Shumin Lai | Searcharxiv

arXiv2026

Characterizations of quotient spaces for Lindel\"of strongly topological gyrogroups

Let $\mathscr{L}$ be the class of Lindel\"of spaces such that $\mathscr{L}$ is closed under finite products. In this paper, we prove that if $G \in \mathscr{L}$ is a strongly topological gyrogroup, then $G$ is range-metrizable. Furthermore, we prove that if $H$ is a strong subgyrogroup of a strongly topological gyrogroup $G \in \mathscr{L}$, then every compact $G_\delta$-set in the quotient space $G/H$ is Dugundji. Finally, for any strongly topological gyrogroup $G$ and any closed strong subgyrogroup $N$ of $G$, if $G \in \mathscr{L}$ and the quotient space $G/N$ is locally compact, then the inequality $w(G/N) \leq c$ is equivalent to the separability of $G/N$. Our results extend the classical results from topological groups to the class of strongly topological gyrogroups in the literature.

math.GN↗

arXiv2026

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.

math.GN↗