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 finite 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[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 the set 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 : XX 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

Given a set S and a coarse structure X, we say that the maps and are close if is a controlled set. A subset B of X is said to be bounded if is a controlled set.

For coarse structures X and Y, we say that is coarse if for each bounded set B of Y the set is bounded in X and for each controlled set E of X the set is controlled in Y.[1] X and Y are said to be coarsely equivalent if there exists coarse maps and such that is close to and is close to .

Examples

See also

References

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