# topology on smooth vector bundle

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 :

1. 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.

• The accepted answer in the first linked question is correct. It's an a priori different, but definitely a compatible atlas; thus, the smooth structures are the same. What's your definition of an atlas / a smooth structure?
– Ben
Sep 19 at 15:35

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.