Every σ-compact strongly topological gyrogroup satisfies equivalent conditions for being homeomorphic or isomorphic to a substructure inside a separable strongly topological gyrogroup.
Suksumran, The algebra of gyrogroups: Cayley’s theorem, Lagrange’s theorem, and isomorphism theorems, Essays Math
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The separability embedding of $\sigma$-compact strongly topological gyrogroups
Every σ-compact strongly topological gyrogroup satisfies equivalent conditions for being homeomorphic or isomorphic to a substructure inside a separable strongly topological gyrogroup.