The existence of suitable sets in locally compact strongly topological gyrogroups

TL;DR

This paper proves that every locally compact strongly topological gyrogroup contains a suitable set, answering a previously posed question and expanding understanding of the structure of such gyrogroups.

Contribution

It establishes the existence of suitable sets in all locally compact strongly topological gyrogroups, a new result in the field.

Findings

Every such gyrogroup has a suitable set.

The result confirms a conjecture posed by F. Lin et al.

Enhances understanding of the structure of topological gyrogroups.

Abstract

A subset $S$ of a topological gyrogroup $G$ is said to be a {\it suitable set} for $G$ if $S$ is discrete, the gyrogroup generated by $S$ is dense in $G$, and $S\cup \{0\}$ is closed in $G$, where $0$ is the identity element of $G$. In this paper, it is proved that every locally compact strongly topological gyrogroup has a suitable set, which gives an affirmative answer to a question posed by F. Lin, et al. in \cite{key14}.