Topological gyrogroups with Frechet-Urysohn property and omega^{omega}-base

TL;DR

This paper characterizes when topological gyrogroups are metrizable, showing it depends on having an { extomega}{ extomega}-base and the Frechet-Urysohn property, and explores related properties of compact subsets.

Contribution

It establishes a necessary and sufficient condition for metrizability of topological gyrogroups involving an { extomega}{ extomega}-base and the Frechet-Urysohn property, extending the theory of topological groups.

Findings

A topological gyrogroup is metrizable iff it has an { extomega}{ extomega}-base and is Frechet-Urysohn.

Every compact subset in such gyrogroups is strongly Frechet-Urysohn.

The properties of being weakly three-space are preserved under certain conditions in topological gyrogroups.

Abstract

The concept of topological gyrogroups is a generalization of a topological group. In this work, ones prove that a topological gyrogroup G is metrizable iff G has an {\omega}{\omega}-base and G is Frechet-Urysohn. Moreover, in topological gyrogroups, every (countably, sequentially) compact subset being strictly (strongly) Frechet-Urysohn and having an {\omega}{\omega}-base are all weakly three-space properties with H a closed L-subgyrogroup