The character of topological groups, via bounded systems, Pontryagin-van Kampen duality and pcf theory

Cristina Chis, M. Vincenta Ferrer, Salvador Hernández, Boaz Tsaban

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

The Birkhoff-Kakutani Theorem asserts that a topological group is metrizable if, and only if, it has countable character. We develop and apply tools for the estimation of the character for a wide class of nonmetrizable topological groups.We consider abelian groups whose topology is determined by a countable cofinal family of compact sets. These are the closed subgroups of Pontryagin-van Kampen duals of metrizable abelian groups, or equivalently, complete abelian groups whose dual is metrizable. By investigating these connections, we show that also in these cases, the character can be estimated, and that it is determined by the weights of the compact subsets of the group, or of quotients of the group by compact subgroups. It follows, for example, that the density and the local density of an abelian metrizable group determine the character of its dual group. Our main result applies to the more general case of closed subgroups of Pontryagin-van Kampen duals of abelian Čech-complete groups.In the special case of free abelian topological groups, our results extend a number of results of Nickolas and Tkachenko, which were proved using combinatorial methods.In order to obtain concrete estimations, we establish a natural bridge between the studied concepts and pcf theory, that allows the direct application of several major results from that theory. We include an introduction to these results and their use.

Original languageEnglish
Pages (from-to)86-119
Number of pages34
JournalJournal of Algebra
Volume420
DOIs
StatePublished - 5 Dec 2014

Bibliographical note

Publisher Copyright:
© 2014 Elsevier Inc..

Keywords

  • Bounded sets
  • Character of a topological group
  • Cofinality
  • Compact-open topology
  • Dual group
  • Free topological group
  • Locally quasi-convex group
  • Metrizable group
  • Pcf theory
  • Pontryagin van Kampen duality

Fingerprint

Dive into the research topics of 'The character of topological groups, via bounded systems, Pontryagin-van Kampen duality and pcf theory'. Together they form a unique fingerprint.

Cite this