Let $G$ the abelian group generated by commuting homeomorphisms $f_1,\dots,f_q:M\rightarrow M$, where $M$ is a compact metric space. Show that there is $X\subset M$ minimal with respect to the relation of inclusion in the family of the non-empty closed $G$-invariant sets.
2026-04-01 15:52:35.1775058755
Finding a minimal non-empty closed $G$-invariant set of a compact metric space
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Hint: Apply Zorn's lemma to the set of closed subsets fixed by $G$ ordered in reverse by inclusion. For your upper bounds take intersections and remember the nice thing that happens when you take an intersection of closed nested sets in a compact metric space.