# cofinite and cocountable topologies

The *cofinite topology ^{}* on a set $X$
is defined to be the topology

^{}$\mathcal{T}$ where

$$\mathcal{T}=\{A\subseteq X\mid X\setminus A\text{is finite, or}A=\mathrm{\varnothing}\}.$$ |

In other words, the closed sets^{} in the cofinite topology are $X$ and the finite subsets of $X$.

Analogously, the *cocountable topology* on $X$
is defined to be the topology
in which the closed sets are $X$ and the countable^{} subsets of $X$.

The cofinite topology on $X$ is the coarsest ${T}_{1}$ topology (http://planetmath.org/T1Space) on $X$.

The cofinite topology on a finite set^{} $X$ is the discrete topology.
Similarly, the cocountable topology on a countable set $X$ is the discrete topology.

A set $X$ together with the cofinite topology forms a compact^{} topological space.

Title | cofinite and cocountable topologies |
---|---|

Canonical name | CofiniteAndCocountableTopologies |

Date of creation | 2013-03-22 13:03:30 |

Last modified on | 2013-03-22 13:03:30 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 21 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54B99 |

Related topic | FiniteComplementTopology |

Defines | cofinite topology |

Defines | cocountable topology |

Defines | cofinite |

Defines | cocountable |