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

2. Closed under taking subsets:

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

3. Closed under taking inverses:

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

4. Closed under taking unions:

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

5. Closed under composition:

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

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

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

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

The controlled sets are "small" sets, or "negligible sets": a set ${\displaystyle A}${\displaystyle A} such that ${\displaystyle A\times A}${\displaystyle A\times A} is controlled is negligible, while a function ${\displaystyle f:X\to X}${\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 ${\displaystyle S}${\displaystyle S} and a coarse structure ${\displaystyle X,}${\displaystyle X,} we say that the maps ${\displaystyle f:S\to X}${\displaystyle f:S\to X} and ${\displaystyle g:S\to X}${\displaystyle g:S\to X} are close if ${\displaystyle \{(f(s),g(s)):s\in S\}}${\displaystyle \{(f(s),g(s)):s\in S\}} is a controlled set.

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

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

• 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年01月16日.

• General topology
• Metric geometry
• Topology

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