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 imagess of small open sets, or neighborhoods, are themselves open. Large-scale properties of a spacesuch as boundedness, or the degrees of freedom of the spacedo 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 of subsets (therefore falling under the more general categorization of binary relations) called possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:

1. Identity/diagonal:

   1. The diagonalthe identity relation.

   Closed under taking subsets:

2. Closed under taking inverses:

   1. then the inverse (or transposethe inverse relation.

   Closed under taking unions:

   1. then their union

   Closed under composition:

   1. then their productthe composition of relations.

is a .

We define the These are forms of projections.

is said to be a is a controlled set.

Intuition

The controlled sets are "small" sets, or "negligible sets 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

are is a controlled set.

is a are said to be

Examples

• The on a metric space of all subsets is finite. With this structure, the integer lattice-dimensional Euclidean space.

• 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 there is a compact is compact.

• The consists of the diagonal off the diagonal.

• is a topological space then the consists of all are relatively compact is relatively compact.

Bounded sets

is coarsely connected is a bornology is not a bornology. Bounded, discrete and indiscrete coarse structures are coarsely connected.

See also

References

• John Roe, Lectures in Coarse Geometry, University Lecture Series Vol. 31, American Mathematical Society: Providence, Rhode Island, 2003. Corrections to