I am supposed to show that $P^n \mathbb{R} = e_0 \cup\cdots\cup e_n$, where $e_i$ is an $i$-dimensional cell.
I also know that there is a quotient map $S^n \rightarrow S^n/\tilde - = P^{n-1} \mathbb{R},$ where the equivalence relation identifies points on the opposite site of the sphere, but I don't understand how that helps me to build a homeomorphism $P^n \mathbb{R}= e_n \cup P^{n-1} \mathbb{R}$? What is $e_n$ and how is it constructed. Could anybody offer here a few details?