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