In the mathematical fields of geometry and topology, a coarse structure on a set X is a collection of subsets of the cartesian product X × X with certain properties which allow the large-scale structure of metric spaces and topological spaces to be defined. The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. Coarse geometry and coarse topology provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties. Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.

Definition

A on a set X is a collection \mathbf{E} of subsets of X \times X (therefore falling under the more general categorization of binary relations on X) called, and so that \mathbf{E} possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly: A set X endowed with a coarse structure \mathbf{E} is a. For a subset K of X, the set E[K] is defined as We define the of E by x to be the set E[{x}], also denoted E_x. The symbol E^y denotes the set These are forms of projections. A subset B of X is said to be a if B \times B is a controlled set.

Intuition

The controlled sets are "small" sets, or "negligible sets": a set A such that A \times A is controlled is negligible, while a function f : X \to X such that its graph is controlled is "close" to the identity. In the bounded coarse structure, these sets are the bounded sets, and the functions are the ones that are a finite distance from the identity in the uniform metric.

Coarse maps

Given a set S and a coarse structure X, we say that the maps f : S \to X and g : S \to X are if is a controlled set. For coarse structures X and Y, we say that f : X \to Y is a if for each bounded set B of Y the set f^{-1}(B) is bounded in X and for each controlled set E of X the set is controlled in Y. X and Y are said to be if there exists coarse maps f : X \to Y and g : Y \to X such that f \circ g is close to and g \circ f is close to

Examples