# pseudocompact space

A topological space^{} $X$ is said to be *pseudocompact* if every continuous function^{} $f:X\to \mathbb{R}$ has bounded^{} image.

All countably compact spaces (which includes all compact spaces and all sequentially compact spaces) are pseudocompact.
A metric space is pseudocompact if and only if it is compact^{}.
A Hausdorff^{} normal space^{} is pseudocompact if and only if it is countably compact.

Title | pseudocompact space |

Canonical name | PseudocompactSpace |

Date of creation | 2013-03-22 14:20:36 |

Last modified on | 2013-03-22 14:20:36 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 7 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54D30 |

Synonym | pseudo compact space |

Synonym | pseudo-compact space |

Related topic | LimitPointCompact |

Defines | pseudocompact |

Defines | pseudocompactness |

Defines | pseudo-compact |

Defines | pseudo-compactness |

Defines | pseudo compact |

Defines | pseudo compactness |