I want to figure out how to show that:
$\omega$-limit set is not a union of two disjoint closed invariant subsets.
where $\omega(x)=\bigcap\limits_{N\geq 0}\overline{\{f^n(x)|n\geq N\}}$, $X$ is a compact metric space and $f:X\to X$ is a homeomorphism.
The sets $A,B$ are closed and disjoint subsets of the compact set $X$, thus have a positive distance. Set $2\epsilon$ to this distance. Use uniform continuity of $f$ on the compact set $X$ to find the associated $\delta<ϵ$ so that $$ d(x,y)<δ \implies d(f(x),f(y))<ϵ . $$ Then there exists an $N$ so that, with $x_n=f^n(x)$, $$ n>N\implies dist(A\cup B,x_n)<δ $$ Now if $dist(A,x_n)<δ$ for some $n>N$, then using the invariance of $A$ one finds $dist(A,x_{n+1})<ϵ$ and as that implies that $dist(B,x_{n+1})>ϵ>δ$, one also concludes $dist(A,x_{n+1})<δ$. This means, continued via induction, that the forward orbit never comes close to $B$, so $B$ can not be a part of the forward limit set, in contradiction to the assumptions of the claim.
