I was wondering if there is a simple argument showing that the complex projective line defined as $\mathbb{CP^1} = \big(\mathbb{C}^2 \setminus \{0\}\big)/{\mathbb{C}^{\times}}$ is hausdorff when equipped with the quotient topology. So far I was picturing this scenario by analogy with $\big(\mathbb{R}^3 \setminus 0\big)/\mathbb{R}^{\times}$ and the 2-sphere therein. Imagining open, disjoint double cones surrounding distinct lines the asseriton seems clear. I also know that it suffices to show that the action of $\mathbb{C}^{\times}$ on $\mathbb{C}^2 \setminus \{0\}$ is proper (which I also don't see). But since I want to avoid defining continuous and proper group actions I was hoping for a more elementary approach.
Optimally I'd like to see a way of directly establishing that $\mathbb{CP^1}$, defined this way, is homeomorphic to the 2-sphere $\mathbb{S^2}$. However, I'm looking for an answer which is short and preferably doesn't use (continuous) group actions. Maybe there is no way around that?