I was trying to prove there exist a meromorphic function on the complex projective line $\Bbb{CP^1}$ which has single simple pole.(therefore it biholomorphic to the Riemann sphere).
In order to let it well defined, the map $f:\Bbb{CP}^1 \to \Bbb{C}$ needs to be homogeneous, for example $$f:\Bbb{CP^1} \to \Bbb{C}\\ [z_1:z_2]\mapsto z_1/z_2$$
however, this does not satisfy the requirement.I can prove it by first construct the coordiante function $z\mapsto z$ on the Riemann sphere, which has a single simple pole and single zero point.Then if we can find the biholomorphic map between $\Bbb{CP}^1$ and $S^2$ then precompose it with the corrdinate function $z\mapsto z$ gives a meromorphic function on $\Bbb{CP}^1$ that satisfies the requirement. Can we construct it directly?