Topological structures of the space of continuous functions on a non-compact space with the Fell topology
type: Item
instance of: scholarly article
zbMATH Open document ID: 1287.54013
author: Zhongqiang Yang
author: Shaorui Hu
author: Guo Wei
publication date: 2013-09-27T00:00:00Z
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 57N20
zbMATH DE Number: 6212824
zbMATH Keywords: Fell topology
zbMATH Keywords: Hilbert cube
zbMATH Keywords: strongly universal space
zbMATH Keywords: upper semi-continuous map
zbMATH Keywords: continuous map
MaRDI profile type: Publication
title: Topological structures of the space of continuous functions on a non-compact space with the Fell topology
published in: Topology Proceedings
review text: Let \(X\) be a metrizable space and \(C(X)\) the set of all continuous real-valued functions from \(X\) to the unit interval \(I.\) Let \(USC(X) = USC(X,I)\) consist of all upper semi-continuous maps from \(X\) to \(I.\) For every \(f \in USC(X)\) let NEWLINE\[NEWLINE\downarrow f = \{(x,t) \in X \times I : t \leq f(x) \} NEWLINE\]NEWLINE be the hypograph of \( f\) and, for every \( A \subset USC(X)\), NEWLINE\[NEWLINE \downarrow A = \{ \downarrow f : f \in A \} NEWLINE\]NEWLINE which is a subset of \( \;Cld(X \times I),\) the hyperspace of all non-empty closed sets of \(X \times I.\) The paper investigates the pair of function spaces \(( \downarrow USC(X), \downarrow C(X))\) with various topologies, but in particular when carrying the Fell topology, in that case denoted as \( USC_{F}(X)\) and \(C_{F}(X),\) respectively. The choice of the Fell topology is justified from its independence of any admissible metric of the underlying space \(X\). The authors perform as main issue the following one:NEWLINENEWLINEIf \(X\) is a non-compact, locally compact, separable and non-dicrete metrizable space with \( { \overline{X}}_{0} = X,\) then NEWLINE\[NEWLINE ( \downarrow USC_{F}(X), \downarrow C_{F}(X)) \thickapprox (Q, c_{0} \cup (Q \setminus \Sigma)),NEWLINE\]NEWLINE where \({\overline{X}}_{0}\) denotes the closure of the set of isolated points in \(X, \;Q = [0,1]^{\omega}\) is the Hilbert cube, \(\Sigma = \{ (x_{n}) \in Q : sup | x_{n}| < 1 \}\) and \(c_{0} = \{ (x_{n}) \in \Sigma : \lim_{n \to \infty}x_{n} = 0 \} \); and, finally, where \((X,Y) \thickapprox (A,B)\) means that there exists a homeomorphism \(h: X \rightarrow A\) such that \(h(Y) = B.\)NEWLINENEWLINEThe authors achieve the main result by means of topological semilattices and by using fine techniques of algebraic topology applied to the players in the triple: NEWLINE\[NEWLINE( \downarrow USC_{F}(X), \downarrow USC_{F}(X) \;\setminus \downarrow C_{0}(X), \downarrow C_{F}(X) \;\setminus \downarrow C_{0}(X))NEWLIN…
reviewed by: Anna Di Concilio
description: scientific article; zbMATH DE number 6212824