Functional boundedness of balleans

TL;DR

This paper explores boundedness properties of functions on balleans and coarse spaces, investigates the invariance of the Bergman property under coarse equivalences, and emphasizes the case of balleans on groups.

Contribution

It introduces open problems on boundedness in balleans, proves the Bergman property is a coarse invariant, and focuses on balleans defined on groups.

Findings

Bergman property is a coarse invariant.

Open problems posed regarding boundedness of functions on balleans.

Special attention given to balleans on groups.

Abstract

We pose some open problems related to boundedness of real-valued functions on balleans and coarse spaces. Also we prove that the Bergman property of groups is a coarse invariant. A special attention is payed to balleans on groups.