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.