QUESTION: Let $X$ be a topologically complete metric space and $T:X\to X$ a continuous map. Let $x\in X$ be a point whose orbit is a closed set. Show that either $x$ has an iterated that is periodic or $\omega_T(x)=\emptyset$.
I tried to do it by exclusion: if $\omega_T(x)=\emptyset$, then no iterated of $x$ is periodic and, if $\omega_T(x)\neq\emptyset$, I want to prove that an iterated of $x$ is periodic, but I couldn't do it.
Can someone help me?
We have $$ \omega(x)=\bigcap_{n\in\mathbb N}\overline{\{T^k(x):k\ge n\}}=\bigcap_{n\in\mathbb N}\{T^k(x):k\ge n\}, $$ since the forward orbit of $x$ is closed. If $y\in\omega(x)$, then there exists $k$ such that $T^k(x)=y$. But one can replace the intersection by $\bigcap_{n>k}$ and so there exists $m>k$ such that $T^m(x)=y$. Hence, $T^m(x)=T^k(x)$ and so $T^{m-k}(z)=z$, where $z=T^k(x)$. In other words, $z$ is periodic.