topology on smooth vector bundle

Question

Let $\pi: E\to M$ be a smooth vector bundle. Here $E$ and $M$ are smooth manifolds.

Wikipedia (and many others) says that the maps $\varphi: U\times \mathbb{R}^k\to \pi^{-1}(U)$ are the local trivialization of $E$.

I have a couple of questions about the topology of $E$ as manifold.

1. Does the topology on $E$ distinguishes two points in the same vector space $\pi^{-1}(x)$ ?

2. If $O$ is open in $\mathbb{R}^k$ and $U$ is open in $M$, is $\varphi(U\times O)$ open in $E$ ?

In my opinion, the two answers are "yes", and the functions $\varphi$ are not the atlas of $E$. If not, a smooth vector field could "jump" from one vector to an other one very far away.

This brings me to an other question :

3. Why do people call them "local trivialization" ?

Possible answer of question 3 : the maps $\varphi$ are local trivializations, but they are not sufficient to be an atlas. In other words, an atlas of $E$ contain, among others, the maps $\varphi$.

Note : This question is merely a duplicate of this one. The difference is that here, the question is more specific about the topology (and, as far as I believe, the accepted answer there is wrong -- but I am far from being sure). The accepted answer here does not answer my specific questions neither.

Answer 1

1. Since $E$ is assumed to be a smooth manifold, it is Hausdorff so any two points in $E$ have disjoint open neighborhoods.

2. The maps $\varphi:\pi^{-1}(U)\to U\times\mathbb R^k$ are assumed to be diffeomorphisms and hence homoeomorphisms. For $V\subset U\subset M$ open and $O\subset\mathbb R^k$ open, $V\times O$ is open in $U\times\mathbb R^k$ and hence $\varphi^{-1}(V\times O)$ is open in $\pi^{-1}(U)$ and hence in $E$.

3. The standard terminology is that a bundle is called trivial if it is isomorphic to a product (bundle). So the map $\varphi$ shows that the restriction of $E$ to $U\subset M$ is a trivial bundle, whence it is called a local trivialization. The maps $\varphi$ are not charts in the usual sense, since their values do not lie in $\mathbb R^N$ for some $N$ but in $M\times\mathbb R^k$. You can "combine" them (in an obvious way) with local charts for $M$ to obtain local charts for $E$. If you start from a vector bundle atlas for $E$ the resulting charts define an atlas for the manifold $E$ although not a maximal one.