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Successors of locally compact topological group topologies on abelian groups

Tom 249 / 2020

Dekui Peng, Wei He, Mikhail Tkachenko, Zhiqiang Xiao Fundamenta Mathematicae 249 (2020), 71-93 MSC: Primary 22A05, 54A25; Secondary 54H11, 54A35. DOI: 10.4064/fm680-6-2019 Opublikowany online: 29 November 2019

Streszczenie

For a group $G$, let $\mathcal{G} (G)$ be the lattice of all topological group topologies on $G$. We prove that if $G$ is abelian, $\tau ,\sigma \in \mathcal{G} (G)$ and $\sigma $ is a successor of $\tau $ in $\mathcal{G} (G)$, then $\sigma $ is precompact iff $\tau $ is precompact. This fact is used to show that if a divisible or connected topological abelian group $(G,\tau )$ contains a discrete subgroup $N$ such that $G/N$ is compact, then $\tau $ does not have successors in $\mathcal {G}(G)$. In particular, no compact Hausdorff topological group topology on a divisible abelian group $G$ has successors in $\mathcal {G}(G)$ and the usual interval topology on $\mathbb {R}$ has no successors in $\mathcal {G}(\mathbb {R})$.

We also prove that a compact Hausdorff topological group topology $\tau $ on an abelian group $G$ has a successor in $\mathcal{G} (G)$ if and only if there exists a prime number $p$ such that $G/pG$ is infinite. Therefore, the usual compact topological group topology of the group $\mathbb Z _p$ of $p$-adic integers does not have successors in $\mathcal{G} (\mathbb Z _p)$.

Our results solve two problems posed by different authors in the years 2006–2018.

Autorzy

  • Dekui PengInstitute of Mathematics
    Nanjing Normal University
    Nanjing 210046, China
    e-mail
  • Wei HeInstitute of Mathematics
    Nanjing Normal University
    Nanjing 210046, China
    e-mail
  • Mikhail TkachenkoUniversidad Autónoma Metropolitana
    Av. San Rafael Atlixco 186
    Col. Vicentina, C.P. 09340
    Del. Iztapalapa, Mexico City, Mexico
    e-mail
  • Zhiqiang XiaoBeijing International Center
    for Mathematical Research (BICMR)
    Beijing University
    No. 5 Yiheyuan Road
    Haidian District
    Beijing 100871, China
    and
    Institute of Mathematics
    Nanjing Normal University
    Nanjing 210046, China
    e-mail
    e-mail

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