New results on totally bounded groups
type: Item
instance of: scholarly article
title: New results on totally bounded groups
zbMATH Open document ID: 0960.22002
DOI: 10.1007/BF02675085
author: Viatcheslav I. Malykhin
published in: Mathematical Notes
publication date: 2001-05-07T00:00:00Z
review text: A countable infinite disjoint family of finite subsets of a topological space is called weak sequence convergent to a point if each neighborhood of this point intersects almost all members of the family. The paper contains results concerning the topology of arbitrary totally bounded topological groups: Theorem 2. In an infinite group, for any point, there is a weak sequence that is convergent to this point in an arbitrary totally bounded group topology. Theorem 3. In a countably infinite totally bounded group, there exists a countable infinite disjoint family \(\mathfrak{F}\) of finite subsets such that, for any totally bounded group topology, each nonempty open subset with respect to this topology intersects almost all members of \(\mathfrak{F.}\) Hence \(\mathfrak{F}\) is a weak sequence that simultaneously converges in this topology to all points of this group. It is proved that in any countable infinite totally bounded group there exists an almost disjoint family, of cardinality 2\(^{\aleph _{0}}\) , of subsets that are dense in the arbitrary totally bounded group topology.
Mathematics Subject Classification ID: 22A05
zbMATH DE Number: 1470984
zbMATH Keywords: group topology
zbMATH Keywords: totally bounded group
zbMATH Keywords: weak sequence
zbMATH Keywords: superdecomposability
reviewed by: Mihail I. Ursul
MaRDI profile type: Publication
cites work: The maximal totally bounded group topology on G and the biggest minimal G-space, for Abelian groups G
description: scientific article; zbMATH DE number 1470984