, Volume 121, Issue 1, pp 85-91

Cantor-Bendixson degrees and convexity in ℝ2

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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 setS is defined by means of topological complexity of 3-cliques in the set. A 3-clique in a setS is a subset ofS all of whose unordered 3-tuples fail to have their convex hull inS. Similarly, δn (S) is defined for alln>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.