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  • Quasi-convexly dense and suitable sets in the arc component of a compact group
    Dikranjan, Dikran N., 1950- ; Shakhmatov, Dmitri
    Let ▫$G$▫ be an abelian topological group. The symbol ▫$\widehat{G}$▫ denotes the group of all continuous characters ▫$\chi \colon G \to \mathbb{T}$▫ endowed with the compact open topology. A subset ... $E$ of ▫$G$▫ is said to be qc-dense in $G$ provided that ▫$\chi(E) \subseteq \varphi([- 1/4, 1/4])$▫ holds only for the trivial character ▫$\chi \in \widehat{G}$▫, where ▫$\varphi \colon \mathbb{R} \to \mathbb{T} = \mathbb{R}/\mathbb{Z}$▫ is the canonical homomorphism. A super-sequence is a non-empty compact Hausdorff space ▫$S$▫ with at most one non-isolated point (to which ▫$S$▫ converges). We prove that an infinite compact abelian group ▫$G$▫ is connected if and only if its arc component ▫$G_a$▫ contains a super-sequence converging to 0 that is qc-dense in ▫$G$▫. This gives as a corollary a recent theorem of Außenhofer: For a connected locally compact abelian group ▫$G$▫, the restriction homomorphism ▫$r \colon \widehat{G} \to \widehat{G}_a$▫ defined by ▫$r(\chi) = \chi \upharpoonright_{G_a}$▫ for ▫$\chi \in \widehat{G}$▫, is a topological isomorphism. We show that an infinite compact group ▫$G$▫ is connected if and only if its arc component ▫$G_a$▫ contains a super-sequence converging to the identity that is qc-dense in ▫$G$▫ and generates a dense subgroup of ▫$G$▫. We also offer a short alternative proof of the result of Hofmann and Morris on the existence of suitable sets of minimal size in the arc component of a compact connected group.
    Vir: Mathematische Nachrichten. - ISSN 0025-584X (Vol. 285, no. 4, 2012, str. 476-485)
    Vrsta gradiva - članek, sestavni del
    Leto - 2012
    Jezik - angleški
    COBISS.SI-ID - 16234841

    Povezava(-e):

    http://dx.doi.org/10.1002/mana.201010013

vir: Mathematische Nachrichten. - ISSN 0025-584X (Vol. 285, no. 4, 2012, str. 476-485)

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