# first countable

Let $X$ be a topological space^{} and let $x\in X$. $X$ is said to be * at $x$* if there is a sequence ${({B}_{n})}_{n\in \mathbb{N}}$ of open sets such that whenever $U$ is an open set containing $x$, there is $n\in \mathbb{N}$ such that $x\in {B}_{n}\subseteq U$.

The space $X$ is said to be if for every $x\in X$, $X$ is first countable at $x$.

Remark. Equivalently, one can take each ${B}_{n}$ in the sequence to be open neighborhood of $x$.

Title | first countable |
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Canonical name | FirstCountable |

Date of creation | 2013-03-22 12:23:33 |

Last modified on | 2013-03-22 12:23:33 |

Owner | Evandar (27) |

Last modified by | Evandar (27) |

Numerical id | 5 |

Author | Evandar (27) |

Entry type | Definition |

Classification | msc 54D99 |

Synonym | first axiom of countability |

Related topic | SecondCountable |

Related topic | TestingForContinuityViaNets |