Quasi-convex density and determining subgroups of compact abelian groups
Abstract
For an abelian topological group G let G^* 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 neighbourhood U of 0 in the circle group, we show that the set of all characters which send X into U has the same size as G^*. (Here, w(G) denotes the weight of G.) A subgroup D of G determines G if the restriction homomorphism G^* --> D^* is an isomorphism between G^* and D^*. We prove that w(G) = min {|D|: 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 questions of Comfort, Hernandez, Macario, Raczkowski and Trigos-Arrieta. As an application, we furnish a short elementary proof of the result from [13] that a compact abelian group G is metrizable provided that every dense subgroup of G determines G.
Cite
@article{arxiv.0807.3846,
title = {Quasi-convex density and determining subgroups of compact abelian groups},
author = {Dikran Dikranjan and Dmitri Shakhmatov},
journal= {arXiv preprint arXiv:0807.3846},
year = {2009}
}
Comments
11 pages. The proof of Lemma 3.6 (from version 2) has been significantly simplified and shorten. 2 lemmas and 4 references have been added. The order of the material has been substantially changed as well