So I did some algebraic topology at university, including homotopy theory and basic simplicial homology, as well as some differential geometry; and now I'm coming back to the subject for fun via Hatcher's textbook. A problem I had in the past and still have now is how to understand projective space RP^n - I just can't visualise it or think about it in any concrete way. Any ideas?

edit: Essentially RP^n is always the example I don't understand. So when for example Hatcher says that S^n is of a CW complex with two cells e^0 and e^n, I can picture what's going on because I know what spheres look like and I can imagine the attachment in some concrete-ish way. But when he says "we see that RP^n is obtained from RP^{n-1} by attaching an n-cell [...] it follows by induction on n that RP^n has a cell complex structure e^0 U e^1 U ... e^n" - my brain just gives up.