Cardinality constraints for pseudocompact and for totally dense subgroups of compact topological groups
type: Item
instance of: scholarly article
title: Cardinality constraints for pseudocompact and for totally dense subgroups of compact topological groups
zbMATH Open document ID: 0592.22005
author: Lewis C. Robertson
author: W. W. Comfort
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: For a compact Hausdorff group K, let m(K) be the minimum cardinality of a dense pseudocompact subgroup of K. Then: m(K) is a function of \(w(K)=\alpha\); \(m(K)<| K|\) if \((\log\alpha)^{\omega} < 2^{\alpha}\); and, assuming SCH, \(m(K)=(\log \alpha)^{\omega}\), whence \(m(K)=| K|\) if \((\log\alpha)^{\omega}= 2^{\alpha}\). Cardinals \(\alpha\) such that \((\log\alpha)^{\omega}\neq 2^{\alpha}\) are characterized and shown to abound in all models of ZFC. A subgroup G of K is totally dense if, for each closed subgroup H of K, \(G\cap H\) is dense in H. If K is connected, every totally dense subgroup D of K satisfies \(| D| =| K|\). Some K with \(w(K)=\alpha\) has a totally dense subgroup D with \(| D| <| K|\) if and only if \(cf(\alpha)=\omega\) and \(\alpha =\log (2^{\alpha})\); w.l.o.g., K is totally disconnected. The characterization of these two classes of cardinals are incompatible in some models of ZFC, compatible in others. The existence of a compact Hausdorff group K with a totally dense pseudocompact subgroup D with \(| D| <| K|\) is therefore undecidable in ZFC.
Mathematics Subject Classification ID: 22C05
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 22A05
zbMATH DE Number: 3950869
zbMATH Keywords: compact Hausdorff group
zbMATH Keywords: pseudocompact
zbMATH Keywords: SCH
zbMATH Keywords: totally dense
zbMATH Keywords: models of ZFC
zbMATH Keywords: undecidable
reviewed by: Douglas L. Grant
MaRDI profile type: Publication
full work available at URL: pjm.1985.119.265
OpenAlex ID: W2015799955
DOI: 10.2140/PJM.1985.119.265
description: scientific article; zbMATH DE number 3950869