The character of topological groups, via Pontryagin-van Kampen duality
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Other documents of the author: Chis, Cristina; Ferrer González, María Vicenta; Hernández, Salvador; Tsaban, Boaz
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Title
The character of topological groups, via Pontryagin-van Kampen dualityDate
2011-06Bibliographic citation
arXiv:1005.0577v3Type
info:eu-repo/semantics/articleVersion
info:eu-repo/semantics/sumittedVersionSubject
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 nonmetriza ... [+]
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. [-]
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