Israel Journal of Mathematics

, Volume 121, Issue 1, pp 85–91

Cantor-Bendixson degrees and convexity in ℝ2

  • Menachem Kojman

DOI: 10.1007/BF02802497

Cite this article as:
Kojman, M. Isr. J. Math. (2001) 121: 85. doi:10.1007/BF02802497


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.

Copyright information

© Hebrew University 2001

Authors and Affiliations

  • Menachem Kojman
    • 1
  1. 1.Department of Mathematics and Computer ScienceBen Gurion University of the NegevBeer ShevaIsrael