Are inversion and multiplicaton open? If $G$ is a topological group, are inversion $G \to G$ and multiplication $G\times G \to G$ open mappings? More concretely, I try to show that division of complex numbers
$$\{(z,w) \in \mathbb{C}^2;\; w \neq 0\} \to \mathbb{C},\; (z,w) \mapsto \tfrac{z}{w}$$
is an open mapping. I want to use this to construct charts on $\mathbb{CP}^1 = (\mathbb{C}^2\setminus\{0\})/\mathbb{C}^{\times}$.
I don't know where to begin.

Take $O_1$ and $O_2$ two open subsets ot $G$. As the inversion $i$ is an homeorphism and an involution, $i(O_1)=i^{-1}(O_1)$ is open. Denote $m$ the multiplication map. Then $$m(0_1\times O_2)=\bigcup_{y\in O_2}O_1y$$
is open. We deal with the general case using the definition of product topology.
