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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.00024 |
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Table of 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 $V \inγ$, $\bigcap_{V\inγ} (V\boxminus V)\subseteq U$.