Relations approximated by continuous functions in the Vietoris topology

L'. Holá, and R. A. McCoy. "Relations approximated by continuous functions in the Vietoris topology." Fundamenta Mathematicae 195.3 (2007): 205-219. <http://eudml.org/doc/283109>.

@article{L2007,
abstract = {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.},
author = {L'. Holá, R. A. McCoy},
journal = {Fundamenta Mathematicae},
keywords = {set-valued mappings; Vietoris topology; upper semicontinuous multifunction; locally semibounded multifunction; continuous function},
language = {eng},
number = {3},
pages = {205-219},
title = {Relations approximated by continuous functions in the Vietoris topology},
url = {http://eudml.org/doc/283109},
volume = {195},
year = {2007},
}

TY - JOUR
AU - L'. Holá
AU - R. A. McCoy
TI - Relations approximated by continuous functions in the Vietoris topology
JO - Fundamenta Mathematicae
PY - 2007
VL - 195
IS - 3
SP - 205
EP - 219
AB - 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.
LA - eng
KW - set-valued mappings; Vietoris topology; upper semicontinuous multifunction; locally semibounded multifunction; continuous function
UR - http://eudml.org/doc/283109
ER -