at.algebraic topology - How should I visualise RP^n?

I find it easiest to get $mathbb RP^n$ from $S^n$ with two cells in every dimension. That is to say, you start with two points for $S^0$, attach two one-cells oriented opposite ways for $S^1$, attach one 2-cell on top and one on the bottom for $S^2$, and so on until you have $S^n$. Then the antipodal map actually switches the two cells in each dimension, so it defines a free $mathbb Z/2$ action on $S^n$. If you quotient out by that action, then you get a cell structure on $mathbb RP^n$ with one cell in each dimension, and you have a universal double cover ($n$>$1$) for it.

I like this construction for a couple of reasons. First, it displays the (co)boundary maps in cellular (co)homology: the two $k$-cells share the same boundary (up to sign $(-1)^k$) which is the two $k-1$-cells. Because it's a covering space you can sort of see how after quotienting you get one $k$-cell whose boundary folds twice onto the $k-1$-cell. So that also shows you why the maps alternate between the 0 map and the multiplication by 2, or are all the 0 map in $mathbb Z/2$ coefficients.

Also, this generalizes to a construction of $S^infty$ and thus $mathbb RP^infty$, with $mathbb RP^n$ inside it just by taking the $n$-skeleton. It looks like you could modify Qiaochu's definition to account for the infinite case by taking a $k$-cell to be the points that vanish after the first $k+1$ dimensions and are scaled to 1 in the $k+1$st.

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