Ultrametric

In analysis and topology , an ultrametric is a metric on a set that is derived from the metric axioms ${\displaystyle d\colon S\times S\to \mathbb {R} }$ ${\displaystyle S}$

1. ${\displaystyle d\left(a,b\right)\geq 0}$
2. ${\displaystyle d\left(a,b\right)=0\Leftrightarrow a=b}$
3. ${\displaystyle d\left(a,b\right)=d(b,a)}$ (Symmetry)
4. ${\displaystyle d\left(a,c\right)\leq d(a,b)+d(b,c)}$( Triangle inequality )

for all the last, the triangular inequality, in its tightened form ${\displaystyle a,b,c\in S,}$

${\displaystyle d(a,c)\leq \max\{d(a,b),d(b,c)\}}$

Fulfills. A room provided with an ultrametric is called an ultrametric room .

Examples

The discrete metric ( for , otherwise ) on a non-empty set is an ultrametric . ${\displaystyle d(a,b)=1}$ ${\displaystyle a\neq b}$ ${\displaystyle 0}$

The p-adic metric on and on the field of the p-adic numbers is an ultrametric. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} _{p}}$

If there is an arbitrary non-empty set, then the set of all sequences in can be made into a metric space by setting the distance between two different sequences to the value , where is the smallest index for which is different from , and the distance of a sequence to puts himself on . This metric space is then complete and ultrametric. The resulting induced topology consistent with the countable product topology of the discrete topology on the same. Important examples for spaces constructed in this way are the Baire space ( countably infinite) and the Cantor space ( finite with at least two elements). ${\displaystyle S}$ ${\displaystyle S^{\mathbb {N} }}$ ${\displaystyle S}$ ${\displaystyle (x_{n}),(y_{n})}$ ${\displaystyle 1/N}$ ${\displaystyle N}$ ${\displaystyle x_{N}}$ ${\displaystyle y_{N}}$ ${\displaystyle 0}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle S}$

properties

Every triangle of points in an ultrametric space is equilateral or isosceles with a shorter base. As proof: are , , the distances of the three vertices ( and so on), then either ( same side) or one side is shorter than another, without limitation, we assume that . Then one can deduce from the tightened triangle inequality that must be (it is , so , and ), so then is isosceles with a shorter base . ${\displaystyle ABC}$ ${\displaystyle S}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle a=d(B,C)}$ ${\displaystyle a=b=c}$ ${\displaystyle ABC}$ ${\displaystyle a<b}$ ${\displaystyle c=b}$ ${\displaystyle a<b\leq \max\{a,c\}}$ ${\displaystyle b\leq c}$ ${\displaystyle c\leq \max\{a,b\}=b}$ ${\displaystyle ABC}$ ${\displaystyle BC}$

Every sphere with a strictly positive radius is both closed and open (but not necessarily an open and closed sphere). (Schikhof, 1984)

Every point in a sphere (open or closed) is the center of this sphere, and the diameter is less than or equal to its radius. ( Marc Krasner , 1944)

Two spheres are either foreign to the element ( disjoint ) or one is completely contained in the other.

A sequence in , in the intervals of contiguous elements from 0 converge, is a Cauchy sequence, because for each there is then a with for all , and thus is considered due to the intensified triangular inequality for all : . ${\displaystyle (a_{n})}$ ${\displaystyle S}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle N}$ ${\displaystyle d(a_{n},a_{n+1})<\varepsilon }$ ${\displaystyle n\geq N}$ ${\displaystyle m>n\geq N}$ ${\displaystyle d(a_{n},a_{m})\leq \max\{d(a_{n},a_{n+1}),\ldots ,d(a_{m-1},a_{m})\}<\varepsilon }$

In an Abelian topological group , the topology of which is generated by a translation-invariant ultrametric (e.g. an ultrametric body like ), an infinite series is a Cauchy sequence if and only if the summands form a zero sequence . If the group is complete , then the series converges in this case. ${\displaystyle \mathbb {Q} _{p}}$

An ultrametric room is totally incoherent .

application

There are applications, for example, in the theory of so-called spin glasses in physics, namely in the replica theory of Giorgio Parisi .