Vector bundle and its definition

Question

Take this definition of vector bundle: its a triple $(V,\pi,M)$ where $V$ is a set, $\pi:V \rightarrow M$ a sutjective map such that for every $m \in M$ the set $\pi^{-1}(m)$ has the structure of real vector space o dimension $r$ and $m$ a real differential variety of finite dimension with a structure of maximal atlas on $V$. Pay attention that i don't assume that $V$ has a topological structure.
Using this assumptions, can i make $V$ a topological space? For example saing that a subset of $V$ is open if its image with $\pi$ is open in $M$.

Answer 1

This kind of approach (have the topologisation of manifold descend from a differential atlas) is followed in quite a rich-in-detail way by the Italian book Abate, Tovena, Geometria Differenziale (which I presume you can read). There you can also check the correctness of your requirements. To quote some lemmas and definitions:

• page 62 (on the topologisation of manifolds):

  A $n$-dimensional atlas $\mathcal{A}=\{(U_\alpha,\varphi_\alpha)\}$ on a set $M$ induces a topology in which a subset $A\subseteq M$ is open if and only if $\ \forall\alpha\ \ \varphi_\alpha\left(A\cap U_\alpha\right)$ is open. Moreover, it is the only topology allowing all the $U_\alpha$ to be open in $M$ and all the $\varphi_\alpha$ to be topological embeddings.

• page 134 (definition of vector bundle):

  A vector bundle of rank $r$ on a manifold $M$ is a manifold $E$ provided with a surjective differential map $\pi:E\to M$ such that:

  1. $\forall p \in M,\ \pi^{-1}(p)$ is endowed with a structure of $r$-dimensional vector space.

  2. $$\forall p \in M\ \exists\ U\ni p\text{ open set },\ \exists\ \chi:\pi^{-1}(U)\to U\times \mathbb{R}^r\ \text{ diffeomorphism s.t.}\\ \forall p\in U\ \pi_2\circ\chi|_{\pi^{-1}(p)}\text{ is a linear isomorphism } \pi^{-1}(p)\to \pi_2\left(\{p\}\times \mathbb{R}^r\right) $$ $\pi_1$ and $\pi_2$ being the projections on the factors of $U\times \mathbb{R}^r$

• page 136 (on the conditions determining a vector bundle)

  Let $M$ a manifold, $E$ a set, $\pi:E\to M$ a surjective function, $\mathcal{A}=\{(U_\alpha,\varphi_\alpha)\}$ an atlas of $M$ and $\chi_\alpha$ maps $\pi^{-1}(U_\alpha)\to U_\alpha\times \mathbb{R}^r$ such that:

  1. $\chi_\alpha(\pi^{-1}(p))\subseteq \{p\}\times\mathbb{R}^r$

  2. $$\forall U_\alpha\cap U_\beta\neq\emptyset\ \exists g_{\alpha\beta}:U_\alpha\cap U_\beta\to \text{GL}(r,\mathbb{R})\text{ differentiable map s.t. }\\\chi_\alpha\circ\chi_\beta^{-1}:\left(\alpha\cap U_\beta\right)\times\mathbb{R}^r\to\left(\alpha\cap U_\beta\right)\times\mathbb{R}^r\text{ satisfies }\\\chi_\alpha\circ\chi_\beta^{-1}(p,x)=(p,g_{\alpha\beta}(x))$$

  Then, $E$ admits exactly one stucture of rank-$r$ vector bundle on $M$ such that $\{(U_\alpha,\phi_\alpha,\chi_\alpha)\}$ satisfies the condition of the point (2) in the definition.

So, as you see, your idea is right in some sense, but the requirements to induce a vector bundle seem quite sharper. The book I'm quoting takes its time (quite the time, actually) to thoroughly deal with these technical lemmas, which seems to me to be useful to your purpose. Of course, I assume it's not the only one in the world.

Added: I don't think I did, but I might have made some mistakes in copying and translating the text.

Added 2: Specifically: let $E, M, U_\alpha,\varphi_\alpha,\chi_\alpha$ as in page 136. Let $\rho_\alpha:\pi^{-1}(U_\alpha)\to \varphi_\alpha(U_\alpha)\times\mathbb{R}^r$ defined by $\rho_\alpha(x)=(\varphi_\alpha\pi_1\chi_\alpha(x),\,\pi_2\chi_\alpha(x))=(\varphi_\alpha\pi(x),\,\pi_2\chi_\alpha(x))$.

$\mathcal{B}=\{(\pi^{-1}(U_\alpha),\rho_\alpha,\varphi_\alpha(U_\alpha)\times\mathbb{R}^r)\}$ is an atlas on the manifold $E$, therefore it induces one and only one topology as of the result in page 62.

Answer 2

What is missing from your definition is not only the topological structure of $V$, but the much more important locally-trivial character: there is a covering of $M$ with open sets $U$ such that $\pi^{-1} (U) \simeq U \times \mathbb R ^r$.