On hyperballeans of bounded geometry

TL;DR

This paper studies the structure of hyperballeans, a type of coarse geometric object, focusing on those with bounded geometry, and describes their properties and classifications.

Contribution

It characterizes all balleans whose hyperballeans have bounded geometry and analyzes the structure of these hyperballeans.

Findings

Characterization of balleans with hyperballeans of bounded geometry

Structural analysis of hyperballeans in these cases

Descriptions of the properties of such hyperballeans

Abstract

A ballean (or coarse structure) is a set endowed with some family of subsets, the balls, is such a way that balleans with corresponding morphisms can be considered as asymptotic counterparts of uniform topological spaces. For a ballean $\mathcal{B}$ on a set $X$, the hyperballean $\mathcal{B}^{\flat}$ is a ballean naturally defined on the set $X^{\flat}$ of all bounded subsets of $X$. We describe all balleans with hyperballeans of bounded geometry and analyze the structure of these hyperballeans.