  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.
