Quasi-convex density and determining subgroups of compact abelian groups

For an abelian topological group G, let Ĝ 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 T, we show that |{χ ∈ Ĝ : χ(X) ⊆ U}| = |Ĝ|. (Here, w(G) denotes the weight of G.) A subgroup D of G determines G if the map r : Ĝ → D̂ defined by r(χ) = χ ↾D for χ ∈ Ĝ, is an isomorphism between Ĝ and D̂. 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, Hernández, Macario, Raczkowski and Trigos-Arrieta from [5, 6, 13]. 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. All topological groups are assumed to be Hausdorff, and all topological spaces are assumed to be Tychonoff. As usual, R denotes the group of real numbers (with the usual topology), Z denotes the group of integer numbers, T = R/Z denotes the circle group (with the usual topology), N denotes the set of natural numbers, P denotes the set of prime numbers, ω denotes the first infinite cardinal, and w(X) denotes the weight of a space X. If A is a subset of a space X, then A denotes the closure of A in X. 1 Preliminaries and background In this section we give necessary definitions and collect five facts that will be needed later. These facts are either known or part of the folklore. However, to make this manuscript self-contained, we provide their proofs in Section 5 for the reader’s convenience. For spaces X and Y , we denote by C(X,Y ) the space of all continuous functions from X to Y endowed with the compact open topology , that is, the topology generated by the family {[K,U ] : K is a compact subset of X and U is an open subset of Y } Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy; email : [email protected]; the first author was partially supported by MEC.MTM2 006-02036 and FEDER FUNDS. Graduate School of Science and Engineering, Division of Mathematics, Physics and Earth Sciences, Ehime University, Matsuyama 790-8577, Japan; e-mail : [email protected]; the second author was partially supported by the Grant-in-Aid for Scientific Research no. 19540092 by the Japan Society for the Promotion of Science (JSPS).