Completeness type properties on \(C_{p}(X,Y)\) spaces
type: Item
instance of: scholarly article
title: Completeness type properties on \(C_{p}(X,Y)\) spaces
zbMATH Open document ID: 1384.54008
author: R. Rojas-Hernández
author: Salvador Garcia-Ferreira
author: Angel Tamariz-Mascarúa
published in: Topology and its Applications
publication date: 2017-02-14T00:00:00Z
review text: The authors present results related to characterizations of several kinds of pseudo-completeness and compactness properties in spaces of continuous functions of the form \(C_p(X,Y)\). In particular, they prove that if \(X\) is Tychonoff, and \(G\) is a separable metrizable topological group, for which \(C_p(X,G)\) is dense in \(G^X\), then \(C_p(X,G)\) is weakly \(\alpha\)-favorable if and only if \(X\) is \(u_G\)-discrete (i.e. every countable \(C\subset X\) is discrete and each continuous \(f:C\to G\) is continuously extendable over \(X\)). This result yields two generalizations of a theorem of \textit{V. V. Tkachuk} [Sov. Math., Dokl. 27, 681--684 (1983; Zbl 0539.54001); translation from Dokl. Akad. Nauk SSSR 270, 795--798 (1983)]: (1) Let \(G\) be a separable completely metrizable topological group and \(X\) a set. If \(H\) is a dense subgroup of \(G^X\) which is homeomorphic to \(G^Y\) for some set \(Y\), then \(H = G^X\) (e.g. when \(X\) is Tychonoff, and \(H = C_p(X,G)\)). (2) Let \(G\) be a realcompact Čech-complete topological group with countable pseudocharacter, and \(X\) be regular \(C^G_{<\omega}\)-discrete (i.e. whenever \(F\subset X\) is closed, \(n<\omega\), \((x_1,\dots,x_n)\in (X\setminus F)^n\), \((y_0,\dots,y_n)\in G^{n+1}\), there is a continuous \(f:X\to G\) with \(f(x) = y_0\) for every \(x \in F\) and \(f(x_i) = y_i\) for all \(1\leq i \leq n\)). Then, \(C_p(X,G)\cong G^\kappa\) if and only if \(X\) is a discrete space of cardinality \(\kappa\). Properties related to weak \(\alpha\)-favorability, in particular, weak pseudo-compactness of \(C_p(X,G)\) are studied when \(G\) is a topological group; several open problems are posed.
reviewed by: Laszlo Zsilinszky
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54C45
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54E52
zbMATH DE Number: 6684523
zbMATH Keywords: space of continuous functions
zbMATH Keywords: weakly \(\alpha\)-favorable space
zbMATH Keywords: weakly pseudocompact space
zbMATH Keywords: Oxtoby complete space
zbMATH Keywords: Todd complete space
zbMATH Keywords: \(\omega\)-discrete space
zbMATH Keywords: \(C_\omega\)-discrete space
zbMATH Keywords: \(u\)-discrete space
MaRDI profile type: Publication
full work available at URL: j.topol.2017.01.007
OpenAlex ID: W2572532036
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DOI: 10.1016/J.TOPOL.2017.01.007
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description: scientific article; zbMATH DE number 6684523