Size levels for simple closed curves (Q1868036)

Let \(X\) be a continuum with hyperspace of nonempty subcontinua \(C(X)\). A size map is a continuous function \(\sigma: C(X)\to [0,+\infty)\) such that \(\sigma(\{x\})=0\) and if \(A\subseteq B\) then \(\sigma(A)\leq \sigma(B)\). This is a generalization of a Whitney map. In this paper for a simple closed curve \(X\) the author gives among others necessary and sufficient conditions for a space to be a size level in \(C(X)\) where the size is larger than or equal to 0 but less than the size of \(X\). This generalizes in a nontrivial way similar known results for arcs.