A supplement on feathered gyrogroups

TL;DR

This paper investigates the properties of feathered topological gyrogroups, establishing conditions under which they are metrizable or sequential, and characterizing their compact resolutions and quotient spaces.

Contribution

It provides new metrization criteria for feathered strongly topological gyrogroups based on $cs^{*}$-character and compact resolutions, extending the theory of topological gyrogroups.

Findings

Compact subsets of certain gyrogroups are metrizable.

Feathered strongly topological gyrogroups with countable $cs^{*}$-character are metrizable.

Existence of a compact resolution relates to the presence of a compact $L$-subgyrogroup with a Polish quotient.

Abstract

A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. It is shown that each compact subset of a topological gyrogroup with an $\omega^{\omega}$-base is metrizable, which deduces that if $G$ is a topological gyrogroup with an $\omega^{\omega}$-base and is a $k$-space, then it is sequential. Moreover, for a feathered strongly topological gyrogroup $G$, based on the characterization of feathered strongly topological gyrogroups, we show that if $G$ has countable $cs^{*}$-character, then it is metrizable; and it is also shown that $G$ has a compact resolution swallowing the compact sets if and only if $G$ contains a compact $L$-subgyrogroup $H$ such that the quotient space $G/H$ is a Polish space.