Separability in (strongly) topological gyrogroups

Abstract

 Separability is one of the most basic and important topological properties. In this paper, the separability in (strongly) topological gyrogroups is studied. It is proved that every fifirst-countable left ω-narrow strongly topological gyrogroup is separable. Furthermore, it is shown that if a feathered strongly topological gyrogroup G is isomorphic to a subgyrogroup of a separable strongly topological gyrogroup, then G is separable. Therefore, if a metrizable strongly topological gyrogroup G is isomorphic to a subgyrogroup of a separable strongly topological gyrogroup, then G is separable, and if a locally compact strongly topological gyrogroup G is isomorphic to a subgyrogroup of a separable strongly topological gyrogroup, then G is separable.