Balleans, hyperballeans and ideals

A ballean 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 X) defined in such a way that 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.