The separability embedding of $σ$-compact strongly topological gyrogroups
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866911708326395904 |
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| author | Lai, Shumin Lin, Fucai |
| author_facet | Lai, Shumin Lin, Fucai |
| contents | In this paper, it is shown that every right $ω$-narrow strongly topological gyrogroup $G$ is right $ω$-balanced by applying the gyrosemidirect product groups. Then we investigate the class of $σ$-compact strongly topological gyrogroups, and conclude that every $σ$-compact strongly topological gyrogroup is range-metrizable. By applying these results, we discuss the separability embedding of $σ$-compact strongly topological gyrogroups, and claim that the following three statements (a)-(c) are equivalent for any $σ$-compact strongly topological gyrogroup $G$:
\smallskip (a) $G$ is homeomorphic to a subspace of a separable regular space;
\smallskip (b) $G$ is topologically gyrogroup isomorphic to a subgyrogroup of a separable strongly topological gyrogroup;
\smallskip (c) $G$ is topologically gyrogroup isomorphic to a closed subgyrogroup of a separable path-connected, locally path-connected strongly topological gyrogroup.
The above results extend the classical results from topological groups to the class of strongly topological gyrogroups in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23441 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The separability embedding of $σ$-compact strongly topological gyrogroups Lai, Shumin Lin, Fucai General Topology Group Theory 22A15, 54D45, 54H11, 54H99 In this paper, it is shown that every right $ω$-narrow strongly topological gyrogroup $G$ is right $ω$-balanced by applying the gyrosemidirect product groups. Then we investigate the class of $σ$-compact strongly topological gyrogroups, and conclude that every $σ$-compact strongly topological gyrogroup is range-metrizable. By applying these results, we discuss the separability embedding of $σ$-compact strongly topological gyrogroups, and claim that the following three statements (a)-(c) are equivalent for any $σ$-compact strongly topological gyrogroup $G$: \smallskip (a) $G$ is homeomorphic to a subspace of a separable regular space; \smallskip (b) $G$ is topologically gyrogroup isomorphic to a subgyrogroup of a separable strongly topological gyrogroup; \smallskip (c) $G$ is topologically gyrogroup isomorphic to a closed subgyrogroup of a separable path-connected, locally path-connected strongly topological gyrogroup. The above results extend the classical results from topological groups to the class of strongly topological gyrogroups in the literature. |
| title | The separability embedding of $σ$-compact strongly topological gyrogroups |
| topic | General Topology Group Theory 22A15, 54D45, 54H11, 54H99 |
| url | https://arxiv.org/abs/2605.23441 |