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Xiao, Zhiqiang; Peng, Dekui

doi:10.1016/j.topol.2021.107668

Topology and its Applications

2021

DOI: 10.1016/j.topol.2021.107668

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Predecessors and successors of the Euclidean topology on a subgroup of GL(2,R)

Zhiqiang Xiao

Dekui Peng

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“…• (Theorem 2.13) If G is solvable, then τ admits no successors. We would like to note that in [16], the authors proved that the topology of locally compact minimal group R ⋊ R + has no successors, where R + is the multiplicative group of positive real numbers and the action is non-trivial. Since this group is connected and solvable, it is an immediate corollary of our new result.…”

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confidence: 99%

Successors of locally compact topological group topologies on abelian groups

Peng

Tkachenko

et al. 2020

Fund. Math.

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Let G be a group and σ, τ be topological group topologies on G. We say that σ is a successor of τ if σ is strictly finer than τ and there is not a group topology properly between them. In this note, we explore the existence of successor topologies in topological groups, particularly focusing on non-abelian connected locally compact groups. Our main contributions are twofold: for a connected locally compact group (G, τ ), we show that (1) if (G, τ ) is compact, then τ has a precompact successor if and only if there exists a discontinuous homomorphism from G into a simple connected compact group with dense image, and (2) if G is solvable, then τ has no successors. Our work relies on the previous characterization of locally compact group topologies on abelian groups processing successors.

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confidence: 99%

Successors of locally compact topological group topologies on abelian groups

Peng

Tkachenko

et al. 2020

Fund. Math.

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Predecessors and successors of the Euclidean topology on a subgroup of GL(2,R)

Cited by 1 publication

References 11 publications

Successors of locally compact topological group topologies on abelian groups

Successors of locally compact topological group topologies on abelian groups

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