﻿ Ultrametric space

# Ultrametric space

In the analysis, and topology is referred to as ultra- metric, a metric to an amount other than the metric axioms

For all even the stricter triangle inequality

Met. A metric space with an ultra- metric is referred to as ultra metric space.

## Examples

The discrete metric ( for, otherwise ) on a non-empty set is an ultrametric.

The p- adic metric and that is on the body of the p-adic number ultra metric.

If any non-empty set, then you can make the set of all sequences in a metric space by setting the distance between two different sequences to the value with the smallest index for which is different from, and the distance of a sequence to itself relies on. This metric space is complete and then ultrametrisch. The thus induced topology coincides with the product topology of the discrete countable topology on. Important examples of such spaces are constructed of Baire space ( countably infinite ) and the Cantor space (finally with at least two elements).

## Properties

Each ball is both complete and open ( but not necessarily an open and closed ball ). ( Schikhof, 1984)

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

Two balls are either disjoint ( disjoint ), or is entirely contained in the other.

A sequence in, in the distances converge directly successive links to 0, is a Cauchy sequence, because for each there is then a with for all, and thus applies the tightening of triangle inequality for all.

In an abelian topological group whose topology is generated by a translation invariant ultrametric (for example, an ultra- metric body like ) is an infinite series is a Cauchy sequence if and only if the summands form a null sequence. The group is complete, then the series converges in this case.

A ultrametrischer space is totally disconnected.

## Application

Applications, there is for example in the theory of so-called spin glasses in physics, in the replica theory of Giorgio Parisi.

• Analysis
• Geometry
• Topology
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