# second countable

A topological space^{} is said to be *second * if it has a countable basis (http://planetmath.org/BasisTopologicalSpace).
It can be shown that a space is both Lindelöf and separable, although the converses fail. For instance, the lower limit topology on the real line is both Lindelöf and separable, but not second countable.

Title | second countable |

Canonical name | SecondCountable |

Date of creation | 2013-03-22 12:05:06 |

Last modified on | 2013-03-22 12:05:06 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 17 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 54D70 |

Synonym | second axiom of countability |

Synonym | completely separable |

Synonym | perfectly separable |

Synonym | second-countable |

Related topic | Separable |

Related topic | Lindelof |

Related topic | EverySecondCountableSpaceIsSeparable |

Related topic | LindelofTheorem |

Related topic | UrysohnMetrizationTheorem |

Related topic | FirstAxiomOfCountability |

Related topic | LocallyCompactGroupoids |

Related topic | FirstCountable |