# when are balls separated

Let $(X,d)$ be a metric space, and let ${B}_{r}(x)$ be the $x$-centered open ball of radius $r$. If $d(x,y)\ge r+s$, then the balls ${B}_{r}(x)$ and ${B}_{s}(y)$ are separated.

To prove this, suppose that ${B}_{r}(x)$ and ${B}_{s}(y)$ are not separated. Then there exists a $z\in X$ such that either

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or

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In either case,

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Title | when are balls separated |
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Canonical name | WhenAreBallsSeparated |

Date of creation | 2013-03-22 15:16:40 |

Last modified on | 2013-03-22 15:16:40 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 6 |

Author | matte (1858) |

Entry type | Example |

Classification | msc 54-00 |

Classification | msc 54D05 |