Inverse limits with upper semi-continuous bonding functions problem 6.1

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Problem 6.1 (ILWUSCBF): For a given compact Hausdorff space $X$, which compact sets are not homeomorphic to inverse limits with a single upper semi-continuous function $f \colon X \rightarrow 2^X$ ($f \colon X \rightarrow C(X)$)?

Answer: Unknown