Definition:Fiber Bundle

Definition
Let $M, E, F$ be topological spaces.

Let $\pi: E \to M$ be a continuous surjection.

Let $\mathcal U := \left\{ {U_\alpha \subseteq M: \alpha \in I} \right\}$ be an open cover of $M$ with index set $I$.

Let $\operatorname{pr}_{1, \alpha} : U_\alpha \times F \to U_\alpha$ be the first projection on $U_\alpha \times F$.

Let there exist homeomorphisms:


 * $\chi_\alpha : \pi^{-1} \left({U_\alpha}\right) \to U_\alpha \times F$

such that for all $\alpha \in I$:


 * $\pi {\restriction}_{U_\alpha} = \operatorname{pr}_{1, \alpha} \mathop \circ \chi_\alpha$

where $\pi {\restriction}_{U_\alpha}$ is the restriction of $\pi$ to $U_\alpha \in \mathcal U$.

Then the ordered tuple $\left({E, M, \pi, F}\right)$ is called a fiber bundle over $M$.

Remarks
By an abuse of language, it is common to say that $E$ is a fiber bundle over $M$. One also finds the formulation Let $E \overset{\pi}{\to} M$ be a fiber bundle in the literature.

Linguistic Note
In British English, the word fibre is used instead of fiber.

Also see

 * Definition:Local Trivialization
 * Definition:Transition Mapping
 * Definition:Fiber (Relation)
 * Definition:Section (Topology)
 * Definition:Smooth Fiber Bundle
 * Definition:Vector Bundle