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 $X\times X$ (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 $\Delta =\{(x,x):x\in X\}$ is a member of $𝐄$—the identity relation.

2. Closed under taking subsets:

   If $E\in 𝐄$ and $F\subseteq E,$ then $F\in 𝐄.$

3. Closed under taking inverses:

   If $E\in 𝐄$ 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 $E,F\in 𝐄$ then their union $E\cup F$ is a member of$𝐄.$

5. Closed under composition:

   If $E,F\in 𝐄$ then their product $E\circ F=\{(x,y):\text{ there exists }z\in X\text{ such that }(x,z)\in E\text{ and }(z,y)\in F\}$ is a member of $𝐄$—the composition of relations.

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

For a subset $K$ of $X,$ the set $E[K]$ is defined as $\{x\in X:(x,k)\in E\text{ for some }k\in K\}.$ 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.

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

Intuition

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

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.

For coarse structures $X$ and $Y,$ we say that $f:X\to Y$ is a coarse map if for each bounded set $B$ of $Y$ the set $f^{-1}(B)$ is bounded in $X$ and for each controlled set $E$ of $X$ the set $(f\times f)(E)$ is controlled in $Y.$ $X$ 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 ${\mathrm{id}}_Y$ and $g\circ f$ is close to ${\mathrm{id}}_X.$

Examples

• The bounded coarse structure on a metric space $(X,d)$ is the collection $𝐄$ of all subsets $E$ of $X\times X$ such that $\underset{(x,y)\in E}{\sup }d(x,y)$ is finite. With this structure, the integer lattice ${\mathbb{Z}}^n$ is coarsely equivalent to $n$-dimensional Euclidean space.
• A space $X$ where $X\times X$ 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 $C_0$ coarse structure on a metric space $(X,d)$ is the collection of all subsets $E$ of $X\times X$ such that for all $\epsilon >0$ there is a compact set $K$ of $E$ such that $d(x,y)<\epsilon$ for all $(x,y)\in E\setminus K\times K.$ Alternatively, the collection of all subsets $E$ of $X\times X$ such that $\{(x,y)\in E:d(x,y)\ge \epsilon \}$ is compact.
• The discrete coarse structure on a set $X$ consists of the diagonal $\Delta$ together with subsets $E$ of $X\times X$ 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 $X\times X,$ meaning all subsets $E$ such that $E[K]$ and $E^{-1}[K]$ are relatively compact whenever $K$ is relatively compact.

See also

• Bornology – Mathematical generalization of boundedness
• Quasi-isometry – Function between two metric spaces that only respects their large-scale geometry
• Uniform space – Topological space with a notion of uniform properties

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