Function spaces jointly metrizable on compacta
type: Item
instance of: scholarly article
title: Function spaces jointly metrizable on compacta
zbMATH Open document ID: 1329.54021
author: Vladimir Tkachuk
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-09-04T00:00:00Z
review text: A topological space \(X\) is jointly metrizable on compacta (a JCM space) if there exists a metric \(d\) on \(X\) such that the topology generated by \(d\) restricted to \(K\) coincides with the subspace topology on \(K\) whenever \(K\) is a compact subspace of \(X\). For a space \(X\), let \(C_{p}(X)\) be the set of all real-valued continuous functions on \(X\) endowed with the topology of pointwise convergence. The author studies when \(C_{p}(X)\) has the JCM property. The main results are: (1) If \(C_p(X)\) is a JCM space then the point-finite cellularity of \(X\) is countable and hence \(X\) has the Souslin property. (2) The JCM property of \(C_p(X)\) need not imply that \(\omega_1\) is a caliber of \(X\). (3) If \(X\) is either submetrizable or a P-space then \(C_p(C_p(X))\) is jointly metrizable on compacta. (4) If \(X\) is a dyadic compact space, then \(C_p(X)\) is jointly metrizable on compacta. Therefore, the JCM property for a compact space does not imply that \(X\) is separable. This result solves Problem 4.12 of the paper [\textit{A. V. Arhangel'skii} et al., Topology Appl. 169, 2--15 (2014; Zbl 1294.54005)] in the negative. Open questions are given in Section 4 of the paper.
reviewed by: Abderrahmane Bouchair
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54B05
zbMATH DE Number: 6478642
zbMATH Keywords: topology of pointwise convergence
zbMATH Keywords: joint metrizability on compacta
zbMATH Keywords: JCM property
zbMATH Keywords: caliber
zbMATH Keywords: compact space
zbMATH Keywords: dyadic space
zbMATH Keywords: \(P\)-space
MaRDI profile type: Publication
full work available at URL: j.jmaa.2015.07.038
OpenAlex ID: W1040510456
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DOI: 10.1016/J.JMAA.2015.07.038
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description: scientific article; zbMATH DE number 6478642