According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a
compact abelian group G determines G if the restriction homomorphism G^ --> D^
of the dual groups is a topological isomorphism. We introduce four conditions
on D that are necessary for it to determine G and we resolve the following
question: If one of these conditions holds for every dense (or G_\delta-dense)
subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a
compact abelian group determined by all its G_\delta-dense subgroups is
metrizable, thereby resolving a question by Hernandez, Macario and
Trigos-Arrieta. (Under the additional assumption of the Continuum Hypothesis
CH, the same statement was proved recently by Bruguera, Chasco, Dominguez,
Tkachenko and Trigos-Arrieta.) As a tool, we develop a machinery for building
G_\delta-dense subgroups without uncountable compact subsets in compact groups
of weight \omega_1 (in ZFC). The construction is delicate, as these subgroups
must have non-trivial convergent sequences in some models of ZFC.Comment: The exposition has substantially improved. Remarks 5.6 and 5.7 are
new. Three references adde
