# hyperconnected space

A topological space^{} $X$ is said to be *hyperconnected* if no pair of nonempty open sets of $X$ is disjoint (or, equivalently, if $X$ is not the union of two proper closed sets^{}).
Hyperconnected spaces are sometimes known as irreducible sets (http://planetmath.org/IrreducibleClosedSet).

All hyperconnected spaces are connected^{}, locally connected, and pseudocompact.

Any infinite set^{} with the cofinite topology^{} is an example of a hyperconnected space.
Similarly, any uncountable set with the cocountable topology is hyperconnected.
Affine spaces and projectives spaces over an infinite field, when endowed with the Zariski topology^{}, are also hyperconnected.

Title | hyperconnected space |
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Canonical name | HyperconnectedSpace |

Date of creation | 2013-03-22 14:20:30 |

Last modified on | 2013-03-22 14:20:30 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 10 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54D05 |

Synonym | hyper-connected space |

Related topic | UltraconnectedSpace |

Related topic | IrreducibleClosedSet |

Defines | hyperconnected |

Defines | hyper-connected |