Quasi-convex density and determining subgroups of compact abelian groups

TL;DR

This paper investigates the relationship between subgroup properties and the structure of infinite compact abelian groups, establishing that the minimal determining subgroup size equals the group's weight and providing conditions for metrizability.

Contribution

It proves that the minimal size of a determining subgroup equals the group's weight and shows that groups determined by countable subgroups are metrizable, answering several open questions.

Findings

The minimal size of a determining subgroup equals the group's weight.

Groups determined by countable subgroups are metrizable.

Provides an elementary proof that dense subgroups determining G imply metrizability.

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.