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 **coarse structure** on a set *X* is a collection **E** of subsets of *X* × *X* (therefore falling under the more general categorization of binary relations on *X*) called *controlled sets*, and so that **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 Δ = {(
*x*,*x*) :*x*in*X*} is a member of**E**—the identity relation. - 2. Closed under taking subsets
- If
*E*is a member of**E**and*F*is a subset of*E*, then*F*is a member of**E**. - 3. Closed under taking inverses
- If
*E*is a member of**E**then the**inverse**(or**transpose**)*E*^{−1}= {(*y*,*x*) : (*x*,*y*) in*E*} is a member of**E**—the inverse relation. - 4. Closed under taking unions
- If
*E*and*F*are members of**E**then the**union**of*E*and*F*is a member of**E**. - 5. Closed under composition
- If
*E*and*F*are members of**E**then the**product***E*o*F*= {(*x*,*y*) : there is a*z*in*X*such that (*x*,*z*) is in*E*, (*z*,*y*) is in*F*} is a member of**E**—the composition of relations.

A set *X* endowed with a coarse structure **E** is a *coarse space*.

The set *E*[*K*] is defined as {*x* in *X* : there is a *y* in *K* such that (*x*, *y*) is in *E*}. We define the *section* 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.

## Intuition

The controlled sets are "small" sets, or "negligible sets": a set *A* such that *A* × *A* is controlled is negligible, while a function *f* : *X* → *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 and are *close* if is a controlled set. A subset *B* of *X* is said to be *bounded* if is a controlled set.

For coarse structures *X* and *Y*, we say that is *coarse* if for each bounded set *B* of *Y* the set is bounded in *X* and for each controlled set *E* of *X* the set is controlled in *Y*.^{[1]} *X* and *Y* are said to be *coarsely equivalent* if there exists coarse maps and such that is close to and is close to .

## Examples

- The
*bounded coarse structure*on a metric space (*X*,*d*) is the collection**E**of all subsets*E*of*X*×*X*such that sup{*d*(*x*,*y*) : (*x*,*y*) is in*E*} is finite.- With this structure, the integer lattice
**Z**^{n}is coarsely equivalent to*n*-dimensional Euclidean space.

- With this structure, the integer lattice
- A space
*X*where*X*×*X*is controlled is called a**bounded space.**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
*C*_{0}*coarse structure*on a metric space*X*is the collection of all subsets*E*of*X*×*X*such that for all ε > 0 there is a compact set*K*of*X*such that*d*(*x*,*y*) < ε for all (*x*,*y*) in*E*−*K*×*K*. Alternatively, the collection of all subsets*E*of*X*×*X*such that {(*x*,*y*) in*E*:*d*(*x*,*y*) ≥ ε} is compact. - The
*discrete coarse structure*on a set*X*consists of the diagonal together with subsets*E*of*X*×*X*which contain only a finite number of points (*x*,*y*) off the diagonal. - If
*X*is a topological space then the*indiscrete coarse structure*on*X*consists of all*proper*subsets of*X*×*X*, meaning all subsets*E*such that*E*[*K*] and*E*^{−1}[*K*] are relatively compact whenever*K*is relatively compact.

## See also

## References

- John Roe, Lectures in Coarse Geometry, University Lecture Series Vol. 31, American Mathematical Society: Providence, Rhode Island, 2003. Corrections to
*Lectures in Coarse Geometry* - Roe, John (June–July 2006). "What is...a Coarse Space?" (PDF).
*Notices of the American Mathematical Society*.**53**(6): 669. Retrieved 2008-01-16.