fibre

Given a function $f\colon X\longrightarrow Y$ , 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 $f\colon\mathbb{R}^{2}\longrightarrow\mathbb{R}$ 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 $\pi\colon TM\to M$ is the canonical projection from the tangent bundle $T M$ 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