# countable complement topology

Let $X$ be an infinite set^{}. We define the *countable complement topology* on $X$ by declaring the empty set^{} to be open, and a non-empty subset $U\subset X$ to be open if $X\backslash U$ is countable^{}.

If $X$ is countable, then the countable complement topology is just the discrete topology, as the complement of *any* set is countable and thus open.

Though defined similarly to the finite complement topology^{}, the countable complement topology lacks many of the strong compactness properties of the finite complement topology. For example, the countable complement topology on an uncountable set gives an example of a topological space that is not weakly countably compact (but *is* pseudocompact).

Title | countable complement topology |
---|---|

Canonical name | CountableComplementTopology |

Date of creation | 2013-03-22 14:37:56 |

Last modified on | 2013-03-22 14:37:56 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 54A05 |

Synonym | cocountable topology |