$\aleph_1$-free abelian non-Archimedean Polish groups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910077125918720 |
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| author | Paolini, Gianluca Shelah, Saharon |
| author_facet | Paolini, Gianluca Shelah, Saharon |
| contents | An uncountable $\aleph_1$-free group cannot admit a Polish group topology but an uncountable $\aleph_1$-free abelian group can, as witnessed, for example, by the Baer-Specker group $\mathbb{Z}^ω$; more strongly, $\mathbb{Z}^ω$ is separable. In this paper we investigate $\aleph_1$-free abelian non-Archimedean Polish groups. We prove two main results. The first is that there are continuum many separable (and so torsionless, and so $\aleph_1$-free) abelian non-Archimedean Polish groups which are pairwise not topologically isomorphic. The second is that the following four properties are complete co-analytic subsets of the space of closed abelian subgroups of $S_\infty$: separability, torsionlessness, $\aleph_1$-freeness and $\mathbb{Z}$-homogeneity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_02485 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\aleph_1$-free abelian non-Archimedean Polish groups Paolini, Gianluca Shelah, Saharon Logic Group Theory 03E15, 20F65, 20B27 An uncountable $\aleph_1$-free group cannot admit a Polish group topology but an uncountable $\aleph_1$-free abelian group can, as witnessed, for example, by the Baer-Specker group $\mathbb{Z}^ω$; more strongly, $\mathbb{Z}^ω$ is separable. In this paper we investigate $\aleph_1$-free abelian non-Archimedean Polish groups. We prove two main results. The first is that there are continuum many separable (and so torsionless, and so $\aleph_1$-free) abelian non-Archimedean Polish groups which are pairwise not topologically isomorphic. The second is that the following four properties are complete co-analytic subsets of the space of closed abelian subgroups of $S_\infty$: separability, torsionlessness, $\aleph_1$-freeness and $\mathbb{Z}$-homogeneity. |
| title | $\aleph_1$-free abelian non-Archimedean Polish groups |
| topic | Logic Group Theory 03E15, 20F65, 20B27 |
| url | https://arxiv.org/abs/2410.02485 |