Relations approximated by continuous functions in the Vietoris topology
Volume 195 / 2007
Abstract
Let $X$ be a Tikhonov space, $C(X)$ be the space of all continuous real-valued functions defined on $X$, and ${\rm CL}(X \times {{\mathbb R}})$ be the hyperspace of all nonempty closed subsets of $X\times {{\mathbb R}}$. We prove the following result: Let $X$ be a locally connected locally compact paracompact space, and let $F \in {\rm CL}(X \times {{\mathbb R}})$. Then $F$ is in the closure of $C(X)$ in ${\rm CL}(X \times {{\mathbb R}})$ with the Vietoris topology if and only if: (1) for every $x \in X$, $F(x)$ is nonempty; (2) for every $x \in X$, $F(x)$ is connected; (3) for every isolated $x \in X$, $F(x)$ is a singleton set; (4) $F$ is upper semicontinuous; and (5) $F$ forces local semiboundedness. This gives an answer to Problem 5.5 in [HM] and to Question 5.5 in [Mc2] in the realm of locally connected locally compact paracompact spaces.
