\begin{definition}[Definition:Local Trivialization]
Let $\struct {E, M, \pi, F}$ be a fiber bundle. 
Let $\pr_1: M \times F \to M$ be the first projection on $M \times F$.
By definition of fiber bundle, for every point $m \in M$ there exists an open neighborhood $U$ of $m$ and a homeomorphism: 
:$\chi: \pi^{-1} \sqbrk U \to U \times F$ 
such that: 
:$\pi {\restriction}_U = \pr_1 \mathop \circ \chi$
where $\pi {\restriction}_U$ is the restriction of $\pi$ to $U$.
Then the ordered pair $\struct {U, \chi}$ is called a '''local trivialization''' of $E$ over $U$.
\end{definition}