Coarse structure

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—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 is a collection of subsets of (therefore falling under the more general categorization of binary relations on ) called , and so that possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:
Identity/diagonal:
The diagonal is a member of —the identity relation.
Closed under taking subsets:
If and then
Closed under taking inverses:
If then the inverse (or transpose) is a member of —the inverse relation.
Closed under taking unions:
If then their union is a member of
Closed under composition:
If then their product is a member of —the composition of relations.
A set endowed with a coarse structure is a .
For a subset of the set is defined as We define the of by to be the set also denoted The symbol denotes the set These are forms of projections.
A subset of is said to be a if is a controlled set.
Intuition
The controlled sets are "small" sets, or "negligible sets": a set such that is controlled is negligible, while a function 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 and a coarse structure we say that the maps and are if is a controlled set.
For coarse structures and we say that is a if for each bounded set of the set is bounded in and for each controlled set of the set is controlled in and are said to be if there exists coarse maps and such that is close to and is close to
Examples
The on a metric space is the collection of all subsets of such that is finite. With this structure, the integer lattice is coarsely equivalent to -dimensional Euclidean space.
A space where 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 is the collection of all subsets of such that for all there is a compact set of such that for all Alternatively, the collection of all subsets of such that is compact.
The on a set consists of the diagonal together with subsets of which contain only a finite number of points off the diagonal.
If is a topological space then the on consists of all subsets of meaning all subsets such that and are relatively compact whenever 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
General topology
Metric geometry
Topology