Coincidence of function space topologies
type: Item
instance of: scholarly article
title: Coincidence of function space topologies
zbMATH Open document ID: 1187.54016
author: Francis Jordan
published in: Topology and its Applications
publication date: 2009-12-15T00:00:00Z
review text: Given topological spaces \(X\) and \(Y\), let \(C(X,Y)\) be the set of all continuous functions from \(X\) to \(Y\). It is known that \(C(X,Y)\) can be equipped with various topologies such as the compact-open topology \(\tau_{co}\), the Isbell topology \(\tau_{is}\), and the topology \(\tau(c)\) of continuous convergence. In general, we have \(\tau_{co} \subseteq \tau_{is} \subseteq \tau(c)\) on \(C(X,Y)\). If \(\tau_{co}=\tau_{is}\) on \(C(X,Y)\), then \(X\) is called \textit{\(Y\)-consonant}; if \(\tau_{is} = \tau(c)\) on \(C(X,Y)\), then \(X\) is called \textit{\(Y\)-concordant}; and finally if \(\tau_{co} =\tau(c)\) on \(C(X,Y)\), then \(X\) is called \textit{\(Y\)-harmonic}. In this paper, the author studies when these topologies coincide on \(C(X,{\mathbb R})\), and investigates \(\mathbb R\)-concordant and \(\mathbb R\)-harmonic spaces. It is shown that a metric space is locally compact if and only if it is \(\mathbb R\)-concordant. It is also shown that the countable sequential fan is a \(\mathbb R\)-harmonic and non-locally compact space, which answers part of Problem 13 in [\textit{E. Pearl}, Open problems in topology. II. (pp. 15-22), Amsterdam: Elsevier. (2007; Zbl 1158.54300)]. Assuming \(\omega_1 <\mathfrak b\), the author constructs a \(\mathbb R\)-concordant non-consonant (thus, non-\(\mathbb R\)-harmonic) space, which provides a consistent answer to Problem 15 in the same reference mentioned previously.
reviewed by: Jiling Cao
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 5647954
zbMATH Keywords: function space
zbMATH Keywords: compact-open topology
zbMATH Keywords: Isbell topology
zbMATH Keywords: natural topology
zbMATH Keywords: upper semicontinuous
zbMATH Keywords: lower semicontinuous
zbMATH Keywords: consonant
zbMATH Keywords: concordant
zbMATH Keywords: harmonic
MaRDI profile type: Publication
full work available at URL: j.topol.2009.09.002
OpenAlex ID: W1988680638
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DOI: 10.1016/J.TOPOL.2009.09.002
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description: scientific article; zbMATH DE number 5647954