Topological Gyrogroups with Fréchet–Urysohn Property and ωω-Base

The concept of topological gyrogroups is a generalization of a topological group. In this work, ones prove that a topological gyrogroup Gis metrizable iff Ghas an ωω-base and Gis Fréchet–Urysohn. Moreover, in topological gyrogroups, every (countably, sequentially) compact subset being strictly (strongly) Fréchet–Urysohn and having an ωω-base are all weakly three-space properties with Ha closed L-subgyrogroup.