I'd like a hint to prove that this function is a homeomorphism: $$f[z:w]=\left(\frac{\operatorname{Re}( w \bar{z})}{|w|^2 + |z|^2}, \frac{\operatorname{Im}(w\bar{z})}{|w|^2 + |z|^2},\frac{|w|^2-|z|^2}{|w|^2+|z|^2}\right)$$ of $\mathbb{P}^1$ onto $\mathbb{S}^2$. Thanks.
ADDED(06/27/12): The previous definition of $f$ was wrong, this new one seems to work...