In
mathematics, an ultrametric space is 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 sett ...
in which the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, bu ...
is strengthened to
. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications.
Formal definition
An ultrametric on a
set is a
real-valued function
:
(where denote the
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s), such that for all :
# ;
# (''symmetry'');
# ;
# if then ;
# (strong triangle inequality or ultrametric inequality).
An ultrametric space is a pair consisting of a set together with an ultrametric on , which is called the space's associated distance function (also called a
metric).
If satisfies all of the conditions except possibly condition 4 then is called an ultrapseudometric on . An ultrapseudometric space is a pair consisting of a set and an ultrapseudometric on .
In the case when is an Abelian group (written additively) and is generated by a
length function (so that
), the last property can be made stronger using the
Krull sharpening to:
:
with equality if
.
We want to prove that if
, then the equality occurs if
.
Without loss of generality, let us assume that
. This implies that
. But we can also compute
. Now, the value of
cannot be
, for if that is the case, we have
contrary to the initial assumption. Thus,
, and
. Using the initial inequality, we have
and therefore
.
Properties

From the above definition, one can conclude several typical properties of ultrametrics. For example, for all
, at least one of the three equalities
or
or
holds. That is, every triple of points in the space forms an
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
, so the whole space is an
isosceles set.
Defining the
(open) ball of radius
centred at
as
, we have the following properties:
* Every point inside a ball is its center, i.e. if