# local compactness is hereditary for locally closed subspaces

Theorem - Let $X$ be a locally compact space and $Y\subseteq X$ a subspace^{}. If $Y$ is locally closed in $X$ then $Y$ is also locally compact.

The converse^{} of this theorem is also true with the additional assumption^{} that $X$ is Hausdorff^{}.

Theorem 2 - Let $X$ be a locally compact Hausdorff space^{} (http://planetmath.org/LocallyCompactHausdorffSpace) and $Y\subseteq X$ a subspace. If $Y$ is locally compact then $Y$ is locally closed in $X$.

Title | local compactness is hereditary for locally closed subspaces |
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Canonical name | LocalCompactnessIsHereditaryForLocallyClosedSubspaces |

Date of creation | 2013-03-22 17:36:33 |

Last modified on | 2013-03-22 17:36:33 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 5 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 54D45 |

Related topic | LocallyCompactHausdorffSpace |