# finite complement topology

Let $X$ be a set. We can define the *finite complement ^{} topology^{}* on $X$ by declaring a subset $U\subset X$ to be open if $X\backslash U$ is finite, or if $U$ is all of $X$ or the empty set

^{}. Note that this is equivalent

^{}to defining a topology by defining the closed sets

^{}in $X$ to be all finite sets

^{}(and $X$ itself).

If $X$ is finite, the finite complement topology on $X$ is clearly the discrete topology, as the complement of *any* subset is finite.

If $X$ is countably infinite^{} (or larger), the finite complement topology gives a standard example of a space that is not Hausdorff^{} (each open set must contain all but finitely many points, so any two open sets must intersect).

In general, the finite complement topology on an infinite set^{} satisfies strong compactness conditions (compact^{}, $\sigma $-compact (http://planetmath.org/SigmaCompact), sequentially compact, etc.) since each open set in a cover contains ”almost all” of the points of $X$. On the other hand, the finite complement topology fails all but the simplest of separation axioms^{} since, as above, $X$ is hyperconnected under this topology.

The finite complement topology is the coarsest T1-topology on a given set.

Title | finite complement topology |
---|---|

Canonical name | FiniteComplementTopology |

Date of creation | 2013-03-22 14:37:54 |

Last modified on | 2013-03-22 14:37:54 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 8 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 54A05 |

Synonym | cofinite topology^{} |

Related topic | CofiniteTopology |

Related topic | CofiniteAndCocountableTopology |