Metrization criteria for compact groups in terms of their dense subgroups

Dikran Dikranjan, and Dmitri Shakhmatov. "Metrization criteria for compact groups in terms of their dense subgroups." Fundamenta Mathematicae 221.2 (2013): 161-187. <http://eudml.org/doc/282736>.

@article{DikranDikranjan2013,
abstract = {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 $G_δ$-dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its $G_δ$-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 $G_δ$-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.},
author = {Dikran Dikranjan, Dmitri Shakhmatov},
journal = {Fundamenta Mathematicae},
keywords = {determined group; pseudocompact; countably compact; -bounded; Bernstein set; all compact subsets are countable; variety of groups; free group in a variety},
language = {eng},
number = {2},
pages = {161-187},
title = {Metrization criteria for compact groups in terms of their dense subgroups},
url = {http://eudml.org/doc/282736},
volume = {221},
year = {2013},
}

TY - JOUR
AU - Dikran Dikranjan
AU - Dmitri Shakhmatov
TI - Metrization criteria for compact groups in terms of their dense subgroups
JO - Fundamenta Mathematicae
PY - 2013
VL - 221
IS - 2
SP - 161
EP - 187
AB - 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 $G_δ$-dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its $G_δ$-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 $G_δ$-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.
LA - eng
KW - determined group; pseudocompact; countably compact; -bounded; Bernstein set; all compact subsets are countable; variety of groups; free group in a variety
UR - http://eudml.org/doc/282736
ER -