# ultrametric space

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

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

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 |