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.