\(D\)-property, monotone monolithicity and function spaces
type: Item
instance of: scholarly article
title: \(D\)-property, monotone monolithicity and function spaces
zbMATH Open document ID: 1258.54004
author: R. Rojas-Hernández
author: Angel Tamariz-Mascarúa
published in: Topology and its Applications
publication date: 2012-10-11T00:00:00Z
review text: In the papers [\textit{O. T. Alas, V. V. Tkachuk} and \textit{R. G. Wilson}, Acta Math. Hungar. 125(4), 369--385 (2009)]; [\textit{V. V. Tkachuk}, Topology Appl. 156, No. 4, 840--846 (2009; Zbl 1165.54009); ibid. 159, No. 6, 1529--1535 (2012; Zbl 1245.54022)] several interesting questions, related to monotonically monolithic spaces and spaces having the Collins-Roscoe property, were raised. In order to answer these questions, the authors prove the following interesting results in this paper. 1. \(C_p(C_p(X))\) is monotonically monolithic if and only if \(X\) is so. 2. If \(X\) is a \(\Sigma_{\kappa}\)-product of a family of spaces with countable network weight, then \(C_p(X)\) is monotonically monolithic. 3. If \(X\) is the product of Lindelöf \(\Sigma\)-spaces, then \(C_p(X)\) is monotonically monolithic. 4. If \(C_p(X)\) is monotonically monolithic and \(Y\) is a cozero subset of \(X\), then \(C_p(Y)\) is monotonically monolithic. 5. Every compact \(\omega\)-monolithic space is monotonically monolithic. 6. If \(X\) is monotonically \(< \kappa\)-monolithic with \(\kappa \geq nw(X)\), then \(X\) is hereditarily D. 7. If \(X\) is pseudocompact and \(nw(X) \leq \omega_1\), then \(C_p(X)\) is hereditarily D. This is a difficult, but highly interesting paper for all researchers interested in \(C_p\)-theory.
reviewed by: S. Kundu
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54B10
zbMATH DE Number: 6092983
zbMATH Keywords: \(D\)-space
zbMATH Keywords: monotonically monolithic space
zbMATH Keywords: monotonically \(\kappa\)-monolithic space
zbMATH Keywords: monotonically \(< \kappa\)-monolithic space
zbMATH Keywords: continuous function space
zbMATH Keywords: Lindelöf \(\Sigma\)-space
zbMATH Keywords: \(\Sigma_{\kappa}\)-product
zbMATH Keywords: pseudocompact space
MaRDI profile type: Publication
full work available at URL: j.topol.2012.07.007
OpenAlex ID: W1981132129
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DOI: 10.1016/J.TOPOL.2012.07.007
description: scientific article; zbMATH DE number 6092983