Simply connected compact subsets of the plane (Q1306867)

For a compact subset \(A\) of the plane, it is said that \(A\) is a set with no holes provided that whenever \(\lambda\) is a Jordan curve in \(A\) we have that \(\operatorname {Ins}(\lambda) \subseteq A\). A simply connected set is a path-connected set with no holes. Denote by NH and SC the collections of all sets with no holes and all simply connected sets, respectively. In this paper the author presents a rank function for the set NH which measures the complexity of sets by looking at trees of sequences of Jordan arcs which in some sense approximate holes. He presents properties of the rank functions, characterizes the sets of rank 1, and shows that the rank function is unbounded in \(\omega_1\) on the set SC. The last result is then used to show that SC and NH can not be analytic sets in the complete separable metric space \(K(\mathbb{R}^2)\) of all compact subsets of the plane with the Hausdorff metric.