The Protasov-Zelenyuk topology and ideal convergence
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| Format: | Preprint |
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2025
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| _version_ | 1866914178861629440 |
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| author | Außenhofer, Lydia Dikranjan, Dikran Bruno, Anna Giordano |
| author_facet | Außenhofer, Lydia Dikranjan, Dikran Bruno, Anna Giordano |
| contents | The so-called $T$-sequences $\mathbf u$ in a group $G$, and the related finest Hausdorff group topology $T_\mathbf u$ on $G$ that makes $\mathbf u$ a null sequence, were introduced by Protasov and Zelenyuk 35 years ago and since then they became a fundamental tool in the field of topological groups. More recently, in the abelian case, the subfamily of $T$-sequences called $TB$-sequences was introduced, as well as the finest precompact group topology $T_\mathbf{pu}$ that makes $\mathbf u$ a null sequence. Here we study the counterpart of all these notions with respect to ideal convergence in place of the classical notion of convergence of a sequence. Also, we study their relation to the already established field of $I$-characterized subgroups of compact abelian groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03885 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Protasov-Zelenyuk topology and ideal convergence Außenhofer, Lydia Dikranjan, Dikran Bruno, Anna Giordano Group Theory General Topology 11B05, 20K45, 22A05, 22C05, 54H11 The so-called $T$-sequences $\mathbf u$ in a group $G$, and the related finest Hausdorff group topology $T_\mathbf u$ on $G$ that makes $\mathbf u$ a null sequence, were introduced by Protasov and Zelenyuk 35 years ago and since then they became a fundamental tool in the field of topological groups. More recently, in the abelian case, the subfamily of $T$-sequences called $TB$-sequences was introduced, as well as the finest precompact group topology $T_\mathbf{pu}$ that makes $\mathbf u$ a null sequence. Here we study the counterpart of all these notions with respect to ideal convergence in place of the classical notion of convergence of a sequence. Also, we study their relation to the already established field of $I$-characterized subgroups of compact abelian groups. |
| title | The Protasov-Zelenyuk topology and ideal convergence |
| topic | Group Theory General Topology 11B05, 20K45, 22A05, 22C05, 54H11 |
| url | https://arxiv.org/abs/2512.03885 |