§ Coarse structures

A coarse structure on the set XX is a collection of relations on XX: E2X×XE \subseteq 2^{X \times X} (called as controlled sets / entourages) such that:
  • (δ{(x,x):xX})E(\delta \equiv \{ (x, x) : x \in X \}) \in E.
  • Closed under subsets: eE,fe    fE\forall e \in E, f \subset e \implies f \in E.
  • Closed under transpose: if eEe \in E then (eT{(y,x):(x,y)e})E(e^T \equiv \{ (y, x) : (x, y) \in e \}) \in E.
  • Closed under finite unions.
  • Closed under composition: e,fE,efE\forall e, f \in E, e \circ f \in E, where \circis composition of relations.
The sets that are controlled are "small" sets. The bounded coarse structure on a metric space (X,d)(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.
(eX×X)E    δR,(x,y)E,d(x,y)<δ (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:ZR,f(x)xf: \mathbb Z \rightarrow \mathbb R, f(x) \equiv x and
  • g:RZ,g(x)xg: \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..?

§ References