The discussion revolves around the proof of the Knaster-Tarski Theorem, specifically focusing on the properties of a monotone function \( F \) and the union of sets whose images are invariant under \( F \). Participants are examining the correctness of a proof presented in an image format.
Participants have not reached a consensus on the correctness of the proof, and there are questions and clarifications being sought regarding the details of the argument.
The discussion highlights potential ambiguities in the proof, particularly concerning the definitions and implications of the sets involved, as well as the assumptions made about the function \( F \).
Evgeny.Makarov said:What is your question?
So, if I understand correctly, $$C=\bigcup_{X\subseteq A}F(F(X))=F(X)$$. But then it is not clear why $B\in C$ implies $B\in X$ for some $X\subseteq A$ such that $X\subseteq F(X)$.Fermat said:$C$ be the union of sets whose image is invariant under F