\(\mathbb{Q}\) is not a Mackey group
type: Item
instance of: scholarly article
title: \(\mathbb{Q}\) is not a Mackey group
zbMATH Open document ID: 1310.54025
author: Daniel de la Barrera Mayoral
published in: Topology and its Applications
publication date: 2014-11-14T00:00:00Z
review text: The letters \(\mathbb{Z}\), \(\mathbb Q\) and \(\mathbb R\) stand for the integers, rationals and real numbers respectively with their usual topology and group structure. The symbol \(\mathbb T\) denotes the quotient group \(\mathbb R/\mathbb Z\) and \(\mathbb T_+=\{x+\mathbb Z\in \mathbb T: x\in [-\frac14,\frac14]\}\). Given a topological group \((G,\tau)\), its dual \((G,\tau)^\wedge\) or simply \(G^\wedge\) is the group of all continuous homomorphisms from \(G\) to \(\mathbb T\). Two group topologies \(\tau\) and \(\mu\) on a group \(G\) are called \textit{compatible} if \((G, \tau)^\wedge = (G, \mu)^\wedge\). A subset \(A\) of a topological group \(G\) is said to be quasi-convex if for any \(x\notin A\), there exists \(\chi\in G^\wedge\) such that \(\chi(A)\subset \mathbb T_+\) and \(\chi(x) \notin \mathbb T_+\). The group \(G\) is \textit{locally quasi-convex} if it has a local neighborhood base consisting of quasi-convex sets. A locally quasi-convex group \((G, \tau)\) is a \textit{Mackey group} if for any locally quasi-convex group topology \(\nu\) on \(G\), if \(\nu\) is compatible with \(\tau\), then \(\nu\subset \tau\). The main result of the paper is Theorem 1.1 which states that the group \(\mathbb Q\) of rational numbers is not a Mackey group.
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 22A25
Mathematics Subject Classification ID: 33D80
Mathematics Subject Classification ID: 11F85
zbMATH DE Number: 6369794
zbMATH Keywords: locally quasi-convex
zbMATH Keywords: Mackey topology
zbMATH Keywords: dual group
reviewed by: Vladimir Tkachuk
MaRDI profile type: Publication
full work available at URL: j.topol.2014.10.004
OpenAlex ID: W2000541114
cites work: Q4484934
cites work: Linear topologies on \(\mathbb Z\) are not Mackey topologies
cites work: Pontryagin duality for metrizable groups
cites work: Q4237891
cites work: Characterizing sequences for precompact group topologies
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DOI: 10.1016/J.TOPOL.2014.10.004
description: scientific article; zbMATH DE number 6369794