Metrization criteria for compact groups in terms of their dense subgroups

Dikran Dikranjan¹ ; Dmitri Shakhmatov²

¹ Dipartimento di Matematica e Informatica Università di Udine Via delle Scienze 206 33100 Udine, Italy
² Division of Mathematics, Physics and Earth Sciences Graduate School of Science and Engineering Ehime University Matsuyama 790-8577, Japan

Fundamenta Mathematicae, Tome 221 (2013) no. 2, pp. 161-187.

Résumé

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.

Détail

DOI : 10.4064/fm221-2-3

Keywords: according comfort raczkowski trigos arrieta dense subgroup compact abelian group determines restriction homomorphism widehat widehat dual groups topological isomorphism introduce conditions necessary determine resolve following question these conditions holds every dense delta dense subgroup metrizable particular prove zfc compact abelian group determined its delta dense subgroups metrizable thereby resolving question hern ndez macario trigos arrieta under additional assumption continuum hypothesis statement proved recently bruguera chasco dom nguez tkachenko trigos arrieta tool develop machinery building delta dense subgroups without uncountable compact subsets compact groups weight omega zfc construction delicate these subgroups have non trivial convergent sequences models zfc