Compactness of complex projective space, references

Question

I know that the complex projective space is compact, but I have to justify it for my thesis, so I need references for this fact. I search the web, but I didn't find anything. Does someone knows a book (or an article) where this fact in shown?

Thanks

Answer 1

The projection map $\pi\colon\mathbb{S}^{2n+1}\to \mathbb{P}^n(\mathbb{C})$ is continuous and surjective, since $\mathbb{S}^{2n+1}$ is compact, we have that: $\mathbb{P}^n(\mathbb{C})=\pi(\mathbb{S}^{2n+1})$ is compact. I think that you can find a similar argument in "Topologia" of M. Manetti or "Geometria 2" of E. Sernesi.