# balls in ultrametric spaces are clopen subsets

In an ultrametric space, both open and closed balls^{} are clopen subsets.

It is indeed straightforward (exercise!) to show that the set of all open balls of radius $r$,
centered in any of the points of a closed ball of radius $r$, forms a partition^{} of the latter.

Thus, in particular, any point of a closed ball is an interior point, and the same holds for the complement of an open ball.

Title | balls in ultrametric spaces are clopen subsets |
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Canonical name | BallsInUltrametricSpacesAreClopenSubsets |

Date of creation | 2013-03-22 18:20:15 |

Last modified on | 2013-03-22 18:20:15 |

Owner | MFH (21412) |

Last modified by | MFH (21412) |

Numerical id | 5 |

Author | MFH (21412) |

Entry type | Example |

Classification | msc 54D05 |