I'm studying the dynamic of the post-critically finite for my master thesis and my professor gave me the problem concerning generalization of post-critically finite ration map. Concretely, let $f : \mathbb{P}^n \rightarrow \mathbb{P}^n$ be a holomorphic map. The critical set consist of points sucht that $$ C(f)=\{z \in \mathbb{P}^n | \, rank T_z f < n\}$$ The post-critical set of $f$ $PC(f)$ is forward image of $C(f)$ $$ PC(f)= \bigcup\limits_{i=0}^{\infty} f^i (C(f))$$ A map is called post-critically finite if $PC(f)$ is algebraic, at least my professor told me so. It's clear that if $PC(f)$ is algebraic, it is closed. My question is: Is that the converse statement is true, i.e if $f$ has $\overline{PC(f)}$ is algebraic, then $PC(f)$ is algebraic?

At first glance, it would be wrong for the statement that "$\overline{D} \subset \mathbb{P}^n$ is algebraic then $D$ is algebraic". For example, take D is a countable dense subset of $\mathbb{P}^n$. but it is not actually the counter example of my question. Thank you for any suggestion. This question is asked at math.stackexchange.com but there no one interested so I hope MO could help me out. Thanks again