Decomposition of Cellular Balleans

TL;DR

This paper studies balleans, a set with a family of subsets called balls, focusing on their decomposition and classification, especially for homogeneous cellular balleans and their relation to uncountable groups.

Contribution

It introduces a decomposition theorem for homogeneous cellular balleans and applies it to classify balleans of uncountable groups by asymorphism.

Findings

Homogeneous cellular balleans can be decomposed into a direct product of pointed sets.

Balleans of two uncountable groups with the same regular cardinality are asymorphic.

Abstract

A ballean is a set endowed with some family of its subsets which are called the balls. We postulate the properties of the family of balls in such a way that the balleans can be considered as the asymptotic counterparts of the uniform topological spaces. The isomorphisms in the category of balleans are called asymorphisms. Every metric space can be considered as a ballean. The ultrametric spaces are prototypes for the cellular balleans. We prove some general theorem about decomposition of a homogeneous cellular ballean in a direct product of a pointed family of sets. Applying this theorem we show that the balleans of two uncountable groups of the same regular cardinality are asymorphic.