In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an ultralimit is a geometric construction that assigns a limit
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
to a sequence of metric spaces
. The concept captures the limiting behavior of finite configurations in the
spaces employing an
ultrafilter
In the Mathematics, 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 element, maximal Filter (mathematics), filter on P; that is, a proper filter on P th ...
to bypass the need for repeated consideration of subsequences to ensure convergence. Ultralimits generalize
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 distance.
Gromov–Hausdorff distance
The Gromov–Hausdorff dist ...
in metric spaces.
Ultrafilters
An
ultrafilter
In the Mathematics, 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 element, maximal Filter (mathematics), filter on P; that is, a proper filter on P th ...
, denoted as ω, on the set of
natural numbers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
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 also which, given any subset ''X'' of , contains either ''X'' or . An ultrafilter on is ''non-principal'' if it contains no finite set.
Limit of a sequence of points with respect to an ultrafilter
In the following, ''ω'' is a non-principal ultrafilter on
.
If
is a sequence of points in a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
(''X'',''d'') and ''x''∈ ''X'', then the point ''x'' is called ''ω''-''limit'' of ''x''
''n'', denoted as
, if for every
it holds that
:
It is observed that,
* 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 should be non-principal.)
A fundamental 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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
.
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 (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s with specified base-points ''p''
''n'' ∈ ''X''
''n''.
Suppose 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
, then denote the set of all admissible sequences by
.
It follows 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
. One can define a relation
on the set
of all admissible sequences as follows. For
, there is
whenever
This helps 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. A simpler example is equ ...
on
The ultralimit with respect to ''ω'' of the sequence (''X''
''n'',''d''
''n'', ''p''
''n'') is a metric space
defined as follows.
Written as a set,
.
For two
-equivalence classes