The construction of Hartman-Mycielski in topological gyrogroups

TL;DR

This paper extends the construction of Hartman-Mycielski to topological gyrogroups, showing that every Hausdorff topological gyrogroup can be embedded into a path-connected, locally path-connected topological gyrogroup.

Contribution

It generalizes previous results by Wattanapan et al., demonstrating embedding of Hausdorff topological gyrogroups into more connected structures.

Findings

Every Hausdorff topological gyrogroup can be embedded as a closed subgyrogroup.

The embedding results in a path-connected, locally path-connected topological gyrogroup.

Extension of Hartman-Mycielski construction to topological gyrogroups.

Abstract

The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. Recently, Wattanapan et al consider the construction of Hartman-Mycielski in strongly topological gyrogroups. In this paper, we extend their results in topological gyrogroups. We mainly, among other results, prove that every Hausdorff topological gyrogroup $G$ can be embedded as a closed subgyrogroup of a Hausdorff path-connected and locally path-connected topological gyrogroup $G^\bullet$.