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 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:
- Identity/diagonal:
- The diagonal
is a member of
—the identity relation.
- The diagonal
- Closed under taking subsets:
- If
and
then
- If
- Closed under taking inverses:
- If
then the inverse (or transpose)
is a member of
—the inverse relation.
- If
- Closed under taking unions:
- If
then their union
is a member of
- If
- Closed under composition:
- If
then their product
is a member of
—the composition of relations.
- If
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
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
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
- 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
- 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
- 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 2008-01-16.