# locally compact

A topological space^{} $X$ is locally compact at a point $x\in X$ if there exists a compact set $K$ which contains a nonempty neighborhood^{} $U$ of $x$. The space $X$ is locally compact if it is locally compact at every point $x\in X$.

Note that local compactness at $x$ does not require that $x$ have a neighborhood which is actually compact, since compact open sets are fairly rare and the more relaxed condition turns out to be more useful in practice. However, it is true that a space is locally compact at $x$ if and only if $x$ has a precompact neighborhood.

Title | locally compact |
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Canonical name | LocallyCompact |

Date of creation | 2013-03-22 12:38:24 |

Last modified on | 2013-03-22 12:38:24 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 6 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 54D45 |

Related topic | Compact |

Defines | local compactness |