# fibre

Given a function $f:X\u27f6Y$, a *fibre* is an inverse image of an element of $Y$. That is given $y\in Y$, ${f}^{-1}(\{y\})=\{x\in X\mid f(x)=y\}$ is a fibre.

Example:
Define $f:{\mathbb{R}}^{2}\u27f6\mathbb{R}$ by $f(x,y)={x}^{2}+{y}^{2}$. Then the fibres of $f$ consist of concentric circles about the origin, the origin itself, and empty sets^{} depending on whether we look at the inverse image of a positive number, zero, or a negative number respectively.

Example: Suppose $M$ is a manifold, and $\pi :TM\to M$ is the canonical projection from the tangent bundle $TM$ to $M$. Then fibres of $\pi $ are the tangent spaces ${T}_{x}(M)$ for $x\in M$.

Title | fibre |
---|---|

Canonical name | Fibre |

Date of creation | 2013-03-22 12:55:23 |

Last modified on | 2013-03-22 12:55:23 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 8 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 03E20 |

Synonym | fiber |

Related topic | LevelSet |