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  • Metrization criteria for compact groups in terms of their dense subgroups
    Dikranjan, Dikran N., 1950- ; Shakhmatov, Dmitri
    According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup ▫$D$▫ of acompact 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 forevery 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.
    Vir: Fundamenta mathematicae. - ISSN 0016-2736 (Vol. 221, no. 2, 2013, str. 161-187)
    Vrsta gradiva - članek, sestavni del
    Leto - 2013
    Jezik - angleški
    COBISS.SI-ID - 16664153

    Povezava(-e):

    http://dx.doi.org/10.4064/fm221-2-3

vir: Fundamenta mathematicae. - ISSN 0016-2736 (Vol. 221, no. 2, 2013, str. 161-187)
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