Closeness and linkness in balleans

Abstract

A set $X$ endowed with a coarse structure is called ballean or coarse space. For a ballean $(X,\mathcal{E})$, we say that two subsets $A$, $B$ of $X$ are close (linked) if there exists an entourage $E\in \mathcal{E}$ such that $A\subseteq E[B]$, $B\subseteq E[A]$ (either $A,B$ are bounded or contain unbounded close subsets). We explore the following general question: which information about a ballean is contained and can be extracted from the relations of closeness and linkness.

Coarse structure, ballean, coarse space, closeness, linkness, rigidity, stability