I've the following problem that gives a characterization of the perfect sets in the Cantor set.
A subset $P\subseteq 2^{\omega}$ is perfect if and only if it is homeomorphic to $2^{\omega}$.
Show that if $P$ is homeomorphic to $2^{\omega}$ then $P$ is perfect it's simple. But for the other implication I can't think of anything.
Does anyone have any hint?