Hereditarily \(\mathbb R\)-factorizable groups
type: Item
instance of: scholarly article
title: Hereditarily \(\mathbb R\)-factorizable groups
zbMATH Open document ID: 1193.54022
published in: Topology and its Applications
publication date: 2010-05-19T00:00:00Z
review text: The aim of this paper is to indicate classes of hereditarily \(\mathbb R\)-factorizable topological groups and paratopological groups. A Lindelöf \(\Sigma\)-space is a continuous image of a Hausdorff space which admits a perfect mapping onto a separable metrizable space. A topological group \(G\) has the property \(LS(\mathfrak{c})\) if for every \(A\subseteq G\) with \(| A| \leq\mathfrak{c}\) there exists a Lindelöf \(\Sigma\)-space \(Y\) such that \(A\subseteq Y\subseteq G\). The main result of the paper is (Theorem 3.2): Every topological group with property \(LS(\mathfrak{c})\) is hereditarily \(\mathbb{R}\)-factorizable and is an Efimov space, i.e. for every family \(\gamma\) of \(G_\delta\)-sets in \(G\), the closure of \(\cup\gamma\) is a zero-set in \(G\). A similar result is true for paratopological groups (Theorem 3.1). Another class of hereditarily \(\mathbb{R}\)-factorizable is given by Theorem 3.6: Let \(\mathcal C\) be the minimal class of topological groups that contains all Lindelöf \(\Sigma\)-groups and is closed under taking continuous homomorphic images, subgroups, countable products, and forming \(\sigma\)-products. Then every group in this class has countable cellularity, is hereditarily \(\mathbb{R}\)-factorizable and perfectly \(k\)-normal. The paper is finished by four open problems.
reviewed by: Mihail I. Ursul
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 54D45
Mathematics Subject Classification ID: 54B30
zbMATH DE Number: 5710146
zbMATH Keywords: \(\sigma\)-product
zbMATH Keywords: Lindelöf \(\Sigma\)-space
zbMATH Keywords: Efimov space
zbMATH Keywords: \(k\)-normal
zbMATH Keywords: cellularity
zbMATH Keywords: network
zbMATH Keywords: paratopological group
zbMATH Keywords: \(\mathbb R\)-factorizable topological group
zbMATH Keywords: \(z\)-embedded
author: Mikhail Tkachenko
MaRDI profile type: Publication
full work available at URL: j.topol.2008.12.045
OpenAlex ID: W2095509958
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DOI: 10.1016/J.TOPOL.2008.12.045
description: scientific article; zbMATH DE number 5710146