Let $X$ be a perfect metric space $T:X \to X$ be a continuous map. Define $O_T^+(x)=\{T^n(x): n \in \Bbb N\}$. If $\overline{O_T^+(x)}=X$
Prove that $w(x)$ the derived set of $O_T^+(x)$ is $X$.
I have proved that $w(x)$ is closed. Now I was trying to show that for any $y \in X\setminus w(x)$ we'll get $x \in w(x)$.