Prove that $\mathbb{C}\mathbb{P}^n$ is connected

Question

I need to show that the complex projective space $\mathbb{C}\mathbb{P}^n$ is connected. I know that $\mathbb{C}\mathbb{P}^1$ is connected, because there exists a homeomorphism (stereographic projection) between $\mathbb{C}\mathbb{P}^1$ and the (connected) sphere $\mathcal{S}^2$. How can I prove that, in general, $\mathbb{C}\mathbb{P}^n$ is connected for $n\geq 1$? Thanks for your help and stay healthy!

Answer 1

By definition, $\mathbb{CP}^n$ is the quotient of the connected space $\mathbb C^{n+1} - \{0\}$ by some equivalence relation. The continous image of a connected space is connected.