Linearly ordered coarse spaces

TL;DR

This paper studies linearly ordered coarse spaces, proving they are locally convex with asymptotic dimension 0 or 1, and shows that in metrizable cases, right bounded subsets have a selector.

Contribution

It introduces the concept of linearly ordered coarse spaces and establishes their local convexity and asymptotic dimension properties, also addressing selector existence in metrizable cases.

Findings

Linearly ordered coarse spaces are locally convex.

Asymptotic dimension of such spaces is 0 or 1.

In metrizable cases, right bounded subsets have a selector.

Abstract

A coarse space $X$, endowed with a linear order compatible with the coarse structure of $X$, is called linearly ordered. We prove that every linearly ordered coarse space $X$ is locally convex and the asymptotic dimension of $X$ is either $0$ or $1$. If $X$ is metrizable then the family of all right bounded subsets of $X$ has a selector.