Balleans, hyperballeans and ideals

Abstract

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

Hyperballeans of groups

“…Recall that the Hausdorff-Bourbaki hyperspace of a Hausdorff uniform space is not Hausdorff in general. Similarly, as we may have expected, in general, exp B is highly disconnected even if B is connected (see [6] and Proposition 3.34). In order to obtain a more manageable object there are two approaches.…”

“…Among all characterisations of thinness (for example, see [6,18]), let us remind the following. The notion of thinnes may seem too restrictive.…”

“…The definition of exp B was inspired by the theory of uniform spaces. In fact, if X is a uniform space, the Hausdorff-Bourbaki hyperspace is a uniform structure on the power set of X and it extends the uniform hyperspace of X, which is the restriction of the Hausdorff-Bourbaki hyperspace to the family of all closed subsets of X (see, for example, [13] for a discussion about uniform hyperspaces, and [6] for further details about similarities between uniform hyperspaces and hyperballeans).…”

Closeness and linkness in balleans

“…The negation of λ (namely, asymptotic disjointness) is used for definition of normal balleans, see [6] and Section 4. The relation δ is an equivalence and each δ-class is a connected component in the hyperballean of (X, E), see [4], [11].…”

Nonstandard methods in large-scale topology

“…Standard;[9, Theorem 2.2]). For every unbounded connected coarse space X, the following are equivalent:1.…”