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

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 \emph{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 \emph{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.