Let $X$ be a compact in the Polish space (metric, complete, separable) and $G\subseteq X\times X$ is open. For $x\in X$ we define the section of $G$: $ s(x) = \{y\in X|\langle x,y \rangle \in \bar{G}\}. $
The set $A'\subseteq X$ is invariant if for all $x\in A'$ holds $s(x)\subset A'$. How to verify if there are non-empty invariant subsets of given compact $A$? Maybe there are known equivalent problems?
It will be even helpful in the case $X = [0,1]$.
This is reformulated and changed a little bit problem from my previous question: Self-complete set in square