7 research outputs found
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.Comment: Minor corrections. Final version before journal editin