Extremal pseudocompact topological groups
type: Item
instance of: scholarly article
title: Extremal pseudocompact topological groups
zbMATH Open document ID: 1071.54019
published in: Journal of Pure and Applied Algebra
publication date: 2005-04-18T00:00:00Z
review text: A pseudocompact group \(G\) is said to be \(r\)-\textit{extremal} [resp., \(s\)-\textit{extremal}] if \(G\) admits no strictly finer pseudocompact group topology [resp., \(G\) has no dense pseudocompact subgroup]. A pseudocompact group is called \textit{doubly extremal} if it is both \(r\)- and \(s\)-extremal. The starting point of this paper is the following conjecture which remains open: No nonmetrizable pseudocompact Abelian group is neither \(r\)- nor \noindent\(s\)-extremal. The authors develop a technique of strengthening a pseudocompact group topology (by using homomorphisms in the circle group \(\mathbb{T}\)) (Theorem 3.10). It is proved that the following classes of pseudocompact groups are neither \(r\)-extremal nor \(s\)-extremal: i) non-divisible connected pseudocompact Abelian groups (Theorem 4.5); ii) totally disconnected Abelian groups of uncountable weight (Corollary 5.8): iii) pseudocompact Abelian groups for which \(\omega < w(G)\leq 2^{\omega}\) (here \(w(G)\) is the weight of \(G\)) (Theorem 5.10(a)). If in a pseudocompact Abelian group \(G\) of uncountable weight every closed subgroup is pseudocompact and \(G\) contains a nonmetrizable finitely generated subgroup, then \(G\) is not \(s\)-extremal (Theorem 6.9). It is proved that under Luzin's hypothesis \([2^\omega=2^{\omega_1}]\) if in a topological Abelian group \(G\) of uncountable weight every closed subgroup is pseudocompact, then \(G\) is not \(s\)-extremal (Theorem 6.10). Furthermore, it is proved that if \(G\) is a pseudocompact group, \(w(G)>\omega,\) then: i) If every closed subgroup of \(G\) is pseudocompact, then \(G\) is not doubly extremal; ii) if \(G\) is countably compact or torsion-free, then \(G\) is not doubly extremal (Corollary 7.3). The paper contains other interesting results and open questions.
reviewed by: Mihail I. Ursul
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 22A05
zbMATH DE Number: 2157520
zbMATH Keywords: pseudocompact group
zbMATH Keywords: r-extremal group
zbMATH Keywords: s-extremal group
zbMATH Keywords: doubly extremal group
zbMATH Keywords: totally bounded group
zbMATH Keywords: Luzin's hypothesis
author: W. W. Comfort
author: Jorge Galindo
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.08.018
OpenAlex ID: W2066462208
cites work: Cardinal invariants and independence results in the poset of precompact group topologies
cites work: Counting subgroups and topological group topologies
cites work: Pseudocompact topological group refinements of maximal weight
cites work: Q4228833
cites work: On the existence of free topological groups
cites work: Concerning connected, pseudocompact Abelian groups
cites work: Long chains of Hausdorff topological group topologies
cites work: Q4314813
cites work: Pseudocompact refinements of compact group topologies
cites work: Long chains of topological group topologies. --- A continuation
cites work: Cardinality constraints for pseudocompact and for totally dense subgroups of compact topological groups
cites work: Q3481960
cites work: Topologies induced by groups of characters
cites work: Pseudocompactness and uniform continuity in topological groups
cites work: Countably compact groups and finest totally bounded topologies
cites work: Pseudocompact group topologies and totally dense subgroups
cites work: Q4228835
cites work: Examples of locally compact non-compact minimal topological groups
cites work: Q4371037
cites work: Products of minimal abelian groups
cites work: Algebraic structure of pseudocompact groups
cites work: Sequentially complete groups: Dimension and minimality
cites work: Q5609457
cites work: Q4546099
cites work: Dual properties in totally bounded Abelian groups
cites work: Rings of Real-Valued Continuous Functions. I
cites work: Q5723435
cites work: Some properties of connected compact groups
cites work: Q3338458
cites work: The Number of T 2 -Precompact Group Topologies on Free Groups
cites work: Q3471204
cites work: Q3477070
cites work: Q4798686
cites work: Q5770510
cites work: Dense Subgroups of Compact Groups
DOI: 10.1016/J.JPAA.2004.08.018
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description: scientific article; zbMATH DE number 2157520