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2010 Mathematics Subject Classification: Primary: 54E35,16W60 [MSN][ZBL]

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.

The term non-Archimedean is also used.


  • 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
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