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

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 \v{C}ech-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.

💡 Research Summary

The paper tackles the long‑standing problem of estimating the character (the minimal size of a local base) of non‑metrizable topological groups. While the classical Birkhoff–Kakutani theorem tells us that a topological group is metrizable precisely when its character is countable, there has been no systematic method for handling groups whose character is uncountable. The authors introduce the notion of a “bounded system” and focus on abelian groups whose topology is generated by a countable cofinal family of compact subsets. Such groups admit two equivalent descriptions: (1) they are closed subgroups of the Pontryagin–van Kampen dual (\widehat{G}) of a metrizable abelian group (G); equivalently, (2) they are complete abelian groups whose dual group is metrizable.

The central achievement is a precise formula for the character of any group (H) in this class:
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