# sphere (metric space)

The set $\{x\mid d(x,c)=r\}$ is called the *sphere* of radius $r$ with centre $c$. This generalizes the notion of spheres to metric spaces.

Note that the sphere in a metric space need not look like a sphere in Euclidean space. For instance, if we impose the metric $d(x,y)=max\{|{x}_{1}-{y}_{1}|,|{x}_{2}-{y}_{2}|,|{x}_{3}-{y}_{3}|\}$ on ${\mathbb{R}}^{3}$ instead of the Euclidean metric^{}, spheres according to this metric are actually cubes! Even more bizarre situations can occur in general — a sphere might be disconnected, or it may be discrete, or it may even be an empty set^{}.

Title | sphere (metric space) |
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Canonical name | SpheremetricSpace |

Date of creation | 2013-03-22 14:47:38 |

Last modified on | 2013-03-22 14:47:38 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 54E35 |

Synonym | sphere |