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 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 abelianČech-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.

A classification of the cofinal structures of precompacta

“…We apply our results to compute the character of the topological groups C(X, G) of continuous functions from a complete metric space X to a group G containing an arc, equipped with the compact-open topology. These extend results obtained earlier for metric groups [4].…”

“…When the completion is not locally compact, we need to estimate the cofinality of the partially ordered set Fin(κ) N . This cofinality can be expressed by well-studied set theoretic functions [4,Section 8]. The introductory material in this section is adapted from an earlier paper [19].…”

“…For functions f, g ∈ Fin(X) N , define f ≤ g if f (n) ⊆ g(n) for all natural numbers n. Since the partially ordered sets Fin(N) N and N N are cofinally equivalent, a topological group G has an N N -base if, and only if, Fin(N) N N (G). The following generalization and refinement of the notion of N N -base were considered, implicitly, earlier [4]. .…”

“…The density of a topological space X, denoted by d(X), is the minimal cardinality of a dense subset. The local density [4] of a topological group G, denoted by ld(G), is the minimal density of a neighborhood of the identity element (or, by homogeneity, any other group element). Each item in the following theorem implies the subsequent one.…”

On topological groups admitting a base at the identity indexed by $\omega ^{\omega }$

“…For a survey and proofs with references, see [23]. Note that implicitly the idea of calculating the actual cofinality of the neighborhood system at the identity in topological groups was used also in [4].…”