Quotients with respect to strongly $L$-subgyrogroups

TL;DR

This paper investigates quotient structures in topological gyrogroups with respect to strongly $L$-subgyrogroups, establishing openness, continuity, and homogeneity properties, and generalizing classical results from topological group theory.

Contribution

It introduces the concept of strongly $L$-subgyrogroups in topological gyrogroups and proves key properties of the quotient spaces, extending classical topological group results.

Findings

The natural homomorphism to the quotient is open and continuous.

Quotient spaces are homogeneous $T_1$-spaces.

The quotient mapping is locally perfect for locally compact subgyrogroups.

Abstract

A topological gyrogroup is a gyrogroup endowed with a compatible topology such that the multiplication is jointly continuous and the inverse is continuous. In this paper, we study the quotient gyrogroups in topological gyrogroups with respect to strongly $L$-subgyrogroups, and prove that let $(G, \tau,\oplus)$ be a topological gyrogroup and $H$ a closed strongly $L$-subgyrogroup of $G$, then the natural homomorphism $\pi$ from a topological gyrogroup $G$ to its quotient topology on $G/H$ is an open and continuous mapping, and $G/H$ is a homogeneous $T_1$-space. We also establish that for a locally compact strongly $L$-subgyrogroup $H$ of a topological gyrogroup $G$, the natural quotient mapping $\pi$ of $G$ onto the quotient space $G/H$ is a locally perfect mapping. This leads us to some interesting results on how properties of $G$ depend on the properties of $G/H$. Some classical results in topological groups are generalized.