How to show that the quotient map $q : \Bbb C^{n+1}-\{0\} \to \Bbb CP^n $ is a closed map?

Question

How to show that the quotient map $q : \Bbb C^{n+1}-\{0\} \to \Bbb CP^n $ is a closed map? This is my attempt :

Let $K$ be a closed subset of $\Bbb C^{n+1}-0$. We must show that $q^{-1}q(K)$ is closed in $\Bbb C^{n+1}-0$, and I know that $q^{-1}q(K)=\{az : a \in \Bbb C-\{0\}, z \in K \}$. Now how do I have to proceed?

Answer 1

It is not true. The set $K = \{(k,1/k,0,\ldots,0) \mid k \in \mathbb N\} \subset \mathbb C^n \setminus \{ 0 \}$ is closed. The set $q^{-1}q(K)$ contains the points $z_k = 1/k \cdot(k,1/k,0,\ldots,0) = (1,1/k^2,0\ldots,0)$. This sequence converges to $(1,0,\ldots,0)$ which is not contained in $q^{-1}q(K)$.

You can show that $q$ is open.

Answer 2

It's not. For instance, for $n=1$, let $K=\{(x,y)\in\mathbb{C}^2:xy=1\}$. Then $K$ is closed in $\mathbb{C}^2-\{0\}$, but $q(K)$ is all of $\mathbb{C}P^1$ except two points (the points with homogeneous coordinates $[0,1]$ and $[1,0]$), which is not closed.