The character of topological groups, via Pontryagin-van Kampen duality

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 precisely 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 ˇCech-complete groups. Even in the special case of free abelian topological groups, our results extend a number of results of Nickolas and Tkachenko, which were proved using laborious elementary methods. In order to obtain concrete estimations, we establish a natural bridge between the studied concepts and pcf theory, which allows the direct application of several major results from that theory. We include an introduction to these results, their use, and their limitations.