arXiv:0807.3846v2 [math.GN] 10 Aug 2008 Quasi-convex density in compact abelian groups, with applications to determined groups Dikran Dikranjan * and Dmitri Shakhmatov † Dedicated to W. Wistar Comfort on the occasion of his 75th anniversary Abstract For an abelian topological group G let  G be 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 |{π ∈  G : π(X ) ⊆ U }| = |  G|. (Here w(G) denotes the weight of G.) A subgroup D of G determines G if the restriction homomorphism  G →  D of the dual groups is a topological isomorphism. We prove that w(G) = min{|D| : D is a subgroup of G that determines G} for every compact abelian group G. In particular, an infinite compact abelian group determined by its countable subgroup must be metrizable. This gives a negative answer to questions of Comfort, Hern´andez, Macario, Raczkowski and Trigos-Arrieta from [4], [5] and [11]. As an application, we furnish a short elementary proof of the result from [11] that compact determined abelian groups are metrizable. All topological groups are assumed to be Hausdorff, and all topological spaces are assumed to be Tychonoff. As usual, T = R/Z denotes the circle group, N denotes the set of natural numbers and P 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 facts 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 include their easy proofs 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, i.e. the topology generated by the family {[K, U ]: K is a compact subset of X and U is an open subset of Y } as a subbase, where [K, U ]= {g ∈ C (X, Y ): g(K) ⊆ U }. * Dipartimento di Matematica e Informatica, Universit`a di Udine, Via delle Scienze 206, 33100 Udine, Italy; e-mail : dikranja@dimi.uniud.it; 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 : dmitri@dpc.ehime-u.ac.jp; 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). 1