$\mathbb{R}P^n$ and $\mathbb{C}P^n$ can be built as CW complexes.
$\mathbb RP^n = e^0 \cup e^1 \cup \cdots \cup e^n$, $\mathbb CP^n = e^0 \cup e^2 \cup \cdots \cup e^{2n}$.
Is there an analogous construction if we consider over the quaternionic projective space $\mathbb HP^n$?
How about the octonions $\mathbb O$?
Can we give these spaces CW structures? How does one go about this? (This is just out of curiosity).
Related question: Real, Complex, Quaternionic and Octonionic Projective spaces
The CW strucure of quaternionic projective spaces is usually explained in books dealing with such things. In particular, you get the $n$-dimensional space by with one $4k$ cells for each $k$ from $0$ to $n$.
With the octonions, it is more complicated: the projective line works as usual, but already for the plane one has to work quite a bit to even define what one means (see the book Octonions by Conway) because of the lack of associativity: the octonions are an alternative algebra, and that is enough to get by. But there are no higher dimensional octonionic projective spaces.