coarsespace

Coarse space, or coarse space, is a set X equipped with a coarse structure E, a collection of subsets of X×X called entourages that describe large-scale proximity between points. The axioms ensure E behaves like a way to measure distant relationships rather than small-scale details: the diagonal Δ_X = { (x,x) : x ∈ X } is in E, E is closed under supersets and finite unions, and E is closed under inverses and composition of entourages. With these axioms, E defines a notion of coarse boundedness: a subset B ⊆ X is bounded if B×B is contained in some E ∈ E.

Coarse maps and equivalences are central concepts. A map f: X → Y between coarse spaces is coarse (bornologous) if for every E ∈ E_X, the image (f×f)[E] is in E_Y. It is proper if the preimage of every bounded set is bounded. Two maps f, g: X → Y are close if the set { (f(x), g(x)) : x ∈ X } lies in E_Y. A coarse equivalence exists when there are coarse maps f: X → Y and g: Y → X such that gf and fg are close to the respective identity maps; this expresses large-scale sameness between spaces.

Examples and context: Every metric space (X,d) induces a coarse structure generated by the entourages E_R = { (x,y) : d(x,y) ≤ R }. Finite-generated groups with the word metric and graphs with path length provide natural coarse spaces. The subject emerged in coarse geometry, associated with Roe and others in the 1990s, and plays a role in areas such as index theory and geometric group theory, including the coarse Baum–Connes conjecture.