The separability embedding of $\sigma$-compact strongly topological gyrogroups

In this paper, it is shown that every right $\omega$-narrow strongly topological gyrogroup $G$ is right $\omega$-balanced by applying the gyrosemidirect product groups. Then we investigate the class of $\sigma$-compact strongly topological gyrogroups, and conclude that every $\sigma$-compact strongly topological gyrogroup is range-metrizable. By applying these results, we discuss the separability embedding of $\sigma$-compact strongly topological gyrogroups, and claim that the following three statements (a)-(c) are equivalent for any $\sigma$-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.