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Coarse structure - Wikipedia

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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.

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.

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.

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}.}

• 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.

• 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

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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