Problem 1.9 (Complex projective space)

Complex projective n-space, denoted by $ℂ{\mathbb{P}}^n$, is the set of all 1-dimensional complex-linear subspaces of ${ℂ}^{n+1}$, with the quotient topology inherited from the natural projection $\pi :{ℂ}^{n+1}\setminus \{0\}\to ℂ{\mathbb{P}}^n$. Show that ${\mathit{ℂ\mathbb{P}}}^n$ is a compact $2n$-dimensional topological manifold, and show how to give it a smooth structure analogous to the one we constructed for $ℝ{\mathbb{P}}^n$. (We use the correspondence

to identify ${ℂ}^{n+1}$ with ${ℝ}^{2n+2}$.) (Used on pp. 48, 96, 172, 560, 561.)