Julia set has no isolated points

Let $f$ be a rational map of degree of a rational map of the Riemann sphere at least two. Its nonempty Julia set $J$ is infinite: a finite completely invariant closed set of a rational map lies in the Fatou set. Since $J$ is compact, its set $J^{\prime }$ of accumulation points is nonempty. It is closed, contained in $J$, and completely invariant; local openness of a nonconstant holomorphic map gives the inverse-image direction.

Apply the completely invariant closed set of a rational map classification to $J^{\prime }$. Its alternative with at most two points would put the nonempty set $J^{\prime }$ in the Fatou set, contradicting $J^{\prime }\subseteq J$. Hence $J\subseteq J^{\prime }$, so $J=J^{\prime }$ and the Julia set has no isolated points. This supplies the perfect property without assuming it in advance.^[ref]