I try to learn about manifolds and I am working with the two textbooks "An Introduction to Manifolds" by Tu and Lee's book "Introduction to Smooth Manifolds". After comparing the two definitions of Vector Bundles in the books I got a question.
Lee defines on pp.249/250: Let $M$ be a topological space. A real vector bundle of rank $k$ over $M$ is a topological space $E$ with a surjective continuous map $\pi\colon E\to M$ satisfying the conditions:
(i) For each $p∈ M$ the fibre $E_p=\pi^{-1}(p)$ over $P$ is endowed with the structure of a $k$-dimensional real vector space.
(ii) For each $p∈ M$ there exist a neighborhood $U$ of $p$ in $M$ and a homeomorphism $\phi\colon \pi^{-1}(U) \to U\times \mathbb{R}^k$ satisfying the conditions:
- $\pi_U \circ \phi = \pi$ where $\pi_U\colon U\times \mathbb{R}^k\to U$ is the projection)
- for each $q∈U$ the restriction of $\phi$ to $E_q$ is a vectorspace isomorphism from $E_q$ to $\{q\}\times \mathbb{R}^k$.
The definition by Tu looks almost the same (changing "topological spaces" to "manifolds" for $E$ and $M$, "continuous" to "smooth" for $\pi$ and "homeomorphism" to "diffeomorphism" for $\phi$). But Tu omits the condition (ii)(1) -- instead he insists that $\phi$ has to be fibre-preserving. So my question is if these definitions are equivalent. I think it has something to do with the Exercise 12.5 in Tu's book where he asks the reader to prove an equivalent statement for a map to be fibre-preserving. Having this in mind I still don't see why Lee is insisting that $\pi_U$ has to be the projection-map. Do you have an idea? Thank you for your help!