I don't know if "pseudo projective space" is a general accepted term, but I once read a book on general topology where the term was used for $\mathbb{S}^n / (\mathbb{Z}/m\mathbb{Z})$ (where you get the normal $\mathbb{R}\mathbb{P}^n$ when you set $m=2$).
Does anyone know of a good reference as to the (differential) topology and geometry of such beasts?
I think this may be an example of orbifolds, but this isn't my field exactly. I guess you get a bunch of singularities when $m > 2$ with which you have to deal.
Edit:
Reading in Sieradski (Introduction to Topology and Homotopy), where this example was coming from. He is talking about the disc $\mathbb{D}^2/eq$, where $eq$ is the equivalence relation coming from $\phi: \mathbb{S}^1 \rightarrow \mathbb{S}^1:z \mapsto z e^{2\pi/m} $ (a rotation of $2\pi/m$ radians. Thus the identification map $q: \mathbb{D}^2 \rightarrow P_m$ wraps the boundary 1-sphere m-times around its image $q(\mathbb{S}^1)$. This $P_m$ is called a pseudo-projective plane of order m.
My question is to references for geometry and (differential) topology of $P_n$, and higher-dimensional analogues.