Maximal topologies on groups
type: Item
instance of: scholarly article
title: Maximal topologies on groups
zbMATH Open document ID: 0935.22002
DOI: 10.1007/BF02674129
published in: Siberian Mathematical Journal
publication date: 2000-04-16T00:00:00Z
review text: The article is devoted to the notion of a maximal topological group. A topological space without isolated points is called a maximal space if the space has an isolated point in every stronger topology. A topological group is called maximal if the underlying topological space of the group is a maximal space. The problem of the existence of a maximal topological group is a problem of set theory by nature. However, maximal homogeneous spaces and left-topological groups can be easily constructed in \(ZFC\). The author proves that, in \(ZFC\), an arbitrary infinite group can be endowed with a maximal regular left-invariant topology. In addition, the construction of this topology yields a solution to the Hindman-Strauss problem of a regular idempotent [Semigroup Forum 51, No. 3, 299-318 (1995; Zbl 0843.22005)]. Besides, the author proves some other theorems about groups with strongest left-invariant topology.
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54H11
zbMATH DE Number: 1432171
zbMATH Keywords: topological group
zbMATH Keywords: maximal topological group
zbMATH Keywords: homogeneous space
zbMATH Keywords: left-topological group
author: Igor V. Protasov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02674129
OpenAlex ID: W2054275020
description: scientific article; zbMATH DE number 1432171