In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an ultralimit is a geometric construction that assigns to a sequence of
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s ''X
n'' a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces ''X
n'' and uses an
ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...
to avoid the process of repeatedly passing to subsequences to ensure convergence. An ultralimit is a generalization of the notion of
Gromov–Hausdorff convergence
In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.
Gromov–Hausdorff distance
The Gromov–Hausdorff ...
of metric spaces.
Ultrafilters
An
ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...
''ω'' on the set of natural numbers is a set of nonempty subsets of (whose inclusion function can be thought of as a measure) which is closed under finite intersection, upwards-closed, and which, given any subset ''X'' of , contains either ''X'' or . An ultrafilter ''ω'' on is ''non-principal'' if it contains no finite set.
Limit of a sequence of points with respect to an ultrafilter
Let ''ω'' be a non-principal ultrafilter on
.
If
is a sequence of points in a
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
(''X'',''d'') and ''x''∈ ''X'', the point ''x'' is called the ''ω'' -''limit'' of ''x''
''n'', denoted
, if for every
we have:
:
It is not hard to see the following:
* If an ''ω'' -limit of a sequence of points exists, it is unique.
* If
in the standard sense,
. (For this property to hold it is crucial that the ultrafilter be non-principal.)
An important basic fact
states that, if (''X'',''d'') is compact and ''ω'' is a non-principal ultrafilter on
, the ''ω''-limit of any sequence of points in ''X'' exists (and is necessarily unique).
In particular, any bounded sequence of real numbers has a well-defined ''ω''-limit in
(as closed intervals are compact).
Ultralimit of metric spaces with specified base-points
Let ''ω'' be a non-principal ultrafilter on
. Let (''X''
''n'',''d''
''n'') be a sequence of
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s with specified base-points ''p''
''n''∈''X''
''n''.
Let us say that a sequence
, where ''x''
''n''∈''X''
''n'', is ''admissible'', if the sequence of real numbers (''d''
''n''(''x
n'',''p
n''))
''n'' is bounded, that is, if there exists a positive real number ''C'' such that
.
Let us denote the set of all admissible sequences by
.
It is easy to see from the triangle inequality that for any two admissible sequences
and
the sequence (''d''
''n''(''x
n'',''y
n''))
''n'' is bounded and hence there exists an ''ω''-limit
. Let us define a relation
on the set
of all admissible sequences as follows. For
we have
whenever
It is easy to show that
is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
on
The ultralimit with respect to ''ω'' of the sequence (''X''
''n'',''d''
''n'', ''p''
''n'') is a metric space
defined as follows.
As a set, we have
.
For two
-equivalence classes