\(\mathbb R\)-factorizable groups and subgroups of Lindelöf \(P\)-groups.
type: Item
instance of: scholarly article
title: \(\mathbb R\)-factorizable groups and subgroups of Lindelöf \(P\)-groups.
zbMATH Open document ID: 1039.54020
DOI: 10.1016/S0166-8641(03)00217-7
published in: Topology and its Applications
publication date: 2004-02-03T00:00:00Z
review text: The main subject of the paper are \(P\)-groups, that is, the topological groups whose \(G_\delta\)-subsets are open. A topological group \(G\) is said to be \(\mathbb R\)-factorizable if for every continuous function \(f:G \rightarrow \mathbb R\), one can find a continuous homomorphism \(p:G\rightarrow H\) onto a second countable topological group \(H\) and a continuous function \(h:H\rightarrow \mathbb R\) such that \(f=h\circ p\). A space \(X\) is said to be pseudo-\(\omega_1\)-compact if every locally finite family of open sets in \(X\) is countable. A space \(X\) is called \(\tau\)-stable if every continuous image \(Y\) of \(X\) which admits a coarser Tychonoff topology of weight \(\leq \tau\) has network weight of no more that \(\tau\). In the paper under review it is proved that for \(P\)-groups \(\mathbb R\)-factorizability is equivalent to pseudo-\(\omega_1\)-compactness and then that it is equivalent to \(\omega\)-stability. Later it is shown that the direct products of \(\mathbb R\)-factorizable \(P\)-groups as well as continuous homomorphic images of \(\mathbb R\)-factorizable groups are \(\mathbb R\)-factorizable. The author also shows that if \(G\) is a dense subgroup of a direct product of Lindelöf \(\Sigma\)-groups, then every continuous homomorphic image of \(G\) is \(\mathbb R\)-factorizable and perfectly \(\kappa\)-normal.
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54G10
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54G20
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54C45
Mathematics Subject Classification ID: 54D60
zbMATH DE Number: 2037250
zbMATH Keywords: \(\mathbb R\)-factorizable
zbMATH Keywords: \(\tau\)-bounded
zbMATH Keywords: complete, Lindelöf
zbMATH Keywords: \(P\)-space
zbMATH Keywords: realcompact
zbMATH Keywords: Dieudonné-complete
zbMATH Keywords: \(C\)-embedded
zbMATH Keywords: \(z\)-embedded
zbMATH Keywords: weakly Lindelöf
zbMATH Keywords: Lindelöf \(\Sigma\)-space
zbMATH Keywords: pseudo-\(\omega_1\)-compact
zbMATH Keywords: \(\tau\)-stable
zbMATH Keywords: stable
author: Mikhail Tkachenko
reviewed by: S. U. Raczkowski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2037250