\) as a subspace of the hyperspace of all non-empty closed subsets of \(X\times Y\) endowed with the Vietoris topology. In this paper, the author proves that the following are equivalent: \(\downarrow C(X,Y)\) is a Baire space; \(\downarrow C(X,Y)\) is not a \(Z_{\sigma}\)-set in itself; the set of all isolated points of \(X\) is dense.