Define the infinitieth derived set of a set $A$ as the intersection of all n-th derived sets of $A.$
What can the infinitieth derived set of $A$ look like? In particular, I am interested in $A\subset S^1$ compact.
The types of infinitieth derived sets I can think of are (some) disjoint unions of closed intervals* and Cantor sets.
*even here, I'm not sure about the exact statement: can an isolated point be the infinitieth derived set of $A$? My intuition says no, but intuition isn't a proof.
Embedding the countable ordinal $\omega^{\omega}$ as a bounded subset of the real line $\mathbb R$ (and then adding the unique limit point) gives an example of a compact set whose "infinitieth derived set" is a point.