# Quotient map from the $(2n+1)$-dimensional sphere into complex projective space is open.

We have a natural quotient map $$\phi\colon S^{2n+1}\rightarrow (\mathbb{C}^{n+1}\setminus{\{0\}})/\mathbb{C}^{*}=\mathbb{P}^n\mathbb{C}$$ and I want to see, that it's open. Denote by $$\iota\colon S^{2n+1}\hookrightarrow\mathbb{C}^{n+1}\setminus{\{0\}}$$ the inclusion and by $$\pi\colon \mathbb{C}^{n+1}\setminus{\{0\}}\rightarrow(\mathbb{C}^{n+1}\setminus{\{0\}})/\mathbb{C}^{*}$$ the quotient map.

It holds $\phi=\pi\circ\iota$. Let's take an open $U\subset S^{2n+1}$, so we find a open $V\subset (\mathbb{C}^{n+1}\setminus{\{0\}})/\mathbb{C}^{*}$, such that $U=V\cap S^{2n+1}$.

$\phi(U)$ is open iff $\pi^{-1}(\pi(V\cap S^{2n+1}))=\bigcup_{\lambda\in\mathbb{C}^{*}}\lambda(V\cap S^{2n+1})$ is open in $\mathbb{C}^{n+1}\setminus{\{0\}}$, but I don't see, why this holds.