Subgyrogroups within the product spaces of paratopological gyrogroups

Abstract

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 $T_1$, $\omega$-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 $\gamma$ of neighborhoods of 0 such that for all $V \in\gamma$, $\bigcap_{V\in\gamma} (\ominus V)\subseteq U$. 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, $\omega$-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 $\gamma$ of neighborhoods of 0 such that for all $V \in\gamma$, $\bigcap_{V\in\gamma} (V\boxminus V)\subseteq U$.