# Alexandrov one-point compactification

The *Alexandrov one-point compactification* of a non-compact topological space^{} $X$ is obtained by adjoining a new point $\mathrm{\infty}$ and defining the topology on $X\cup \{\mathrm{\infty}\}$ to consist of the open sets of $X$ together with the sets of the form $U\cup \{\mathrm{\infty}\}$, where $U$ is an open subset of $X$ with compact^{} complement.

With this topology, $X\cup \{\mathrm{\infty}\}$ is always compact.
Furthermore, it is Hausdorff^{} if and only if $X$ is Hausdorff and locally compact.

Title | Alexandrov one-point compactification |

Canonical name | AlexandrovOnepointCompactification |

Date of creation | 2013-03-22 13:47:54 |

Last modified on | 2013-03-22 13:47:54 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 9 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54D35 |

Synonym | one-point compactification |

Synonym | Alexandroff one-point compactification |

Synonym | Aleksandrov one-point compactification |

Synonym | Alexandrov compactification |

Synonym | Aleksandrov compactification |

Synonym | Alexandroff compactification |

Related topic | Compactification |