I am reading this MO post here about the classification theorem of vector bundles over $\Bbb{P}^1$. However, I am mainly interested in the case of just line bundles. Now if the general definition of a topological line bundle (over $\Bbb{P}^1$ say), the local trivialization condition states:
"There is an open cover $\{U_\alpha\}$ of $\Bbb{P}^1$ over which we have trivializations...."
However, in the MO post above it seems to be stated that given any line bundle over $\Bbb{P}^1$, it is enough to assume the trivializations happen over the standard open cover $U_0,U_1$.
My question is: Why is it enough to assume the cover is just given by the standard affine opens $U_0$ and $U_1$?
On Spec $A$, locally free sheaves of rank one correspond to projective $A$-modules of rank one. But $U_i \cong $ Spec $k[x]$, and $k[x]$ is a PID. Thus projective modules over it are in fact free, thus a projective module of rank one is just isomorphic to $k[x]$ itself, and so the associated invertible sheaf is just isomorphic to the structure sheaf.
In short, any line bundle on the affine line is automatically trivial. (Replacing "one" by "$n$" in the above, we see that the same is true for vector bundles. And in fact the same is true for vector bundles over affine space of any dimension; in the line bundle case this just uses that $k[x_1,\ldots,x_d]$ is a UFD, while for higher rank bundles it was conjectured by Serre and eventually proved (independently) by Quillen and Suslin.)