Metrization criteria for compact groups in terms of their dense subgroups
Volume 221 / 2013
Abstract
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.
