Vector bundle

Definition

A vector bundle over a topological space $B$ is a topological space $E$ with a bundle map $p:E\to B$, along with an open cover $U_i$ of $B$ such that:

1. For every $i$, the projection map from $p^{-1}(U_i)$ to $U_i$ looks like a coordinate projection from $U_i\times {\mathbb{R}}^n$ to $U_i$
2. The transition maps between the different coordinatizations are linear in the fiber over every point.