Some generalized metric properties of $n$-semitopological groups
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arXiv
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2024
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| author | Lin, Fucai Qi, Xixi |
| author_facet | Lin, Fucai Qi, Xixi |
| contents | A semitopological group $G$ is called {\it an $n$-semitopological group}, if for any $g\in G$ with $e\not\in\overline{\{g\}}$ there is a neighborhood $W$ of $e$ such that $g\not\in W^{n}$, where $n\in\mathbb{N}$. The class of $n$-semitopological groups ($n\geq 2$) contains the class of paratopological groups and Hausdorff quasi-topological groups. Fix any $n\in\mathbb{N}$. Some properties of $n$-semitopological groups are studied, and some questions about $n$-semitopological groups are posed. Some generalized metric properties of $n$-semitopological groups are discussed, which contains mainly results are that (1) each Hausdorff first-countable 2-semitopological group admits a coarsersemi-metrizable topology; (2) each locally compact, Baire and $σ$-compact 2-semitopological group is a topological group; (3) the condensation of some kind of 2-semitopological groups topologies are given. Finally, some cardinal invariants of $n$-semitopological groups are discussed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_06927 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some generalized metric properties of $n$-semitopological groups Lin, Fucai Qi, Xixi Group Theory General Topology Primary 54H11, 22A05, secondary 54A25, 54B15, 54E35 A semitopological group $G$ is called {\it an $n$-semitopological group}, if for any $g\in G$ with $e\not\in\overline{\{g\}}$ there is a neighborhood $W$ of $e$ such that $g\not\in W^{n}$, where $n\in\mathbb{N}$. The class of $n$-semitopological groups ($n\geq 2$) contains the class of paratopological groups and Hausdorff quasi-topological groups. Fix any $n\in\mathbb{N}$. Some properties of $n$-semitopological groups are studied, and some questions about $n$-semitopological groups are posed. Some generalized metric properties of $n$-semitopological groups are discussed, which contains mainly results are that (1) each Hausdorff first-countable 2-semitopological group admits a coarsersemi-metrizable topology; (2) each locally compact, Baire and $σ$-compact 2-semitopological group is a topological group; (3) the condensation of some kind of 2-semitopological groups topologies are given. Finally, some cardinal invariants of $n$-semitopological groups are discussed. |
| title | Some generalized metric properties of $n$-semitopological groups |
| topic | Group Theory General Topology Primary 54H11, 22A05, secondary 54A25, 54B15, 54E35 |
| url | https://arxiv.org/abs/2406.06927 |