Reflexivity of prodiscrete topological groups

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:384

Article

Galindo, J.¹  Recoder-Nunez, L.²  Tkachenko, M.³ 

[1] Univ Jaume 1, Dept Matemat, Inst Matemat & Aplicac IMAC, E-12071 Castellon de La Plana, Spain

[2] Cent Connecticut State Univ, Dept Math Sci, New Britain, CT 06050 USA

[3] Univ Autonoma Metropolitana Iztapalapa, Dept Matemat, Mexico City 09340, DF, Mexico

关键词: Pontryagin's duality;    Reflexive;    Protodiscrete;    Prodiscrete;    P-group;    P-modification;    omega-Bounded;    Direct product;    Sigma-product;    Compact;    Complete;   

DOI  :  10.1016/j.jmaa.2011.05.063

来源: Elsevier

【 摘 要 】

We study. the duality properties of two rather different classes of subgroups of direct products of discrete groups (protodiscrete groups): P-groups, i.e., topological groups such that countable intersections of its open subsets are open, and protodiscrete groups of countable pseudocharacter (topological groups in which the identity is the intersection of countably many open sets). It was recently shown by the same authors that the direct product Pi of an arbitrary family of discrete Abelian groups becomes reflexive when endowed with the omega-box topology. This was the first example of a non-discrete reflexive P-group. Here we present a considerable generalization of this theorem and show that every product of feathered (equivalently, almost metrizable) Abelian groups equipped with the P-modified topology is reflexive. In particular, every locally compact Abelian group with the P-modified topology is reflexive. We also examine the reflexivity of dense subgroups of products Pi with the P-modified topology and obtain the first examples of non-complete reflexive P-groups. We find as well that the better behaved class of prodiscrete groups (complete protodiscrete groups) of countable pseudocharacter contains non-reflexive members-any uncountable bounded torsion Abelian group G of cardinality 2(omega) supports a topology tau such that (G, tau) is a non-reflexive prodiscrete group of countable pseudocharacter. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

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