Balleans, hyperballeans and ideals

TL;DR

This paper investigates the structure of balleans and hyperballeans derived from ideals in Boolean algebras, focusing on their connectedness and the number of connected components.

Contribution

It introduces and analyzes specific balleans and hyperballeans constructed from ideals, emphasizing their connectedness properties and component counts.

Findings

Characterization of connected components in these balleans

Analysis of hyperballeans derived from Boolean algebra ideals

Results on the number of connected components

Abstract

A ballean $\mathcal{B}$ (or a coarse structure) on a set $X$ is a family of subsets of $X$ called balls (or entourages of the diagonal in $X\times X$) defined in such a way that $\mathcal{B}$ can be considered as the asymptotic counterpart of a uniform topological space. The aim of this paper is to study two concrete balleans defined by the ideals in the Boolean algebra of all subsets of $X$ and their hyperballeans, with particular emphasis on their connectedness structure, more specifically the number of their connected components.