Binary coronas of balleans

Abstract

A ballean \(\mathbb B\) is a set \(X\) endowed with some family of subsets of \(X\) which are called the balls. We postulate the properties of the family of balls in such a way that a ballean can be considered as an asymptotic counterpart of a uniform topological space. Using slow oscillating functions from \(X\) to \(\{0,1\}\), we define a zero-dimensional compact space which is called a binary corona of \(\mathbb B\).  We define a class of binary normal ballean and, for every ballean from this class, give an intrinsic characterization of its binary corona. The class of binary normal balleans contains all balleans of graph. We show that a ballean of graph is a projective limit of some sequence of \(\breve{C}\)ech-Stone compactifications of discrete spaces. The obtained results witness that a binary corona of balleans can be interpreted as a "generalized space of ends" of ballean.