Coarse structure

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 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.

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 $f:S\to X$ and $g:S\to X$ are close if $\{(f(s),g(s))|s\in S\}$ is a controlled set. A subset B of X is said to be bounded if $B\times B$ is a controlled set.

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

Examples

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.

References

Hoffland, Christian Stuart. Course structures and Higson compactification. OCLC 76953246.

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.

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[1] Hoffland, Christian Stuart. Course structures and Higson compactification. OCLC 76953246.