Coarse structure

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&mdash;such as boundedness, or the degrees of freedom of the space&mdash;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:


 * 1) Identity/diagonal:
 * The diagonal $$\Delta = \{(x, x) : x \in X\}$$ is a member of $$\mathbf{E}$$&mdash;the identity relation.
 * 1) Closed under taking subsets:
 * If $$E \in \mathbf{E}$$ and $$F \subseteq E,$$ then $$F \in \mathbf{E}.$$
 * 1) Closed under taking inverses:
 * If $$E \in \mathbf{E}$$ then the inverse (or transpose) $$E^{-1} = \{(y, x) : (x, y) \in E\}$$ is a member of $$\mathbf{E}$$&mdash;the inverse relation.
 * 1) Closed under taking unions:
 * If $$E, F \in \mathbf{E}$$ then their union $$E \cup F$$ is a member of$$\mathbf{E}.$$
 * 1) Closed under composition:
 * If $$E, F \in \mathbf{E}$$ then their product $$E \circ F = \{(x, y) : \text{ there exists } z \in X \text{ such that } (x, z) \in E \text{ and } (z, y) \in F\}$$ is a member of $$\mathbf{E}$$&mdash;the composition of relations.

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 $$\{x \in X : (x, k) \in E \text{ for some } k \in K\}.$$ We define the of $$E$$ by $$x$$ to be the set $$E[\{x\}],$$ also denoted $$E_x.$$ The symbol $$E^y$$ denotes the set $$E^{-1}[\{y\}].$$ 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 $$\{(f(s), g(s)) : s \in S\}$$ 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 $$(f \times f)(E)$$ 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 $$\operatorname{id}_Y$$ and $$g \circ f$$ is close to $$\operatorname{id}_X.$$

Examples

 * The on a metric space $$(X, d)$$ is the collection $$\mathbf{E}$$ of all subsets $$E$$ of $$X \times X$$ such that $$\sup_{(x, y) \in E} d(x, y)$$ is finite. With this structure, the integer lattice $$\Z^n$$ is coarsely equivalent to $$n$$-dimensional Euclidean space.
 * A space $$X$$ where $$X \times X$$ is controlled is called a . Such a space is coarsely equivalent to a point. A metric space with the bounded coarse structure is bounded (as a coarse space) if and only if it is bounded (as a metric space).
 * The trivial coarse structure only consists of the diagonal and its subsets. In this structure, a map is a coarse equivalence if and only if it is a bijection (of sets).
 * The on a metric space $$(X, d)$$ is the collection of all subsets $$E$$ of $$X \times X$$ such that for all $$\varepsilon > 0$$ there is a compact set $$K$$ of $$E$$ such that $$d(x, y) < \varepsilon$$ for all $$(x, y) \in E \setminus K \times K.$$ Alternatively, the collection of all subsets $$E$$ of $$X \times X$$ such that $$\{(x, y) \in E : d(x, y) \geq \varepsilon\}$$ is compact.
 * The on a set $$X$$ consists of the diagonal $$\Delta$$ together with subsets $$E$$ of $$X \times X$$ which contain only a finite number of points $$(x, y)$$ off the diagonal.
 * If $$X$$ is a topological space then the on $$X$$ consists of all  subsets of $$X \times X,$$ meaning all subsets $$E$$ such that $$E[K]$$ and $$E^{-1}[K]$$ are relatively compact whenever $$K$$ is relatively compact.