A coarse structure on a set X is a collection E of subsets of X × X called controlled sets, and so that E possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly: ;1. Identity/diagonal: The diagonal Δ = is a member of E—the identity relation. ;2. Closed under taking subsets: If E is a member of E and F is a subset of E, then F is a member of E. ;3. Closed under taking inverses: If E is a member of E then the inverseE−1 = is a member of E—the inverse relation. ;4. Closed under taking unions: If E and F are members of E then the union of E and F is a member of E. ;5. Closed under composition: If E and F are members of E then the productE o F = is a member of E—the composition of relations. A set Xendowed with a coarse structure E is a coarse space. The set E is defined as. We define the section of E by x to be the set E, also denoted Ex. The symbol Ey denotes the set E−1. These are forms of projections.
Intuition
The controlled sets are "small" sets, or "negligible sets": a set A such that A × A is controlled is negligible, while a function f : X → 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.
Coarse Maps
Given a set S and a coarse structure X, we say that the maps and are close if is a controlled set. A subset B of X is said to be bounded if is a controlled set. For coarse structures X and Y, we say that is coarse if for each bounded setB of Y the set is bounded in X and for each controlled set E of X the set is controlled in Y. X and Y are said to be coarsely equivalent if there exists coarse maps and such that is close to and is close to.
Examples
The bounded coarse structure on a metric space is the collection E of all subsets E of X × X such that sup is finite.
A spaceX where X × X is controlled is called a bounded space. Such a space is coarsely equivalent to a point. A metric space with the bounded coarse structure is bounded if and only if it is bounded.
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.
The C0coarse structure on a metric space X is the collection of all subsets E of X × X such that for all ε > 0 there is a compact setK of X such that d < ε for all in E − K × K. Alternatively, the collection of all subsets E of X × X such that is compact.
The discrete coarse structure on a set X consists of the diagonal together with subsets E of X × X which contain only a finite number of points off the diagonal.
If X is a topological space then the indiscrete coarse structure on X consists of all proper subsets of X × X, meaning all subsets E such that E and E−1 are relatively compact whenever K is relatively compact.
