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. 



How should I visualise RP^n?
You can "visualize" the cell structure on $\mathbb{R}P^n$ rather explicitly as follows. The set of tuples $(x\_0, ... x\_n) \in \mathbb{R}^{n+1}$, not all equal to zero, under the equivalence relation where we identify two tuples that differ by multiplication by a nonzero real number, can be broken up into pieces depending on which of the $x\_i$ are equal to zero.


* If $x\_0 \neq 0$, the corresponding points can be written $(1, x\_1, ... x\_n)$, and they form a subspace isomorphic to $\mathbb{R}^n$.
* If $x\_0 = 0$ and $x\_1 \neq 0$, the corresponding points can be written $(0, 1, x\_2, ... x\_n)$, and they form a subspace isomorphic to $\mathbb{R}^{n-1}$.


And so forth. One way to say this is that the tuples where $x\_0 \neq 0$ form an affine slice or affine cover of $\mathbb{R}P^n$ and the tuples where $x\_0 = 0$ constitute the "points at infinity," which themselves form a copy of $\mathbb{R}P^{n-1}$. 


