Refining connected topological group topologies on Abelian torsion groups
type: Item
instance of: scholarly article
title: Refining connected topological group topologies on Abelian torsion groups
zbMATH Open document ID: 0930.54030
DOI: 10.1016/S0166-8641(97)00084-9
published in: Topology and its Applications
publication date: 1999-07-26T00:00:00Z
review text: There is a vast literature, to which the authors' List of References provides a substantial guide, dealing with questions of the following type: Given topological properties \(\mathbb{P}\) and \(\mathbb{Q}\), does every topological group \((G,{\mathcal T})\in\mathbb{P}\) admit a topological group topology \({\mathcal U}\) such that \({\mathcal U}\supseteq {\mathcal T}\), \({\mathcal U}\neq {\mathcal T}\), and \((G,{\mathcal U})\in \mathbb{Q}\)? Continuing their earlier work concerning Abelian torsion-free groups [J. Pure Appl. Algebra 124, No. 1-3, 281-288 (1998; Zbl 0895.54023)], the authors here establish positive answers to the above-cited questions for every (torsion) Abelian group of bounded exponent in each of the following two cases: (A) \(\mathbb{P}= \mathbb{Q}=\) the class of connected c.c.c. groups \((G,{\mathcal T})\) with \(w(G,{\mathcal T})\leq{\mathfrak c}\). (B) \(\mathbb{P}= \mathbb{Q}=\) the class of connected, separable groups. Several interesting related unsolved problems are posed.
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54D05
zbMATH DE Number: 1308020
zbMATH Keywords: refinement
zbMATH Keywords: connected group topology
zbMATH Keywords: separable topological group
author: Mikhail Tkachenko
author: Luis Miguel Villegas Silva
reviewed by: W. W. Comfort
MaRDI profile type: Publication
cites work: On countably compact topologies on compact groups and on dyadic compacta
cites work: Q3741814
cites work: Q4281178
cites work: Some topological groups with, and some without, proper dense subgroups
cites work: Q3949653
cites work: Pseudocompact refinements of compact group topologies
cites work: Proper Pseudocompact Extensions of Compact Abelian Group Topologies
cites work: Q3481960
cites work: Q4294101
cites work: Pseudocompactness and uniform continuity in topological groups
cites work: The Product of Two Countably Compact Topological Groups
cites work: Q4150804
cites work: Q5730943
cites work: Q3934622
cites work: Q4843641
cites work: Connected and disconnected plane sets and the functional equation 𝑓(𝑥)+𝑓(𝑦)=𝑓(𝑥+𝑦)
cites work: Q3938195
cites work: Q5840192
cites work: Q5840186
cites work: Q3699945
cites work: Q3961021
cites work: Q4064169
cites work: Q4276162
cites work: Refining connected topological group topologies on Abelian torsion-free groups
full work available at URL: s0166-8641(97)00084-9
OpenAlex ID: W2088533386
description: scientific article; zbMATH DE number 1308020