For every countable group G, there are 2 ω distinct classes of coarsely equivalent subsets of G.2010 MSC: 54E15; 20F69.

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.MSC :54E15

Coarse geometry is the study of large-scale properties of spaces. In this paper we study group coarse structures (i.e., coarse structures on groups that agree with the algebraic structures), by using group ideals. We introduce a large class of examples of group coarse structures induced by cardinal invariants. In order to enhance the categorical treatment of the subject, we use quasihomomorphisms, as a large-scale counterpart of homomorphisms. In particular, the localisation of a category plays a fundamental role. We then define the notion of functorial coarse structures and we give various examples of those structures.

Coarse geometry, the branch of topology that studies the global properties of spaces, was originally developed for metric spaces and then Roe introduced coarse structures ([21]) as a large-scale counterpart of uniformities. In the literature, there are very important generalisations of uniform spaces, such as semi-uniform and quasi-uniform spaces. In this paper, we introduce and start to study their large-scale counterparts, which generalise coarse spaces: semi-coarse spaces and quasi-coarse spaces. * MSC: 18B99, 54E15, 54E25, 54E99, 54A99.

We study the combinatorial size of subsets of a ballean, as defined in [19,23] (largeness, smallness, extralargeness, etc.), paying particular attention to the preservation of these properties under taking images and inverse images along various classes of maps (bornologous, effectively proper, (weakly) soft, coarse embeddings, canonical projections of products, canonical inclusions of co-products, etc.). We show by appropriate examples that many of the properties describing the size are not preserved under coarse equivalences (even injective or surjective ones), whereas largeness and smallness are preserved under arbitrary coarse equivalences