Answer to Raczkowski's questions on convergent sequences of integers
type: Item
instance of: scholarly article
title: Answer to Raczkowski's questions on convergent sequences of integers
zbMATH Open document ID: 1022.22001
DOI: 10.1016/S0166-8641(02)00366-8
author: Giuseppina Barbieri
author: Dikran Dikranjan
author: Chiara Milan
author: Hans Weber
published in: Topology and its Applications
publication date: 2003-08-07T00:00:00Z
review text: Denote by \(T\) the multiplicative group of complex numbers with modulus 1 equipped with the topology inherited from the complex plane and equipped with its usual Lebesgue measure. It is well known that if \(Z\) denotes the group of integers, the set Hom\((Z,T)\) of homomorphisms from \(Z\) into \(T\) becomes a group topologically isomorphic to \(T\) with operation defined pointwise and with the topology of pointwise convergence. Under this identification it can be proved that given a (dense) subgroup \(H\) of \(T\), the weakest topology \(\tau_H\) on \(Z\) that makes the elements of \(H\) continuous is a (Hausdorff) precompact group topology. That every such topology is given as some \(\tau_H\) follows from a theorem of \textit{W. W. Comfort} and \textit{K. A. Ross} [Fundam. Math. 55, 284-291 (1964; Zbl 0138.02905)]. Thus it is natural to ask how properties of the subgroups of \(T\) influence the properties of the precompact topologies on \(Z\). For example, every \(\tau_T\)-compact subset of \(Z\) must be finite [\textit{H. Leptin}, Abh. Math. Sem. Univ. Hamburg 19, 244-263 (1955; Zbl 0065.01501); \textit{I. Glicksberg}, Can. J. Math. 14, 269-276 (1962; Zbl 0109.02001)], thus in particular there are no non-trivial \(\tau_T\)-convergent sequences [\textit{P. Flor}, Math. Scand. 23, 169-170 (1968; Zbl 0182.36003)]. \textit{W. W. Comfort, T. S. Wu} and the reviewer [Fundam. Math. 143, 119-136 (1993; Zbl 0812.22001); Correction, same journal 152, 97-98] proved that given a sequence \(\sigma:=\langle x_n \rangle\) in \(Z\), the group \(A_\sigma:=\{t \in T: t(x_n) \rightarrow 1\}\) must have measure 0, hence if \(H\) is a non-measurable subgroup of \(T\), then there are no non-trivial \(\tau_H\)-convergent sequences. \textit{S. U. Raczkowski} [Topology Appl. 121, 63-74 (2002; Zbl 1007.22003)] generalized these results as follows: (i) There are \(2^c (c:=2^{\aleph_0})\) non-measurable subgroups of \(T\), hence \(2^c\) (Hausdorff) precompact group topologies on \(Z\) with…
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 11B05
Mathematics Subject Classification ID: 22B05
zbMATH DE Number: 1961619
zbMATH Keywords: topologically torsion element
zbMATH Keywords: convergent sequence of integers
zbMATH Keywords: precompact topology
reviewed by: F. Javier Trigos-Arrieta
MaRDI profile type: Publication
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full work available at URL: s0166-8641(02)00366-8
OpenAlex ID: W2011531237
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description: scientific article; zbMATH DE number 1961619