Pseudodiscrete balleans

Abstract

A ballean \(\mathcal{B}\) is a set \(X\) endowed with some family of subsets of \(X\) which are called the balls. The properties of the balls are postulated in such a way that a ballean can be considered as an asymptotic counterpart of a uniform topological space. A ballean is called pseudodiscrete if "almost all" balls of every pregiven radius are singletons. We give a filter characterization of pseudodiscrete balleans and their classification up to quasi-asymorphisms. It is proved that a ballean is pseudodiscrete if and only if every real function defined on its support is slowly oscillating. We show that the class of irresolvable balleans are tightly connected with the class of pseudodiscrete balleans.