Balleans, hyperballeans and ideals

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.