Boundaries of coarse proximity spaces and boundaries of compactifications

TL;DR

This paper introduces a new boundary concept for coarse proximity spaces, demonstrating its properties and connections to various well-known compactification boundaries, unifying several boundary types in coarse geometry.

Contribution

It defines the boundary of a coarse proximity space and shows its relation to existing compactification boundaries, including Gromov, visual, Higson, and Freudenthal boundaries.

Findings

The boundary $X$ is compact and Hausdorff.

Every compactification of a locally compact Hausdorff space induces a coarse proximity structure.

Boundaries of well-known compactifications are realized as boundaries of coarse proximity spaces.

Abstract

In this paper, we introduce the boundary $\mathcal{U}X$ of a coarse proximity space $(X,\mathcal{B},{\bf b}).$ This boundary is a subset of the boundary of a certain Smirnov compactification. We show that $\mathcal{U}X$ is compact and Hausdorff and that every compactification of a locally compact Hausdorff space induces a coarse proximity structure whose corresponding boundary is the boundary of the compactification. We then show that many boundaries of well-known compactifications arise as boundaries of coarse proximity spaces. In particular, we give four coarse proximity structures whose boundaries are the Gromov, visual, Higson, and Freudenthal boundaries.