Coarse structure
Encyclopedia
In the mathematical fields of geometry
and topology
, a coarse structure on a set X is a collection of subset
s of the cartesian product
X × X with certain properties which allow the large-scale structure of metric space
s and topological space
s 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 image
s of small open set
s, 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
.
s of X × X (therefore falling under the more general categorization of binary relation
s 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.
s": 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.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a coarse structure on a set X is a collection of subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of the cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
X × X with certain properties which allow the large-scale structure of metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s and topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s to be defined.
The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
depend on whether the inverse image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
s of small open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s, or neighborhoods
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
, are themselves open. Large-scale properties of a space—such as boundedness
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
, or the degrees of freedom
Degrees of freedom (statistics)
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...
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
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
or a topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
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
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
.
Definition
A coarse structure on a set X is a collection E of subsetSubset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of X × X (therefore falling under the more general categorization of binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
s 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
Composition of relations
In mathematics, the composition of binary relations is a concept of forming a new relation from two given relations R and S, having as its most well-known special case the composition of functions.- Definition :...
. Explicitly:
1. Identity/diagonal: The diagonal
Diagonal
A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...
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
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
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 setNegligible set
In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose.As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function.Negligible sets define...
s": 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 spaceMetric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean 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 latticeInteger latticeIn mathematics, the n-dimensional integer lattice , denoted Zn, is the lattice in the Euclidean space Rn whose lattice points are n-tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. Zn is the simplest example of a root lattice...
Zn is coarsely equivalent to n-dimensional Euclidean spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
.
- With this structure, the integer lattice
- 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 compactCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
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 diagonalDiagonalA diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...
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 spaceTopological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
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 compactRelatively compact subspaceIn mathematics, a relatively compact subspace Y of a topological space X is a subset whose closure is compact....
whenever K is relatively compact.