Coarse structure

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"Coarse space" redirects here. For the use of "coarse space" in numerical analysis, see coarse problem.

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 coarse structure on a set X is a collection E of subsets of X × X (therefore falling under the more general categorization of binary relations on X) called controlled sets, and so that E possesses the identity relation, is closed under taking subsets, inverses, and unions, and is closed under composition of relations. Explicitly:

1. Identity/diagonal
The diagonal Δ = {(x, x) : x in X} is a member of E—the identity relation.
2. Closed under taking subsets
If E is a member of E and F is a subset of E, then F is a member of E.
3. Closed under taking inverses
If E is a member of E then the inverse (or transpose) E −1 = {(y, x) : (x, y) in E} is a member of E—the inverse relation.
4. Closed under taking unions
If E and F are members of E then the union of E and F is a member of E.
5. Closed under composition
If E and F are members of E then the product E o F = {(x, y) : there is a z in X such that (x, z) is in E, (z, y) is in F} is a member of E—the composition of relations.

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

The set E −1 [K] is defined as {x in X : there is a y in K such that (x, y) is in E}. We define the section of E by x to be the set E[{x}], also denoted E x. The symbol Ey denotes E −1[{y}]. These are forms of projections.

Intuition

The controlled sets are "small" sets, or "negligible sets": a set A such that A × A is controlled is negligible, while a function f : X → 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.

Examples

  • The bounded coarse structure on a metric space (X, d) is the collection E of all subsets E of X × X such that sup{d(x, y) : (x, y) is in E} is finite.
    With this structure, the integer lattice Zn is coarsely equivalent to n-dimensional Euclidean space.
  • A space X where X × 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 C0 coarse structure on a metric space X is a the collection of all subsets E of X × X such that for all ε > 0 there is a compact set K of X such that d(x, y) < ε for all (x, y) in E − K × K. Alternatively, the collection of all subsets E of X × X such that {(x, y) in E : d(x, y) ≥ ε} is compact.
  • The discrete coarse structure on a set X consists of the diagonal together with subsets E of X × X which contain only a finite number of points (x, y) off the diagonal.
  • If X is a topological space then the indiscrete coarse structure on X consists of all proper subsets of X × X, meaning all subsets E such that E [K] and E −1[K] are relatively compact whenever K is relatively compact.

References

See also

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