Metrization criteria for compact groups in terms of their dense subgroups

• Autorzy: Dikran Dikranjan , Dmitri Shakhmatov
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 54D65: Separability
• 54E35: Metric spaces, metrizability
• 54D30: Compactness
• 22C05: Compact groups
• 22D35: Duality theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2013
• Tom: 221
• Numer: 2
• Strony: 161-187

Daty

wydano

2013

Twórcy

autor

Dikran Dikranjan

• Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy

autor

Dmitri Shakhmatov

• Division of Mathematics, Physics and Earth Sciences, Graduate School of Science and Engineering, Ehime University, Matsuyama 790-8577, Japan

Identyfikatory

DOI

10.4064/fm221-2-3