Generalisations of coarse spaces

TL;DR

This paper introduces and begins the study of semi-coarse and quasi-coarse spaces, extending coarse geometry to broader classes of spaces and exploring their large-scale properties beyond traditional metric and uniform spaces.

Contribution

It proposes new classes of spaces, semi-coarse and quasi-coarse spaces, as large-scale generalizations of coarse spaces, semi-uniform, and quasi-uniform spaces.

Findings

Introduction of semi-coarse spaces as large-scale counterparts.

Introduction of quasi-coarse spaces as further generalizations.

Initial properties and potential applications outlined.

Abstract

Coarse geometry, the branch of topology that studies the global properties of spaces, was originally developed for metric spaces and then Roe introduced coarse structures as a large-scale counterpart of uniformities. In the literature, there are very important generalisations of uniform spaces, such as semi-uniform and quasi-uniform spaces. In this paper, we introduce and start to study their large-scale counterparts, which generalise coarse spaces: semi-coarse spaces and quasi-coarse spaces.