A class of quotient spaces in strongly topological gyrogroups

TL;DR

This paper investigates conditions under which quotient spaces of strongly topological gyrogroups are first-countable or metrizable, establishing equivalences and properties related to neutrality and base structures.

Contribution

It provides new characterizations of when quotient spaces of strongly topological gyrogroups are metrizable or first-countable, especially involving neutral subgyrogroups and base conditions.

Findings

G/H is first-countable iff it is metrizable under certain conditions.

If H is neutral and G/H is Frechet-Urysohn with an ωω-base, then G/H is first-countable.

For neutral H, the pi-character equals the character, and the pi-weight equals the weight of G/H.

Abstract

Quotient space is a class of the most important topological spaces in the research of topology. In this paper, we show that if G is a strongly topological gyrogroup with a symmetric neighborhood base U at 0 and H is an admissible subgyrogroup generated from U , then G/H is first-countable if and only if it is metrizable. Moreover, if H is neutral and G/H is Frechet-Urysohn with an {\omega}{\omega}-base, then G/H is first-countable. Therefore, we obtain that if H is neutral, then G/H is metrizable if and only if G/H is Frechet-Urysohn with an {\omega}{\omega}-base. Finally, it is shown that if H is neutral, {\pi}\c{hi}(G/H) = \c{hi}(G/H) and {\pi}{\omega}(G/H) = {\omega}(G/H).