Coarse Space

A coarse space is a collection of subsets of $X\times X$ that allow one to study large-scale structure of Metric Spaces and Topological Spaces. These are roughly dual to Uniform Spaces.

Explicitly, a coarse space is a set $E\subseteq\mathcal{P}(X\times X)$ of "controlled sets" such that:

• The Identity Relation $\Delta$ is a member of $E$.
• $E$ is closed under Converses: if $R\in E$, then $R^{\dagger}\in E$.
• $E$ is closed under Relational Composition: if $R\in E$ and $S\in E$, then $R\circ S\in E$.
• $E$ is downwards closed: if $R\in E$ and $S\subseteq R$, then $S\in E$.
• $E$ is closed under unions: if $R\in E$ and $S\in E$, then $R\cup S\in E$.