I want to understand the proof of the following basic theory related to dicrete dynamical system. $T$ is continuous, $\omega(x)$ is the limit set of $T^nx$, that is to say, $\omega(x)=\{y|T^{n_i}(x)\to y\}$. An invariant set means that $T(D)=D$, and invariantly connected means it can't be written as the union of two closed invariant subset.
If $T^nx$ is bounded. Then $\omega(x)$ is nonemply, compact, invariant, invariantly connected. And is the smallest closed set Ihat $T^nx$ approaches, which means $d(T^nx,\omega(x))\to 0$.
The difficulty I'm faced with is how to prove invariantly connected.
