Closeness and linkness in balleans

TL;DR

This paper investigates the concepts of closeness and linkness in balleans, aiming to understand what structural information about these coarse spaces can be derived from these relations.

Contribution

It introduces the notions of closeness and linkness in balleans and explores how these relations reveal structural properties of coarse spaces.

Findings

Defined closeness and linkness in balleans

Analyzed the extent of information these relations encode

Provided insights into the structure of coarse spaces

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.