I've read the following definition of a vector bundle.
A rank $n$ vector bundle on a manifold $M$ is a map $\pi:V\to M$, where for each point $x\in M$, the fibre $\pi^{-1}(x)$ is endowed with the structure of an $n$-dimensional vector space. Each point $x\in M$ has a neighbourhood $U$ such that $\pi^{-1}(U)$ is homeomorphic to $U\times \Bbb R^n$, such that the diagram given by $\pi^{-1}(U)\to U$ and $\pi^{-1}(U)\to U\times \Bbb R^n \stackrel{p_1}\to U$ is commutative. Additionally the homeomorphism must restrict to fibres and give vector space isomorphisms $\pi^{-1}(x)\to \{x\}\times \Bbb R^n$.
Now, in this definition, we say that $U\times \Bbb R^n$ is homeomorphic to $\pi^{-1}(U)$. Are we endowing $U\times \Bbb R^n$ with the product topology? Are we endowing the total space $V$ with the topology that this is inducing? What structure does $V$ have thus?