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Order- and graph-theoretic investigation of dimensions of finite topological spaces and Alexandroff spaces

  • Rudolf Berghammer
  • Michael WinterEmail author
Article
  • 17 Downloads

Abstract

Using concepts from order theory and graph theory we investigate the dimensions \( ind \), \( Ind \) and \( dim \) of finite topological spaces and Alexandroff spaces. We present specifications of them by means of specialisation pre-orders and algorithms for their computation. For finite spaces we give sharp upper bounds, characterisations of maximal-dimensional spaces via specialisation pre-orders and determine the number of maximal-dimensional spaces on a given set and whether these spaces are homeomorphic. These questions are also investigated for zero-dimensional Alexandroff spaces. We also consider relationships between the dimensions \( ind \), \( Ind \) and \( dim \).

Keywords

Topological space Alexandroff space Dimension Specialisation pre-order Linear order Flat order Hasse-diagram Linear directed binary tree 

Mathematics Subject Classification

54F45 06A06 

Notes

References

  1. 1.
    Alexandroff, P.: Diskrete Räume. Matematicheskij Sbornik NS 2, 501–518 (1937)zbMATHGoogle Scholar
  2. 2.
    Arenas, F.G.: Alexandroff spaces. Acta Mathematica Universitatis Comenianae 68, 17–25 (1999)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Barmak, J.A.: Algebraic Topology of Finite Topological Spaces and Applications. Lecture Notes in Mathematics, vol. 2032. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bass, D.W.: Dimension and finite spaces. J. Lond. Math. Soc. 44, 159–162 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bass, D.W.: Covering dimension and quasi-order spaces. J. Lond. Math. Soc. s2–1(1), 505–508 (1969)Google Scholar
  6. 6.
    Berghammer, R., Leoniuk, B., Milanese, U.: Implementation of relation algebra using binary decision diagrams. In: de Swart, H. (ed.) Relational Methods in Computer Science. Lecture Notes in Computer Science, vol. 2561, pp. 241–257. Springer, New York (2002)CrossRefGoogle Scholar
  7. 7.
    Berghammer, R., Neumann, F.: RelView—an OBDD-based Computer Algebra system for relations. In: Gansha, V.G., Mayr, E.W., Vorozhtsov, E. (eds.) Computer Algebra in Scientific Computing. Lecture Notes in Computer Science, vol. 3718, pp. 40–51. Springer, New York (2005)CrossRefGoogle Scholar
  8. 8.
    Berghammer, R., Winter, M.: Solving computational tasks on finite topologies by means of relation algebra and the RelView tool. J. Log. Algebraic Methods Program. 88, 1–25 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berghammer, R.: Tool-based relational investigation of closure-interior-relatives for finite topologies. In: Höfner, P., Pous, D., Struth, G. (eds.) Relational and Algebraic Methods in Computer Science. Lecture Notes in Computer Science, vol. 10226, pp. 60–77. Springer, New York (2017)Google Scholar
  10. 10.
    Birkhoff, G.: Lattice theory, revised edn. AMS Colloquium Publications, vol. XXV, American Mathematical Society, Providence (1961)Google Scholar
  11. 11.
    Bourbaki, N.: Elements of Mathematics: General Topology. Addison-Wesley, Boston (1966)zbMATHGoogle Scholar
  12. 12.
    Brouwer, L.G.J.: Über den natürlichen Dimensionsbegriff. Journal für die reine und angewandte Mathematik 142, 146–152 (1913)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Čech, E.: Sur la the’orie de la dimension. Comptes Rendus de l’Académie des Sciences 193, 976–977 (1931)zbMATHGoogle Scholar
  14. 14.
    Čech, E.: Přispěvek k theorii dimense. Časopis pro pěstováni matematiky a fysiky 62, 277–291 (1933). French translation in: E. Čech: Topological papers, Academia Publishing House of the Czechoslovak Academy of Sciences, pp. 129–142 (1968)Google Scholar
  15. 15.
    Coornaert, M.: Topological Dimension and Dynamical Systems. Springer, New York (2015)CrossRefzbMATHGoogle Scholar
  16. 16.
    Dushnik, B., Miller, E.W.: Partially ordered sets. Am. J. Math. 63(3), 600–610 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Engelking, R.: Dimension Theory. North-Holland Mathematical Library, vol. 19. North-Holland, Amsterdam (1978)Google Scholar
  18. 18.
    Erné, M., Stege, K.: Counting finite posets and topologies. Order 8, 247–265 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Evaco, A.V., Kopperman, R., Mukhin, Y.V.: Dimensional properties of graphs and digital spaces. J. Math. Imaging Vis. 6, 109–119 (1996)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Georgiou, D.N., Megaritis, A.C.: Covering dimension and finite spaces. Appl. Math. Comput. 218, 3122–3130 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Georgiou, D.N., Megaritis, A.C.: An algorithm of polynomial order for computing the covering dimension of a finite space. Appl. Math. Comput. 231, 276–283 (2014)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Georgiou, D.N., Megaritis, A.C., Moshokoa, S.P.: Small inductive dimension and Alexandroff topological spaces. Topol. Appl. 168, 103–119 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Georgiou, D.N., Megaritis, A.C., Moshokoa, S.P.: A computing procedure for the small inductive dimension of a finite \(T_0\)-space. Comput. Appl. Math. 34, 401–415 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Georgion, D.N., Megaritis, A.C., Moshokoa, S.P.: Finite spaces: a reduction algorithm for the computation of the small inductive dimension of a finite \(T_0\)-space. Comput. Appl. Math. 36, 791–803 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hausdorff, F.: Dimension und äußeres Maß. Mathematische Annalen 79, 157–179 (1919)CrossRefzbMATHGoogle Scholar
  26. 26.
    Kelley, J.L.: General Topology. Springer, Berlin (1975)zbMATHGoogle Scholar
  27. 27.
    Lebesque, H.: Sur la non applicabilité de deux domaines appartenant respectivement á des espaces, de \(n\) et \(n+p\) dimensions. Mathematische Annalen 70, 166–168 (1911)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    May, J.P.: Finite Topological Spaces. University of Chicago, Notes for REU (2003)Google Scholar
  29. 29.
    Menger, K.: Über die Dimensionalität von Punktmengen. Monatshefte für Mathematik und Physik 33, 148–160 (1923)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Menger, K.: Über die Dimension von Punktmengen, II. Teil. Monatshefte für Mathematik und Physik 34, 137–161 (1926)CrossRefzbMATHGoogle Scholar
  31. 31.
    Stong, R.E.: Finite topological spaces. Trans. Am. Math. Soc. 123, 325–340 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1, 146–160 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Uryson, P.: Les multiplicités cantoriennes. Comptes Rendus de l’Académie des Sciences 175, 440–442 (1922)zbMATHGoogle Scholar
  34. 34.
    Wiederhold, P., Wilson, R.G.: Dimension for Alexandrov spaces. In: Melter, R.A., Wu, A.Y. (eds.): Vision Geometry Proceedings of Society Photo-Optical Instrumentation Engineers, vol. 1832, pp. 13–22 (1992)Google Scholar
  35. 35.

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für InformatikChristian-Albrechts-Universität zu KielKielGermany
  2. 2.Department of Computer ScienceBrock UniversitySaint CatharinesCanada

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