The Baire property of certain hypo-graph spaces

Abstract

Let $X$ be a compact metrizable space and $Y$ be a nondegenerate dendrite with an end point 0. For each continuous function $f : X \rightarrow Y$, we define the hypo-graph $\downarrow f = \cup_{x \in X \{x} \times [0, f(x]$ of $f$, where $[0, f(x)]$ is the unique path from 0 to $f(x)$ in $Y$. Then we can regard $\downarrow \mathrm{C}(X,Y) = \{\downarrow f | f : X \rightarrow Y$ is continuous} as a subspace of the hyperspace consisting of non-empty closed sets in $X \times Y$ equipped with the Vietoris topology. In this paper, we prove that $\downarrow \mathrm{C}(X,Y)$ is a Baire space if and only if the set of isolated points of $X$ is dense.