# metric equivalence

Let $X$ be a set equipped with two metrics $\rho $ and $\sigma $. We say that $\rho $ is *equivalent ^{}* to $\sigma $ (on $X$) if the identity map

^{}on $X$, is a homeomorphism

^{}between the metric topology

^{}on $X$ induced by $\rho $ and the metric topology on $X$ induced by $\sigma $.

For example, if $(X,\rho )$ is a metric space, then the function $\sigma :X\to \mathbb{R}$ defined by

$$\sigma (x,y):=\frac{\rho (x,y)}{1+\rho (x,y)}$$ |

is a metric on $X$ that is equivalent to $\rho $. This shows that every metric is equivalent to a bounded^{} metric.

Title | metric equivalence |
---|---|

Canonical name | MetricEquivalence |

Date of creation | 2013-03-22 19:23:11 |

Last modified on | 2013-03-22 19:23:11 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 6 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 54E35 |

Defines | equivalent |