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## Fundamenta Mathematicae

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## Successors of locally compact topological group topologies on abelian groups

### Tom 249 / 2020

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
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
Haidian District
Beijing 100871, China
and
Institute of Mathematics
Nanjing Normal University
Nanjing 210046, China
e-mail
e-mail

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