Digital Jordan Curve Theorems

  • Christer O. Kiselman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1953)


Efim Khalimsky’s digital Jordan curve theorem states that the complement of a Jordan curve in the digital plane equipped with the Khalimsky topology has exactly two connectivity components. We present a new, short proof of this theorem using induction on the Euclidean length of the curve. We also prove that the theorem holds with another topology on the digital plane but then only for a restricted class of Jordan curves.


Topological Space Jordan Curve Euclidean Plane Adjacent Point Connected Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Aleksandrov, P. S. 1937, Diskrete Räume. Mat. Sb. 2 (44), 501–519.Google Scholar
  2. 2.
    Bourbaki, Nicolas 1961, Topologie générale. Éléments de mathématique, première partie, livre III, chapitres 1 & 2. Third edition. Paris: Hermann.Google Scholar
  3. 3.
    Halimskiı, E. D. 1970, Applications of connected ordered topological spaces in topology. Conference of Math. Departments of Povolsia.Google Scholar
  4. 5.
    Khalimsky, Efim; Kopperman, Ralph; Meyer, Paul R. 1990, Computer graphics and connected topologies on finite ordered sets. Topology Appl. 36, 1–17.Google Scholar
  5. 6.
    Kong, Yung; Kopperman, Ralph; Meyer, Paul R. 1991, A topological approach to digital topology. Amer. Math. Monthly 98, 901–917.Google Scholar
  6. 7.
    Rosenfeld, Azriel 1979, Digital topology. Amer. Math. Monthly 86, 621–630.Google Scholar
  7. 8.
    Wyse, Frank, et al. 1970, Solution to problem 5712. Amer. Math. Monthly 77, 1119.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Christer O. Kiselman
    • 1
  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden

Personalised recommendations