Locally Compact Groups with All Dense Subgroups Separable
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909425834393600 |
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| author | Peng, Dekui |
| author_facet | Peng, Dekui |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_06426 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Locally Compact Groups with All Dense Subgroups Separable Peng, Dekui Group Theory General Topology 22C05, 22D05, 54A25 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. |
| title | Locally Compact Groups with All Dense Subgroups Separable |
| topic | Group Theory General Topology 22C05, 22D05, 54A25 |
| url | https://arxiv.org/abs/2303.06426 |