# compact

A topological space^{} $X$ is compact^{} if, for every collection^{} ${\{{U}_{i}\}}_{i\in I}$ of open sets in $X$ whose union is $X$, there exists a finite subcollection ${\{{U}_{{i}_{j}}\}}_{j=1}^{n}$ whose union is also $X$.

A subset $Y$ of a topological space $X$ is said to be compact if $Y$ with its subspace topology is a compact topological space.

Note: Some authors require that a compact topological space be Hausdorff^{} as well, and use the term quasi-compact to refer to a non-Hausdorff compact space. The modern convention seems to be to use compact in the sense given here, but the old definition is still occasionally encountered (particularly in the French school).

Title | compact |

Canonical name | Compact |

Date of creation | 2013-03-22 11:53:35 |

Last modified on | 2013-03-22 11:53:35 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 11 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 54D30 |

Classification | msc 81-00 |

Classification | msc 83-00 |

Classification | msc 82-00 |

Classification | msc 46L05 |

Classification | msc 22A22 |

Related topic | QuasiCompact |

Related topic | LocallyCompact |

Related topic | HeineBorelTheorem |

Related topic | TychonoffsTheorem |

Related topic | Compactification |

Related topic | SequentiallyCompact |

Related topic | Lindelof |

Related topic | NoetherianTopologicalSpace |

Defines | compact set |

Defines | compact subset |