Quasi-convex density and determining subgroups of compact abelian groups

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:363

Article

Dikranjan, Dikran²  Shakhmatov, Dmitri¹ 

[1] Ehime Univ, Div Math Phys & Earth Sci, Grad Sch Sci & Engn, Matsuyama, Ehime 7908577, Japan

[2] Univ Udine, Dipartimento Matemat & Informat, I-33100 Udine, Italy

关键词: Dual group;    Pontryagin-van Kampen duality;    Compact open topology;    Compact group;    Abelian group;    Quasi-convex set;    Quasi-convex density;    Aussenhofer-Chasco theorem;    Dense subgroup;    Determined group;    Convergent sequence;    Super-sequence;    Hofmann-Morris theorem;    Suitable set;   

DOI  :  10.1016/j.jmaa.2009.07.038

来源: Elsevier

【 摘 要 】

For an abelian topological group G, let (G) over cap denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X) < w(G), and an open neighborhood U of 0 in T, we show that vertical bar{chi is an element of <(G)over cap>: chi(X) subset of U}vertical bar = vertical bar(G) over cap vertical bar. (Here, w(G) denotes the weight of G.) A subgroup D of G determines G if the map r: (G) over cap -> (D) over cap defined by r(chi) = chi (sic)D for chi is an element of(G) over cap, is an isomorphism between (G) over cap and (D) over cap. We prove that w(G) = min{vertical bar D vertical bar: D is a subgroup of G that determines G} for every infinite compact abelian group G. In particular, an infinite compact abelian group determined by a countable subgroup is metrizable. This gives a negative answer to a question of Comfort, Raczkowski and Trigos-Arrieta (repeated by Hernandez, Macario and Trigos-Arrieta). As an application, we furnish a short elementary proof of the result from [S. Hernandez, S. Macario, FJ. Trigos-Arrieta, Uncountable products of determined groups need not be determined, J. Math. Anal. Appl. 348 (2008) 834-842] that a compact abelian group G is metrizable provided that every dense subgroup of G determines G. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free