The hyperspace of the regions below continuous maps with the Fell topology
type: Item
instance of: scholarly article
title: The hyperspace of the regions below continuous maps with the Fell topology
zbMATH Open document ID: 1270.54016
author: Bao Can Zhang
author: Zhongqiang Yang
publication date: 2013-03-14T00:00:00Z
review text: For a Tychonoff space \(X\) and \(\mathbf I=[0,1]\), \(\text{Cld}_F(X\times \mathbf I)\) stands for the hyperspace \(\text{Cld}(X\times \mathbf I)\) of all nonempty closed subsets of \(X\times \mathbf I\) endowed with the \textit{Fell topology} having as subbase sets of the form \(\{A\in \text{Cld}(X\times \mathbf I): A\cap U\neq\emptyset\}\), and \(\{A\in \text{Cld}(X\times \mathbf I): A\cap K=\emptyset\}\), where \(U\subseteq X\times \mathbf I\) is open and \(K\subseteq X\times \mathbf I\) is compact. If \(\downarrow USC(X)\), and \(\downarrow C(X)\) stand for the collection of hypographs \(\{(x,\lambda)\in X\times \mathbf I: \lambda \leq f(x)\}\) of upper semicontinuous and continuous maps \(f:X\to\mathbf I\), respectively, then \(\downarrow USC(X)\) and \(\downarrow C(X)\) can be viewed as subspaces of \(\text{Cld}_F(X\times \mathbf I)\). Motivated by analogous results for the so-called Vietoris hyperspace topology, the authors characterize under what conditions on \(X\) there is a homeomorphism \(h\) between \(\downarrow~USC(X)\) and \([-1,1]^{\omega}\) with \(h(\downarrow C(X))=c_0=\{(x_n)\in [-1,1]^{\omega}: \lim_n x_n=0\}\). The main result is that this happens if and only if \(X\) is locally compact, separable, metrizable, and the set of isolated points of \(X\) is not dense in \(X\).
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 6145089
zbMATH Keywords: Fell topology
zbMATH Keywords: hyperspace
zbMATH Keywords: hypograph
zbMATH Keywords: uppers semicontinuous
zbMATH Keywords: continuous
zbMATH Keywords: Hilbert cube
reviewed by: Laszlo Zsilinszky
MaRDI profile type: Publication
full work available at URL: s10114-012-0030-6
OpenAlex ID: W2047569199
cites work: Hyperspaces of Peano continua are Hubert cubes
cites work: The infinite-dimensional topology of function spaces
cites work: Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube.
cites work: The hyperspace of the regions below of all lattice-value continuous maps and its Hilbert cube compactification
cites work: The hyperspace of the regions below of continuous maps is homeomorphic to \(c_{0}\)
cites work: A pair of spaces of upper semi-continuous maps and continuous maps
cites work: A topological position of the set of continuous maps in the set of upper semicontinuous maps
cites work: Q3417775
cites work: Q4277838
cites work: Q3812929
cites work: Absorbing systems in infinite-dimensional manifolds
cites work: The space of infinite-dimensional compacta and other topological copies of \((l_ f^ 2)^ \omega\)
cites work: On some metrics compatible with the Fell-Matheron topology
cites work: The completions of metric ANR's and homotopy dense subsets
DOI: 10.1007/S10114-012-0030-6
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6145089