\begin{definition}[Definition:Fiber Bundle]
Let $M, E, F$ be topological spaces. 
Let $\pi: E \to M$ be a continuous surjection. 
Let $\UU := \set {U_\alpha \subseteq M: \alpha \in I}$ be an open cover of $M$ with index set $I$. 
Let $\pr_{1, \alpha}: U_\alpha \times F \to U_\alpha$ be the first projection on $U_\alpha \times F$.
Let there exist homeomorphisms:
:$\chi_\alpha: \map {\pi^{-1} } {U_\alpha} \to U_\alpha \times F$ 
such that for all $\alpha \in I$:
:$\pi {\restriction}_{U_\alpha} = \pr_{1, \alpha} \circ \chi_\alpha$
where $\pi {\restriction}_{U_\alpha}$ is the restriction of $\pi$ to $U_\alpha \in \UU$.
Then the ordered tuple $\struct {E, M, \pi, F}$ is called a '''fiber bundle over $M$'''.
\end{definition}