This is an exercise from Falconer's book. How can I prove that Julia set $J(f_c)$ is uncountable for all c, $f_c(z) = z^2 +c$？It gives me a hint that perfect set E is uncountable, so I want to prove $J(f_c)$ is a perfect set. But I only know that $J(f_c)$ is closed.
Another solution I tried is to choose a point $z_0$ in the filled-in Julia set K and a circle $C$ outside K and find a boundary point on the line joining $z_0$ and $z \in C$. Is this method feasible?
I'll be very grateful for any help.