Let X be a Tikhonov space, C(X) be the space of all continuous real-valued functions defined on X, and CL(X×ℝ) be the hyperspace of all nonempty closed subsets of X×ℝ. We prove the following result: Let X be a locally connected locally compact paracompact space, and let F ∈ CL(X×ℝ). Then F is in the closure of C(X) in CL(X×ℝ) with the Vietoris topology if and only if: (1) for every x ∈ X, F(x) is nonempty; (2) for every x ∈ X, F(x) is connected; (3) for every isolated x ∈ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm195-3-2,
author = {L'. Hol\'a and R. A. McCoy},
title = {Relations approximated by continuous functions in the Vietoris topology},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {205-219},
zbl = {1130.54008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm195-3-2}
}
L'. Holá; R. A. McCoy. Relations approximated by continuous functions in the Vietoris topology. Fundamenta Mathematicae, Tome 193 (2007) pp. 205-219. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm195-3-2/