According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism Ĝ → D̂ 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 -dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its -dense subgroups is metrizable, thereby resolving a question of Hernández, Macario and Trigos-Arrieta. (Under the additional assumption of the Continuum Hypothesis CH, the same statement was proved recently by Bruguera, Chasco, Domínguez, Tkachenko and Trigos-Arrieta.) As a tool, we develop a machinery for building -dense subgroups without uncountable compact subsets in compact groups of weight ω₁ (in ZFC). The construction is delicate, as these subgroups must have non-trivial convergent sequences in some models of ZFC.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm221-2-3,
author = {Dikran Dikranjan and Dmitri Shakhmatov},
title = {Metrization criteria for compact groups in terms of their dense subgroups},
journal = {Fundamenta Mathematicae},
volume = {220},
year = {2013},
pages = {161-187},
zbl = {1283.22004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm221-2-3}
}
Dikran Dikranjan; Dmitri Shakhmatov. Metrization criteria for compact groups in terms of their dense subgroups. Fundamenta Mathematicae, Tome 220 (2013) pp. 161-187. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm221-2-3/