$\Bbb C^n-\{z_1=0\}$ is connected

Question

I have done a problem that $\Bbb R^n-\{x_1=0\}$ has two connected components. This can be proved by using Jordan–Brouwer separation theorem. However, to prove $\Bbb C^n-\{z_1=0\}$ is connected the theorem can work no longer. I don't know where even to begin.

Any help would be much appreciated.

Answer 1

$$\Bbb C^n\setminus \{z_1=0\}=\Bbb C^*\times \Bbb C^{n-1}$$

Answer 2

You could produce a continuous bijection from $\mathbb{R}^n\times\mathbb {R}^n $ to $\mathbb {C}^n $