Recall that if $V$ is a vector space over some field $k (=\mathbb R,\mathbb C,\mathbb H \text {(quaternions)} $) then a projective space can be thought of as the set of lines through the origin of $V$. Suppose dim$_{R} (k)=d$ then is it true that
$k\mathbb P^n$ can be obtained from $k\mathbb P^{n-1}$ by attaching a $dn$-cell to $k \mathbb P^{n-1}$.
I have proved it in case $k=\mathbb R$ but i am having trouble for other fields $\mathbb R$ and $\mathbb H$.