TY - JOUR

T1 - The character of topological groups, via bounded systems, Pontryagin-van Kampen duality and pcf theory

AU - Chis, Cristina

AU - Ferrer, M. Vincenta

AU - Hernández, Salvador

AU - Tsaban, Boaz

N1 - Publisher Copyright:
© 2014 Elsevier Inc..

PY - 2014/12/5

Y1 - 2014/12/5

N2 - 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.

AB - 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.

KW - Bounded sets

KW - Character of a topological group

KW - Cofinality

KW - Compact-open topology

KW - Dual group

KW - Free topological group

KW - Locally quasi-convex group

KW - Metrizable group

KW - Pcf theory

KW - Pontryagin van Kampen duality

UR - http://www.scopus.com/inward/record.url?scp=84908530441&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2014.06.040

DO - 10.1016/j.jalgebra.2014.06.040

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84908530441

SN - 0021-8693

VL - 420

SP - 86

EP - 119

JO - Journal of Algebra

JF - Journal of Algebra

ER -