# Definition:Fiber Bundle/System of Local Trivializations

## Definition

Let $B = \left({E, M, \pi, F}\right)$ be a fiber bundle.

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

Let $\left({U_\alpha, \chi_\alpha}\right)$ be local trivializations for all $\alpha \in I$.

The set $\left\{ {\left({U_\alpha, \chi_\alpha}\right): \alpha \in I}\right\}$ is called a system of local trivializations of $E$ on $M$.