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| Autori principali: | , , , |
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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2506.00024 |
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| _version_ | 1866908441498353664 |
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| author | Jin, Ying-Ying Sheng, Ye-Qing Wang, Yi-Ting Xie, Li-Hong |
| author_facet | Jin, Ying-Ying Sheng, Ye-Qing Wang, Yi-Ting Xie, Li-Hong |
| contents | We present a characterization of paratopological gyrogroups that can be topologically embedded as subgyrogroups into a product of first-countable $T_{i}$ paratopological gyrogroups for $i = 0, 1, 2$. Specifically, we demonstrate that a strongly paratopological gyrogroup $G$ is topologically isomorphic to a subgyrogroup of a topological product of first-countable $T_1$ strongly paratopological gyrogroups if and only if $G$ is $T_1$, $ω$-balanced and the weakly Hausdorff number of $G$ is countable. This means that for every neighborhood $U$ of the identity 0 in $G$, there exists a countable family $γ$ of neighborhoods of 0 such that for all $V \inγ$, $\bigcap_{V\inγ} (\ominus V)\subseteq U$. Similarly, we prove that a strongly paratopological gyrogroup $G$ is topologically isomorphic to a subgyrogroup of a topological product of first-countable Hausdorff strongly paratopological gyrogroups if and only if $G$ is Hausdorff, $ω$-balanced and the Hausdorff number of $G$ is countable. This means that for every neighborhood $U$ of the identity 0 in $G$, there exists a countable family $γ$ of neighborhoods of 0 such that for all $V \inγ$, $\bigcap_{V\inγ} (V\boxminus V)\subseteq U$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00024 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subgyrogroups within the product spaces of paratopological gyrogroups Jin, Ying-Ying Sheng, Ye-Qing Wang, Yi-Ting Xie, Li-Hong General Topology We present a characterization of paratopological gyrogroups that can be topologically embedded as subgyrogroups into a product of first-countable $T_{i}$ paratopological gyrogroups for $i = 0, 1, 2$. Specifically, we demonstrate that a strongly paratopological gyrogroup $G$ is topologically isomorphic to a subgyrogroup of a topological product of first-countable $T_1$ strongly paratopological gyrogroups if and only if $G$ is $T_1$, $ω$-balanced and the weakly Hausdorff number of $G$ is countable. This means that for every neighborhood $U$ of the identity 0 in $G$, there exists a countable family $γ$ of neighborhoods of 0 such that for all $V \inγ$, $\bigcap_{V\inγ} (\ominus V)\subseteq U$. Similarly, we prove that a strongly paratopological gyrogroup $G$ is topologically isomorphic to a subgyrogroup of a topological product of first-countable Hausdorff strongly paratopological gyrogroups if and only if $G$ is Hausdorff, $ω$-balanced and the Hausdorff number of $G$ is countable. This means that for every neighborhood $U$ of the identity 0 in $G$, there exists a countable family $γ$ of neighborhoods of 0 such that for all $V \inγ$, $\bigcap_{V\inγ} (V\boxminus V)\subseteq U$. |
| title | Subgyrogroups within the product spaces of paratopological gyrogroups |
| topic | General Topology |
| url | https://arxiv.org/abs/2506.00024 |