Coarse structure

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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

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A coarse structure on a set   is a collection   of subsets of   (therefore falling under the more general categorization of binary relations on  ) called controlled sets, and so that   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   is a member of  —the identity relation.
  2. Closed under taking subsets:
    If   and   then  
  3. Closed under taking inverses:
    If   then the inverse (or transpose)   is a member of  —the inverse relation.
  4. Closed under taking unions:
    If   then their union   is a member of 
  5. Closed under composition:
    If   then their product   is a member of  —the composition of relations.

A set   endowed with a coarse structure   is a coarse space.

For a subset   of   the set   is defined as   We define the section of   by   to be the set   also denoted   The symbol   denotes the set   These are forms of projections.

A subset   of   is said to be a bounded set if   is a controlled set.

Intuition

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The controlled sets are "small" sets, or "negligible sets": a set   such that   is controlled is negligible, while a function   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

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Given a set   and a coarse structure   we say that the maps   and   are close if   is a controlled set.

For coarse structures   and   we say that   is a coarse map if for each bounded set   of   the set   is bounded in   and for each controlled set   of   the set   is controlled in  [1]   and   are said to be coarsely equivalent if there exists coarse maps   and   such that   is close to   and   is close to  

Examples

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  • The bounded coarse structure on a metric space   is the collection   of all subsets   of   such that   is finite. With this structure, the integer lattice   is coarsely equivalent to  -dimensional Euclidean space.
  • A space   where   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   coarse structure on a metric space   is the collection of all subsets   of   such that for all   there is a compact set   of   such that   for all   Alternatively, the collection of all subsets   of   such that   is compact.
  • The discrete coarse structure on a set   consists of the diagonal   together with subsets   of   which contain only a finite number of points   off the diagonal.
  • If   is a topological space then the indiscrete coarse structure on   consists of all proper subsets of   meaning all subsets   such that   and   are relatively compact whenever   is relatively compact.

See also

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  • 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

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  1. ^ Hoffland, Christian Stuart. Course structures and Higson compactification. OCLC 76953246.