On hyperballeans of bounded geometry

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 ${\mathcal{B}}^{♭}$ is a ballean naturally defined on the set $X^{♭}$ of all bounded subsets of $X$. We describe all balleans with hyperballeans of bounded geometry and analyze the structure of these hyperballeans.