ultrametric space

The metric space  $(X,d)$  is called an ultrametric space, if its metric $d$ is an ultrametric, i.e. if

$$d(x,z)\leqq \max \{d(x,y),d(y,z)\}\mathit{\hspace{1em}}\forall x,y,z\in X.$$

Example.  The field $\mathbb{Q}$ together with any of its $p$-adic metrics

$$d_p(x,y)={|x-y|}_p,$$

where  $|\cdot {|}_p$  is the $p$-adic valuation (http://planetmath.org/PAdicValuation) of $\mathbb{Q}$,  forms an ultrametric space.

Title	ultrametric space
Canonical name	UltrametricSpace
Date of creation	2013-03-22 14:55:28
Last modified on	2013-03-22 14:55:28
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	6
Author	pahio (2872)
Entry type	Definition
Classification	msc 54E35
Related topic	UltrametricTriangleInequality
Related topic	Ultrametric