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3. Pullback bundle is a bundle

Pullback bundle is a bundle

Theorem : Given a fiber bundle $(E, \pi, M, F)$ and map $f : N \to M$, the pullback bundle $(f^*E, \pi_1, M, F)$ is a fiber bundle.

Proof : By definition, the pullback of the total space is

\begin{eqnarray} f^*E = \left\{ (q, e) \in N \times E | f(q) = \pi(e) \right\} \end{eqnarray}