Relations approximated by continuous functions in the Vietoris topology

Abstract

LetX be a Tikhonov space,C(X) be the space of all continuous real- valued functions defined onX, and CL(X × R) be the hyperspace of all nonempty closed subsets ofX × R. We prove the following result: LetX be a locally connected locally compact paracompact space, and letF 2 CL(X × R). ThenF is in the closure ofC(X) in CL(X × R) with the Vietoris topology if and only if: (1) for every x 2 X,F(x) is nonempty; (2) for everyx 2X,F(x) is connected; (3) for every isolatedx 2X,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.