# weakly countably compact

A topological space^{} $X$ is said to be *weakly countably compact*
(or *limit point compact*)
if every infinite subset of $X$ has a limit point^{}.

Every countably compact space is weakly countably compact.
The converse^{} is true in ${\mathrm{T}}_{1}$ spaces (http://planetmath.org/T1Space).

A metric space is weakly countably compact if and only if it is compact^{}.

An easy example of a space $X$ that is not weakly countably compact is any infinite set with the discrete topology. A more interesting example is the countable complement topology on an uncountable set.

Title | weakly countably compact |

Canonical name | WeaklyCountablyCompact |

Date of creation | 2013-03-22 12:06:46 |

Last modified on | 2013-03-22 12:06:46 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 9 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54D30 |

Synonym | limit point compact |

Synonym | limit-point compact |

Related topic | Compact |

Related topic | CountablyCompact |

Related topic | SequentiallyCompact |

Related topic | PseudocompactSpace |

Defines | limit point compactness |

Defines | weak countable compactness |