A totally bounded Uniformity on coarse metric Spaces

TL;DR

This paper introduces a novel boundary concept for coarse metric spaces, creating a functor that maps these spaces to uniform topological spaces and preserves coarse maps as uniformly continuous maps.

Contribution

It proposes a new boundary construction for coarse spaces and establishes a functorial relationship with uniform topological spaces.

Findings

Defines a new boundary for coarse metric spaces

Establishes a functor from coarse spaces to uniform topological spaces

Shows coarse maps correspond to uniformly continuous maps

Abstract

This paper presents a new version of boundary on coarse spaces. The space of ends functor maps coarse metric spaces to uniform topological spaces and coarse maps to uniformly continuous maps.