Editing Ultrametric space (section)

== Examples ==

* The [[discrete metric]] is an ultrametric.
* The [[p-adic numbers|''p''-adic numbers]] form a complete ultrametric space.
* Consider the [[Formal language|set of words]] of arbitrary length (finite or infinite), Σ<sup>*</sup>, over some alphabet Σ. Define the distance between two different words to be 2<sup>−''n''</sup>, where ''n'' is the first place at which the words differ. The resulting metric is an ultrametric.
* The [[Formal language|set of words]] with glued ends of the length ''n'' over some alphabet Σ is an ultrametric space with respect to the ''p''-close distance. Two words ''x'' and ''y'' are ''p''-close if any substring of ''p'' consecutive letters (''p'' < ''n'') appears the same number of times (which could also be zero) both in ''x'' and ''y''.<ref>{{citation
 | last = Osipov | first = Gutkin
 | issue = 26
 | journal = Nonlinearity
 | volume = 26
 | pages = 177–200
 | title = Clustering of periodic orbits in chaotic systems
 | doi=10.1088/0951-7715/26/1/177
 | year = 2013| bibcode = 2013Nonli..26..177G
 }}.</ref>
* If ''r'' = (''r<sub>n</sub>'') is a sequence of [[real number]]s decreasing to zero, then |''x''|<sub>''r''</sub> := [[lim sup]]<sub>''n''→∞</sub> |''x<sub>n</sub>''|<sup>''r<sub>n</sub>''</sup> induces an ultrametric on the space of all complex sequences for which it is finite. (Note that this is not a [[seminorm]] since it lacks [[homogeneous function|homogeneity]] &mdash; If the ''r<sub>n</sub>'' are allowed to be zero, one should use here the rather unusual convention that [[Zero to the power of zero|0<sup>0</sup>]]&nbsp;=&nbsp;0.)
* If ''G'' is an edge-weighted [[undirected graph]], all edge weights are positive, and ''d''(''u'',''v'') is the weight of the [[widest path problem|minimax path]] between ''u'' and ''v'' (that is, the largest weight of an edge, on a path chosen to minimize this largest weight), then the vertices of the graph, with distance measured by ''d'', form an ultrametric space, and all finite ultrametric spaces may be represented in this way.<ref>{{citation
 | last = Leclerc | first = Bruno
 | mr = 623034
 | issue = 73
 | journal = Centre de Mathématique Sociale. École Pratique des Hautes Études. Mathématiques et Sciences Humaines
 | language = fr
 | pages = 5–37, 127
 | title = Description combinatoire des ultramétriques
 | year = 1981}}.</ref>