Cantor-Bendixson degrees and convexity in ℝ2
- Menachem Kojman
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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.
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- Cantor-Bendixson degrees and convexity in ℝ2
Israel Journal of Mathematics
Volume 121, Issue 1 , pp 85-91
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- Menachem Kojman (1)
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- 1. Department of Mathematics and Computer Science, Ben Gurion University of the Negev, 84105, Beer Sheva, Israel