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  • Cardinal invariants and convergence properties of locally minimal groups
    Dikranjan, Dikran N., 1950- ; Shakhmatov, Dmitri
    If ▫$G$▫ is a locally essential subgroup of a compact abelian group ▫$K$▫, then: (i) ▫$t(G)=w(G)=w(K)$▫, where ▫$t(G)$▫ is the tightness of ▫$G$▫; (ii) if ▫$G$▫ is radial, then ▫$K$▫ must be ... metrizable; (iii) ▫$G$▫ contains a super-sequence ▫$S$▫ converging to 0 such that ▫$|S|=w(G)=w(K)$▫. Items (i)--(iii) hold when ▫$G$▫ is a dense locally minimal subgroup of ▫$K$▫. We show that locally minimal, locally precompact abelian groups of countable tightness are metrizable. In particular, a minimal abelian group of countable tightness is metrizable. This answers a question of O. Okunev posed in 2007. For every uncountable cardinal kappa, we construct a Frechet-Urysohn minimal group ▫$G$▫ of character ▫$\kappa$▫ such that the connected component of ▫$G$▫ is an open normal omega-bounded subgroup (thus, ▫$G$▫ is locally precompact). We also build a minimal nilpotent group of nilpotency class 2 without non-trivial convergent sequences having an open normal countably compact subgroup.
    Vir: Topology and its Applications. - ISSN 0166-8641 (Vol. 272, March 2020, art. 106984 [24 str.])
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2020
    Jezik - angleški
    COBISS.SI-ID - 18861145

    Povezava(-e):

    https://doi.org/10.1016/j.topol.2019.106984
    DOI

vir: Topology and its Applications. - ISSN 0166-8641 (Vol. 272, March 2020, art. 106984 [24 str.])

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