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.

Yes, open maps. The inversion $g\mapsto g^{-1}$ is in fact a homeomorphism, continuous by def. and its inverse is itself.
For the product, if $U,V\subseteq G$ open subsets, then the image of $(U,V)$ under multiplication is the complex product
$$U\cdot V =\{u\cdot v\mid u\in U,v\in V\} =\bigcup_{u\in U}(u\cdot V)$$
is a union of open sets.
