\begin{definition}[Definition:Fiber Bundle/System of Local Trivializations]
Let $B = \struct {E, M, \pi, F}$ be a fiber bundle. 
Let $\UU = \set {U_\alpha \subseteq M: \alpha \in I}$ be an open cover of $M$ with index set $I$. 
Let $\struct {U_\alpha, \chi_\alpha}$ be local trivializations for all $\alpha \in I$. 
The set $\set {\struct {U_\alpha, \chi_\alpha}: \alpha \in I}$ is called a '''system of local trivializations''' of $E$ on $M$.
\end{definition}