Coarse structure explained
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
is a collection
of
subsets of
(therefore falling under the more general categorization of
binary relations on
) called, 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:
is a member of
- the identity relation.
- Closed under taking subsets:
If
and
then
- Closed under taking inverses:
If
then the
inverse (or
transpose)
is a member of
- the inverse relation.
- Closed under taking unions:
If
then their
union
is a member of
- Closed under composition:
If
then their
product E\circF=\{(x,y):thereexistsz\inXsuchthat(x,z)\inEand(z,y)\inF\}
is a member of
- the
composition of relations.
A set
endowed with a coarse structure
is a .
For a subset
of
the set
is defined as
\{x\inX:(x,k)\inEforsomek\inK\}.
We define the 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 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 if
is a controlled set.
For coarse structures
and
we say that
is a 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 if there exists coarse maps
and
such that
is close to
and
is close to
Examples
is the collection
of all
subsets
of
such that
is
finite. With this structure, the
integer lattice
is coarsely equivalent to
-dimensional
Euclidean space.
where
is controlled is called a . 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 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
\{(x,y)\inE:d(x,y)\geq\varepsilon\}
is compact.
consists of the
diagonal
together with subsets
of
which contain only a finite number of points
off the diagonal.
is a
topological space then the on
consists of all subsets of
meaning all subsets
such that
and
are
relatively compact whenever
is relatively compact.
References
Notes and References
- Book: Course structures and Higson compactification. Hoffland, Christian Stuart. 76953246.
