§ Coarse structures

A coarse structure on the set $X$ is a collection of relations on $X$: $E \subseteq 2^{X \times X}$ (called as controlled sets / entourages) such that:
• $(\delta \equiv \{ (x, x) : x \in X \}) \in E$.
• Closed under subsets: $\forall e \in E, f \subset e \implies f \in E$.
• Closed under transpose: if $e \in E$ then $(e^T \equiv \{ (y, x) : (x, y) \in e \}) \in E$.
• Closed under finite unions.
• Closed under composition: $\forall e, f \in E, e \circ f \in E$, where $\circ$is composition of relations.
The sets that are controlled are "small" sets. The bounded coarse structure on a metric space $(X, d)$ is the set of all relations such that there exists a uniform bound such that all related elements are within that bounded distance.
$(e \subset X \times X) \in E \iff \exists \delta \in \mathbb R, \forall (x, y) \in E, d(x, y) < \delta$
We can check that the functions:
• $f: \mathbb Z \rightarrow \mathbb R, f(x) \equiv x$ and
• $g: \mathbb R \rightarrow \mathbb Z, g(x) \equiv \lfloor x \rfloor$
are coarse inverses to each other. I am interested in this because if topology is related to semidecidability, then coarse structures (which are their dual) are related to..?