{{Short description|Concept in geometry and topology}} 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 <math>X</math> is a collection <math>\mathbf{E}</math> of subsets of <math>X \times X</math> (therefore falling under the more general categorization of binary relations on <math>X</math>) called {{em|{{visible anchor|controlled set}}s}}, and so that <math>\mathbf{E}</math> 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 <math>\Delta = \{(x, x) : x \in X\}</math> is a member of <math>\mathbf{E}</math>—the identity relation. # '''Closed under taking subsets''': #: If <math>E \in \mathbf{E}</math> and <math>F \subseteq E,</math> then <math>F \in \mathbf{E}.</math> # '''Closed under taking inverses''': #: If <math>E \in \mathbf{E}</math> then the '''inverse''' (or '''transpose''') <math>E^{-1} = \{(y, x) : (x, y) \in E\}</math> is a member of <math>\mathbf{E}</math>—the inverse relation. # '''Closed under taking unions''': #: If <math>E, F \in \mathbf{E}</math> then their '''union''' <math>E \cup F</math> is a member of<math>\mathbf{E}.</math> # '''Closed under composition''': #: If <math>E, F \in \mathbf{E}</math> then their '''product''' <math>E \circ F = \{(x, y) : \text{ there exists } z \in X \text{ such that } (x, z) \in E \text{ and } (z, y) \in F\}</math> is a member of <math>\mathbf{E}</math>—the composition of relations.
A set <math>X</math> endowed with a coarse structure <math>\mathbf{E}</math> is a {{em|{{visible anchor|coarse space}}}}.
Let <math>E \in \mathbf{E}</math> a controlled set. For a subset <math>K</math> of <math>X,</math> the set <math>E[K]</math> is defined as <math>\{x \in X : (x, k) \in E \text{ for some } k \in K\}.</math> We define the {{em|{{visible anchor|section}}}} of <math>E</math> by <math>x</math> to be the set <math>E[\{x\}],</math> also denoted <math>E_x.</math> The symbol <math>E^y</math> denotes the set <math>E^{-1}[\{y\}].</math> These are forms of projections.
A subset <math>B</math> of <math>X</math> is said to be a {{em|{{visible anchor|bounded set}}}} if <math>B \times B</math> is a controlled set.
===Intuition===
The controlled sets are "small" sets, or "negligible sets": a set <math>A</math> such that <math>A \times A</math> is controlled is negligible, while a function <math>f : X \to X</math> 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 <math>S</math> and a coarse structure <math>X,</math> we say that the maps <math>f : S \to X</math> and <math>g : S \to X</math> are {{em|{{visible anchor|close}}}} if <math>\{(f(s), g(s)) : s \in S\}</math> is a controlled set.
For coarse structures <math>X</math> and <math>Y,</math> we say that <math>f : X \to Y</math> is a {{em|{{visible anchor|coarse map}}}} if for each bounded set <math>B</math> of <math>Y</math> the set <math>f^{-1}(B)</math> is bounded in <math>X</math> and for each controlled set <math>E</math> of <math>X</math> the set <math>(f \times f)(E)</math> is controlled in <math>Y.</math><ref name="Course structures and Higson compactification">{{Cite book|title=Course structures and Higson compactification|author=Hoffland, Christian Stuart|oclc=76953246}}</ref> <math>X</math> and <math>Y</math> are said to be {{em|{{visible anchor|coarsely equivalent}}}} if there exists coarse maps <math>f : X \to Y</math> and <math>g : Y \to X</math> such that <math>f \circ g</math> is close to <math>\operatorname{id}_Y</math> and <math>g \circ f</math> is close to <math>\operatorname{id}_X.</math>
==Examples==
* The {{em|{{visible anchor|bounded coarse structure}}}} on a metric space <math>(X, d)</math> is the collection <math>\mathbf{E}</math> of all subsets <math>E</math> of <math>X \times X</math> such that <math>\sup_{(x, y) \in E} d(x, y)</math> is finite. With this structure, the integer lattice <math>\Z^n</math> is coarsely equivalent to <math>n</math>-dimensional Euclidean space. * A space <math>X</math> where <math>X \times X</math> 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=<math>C_0</math> coarse structure}}}} on a metric space <math>(X, d)</math> is the collection of all subsets <math>E</math> of <math>X \times X</math> such that for all <math>\varepsilon > 0</math> there is a compact set <math>K</math> of <math>E</math> such that <math>d(x, y) < \varepsilon</math> for all <math>(x, y) \in E \setminus K \times K.</math> Alternatively, the collection of all subsets <math>E</math> of <math>X \times X</math> such that <math>\overline{\{(x, y) \in E : d(x, y) \geq \varepsilon\}}</math> is compact. * The {{em|{{visible anchor|discrete coarse structure}}}} on a set <math>X</math> consists of the diagonal <math>\Delta</math> together with subsets <math>E</math> of <math>X \times X</math> which contain only a finite number of points <math>(x, y)</math> off the diagonal. * If <math>X</math> is a topological space then the {{em|{{visible anchor|indiscrete coarse structure}}}} on <math>X</math> consists of all {{em|proper}} subsets of <math>X \times X,</math> meaning all subsets <math>E</math> such that <math>E[K]</math> and <math>E^{-1}[K]</math> are relatively compact whenever <math>K</math> is relatively compact.
==Bounded sets==
Let <math>\mathcal{B} = \{B\subseteq X : B\times B\in\mathbf{E}\}</math> be the collection of all bounded sets of a coarse space <math>X</math>. Say that a coarse structure <math>\mathbf{E}</math> on <math>X</math> is <em>coarsely connected</em><ref name="Course structures and Higson compactification" /> if <math>\{(x, y)\}\in\mathbf{E}</math> for all <math>(x, y)\in X\times X</math>. Then <math>\mathcal{B}</math> is a bornology on <math>X</math> if and only if <math>\mathbf{E}</math> is coarsely connected. For example, if <math>X</math> has at least two points and <math>\mathbf{E}</math> is the trivial coarse structure, then <math>\mathcal{B}</math> is not a bornology. Bounded, discrete and indiscrete coarse structures are coarsely connected.
==See also==
* {{annotated link|Bornology}} * {{annotated link|Quasi-isometry}} * {{annotated link|Uniform space}}
==References==
{{reflist}}
* John Roe, <cite>Lectures in Coarse Geometry</cite>, University Lecture Series Vol. 31, American Mathematical Society: Providence, Rhode Island, 2003. <small>[http://www.personal.psu.edu/jxr57/writings/correction.pdf Corrections to ''Lectures in Coarse Geometry'']</small> * {{ 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 }}
{{Topology|expanded}}
Category:General topology Category:Metric geometry Category:Topology