Israel Journal of Mathematics

, Volume 137, Issue 1, pp 321–354 | Cite as

On the structure ofn-point sets

  • Khalid Bouhjar
  • Jan J. Dijkstra


Letn be an integer greater than one. Our main result, called the “Structure Theorem” is that a set that containsn−1 disjoint continua that are cut by a single line cannot be ann-point set, that is, a set that meets every line in preciselyn points. This theorem unifies and significantly improves upon a number of known theorems. The second part of the paper is devoted to several theorems that address the question when a set that meets every line in at mostn points can be extended to ann-point set. These theorems also highlight the sharpness of the Structure Theorem.


Structure Theorem London Mathematical Society Dense Open Subset Nonempty Open Subset Countable Dense 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Bagemihl,A theorem on intersections of prescribed cardinality, Annals of Mathematics55 (1952), 34–37.CrossRefMathSciNetGoogle Scholar
  2. [2]
    K. Bouhjar,On the Structure of n-Point Sets, Doctoral dissertation, Vrije Universiteit, Amsterdam, 2002.Google Scholar
  3. [3]
    K. Bouhjar, J. J. Dijkstra and R. D. Mauldin,No n-point set is σ-compact, Proceedings of the American Mathematical Society129 (2001), 621–622.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    K. Bouhjar, J. J. Dijkstra and J. van Mill,Three-point sets, Topology and its Applications112 (2001), 215–227.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    J. J. Dijkstra,Generic partial two-point sets are extendable, Canadian Mathematical Bulletin42 (1999), 46–51.zbMATHMathSciNetGoogle Scholar
  6. [6]
    J. J. Dijkstra, K. Kunen and J. van Mill,Hausdorff measures and two-point set extensions, Fundamenta Mathematicae157 (1998), 43–60.zbMATHMathSciNetGoogle Scholar
  7. [7]
    J. J. Dijkstra and J. van Mill,Two point set extensions — a counterexample, Proceedings of the American Mathematical Society125 (1997), 2501–2502.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    J. J. Dijkstra and J. van Mill,On sets that meet every hyperplane in n-space in at most n points, The Bulletin of the London Mathematical Society34 (2002), 361–368.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    R. Dougherty, unpublished.Google Scholar
  10. [10]
    R. Engelking,Dimension Theory, PWN, Warsaw, 1977.Google Scholar
  11. [11]
    D. L. Fearnley, L. Fearnley and J. W. Lamoreaux,Every three-point set is zero-dimensional, Proceedings of the American Mathematical Society131 (2003), 2241–2245.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    J. Kulesza,A two-point set must be zero-dimensional, Proceedings of the American Mathematical Society116 (1992), 551–553.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    C. Kuratowski and St. Ulam,Quelques propriétés topologiques du produit combinatoire, Fundamenta Mathematicae19 (1932), 247–251.zbMATHGoogle Scholar
  14. [14]
    D. G. Larman,A problem of incidence, Journal of the London Mathematical Society43 (1968), 407–409.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    R. D. Mauldin,On sets which meet each line in exactly two points, The Bulletin of the London Mathematical Society30 (1998), 397–403.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    S. Mazurkiewicz,O pewnej mnogości płaskiej, która ma z każda prosta dwa i tylko dwa punkty wspólne (Polish), Comptes Rendus des Séances de la Société des Sciences et Lettres de Varsovie7 (1914), 382–384; French transl.:Sur un ensemble plan qui a avec chaque droite deux et seulement deux points communs, in: Stefan Mazurkiewicz,Traveaux de Topologie et ses Applications (K. Borsuk et al., eds.), PWN, Warsaw, 1969, pp. 46–47.Google Scholar
  17. [17]
    K. Menger,Kurventheorie, Teubner, Leipzig, 1932.Google Scholar
  18. [18]
    M. E. Rudin,Martin’s Axiom, inHandbook of Mathematical Logic (J. Barwise, ed.), North-Holland, Amsterdam, 1977, pp. 491–501.Google Scholar
  19. [19]
    W. Sierpiński,Une généralisation des théorèmes de S. Mazurkiewicz et F. Bagemihl, Fundamenta Mathematicae40 (1953), 1–2.MathSciNetGoogle Scholar

Copyright information

© Hebrew University 2003

Authors and Affiliations

  • Khalid Bouhjar
    • 1
  • Jan J. Dijkstra
    • 2
  1. 1.Afdeling WiskundeVrije UniversiteitAmsterdamThe Netherlands
  2. 2.Afdeling WiskundeVrije UniversiteitAmsterdamThe Netherlands

Personalised recommendations