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 ${\displaystyle X}$ is a collection ${\displaystyle \mathbf {E} }$ of subsets of ${\displaystyle X\times X}$ (therefore falling under the more general categorization of binary relations on ${\displaystyle X}$) called controlled sets, and so that ${\displaystyle \mathbf {E} }$ possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:

A set ${\displaystyle X}$ endowed with a coarse structure ${\displaystyle \mathbf {E} }$ is a coarse space.

For a subset ${\displaystyle K}$ of ${\displaystyle X,}$ the set ${\displaystyle E[K]}$ is defined as ${\displaystyle \{x\in X:(x,k)\in E{\text{ for some }}k\in K\}.}$ We define the section of ${\displaystyle E}$ by ${\displaystyle x}$ to be the set ${\displaystyle E[\{x\}],}$ also denoted ${\displaystyle E_{x}.}$ The symbol ${\displaystyle E^{y}}$ denotes the set ${\displaystyle E^{-1}[\{y\}].}$ These are forms of projections.

A subset ${\displaystyle B}$ of ${\displaystyle X}$ is said to be a bounded set if ${\displaystyle B\times B}$ is a controlled set.

The controlled sets are "small" sets, or "negligible sets": a set ${\displaystyle A}$ such that ${\displaystyle A\times A}$ is controlled is negligible, while a function ${\displaystyle f:X\to X}$ 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.