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 is a collection of subsets of (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.
For a subset of the set is defined as We define the section 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 bounded set if is a controlled set.
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.
Given a set and a coarse structure we say that the maps and are close if is a controlled set.
For coarse structures and we say that is a coarse map if for each bounded set of the set is bounded in and for each controlled set of the set is controlled in [1] and are said to be coarsely equivalent if there exists coarse maps and such that is close to and is close to
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 is a collection of subsets of (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.
For a subset of the set is defined as We define the section 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 bounded set if is a controlled set.
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.
Given a set and a coarse structure we say that the maps and are close if is a controlled set.
For coarse structures and we say that is a coarse map if for each bounded set of the set is bounded in and for each controlled set of the set is controlled in [1] and are said to be coarsely equivalent if there exists coarse maps and such that is close to and is close to