Coarse structure

A coarse structure on a set X is a collection E of subsets of X × X (therefore falling under the more general categorization of binary relations on 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 = {(x, x) : x in X} 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 inverse (or transpose) E ⁻¹ = {(y, x) : (x, y) in E} 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 product E o F = {(x, y) : there is a z in X such that (x, z) is in E, (z, y) is in F} is a member of E—the composition of relations.

A set X endowed with a coarse structure E is a coarse space.

The set E[K] is defined as {x in X : there is a y in K such that (x, y) is in E}. We define the section of E by x to be the set E[{x}], also denoted E _x. The symbol E^y denotes the set E ⁻¹[{y}]. These are forms of projections.

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.

Given a set S and a coarse structure X, we say that the maps ${\displaystyle f:S\to X}$ and ${\displaystyle g:S\to X}$ are close if ${\displaystyle \{(f(s),g(s))|s\in S\}}$ is a controlled set. A subset B of X is said to be bounded if ${\displaystyle B\times B}$ is a controlled set.

For coarse structures X and Y, we say that ${\displaystyle f:X\to Y}$ is coarse if for each bounded set B of Y the set ${\displaystyle f^{-1}(Y)}$ is bounded in X and for each controlled set E of X the set ${\displaystyle (f\times f)(E)}$ is controlled in Y.[1] X and Y are said to be coarsely equivalent if there exists coarse maps ${\displaystyle f:X\to Y}$ and ${\displaystyle g:Y\to X}$ such that ${\displaystyle f\circ g}$ is close to ${\displaystyle \operatorname {id} _{Y}}$ and ${\displaystyle g\circ f}$ is close to ${\displaystyle \operatorname {id} _{X}}$.

• The bounded coarse structure on a metric space (X, d) is the collection E of all subsets E of X × X such that sup{d(x, y) : (x, y) is in E} is finite.

  With this structure, the integer lattice Zⁿ is coarsely equivalent to n-dimensional Euclidean space.

• A space X 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 (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 C₀ coarse 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 set K of X such that d(x, y) < ε for all (x, y) in E − K × K. Alternatively, the collection of all subsets E of X × X such that {(x, y) in E : d(x, y) ≥ ε} 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 (x, y) 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 [K] and E ⁻¹[K] are relatively compact whenever K is relatively compact.

• uniform space
• quasi-isometry

• John Roe, Lectures in Coarse Geometry, University Lecture Series Vol. 31, American Mathematical Society: Providence, Rhode Island, 2003. Corrections to Lectures in Coarse Geometry
• Roe, John (June–July 2006). "What is...a Coarse Space?" (PDF). Notices of the American Mathematical Society. 53 (6): 669. Retrieved 2008-01-16.