I'm trying to study line bundle over $S^2$. In this post was outlined the method based on clutching functions. But now I'm interesting in another approach.
For the sphere there is two maps : upper hemisphere and lower hemisphere with intersection as $[-\epsilon,\epsilon]\times S^1$. For the upper hemisphere and lower hemisphere its well-known that bundles over this spaces is trivial. (Any bundle over a contractible base is trivial). So to prove the fact that line bundle over $S^2$ is trivial we must create continuation of trivialization from upper hemisphere (for example) to the lower hemisphere through "border" $[-\epsilon,\epsilon]\times S^1$.
As I understand it is sufficient to continue trivialization from the "border" to the center of the "disk". (I think here it is possible to use a partition of unity, but I'm not sure).
I can't formalize this reasoning.