Does the metric $d(x,y)=||x-y||^p$ for $0<p<1$ induce the usual topology on $\mathbb{R}^n$?
Yes, it does generate the same topology. The reason? The balls are the same! If $B_1(x, r)$ is the open ball centred at $x$, radius $r$, in this metric space, while $B_2(x, r)$ is the corresponding open ball with respect to the usual Euclidean metric, then $$B_2(x, r) = B_1(x, r^p),$$ as $$\|x - y\| < r \iff \|x - y\|^p < r^p \iff d(x, y) < r^p.$$