View source for Ultrametric

{{TEX|done}}{{MSC|54E35,16W60}}


An ultrametric is a [[metric]] $d$ on a set $X$ satisfying the ''strong triangle inequality'': $d(x,y) \le \max\{d(x,z), d(z,y)\}$.  An ultrametric topology is one induced by an ultrametric.

An ultrametric [[valuation]]  $\Vert{\cdot}\Vert$ on a ring $R$ similarly satisfies the condition: $\Vert x+y \Vert \le \max\{\Vert x \Vert, \Vert y \Vert\}$.  An example is the [[p-adic valuation|$p$-adic valuation]].

The term '''non-Archimedean''' is also used.

====References====
* Natarajan, P. N. "An introduction to ultrametric summability theory". SpringerBriefs in Mathematics. Springer (2014) ISBN 978-81-322-1646-9  {{ZBL|1284.40001}}
* Schikhof, W.H. "Ultrametric calculus. An introduction to $p$-adic analysis". Cambridge Studies in Advanced Mathematics '''4''' Cambridge University Press (1984) ISBN 0-521-24234-7 {{ZBL|0553.26006}}