Prove that the closure of every $T$-orbit is a $T$-invariant subset of X

Question

Let $X$ be a topological space and $T : X → X$ be a homeomorphism. Prove that the closure of every $T$-orbit is a $T$-invariant subset of X (that is, $T(\overline{O_T (x)}) =\overline{O_T (x)})$.

Pf: What I have tried

$O_T(x)=\{T^n(x): n\in \Bbb Z\}$ is a $T$ invariant set. Now $T$ is homeomorphism so $T$ takes closed set to closed set $\Rightarrow \overline{O_T (x)} \subseteq T(\overline{O_T (x)})$.

Similarly, from $O_T(x)=\{T^n(x): n\in \Bbb Z\}$ is a $T^{-1}$ invariant set, we get reverse inclusion. So, I am done right.

Answer 1

Right, exactly.

$O_T(x)= T^{-1}(O_T(x))\subseteq T^{-1}(\overline{O_T(x)})$, so $\ \overline{O_T(x)}\subseteq T^{-1}(\overline{O_T(x)})$, which implies
$T(\overline{O_T(x)})\subseteq \overline{O_T(x)}$.