Surprisingly I didn’t find this question although I am sure it must have been asked.
As the title suggests I want to define CW structure on $\mathbb{C}P^n$. I wanted to generalize the following construction for $\mathbb{R}P^n$.
$\mathbb{R}P^n \cong D^n /\text{~}$ where the relation glues together diametrically opposed points on $\partial D^n$. So there is an obvious attaching map $D^n \to \mathbb{R}P^n$.
For $\mathbb{C}P^n$ it is also true that $\mathbb{C}P^n \cong D^{2n+1}/ \text{~}$ but here the relation is nontrivial not only on $\partial D^{2n+1}$ but everywhere so the map $\operatorname{Int} D^{2n+1} \to \mathbb{C}P^n$ is not injective hence not homeomorphism.
So what do I do?