# presheaf of a topological basis

Let $X$ be a topological space^{} and let $\mathcal{B}$ be a basis of its topology^{}. We can regard $\mathcal{B}$ as a category^{}
with objects being the open sets in $\mathcal{B}$ and arrows/morphisms between $U,V\in \mathcal{B}$ to exists only if $U\subset V$, and
where the *only* element of $\mathcal{B}(U,V)$ is the injection map $U\hookrightarrow V$. Let now $\mathcal{C}$ be a complete
category, we now define the
*presheaf ^{} of $\mathrm{C}$-objects over the basis $\mathrm{B}$ of the topology of $X$* to be a contravariant functor

^{}

$$\mathcal{P}:\mathcal{B}\to \mathcal{C}$$ |

Title | presheaf of a topological basis |
---|---|

Canonical name | PresheafOfATopologicalBasis |

Date of creation | 2013-03-22 16:22:36 |

Last modified on | 2013-03-22 16:22:36 |

Owner | jocaps (12118) |

Last modified by | jocaps (12118) |

Numerical id | 14 |

Author | jocaps (12118) |

Entry type | Definition |

Classification | msc 14F05 |

Classification | msc 54B40 |

Classification | msc 18F20 |

Related topic | site |