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Convenient Closure Operators on \(\mathbb Z^2\)

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Combinatorial Image Analysis (IWCIA 2009)

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We discuss closure operators on \(\mathbb Z^2\) with respect to which some cycles in a certain natural graph with the vertex set \(\mathbb Z^2\) are Jordan curves. We deal with several Alexandroff T 0-pretopologies and topologies and also one closure operator that is not a pretopology.

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  1. Čech, E.: Topological Spaces. In: Topological Papers of Eduard Čech, pp. 436–472. Academia, Prague (1968)

    Google Scholar 

  2. Čech, E.: Topological Spaces (Revised by Frolík, Z., Katětov, M.). Academia, Prague (1966)

    Google Scholar 

  3. Eckhardt, U., Latecki, L.J.: Topologies for the digital spaces ℤ2 and ℤ3. Comput. Vision Image Understanding 90, 295–312 (2003)

    Article  MATH  Google Scholar 

  4. Engelking, R.: General Topology. Państwowe Wydawnictwo Naukowe, Warszawa (1977)

    Google Scholar 

  5. Khalimsky, E.D., Kopperman, R., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topology Appl. 36, 1–17 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Khalimsky, E.D., Kopperman, R., Meyer, P.R.: Boundaries in digital planes. Jour. of Appl. Math. and Stoch. Anal. 3, 27–55 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kiselman, C.O.: Digital Jordan curve theorems. In: Nyström, I., Sanniti di Baja, G., Borgefors, G. (eds.) DGCI 2000. LNCS, vol. 1953, pp. 46–56. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  8. Kong, T.Y., Kopperman, R., Meyer, P.R.: A topological approach to digital topology. Amer. Math. Monthly 98, 902–917 (1991)

    Article  MathSciNet  Google Scholar 

  9. Kopperman, R., Meyer, P.R., Wilson, R.G.: A Jordan surface theorem for three-dimensional digital spaces. Discr. and Comput. Geom. 6, 155–161 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Marcus, D., et al.: A special topology for the integers (Problem 5712). Amer. Math. Monthly 77, 1119 (1970)

    Article  MathSciNet  Google Scholar 

  11. Rosenfeld, A.: Digital topology. Amer. Math. Monthly 86, 621–630 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  12. Rosenfeld, A.: Picture Languages. Academic Press, New York (1979)

    MATH  Google Scholar 

  13. Šlapal, J.: A digital analogue of the Jordan curve theorem. Discr. Appl. Math. 139, 231–251 (2004)

    Article  MATH  Google Scholar 

  14. Šlapal, J.: A quotient-universal digital topology. Theor. Comp. Sci. 405, 164–175 (2008)

    Article  MATH  Google Scholar 

  15. Šlapal, J.: Closure operations for digital topology. Theor. Comp. Sci. 305, 457–471 (2003)

    Article  MATH  Google Scholar 

  16. Šlapal, J.: Digital Jordan curves. Top. Appl. 153, 3255–3264 (2006)

    Article  MATH  Google Scholar 

  17. Šlapal, J.: Relational closure operators. In: Contributions to General Algebra 16, pp. 251–259. Verlag Johannes Heyn, Klagenfurt (2005)

    Google Scholar 

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Šlapal, J. (2009). Convenient Closure Operators on \(\mathbb Z^2\) . In: Wiederhold, P., Barneva, R.P. (eds) Combinatorial Image Analysis. IWCIA 2009. Lecture Notes in Computer Science, vol 5852. Springer, Berlin, Heidelberg.

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10208-0

  • Online ISBN: 978-3-642-10210-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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