# 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 $TM$ to $M$. Then fibres of $\pi$ are the tangent spaces $T_{x}(M)$ for $x\in M$.

Title fibre Fibre 2013-03-22 12:55:23 2013-03-22 12:55:23 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 03E20 fiber LevelSet