Definition:Fiber Bundle

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

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

Let $I$ be some index set and let $\mathfrak U := \left\lbrace U \alpha \subseteq M \middle\vert \alpha \in I \right\rbrace$ be an open cover, such that $\forall \alpha \in I$ there exist homeomorphisms
 * $\displaystyle \chi_\alpha \colon \pi^{-1}\left( U _\alpha \right) \to U_\alpha \times F$

satisfying:


 * 1) $\forall \alpha, \beta \in I$ with $U_\alpha \cap U_\beta \neq \emptyset$ the map $\displaystyle \chi_\beta \circ \chi_\alpha^{-1} \colon U_\alpha \cap U_\beta \times F \to  U_\alpha \cap U_\beta \times F$ is continuous.
 * 2) If $\mathrm{pr}^1_\alpha: U_\alpha  \times  F \to U_\alpha $ is the $1$st projection, then $\pi= \mathrm{pr}^1_\alpha \circ \chi_ \alpha$ on $\pi^{-1} \left(U_\alpha \right)$ for all $\alpha \in I$.

Then the tuple $(E,M,F,\pi)$ is called a fiber bundle.

$E$ is called the total space over the base space $M$.

$F$ is called the fiber.

$\pi$ is called the bundle projection.

$\pi^{-1}\left( m\right)$ is called the fiber over the base point $m \in M$.

$\chi_\alpha$ is called a local trivialization.

$\chi_\beta \circ \chi_\alpha^{-1}$ is called a transition map.