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.
A coarse structure on a set
(therefore falling under the more general categorization of binary relations on
) called controlled sets, and so that
possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations.
Explicitly: A set
endowed with a coarse structure
is a coarse space.
We define the section of
to be the set
denotes the set
These are forms of projections.
is said to be a bounded set if
is a controlled set.
The controlled sets are "small" sets, or "negligible sets": a set
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.
Given a set
and a coarse structure
is a controlled set.
For coarse structures
is a coarse map if for each bounded set
the set
and for each controlled set
are said to be coarsely equivalent if there exists coarse maps