Quasi-convex density and determining subgroups of compact abelian groups

Abstract

For an abelian topological group G, let G ˆ 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 | { χ ∈ G ˆ : χ ( X ) ⊆ U } | = | G ˆ | . (Here, w ( G ) denotes the weight of G.) A subgroup D of G determines G if the map r : G ˆ → D ˆ defined by r ( χ ) = χ ↾ D for χ ∈ G ˆ , is an isomorphism between G ˆ and D ˆ . 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 a question of Comfort, Raczkowski and Trigos-Arrieta (repeated by Hernández, Macario and Trigos-Arrieta). As an application, we furnish a short elementary proof of the result from [S. Hernández, S. Macario, F.J. 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.