CantorBendixson degrees and convexity in ℝ^{2}
 Menachem Kojman
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We present an ordinal rank, δ^{3}, which refines the standard classification of nonconvexity among closed planar sets. The class of closed planar sets falls into a hierarchy of order type ω_{1} + 1 when ordered by δrank.
The rank δ^{3} (S) of a setS is defined by means of topological complexity of 3cliques in the set. A 3clique in a setS is a subset ofS all of whose unordered 3tuples fail to have their convex hull inS. Similarly, δ^{n} (S) is defined for alln>1.
The classification cannot be done using δ^{2}, which considers only 2cliques (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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 Title
 CantorBendixson degrees and convexity in ℝ^{2}
 Journal

Israel Journal of Mathematics
Volume 121, Issue 1 , pp 8591
 Cover Date
 20011201
 DOI
 10.1007/BF02802497
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
 Additional Links
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 Authors

 Menachem Kojman ^{(1)}
 Author Affiliations

 1. Department of Mathematics and Computer Science, Ben Gurion University of the Negev, 84105, Beer Sheva, Israel