An example of a planar group whose quasicomponent does not coincide with its component
type: Item
instance of: scholarly article
title: An example of a planar group whose quasicomponent does not coincide with its component
zbMATH Open document ID: 0598.22005
DOI: 10.1007/BF01158403
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: The quasicomponent of a point in a topological group is the intersection of all clopen (closed-open) sets containing this point, and the component of a point is the usual maximal connected set containing that point. Treating the plane \({\mathbb{R}}^ 2\) as an abelian topological group (usual topology) with respect to addition, the author constructs a subgroup S of \({\mathbb{R}}^ 2\) and shows that the quasicomponent of the zero element (0,0) in S is isomorphic to the group \({\mathbb{Z}}\) of integers and that the component of (0,0) in S is itself this point thus providing an example of a group in which the quasicomponent of a point is distinct from its component. The subgroup in question is totally disconnected.
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54D99
zbMATH DE Number: 3964210
zbMATH Keywords: planar group
zbMATH Keywords: quasicomponent
zbMATH Keywords: topological group
author: Mihail I. Ursul
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf01158403
OpenAlex ID: W2084317576
description: scientific article; zbMATH DE number 3964210