Cantor-Bendixson degrees and convexity in ℝ²

We present an ordinal rank, δ³, which refines the standard classification of non-convexity among closed planar sets. The class of closed planar sets falls into a hierarchy of order type ω₁ + 1 when ordered by δ-rank. The rank δ³ (S) of a set S is defined by means of topological complexity of 3-cliques in the set. A 3-clique in a set S is a subset of 5 all of whose unordered 3-tuples fail to have their convex hull in S. Similarly, δⁿ(S) is defined for all n > 1. The classification cannot be done using δ², which considers only 2-cliques (known in the literature also as "visually independent subsets"), and in dimension 3 or higher the analogous classification is not valid.