# Smirnov metrization theorem

The Smirnov metrization theorem establishes necessary and sufficient conditions for a topological space^{} to be metrizable. The theorem reduces questions of metrizability to paracompactness and a local metrizability condition.

Definition: A space $X$ is *locally metrizable* if every point $x\in X$ has a neighborhood^{} that is metrizable in the subspace topology.

Theorem (Smirnov metrization theorem): A space is metrizable if and only if it is paracompact and locally metrizable.

Title | Smirnov metrization theorem |
---|---|

Canonical name | SmirnovMetrizationTheorem |

Date of creation | 2013-03-22 18:01:00 |

Last modified on | 2013-03-22 18:01:00 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 7 |

Author | rm50 (10146) |

Entry type | Theorem |

Classification | msc 54E35 |

Defines | locally metrizable |