Advertisement

Topological Digital Topology

  • Ralph Kopperman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2886)

Abstract

The usefulness of topology in science and mathematics means that topological spaces must be studied, and computers should be used in this study. We discuss how many useful spaces (including all compact Hausdorff spaces) can be approximated by finite spaces, and these finite spaces are completely determined by their specialization orders. As a special case, digital n-space, used to interpret Euclidean n-space and in particular, the computer screen, is also dealt with in terms of the specialization. Indeed, algorithms written using the specialization are comparable in difficulty, storage usage and speed to those which use the traditional (8,4), (4,8) and (6,6) adjacencies, and are of course completely representative of the spaces.

Keywords

Digital topology general topology T0-space specialization (order) connected ordered topological space (COTS) Alexandroff space Khalimsky line digital n-space metric and polyhedral analogs chaining maps calming maps normalizing maps inverse limit Hausdorff reflection skew (=stable) compactness (graph) path and arc connectedness and components (topological) adjacency Jordan curve robust scene cartoon 

References

  1. 1.
    Alexandroff, P.S.: Untersuchungen über Gestalt und Lage abgeschlossener Mengen beliebeger Dimension. Annals Math. 30, 101–187 (1928-1929)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Alexandroff, P.S.: Diskrete Räume. Mat. Sbornik 2(44), 501–519 (1937)zbMATHGoogle Scholar
  3. 3.
    Flachsmeyer, J.: Zur Spektralentwicklung topologischer Räume. Math. Ann. 144, 253–274 (1961)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Freudenthal, H.: Entwicklungen von Räumen und ihren Gruppen. Compositio Math 4, 154–234 (1937)Google Scholar
  5. 5.
    Herman, G.T.: Geometry of digital spaces. Birkhäuser, Basel (1998)zbMATHGoogle Scholar
  6. 6.
    Hocking, J., Young, G.: Topology. Addison Wesley, Reading (1961)zbMATHGoogle Scholar
  7. 7.
    Khalimsky, E., Kopperman, R.D., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topology and its Appl. 36, 1–17 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Khalimsky, E., Kopperman, R.D., Meyer, P.R.: Boundaries in Digital Planes. Journal of Applied Mathematics and Stochastic Analysis 3, 27–55 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kong, T.Y.: The Khalimsky topologies are precisely those simply-connected topologies on Zn whose connected sets include all 2n-connected sets but no (3n–1)- disconnected sets. Theoretical Computer Science (to appear)Google Scholar
  10. 10.
    Kong, T.Y., Khalimsky, E.: Polyhedral analogs of locally finite topological spaces. In: Shortt, R.M. (ed.) General Topology and Applications: Proceedings of the 1988 Northeast Conference, Marcel Dekker, pp. 153–164 (1990)Google Scholar
  11. 11.
    Kong, T.Y., Kopperman, R.D., Meyer, P.R.: A Topological Approach to Digital Topology. Am. Math. Monthly 98, 901–917 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kong, T.Y., Rosenfeld, A.: “Digital topology: Introduction and survey. Computer Vision, Graphics, and Image Processing 48, 357–393 (1989)CrossRefGoogle Scholar
  13. 13.
    Kopperman, R.D.: The Khalimsky Line as a Foundation for Digital Topology. In: Ying-Lie, O., et al. (eds.) Shape in Picture, vol. F-126, pp. 3–20. Springer, Heidelberg (1994)Google Scholar
  14. 14.
    Kopperman, R.D., Tkachuk, V.V., Wilson, R.G.: The approximation of compacta by finite T0-spaces. Quaestiones Math (to appear)Google Scholar
  15. 15.
    Kopperman, R.D., Wilson, R.G.: Finite approximation of compact Hausdorff spaces. Topology Proceedings 22, 175–201 (1999)MathSciNetGoogle Scholar
  16. 16.
    Kopperman, R.D., Wilson, R.G.: On the role of finite, hereditarily normal spaces and maps in the genesis of compact Hausdorff spaces. In: Topology and its Appl. (to appear)Google Scholar
  17. 17.
    Mardešić, S.: Approximating topological spaces by polyhedra. In: Ferrera, J., López-Gómez, J., del Portal, F.R.R. (eds.) Approximation Theory and its Applications. Nova Science Publishers, New YorkGoogle Scholar
  18. 18.
    Morris, S.A.: Topology Without Tears, Available from author’s web siteGoogle Scholar
  19. 19.
    Simmons, G.F.: Introduction to Topology and Modern Analysis, Krieger, Malabar, FL (1983)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Ralph Kopperman
    • 1
  1. 1.Department of MathematicsCity College of New YorkNew YorkUSA

Personalised recommendations