@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 -
