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 {{em|{{visible anchor|coarse structure}}}} on a set $X$ is a collection $\mathbf{E}$ of subsets of $X \times X$ (therefore falling under the more general categorization of binary relations on $X$ ) called {{em|{{visible anchor|controlled set}}s}}, and so that $\mathbf{E}$ possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:

Identity/diagonal:
: The diagonal $\Delta = \{(x, x) : x \in X\}$ is a member of $\mathbf{E}$ —the identity relation.
Closed under taking subsets:
: If $E \in \mathbf{E}$ and $F \subseteq E,$ then $F \in \mathbf{E}.$
Closed under taking inverses:
: If $E \in \mathbf{E}$ then the inverse (or transpose) $E^{-1} = \{(y, x) : (x, y) \in E\}$ is a member of $\mathbf{E}$ —the inverse relation.
Closed under taking unions:
: If $E, F \in \mathbf{E}$ then their union $E \cup F$ is a member of $\mathbf{E}.$
Closed under composition:
: If $E, F \in \mathbf{E}$ 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 $\mathbf{E}$ —the composition of relations.

A set $X$ endowed with a coarse structure $\mathbf{E}$ is a {{em|{{visible anchor|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 {{em|{{visible anchor|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 {{em|{{visible anchor|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 {{em|{{visible anchor|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 {{em|{{visible anchor|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 $Y.$ {{Cite book|title=Course structures and Higson compactification|author=Hoffland, Christian Stuart|oclc=76953246}} $X$ and $Y$ are said to be {{em|{{visible anchor|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 {{em|{{visible anchor|bounded coarse structure}}}} on a metric space $(X, d)$ is the collection $\mathbf{E}$ of all subsets $E$ of $X \times X$ such that $\sup_{(x, y) \in E} d(x, y)$ is finite. With this structure, the integer lattice $\Z^n$ is coarsely equivalent to $n$ -dimensional Euclidean space.
A space $X$ where $X \times X$ is controlled is called a {{em|{{visible anchor|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 {{em|{{visible anchor|C0 coarse structure|text= $C_0$ coarse structure}}}} on a metric space $(X, d)$ is the collection of all subsets $E$ of $X \times X$ such that for all $\varepsilon > 0$ there is a compact set $K$ of $E$ such that $d(x, y) < \varepsilon$ for all $(x, y) \in E \setminus K \times K.$ Alternatively, the collection of all subsets $E$ of $X \times X$ such that $\{(x, y) \in E : d(x, y) \geq \varepsilon\}$ is compact.
The {{em|{{visible anchor|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 {{em|{{visible anchor|indiscrete coarse structure}}}} on $X$ consists of all {{em|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.

References

John Roe, Lectures in Coarse Geometry, University Lecture Series Vol. 31, American Mathematical Society: Providence, Rhode Island, 2003. [http://www.personal.psu.edu/jxr57/writings/correction.pdf Corrections to Lectures in Coarse Geometry]
{{ cite journal

| last = Roe

| first = John

| title = What is...a Coarse Space?

| journal = Notices of the American Mathematical Society

|date=June–July 2006

| volume = 53

| issue = 6

| pages = 669

| url = https://www.ams.org/notices/200606/whatis-roe.pdf

| accessdate = 2008-01-16 }}

Category:General topology

Category:Metric geometry