2 - Coarse Structure and Metrisability

Summary

In this chapter, we introduce the fundamental concepts investigated, namely coarse structures, coarsely bounded sets and large-scale geometry. We show how all topological groups carry an intrinsic coarse structure and determine when this is metrisable. We furthermore discuss quasi-metric structures and spaces and show how, in many familiar cases, the coarse structure on a topological group admits a refinement to a quasi-metric structure.