Pontryagin duality in the class of precompact Abelian groups and the Baire property

Details Pontryagin duality in the class of precompact Abelian groups and the Baire property Pontryagin duality in the class of precompact Abelian groups and the Baire property

2011-01-24

arxiv.org/abs/1101.4504v1

Mathematics

General Topology

Scientific paper

We present a wide class of reflexive, precompact, non-compact, Abelian topological groups $G$ determined by three requirements. They must have the Baire property, satisfy the \textit{open refinement condition}, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group $G^\wedge$ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compact Abelian group $G$ of weight $\geq 2^\om$, we find proper dense subgroups $H_1$ and $H_2$ of $G$ such that $H_1$ is reflexive and pseudocompact, while $H_2$ is non-reflexive and almost metrizable.

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