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[edit]

A coarse structure on a set X {\displaystyle X} is a collection  E {\displaystyle \mathbf {E} } of subsets of  X × X {\displaystyle X\times X} (therefore falling under the more general categorization of binary relations on  X {\displaystyle X} ) called controlled sets, and so that  E {\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 Δ = { ( x , x ) : x ∈ X } {\displaystyle \Delta =\\{(x,x):x\in X\\}} is a member of  E {\displaystyle \mathbf {E} } --the identity relation.
2. Closed under taking subsets:

If  E ∈ E {\displaystyle E\in \mathbf {E} } and  F ⊆ E , {\displaystyle F\subseteq E,} then  F ∈ E . {\displaystyle F\in \mathbf {E} .}
3. Closed under taking inverses:

If  E ∈ E {\displaystyle E\in \mathbf {E} } then the inverse (or transpose)  E − 1 = { ( y , x ) : ( x , y ) ∈ E } {\displaystyle E^{-1}=\\{(y,x):(x,y)\in E\\}} is a member of  E {\displaystyle \mathbf {E} } --the inverse relation.
4. Closed under taking unions:

If  E , F ∈ E {\displaystyle E,F\in \mathbf {E} } then their union E ∪ F {\displaystyle E\cup F} is a member of E . {\displaystyle \mathbf {E} .}
5. Closed under composition:

If  E , F ∈ E {\displaystyle E,F\in \mathbf {E} } then their product E ∘ F = { ( x , y ) : there exists z ∈ X such that ( x , z ) ∈ E and ( z , y ) ∈ F } {\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  E {\displaystyle \mathbf {E} } --the composition of relations.

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

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

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

### Intuition[edit]

The controlled sets are "small" sets, or "negligible sets": a set  A {\displaystyle A} such that  A × A {\displaystyle A\times A} is controlled is negligible, while a function  f : X -> X {\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[edit]

Given a set  S {\displaystyle S} and a coarse structure  X , {\displaystyle X,} we say that the maps  f : S -> X {\displaystyle f:S\to X} and  g : S -> X {\displaystyle g:S\to X} are close if  { ( f ( s ) , g ( s ) ) : s ∈ S } {\displaystyle \\{(f(s),g(s)):s\in S\\}} is a controlled set.

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

## Examples[edit]

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

## See also[edit]

* 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[edit]

1. ^ 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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