Variations of selective separability
type: Item
instance of: scholarly article
title: Variations of selective separability
zbMATH Open document ID: 1168.54009
author: Angelo Bella
author: Maddalena Bonanzinga
author: Mikhail Matveev
published in: Topology and its Applications
publication date: 2009-04-23T00:00:00Z
review text: Let \(X\) be a Tychonoff space, \({\mathcal D}\) denote the family of dense subsets of \(X\), and \(C_p(X)\) be the space of continuous real valued functions on \(X\) with the topology of pointwise convergence. We say that \(X\) is \(M\)- \((R\)-)separable in case, for every countable sequence \(\{D_n\}\) with \(D_n\in{\mathcal D}\), there is a finite \(F_n\subset D_n\) (point \(p_n\in C_n)\) for each \(n\) such that \(\cup \{F_n\} (\{p_n\})\) belongs to \({\mathcal D}\). In the present paper these and related properties are studied with the object of extending earlier work by \textit{A. Bella, M. Bonanzinga, M. V. Matveev}, and \textit{V. V. Tkachuk} [Topol. Proc. 32, 15--32 (2008; Zbl 1165.54008)]. The authors show, among many other things, that (i) \(C_p(X)\) is separable and has countable fan tightness if, and only if, \(C_p(X)\) is \(M\)-separable; (ii) if \(C_p(X)\) is \(R\)-separable, then \(C_p(X^n)\) is \(R\)-separable for each finite \(n\) and \((C_p(X))^\omega\) is \(R\)-separable; (iii) if we assume the continuum hypothesis and denote its cardinality by \(c\), then \(2^c\) contains a countable dense \(R\)-separable subspace. A number of open questions are also posed.
reviewed by: James V. Whittaker
Mathematics Subject Classification ID: 54D65
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 5546608
zbMATH Keywords: selection principles
zbMATH Keywords: separable space
zbMATH Keywords: M-separable space
zbMATH Keywords: H-separable space
zbMATH Keywords: R-separable space
zbMATH Keywords: GN-separable space
zbMATH Keywords: Menger space
zbMATH Keywords: Hurewicz space
zbMATH Keywords: Rothberger space
zbMATH Keywords: Gerlitz-Nagy property (\(*\))
zbMATH Keywords: \(C_p\) space
MaRDI profile type: Publication
full work available at URL: j.topol.2008.12.029
OpenAlex ID: W2065325103
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DOI: 10.1016/J.TOPOL.2008.12.029
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description: scientific article; zbMATH DE number 5546608