I am trying to understand the definition of a morphism of elliptic curves, and perhaps in doing so something about $\mathbb{P}^1$ as well. We say $\varphi:E_1\rightarrow E_2$ is a morphism from $E_1$ to $E_2$ if for every $P\in E_1$ there exists a neighborhood $U$ of $P$ such that $\varphi|_U(P)=[f_0(P):f_1(P):f_2(P)]$ for homogeneous $f_i\in\overline{F}[x_0,y_0,z_0]$ of the same degree.
My question is what does an open neighborhood of $\mathbb{P}^1$ even look like? Is there an implicit topology?