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 ${\displaystyle X}$ is a collection ${\displaystyle \mathbf {E} }$ of subsets of ${\displaystyle X\times X}$ (therefore falling under the more general categorization of binary relations on ${\displaystyle X}$) called controlled sets, and so that ${\displaystyle \mathbf {E} }$ 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 ${\displaystyle \Delta =\{(x,x):x\in X\}}$ is a member of ${\displaystyle \mathbf {E} }$—the identity relation.

2. Closed under taking subsets:

   If ${\displaystyle E\in \mathbf {E} }$ and ${\displaystyle F\subseteq E,}$ then ${\displaystyle F\in \mathbf {E} .}$

3. Closed under taking inverses:

   If ${\displaystyle E\in \mathbf {E} }$ then the inverse (or transpose) ${\displaystyle E^{-1}=\{(y,x):(x,y)\in E\}}$ is a member of ${\displaystyle \mathbf {E} }$—the inverse relation.

4. Closed under taking unions:

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

5. Closed under composition:

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

A set ${\displaystyle X}$ endowed with a coarse structure ${\displaystyle \mathbf {E} }$ is a coarse space.

For a subset ${\displaystyle K}$ of ${\displaystyle X,}$ the set ${\displaystyle E[K]}$ is defined as ${\displaystyle \{x\in X:(x,k)\in E{\text{ for some }}k\in K\}.}$ We define the section of ${\displaystyle E}$ by ${\displaystyle x}$ to be the set ${\displaystyle E[\{x\}],}$ also denoted ${\displaystyle E_{x}.}$ The symbol ${\displaystyle E^{y}}$ denotes the set ${\displaystyle E^{-1}[\{y\}].}$ These are forms of projections.

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

Intuition

The controlled sets are "small" sets, or "negligible sets": a set ${\displaystyle A}$ such that ${\displaystyle A\times A}$ is controlled is negligible, while a function ${\displaystyle f:X\to X}$ 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 ${\displaystyle S}$ and a coarse structure ${\displaystyle X,}$ 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.

For coarse structures ${\displaystyle X}$ and ${\displaystyle Y,}$ we say that ${\displaystyle f:X\to Y}$ is a coarse map if for each bounded set ${\displaystyle B}$ of ${\displaystyle Y}$ the set ${\displaystyle f^{-1}(B)}$ is bounded in ${\displaystyle X}$ and for each controlled set ${\displaystyle E}$ of ${\displaystyle X}$ the set ${\displaystyle (f\times f)(E)}$ is controlled in ${\displaystyle Y.}$^[1] ${\displaystyle X}$ and ${\displaystyle Y}$ 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 ${\displaystyle \operatorname {id} _{X}.}$

Examples

• The bounded coarse structure on a metric space ${\displaystyle (X,d)}$ is the collection ${\displaystyle \mathbf {E} }$ of all subsets ${\displaystyle E}$ of ${\displaystyle X\times X}$ such that ${\displaystyle \sup _{(x,y)\in E}d(x,y)}$ is finite. With this structure, the integer lattice ${\displaystyle \mathbb {Z} ^{n}}$ is coarsely equivalent to ${\displaystyle n}$-dimensional Euclidean space.
• A space ${\displaystyle X}$ where ${\displaystyle X\times X}$ 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 ${\displaystyle C_{0}}$ coarse structure on a metric space ${\displaystyle (X,d)}$ is the collection of all subsets ${\displaystyle E}$ of ${\displaystyle X\times X}$ such that for all ${\displaystyle \varepsilon >0}$ there is a compact set ${\displaystyle K}$ of ${\displaystyle E}$ such that ${\displaystyle d(x,y)<\varepsilon }$ for all ${\displaystyle (x,y)\in E\setminus K\times K.}$ Alternatively, the collection of all subsets ${\displaystyle E}$ of ${\displaystyle X\times X}$ such that ${\displaystyle \{(x,y)\in E:d(x,y)\geq \varepsilon \}}$ is compact.
• The discrete coarse structure on a set ${\displaystyle X}$ consists of the diagonal ${\displaystyle \Delta }$ together with subsets ${\displaystyle E}$ of ${\displaystyle X\times X}$ which contain only a finite number of points ${\displaystyle (x,y)}$ off the diagonal.
• If ${\displaystyle X}$ is a topological space then the indiscrete coarse structure on ${\displaystyle X}$ consists of all proper subsets of ${\displaystyle X\times X,}$ meaning all subsets ${\displaystyle E}$ such that ${\displaystyle E[K]}$ and ${\displaystyle E^{-1}[K]}$ are relatively compact whenever ${\displaystyle K}$ 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
• 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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