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## Metrization criteria for compact groups in terms of their dense subgroups

### Tom 221 / 2013

Fundamenta Mathematicae 221 (2013), 161-187 MSC: Primary 22C05; Secondary 22D35, 54D30, 54D65, 54E35. DOI: 10.4064/fm221-2-3

#### Streszczenie

According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup $D$ of a compact abelian group $G$ determines $G$ if the restriction homomorphism $\widehat{G}\to \widehat{D}$ 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 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 $G_\delta$-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.

#### Autorzy

• Dikran DikranjanDipartimento di Matematica e Informatica
Università di Udine
Via delle Scienze 206
33100 Udine, Italy
e-mail
• Dmitri ShakhmatovDivision of Mathematics,
Physics and Earth Sciences
Graduate School of Science and Engineering
Ehime University
Matsuyama 790-8577, Japan
e-mail

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