# subbasis

Let $(X,\mathcal{T})$ be a topological space^{}. A subset $\mathcal{A}\subseteq \mathcal{T}$ is said to be a *subbasis* if the collection^{} $\mathcal{B}$ of intersections^{} of finitely many elements of $\mathcal{A}$ is a basis (http://planetmath.org/BasisTopologicalSpace) for $\mathcal{T}$.

Conversely, given an arbitrary collection $\mathcal{A}$ of subsets of $X$, a topology can be formed by first taking the collection $\mathcal{B}$ of finite intersections of members of $\mathcal{A}$ and then taking the topology $\mathcal{T}$ generated by $\mathcal{B}$ as basis. $\mathcal{T}$ will then be the smallest topology such that $\mathcal{A}\subseteq \mathcal{T}$.

Title | subbasis |
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Canonical name | Subbasis |

Date of creation | 2013-03-22 12:10:32 |

Last modified on | 2013-03-22 12:10:32 |

Owner | evin290 (5830) |

Last modified by | evin290 (5830) |

Numerical id | 10 |

Author | evin290 (5830) |

Entry type | Definition |

Classification | msc 54A99 |

Synonym | subbasic |

Synonym | subbasic |

Synonym | subbase |

Related topic | Basis |

Related topic | BasisTopologicalSpace |