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