Vector bundle

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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 \R^n to U_i
  2. The transition maps between the different coordinatizations are linear in the fiber over every point.