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