§ Coarse structures

A coarse structure on the set XX is a collection of relations on XX: E⊆2X×XE \subseteq 2^{X \times X} (called as controlled sets / entourages ) such that:
  • (δ≡{(x,x):x∈X})∈E(\delta \equiv \{ (x, x) : x \in X \}) \in E.
  • Closed under subsets: ∀e∈E,f⊂e  ⟹  f∈E\forall e \in E, f \subset e \implies f \in E.
  • Closed under transpose: if e∈Ee \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,f∈E,e∘f∈E\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.
(e⊂X×X)∈E  ⟺  ∃δ∈R,∀(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:Z→R,f(x)≡xf: \mathbb Z \rightarrow \mathbb R, f(x) \equiv x and
  • g:R→Z,g(x)≡⌊x⌋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..?

§ References