Relations approximated by continuous functions in the Vietoris topology

L'. Holá¹ ; R. A. McCoy²

¹ Mathematical Institute Slovak Academy of Sciences Štefánikova 49 814 73 Bratislava, Slovakia
² Virginia Polytechnic Institute and State University Department of Mathematics Blacksburg, VA 24061, U.S.A.

Fundamenta Mathematicae, Tome 195 (2007) no. 3, pp. 205-219

Résumé

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.

Détail

DOI : 10.4064/fm195-3-2

Keywords: tikhonov space space continuous real valued functions defined times mathbb hyperspace nonempty closed subsets times mathbb prove following result locally connected locally compact paracompact space times mathbb closure times mathbb vietoris topology only every nonempty every connected every isolated singleton set upper semicontinuous forces local semiboundedness gives answer problem question realm locally connected locally compact paracompact spaces