Locally compact groups with all dense subgroups separable
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Dekui Peng
Abstract
By a recent result of Juhász and van Mill, a locally compact topological group whose dense subspaces are all separable is metrizable. In this note we investigate the following question: Is every locally compact group having all dense subgroups separable also metrizable? We give an example to show the answer is negative for locally compact abelian groups, thereby showing that one cannot directly generalize the assertion by replacing “subspaces” with “subgroups”. On the other hand, we prove that the answer is positive for compact groups which are either connected or algebraically abelian; and for locally compact groups containing only separable subgroups. As an application, we obtain a necessary condition for metrizability of pronilpotent groups.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12271258
Award Identifier / Grant number: 12301089
Funding statement: This work is supported by the National Natural Science Foundation of China (Grants No. 12271258 and No. 12301089), and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 23KJB110017).
Acknowledgements
I would like to express my sincere gratitude to the anonymous reviewer for thoughtful and constructive comments, which greatly contributed to improving the quality of this manuscript. The valuable feedback and suggestions were instrumental in enhancing the clarity and depth of the research presented. I also would like to thank Professor Wei He for his numerous suggestions and thank Doctor Víctor Hugo Yañez for helping me improving the earlier version of this paper.
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Articles in the same Issue
- Frontmatter
- Weil representations of twisted loop groups of type A n (2)
- Local Birkhoff decompositions for loop groups and a finiteness result
- Uniform boundedness of oscillatory singular integrals with rational phases
- On the stabilizer of the graph of linear functions over finite fields
- Global solution and blow-up of critical heat equation with nonlocal interaction
- Locally compact groups with all dense subgroups separable
- On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
- On isometry groups of gradient Ricci solitons
- Extensions of a theorem of P. Hall on indexes of maximal subgroups
- A trace formula for Hecke operators on Fuchsian groups
- Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
- Computation of endo-fixed closures in free-abelian times free groups
- Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
- Traces of partition Eisenstein series
- Gabor System based on the unitary dual of the Heisenberg group
- Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
- Paley inequality for the Weyl transform and its applications
