Partitioning Topological Spaces

  • William Weiss
Part of the Algorithms and Combinatorics book series (AC, volume 5)


The study of partitions of topological spaces is a relatively new addition to Ramsey theory, but one which promises interesting things in the future. We partition topological spaces and hope to obtain a homogeneous set which is topologically relevant — for instance, a set homeomorphic to a well known topological space. We thus add something new to the ordinary partition calculus of set theory. We do, however, borrow the arrow notation. We write
$$X\, \to \,(Y)_\lambda ^n$$
to mean the following statement.


Topological Space Order Topology Countable Subset Stationary Subset Partition Relation 
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. Baumgartner, J.E. (1986): Partition relations for countable topological spaces. J. Comb. Theory, Ser. A 43, 178–195 [This includes Theorems 2.6, 3.5, 3.6 and 3.7 as well as other related information]MathSciNetzbMATHCrossRefGoogle Scholar
  2. Bregman, Yu.Kh., Shapirovskii, B.E., Shostak, A.P. (1984): On partition of topological spaces. Cas. Pestovani Mat. 109, 27–53 [This is the proof of Theorem 2.23]zbMATHGoogle Scholar
  3. Elekes, G., Erdos, P., Hajnal, A. (1978): On some partition properties of families of sets. Stud. Sci. Math. Hung. 13, 151–155 [This contains sketches of the proofs of Theorem 2.26 for regular cardinals and of Theorem 2.28, along with some related material]MathSciNetzbMATHGoogle Scholar
  4. Friedman, H. (1974): On closed sets of ordinals. Proc. Am. Math. Soc. 43, 190–192 [This contains a proof of Lemma 2.3 and some remarks concerning k → (top ω1)21]zbMATHCrossRefGoogle Scholar
  5. Friedman, H. (1975): One hundred and two problems in mathematical logic. J. Symb. Logic 40, 113–129 [ω2→ (top ω1)21 is problem number 72]zbMATHCrossRefGoogle Scholar
  6. Hajnal, A., Juhász, I., Shelah, S. (1986): Splitting strongly almost disjoint families. Trans. Am. Math. Soc. 295, 369–387 [This contains a synthesis of the proofs of Theorem 2.15 and Theorem 2.23 and gives connections to other combinatorial problems]zbMATHCrossRefGoogle Scholar
  7. Malyhin, V.I. (1979): On Ramsey spaces. Sov. Math., Dokl. 20, 894–898 [This has an independent proof of Corollary 2.16]MathSciNetGoogle Scholar
  8. Nešetřil, J., Rodl, V. (1977): Ramsey topological spaces. In: Novák, J. (ed.): General topology and its relations to modern analysis and algebra IV, Part B. Society of Czechoslovak Mathematicians and Physicists, Prague, pp. 333–337 [A successful attempt at popularizing Question 4.1]Google Scholar
  9. Nešetřil, J., Rödl, V. (1979): Partition theory and its application. In: Bollobás, B. (ed.): Surveys in combinatorics. Cambridge University Press, Cambridge, pp. 96–156 (Lond. Math. Soc. Lect. Note Ser., Vol. 38) [Discussion of the project of extending Ramsey theory to topological spaces]Google Scholar
  10. Prikry, K., Solovay, R. (1975): On partitions into stationary sets. J. Symb. Logic 40, 75–80 [The proof that V=L implies that for all k, k \(\not \to \) (topω1)21]MathSciNetCrossRefGoogle Scholar
  11. Shelah, S. (1982): Proper forcing. Springer Verlag, Berlin, Heidelberg (Lect. Notes Math., Vol. 940) [This contains the proof of Theorem 2.12, Part (i)]zbMATHCrossRefGoogle Scholar
  12. Todorčevič, S. (1983): Forcing positive partition relations. Trans. Am. Math. Soc. 280, 703–720 [This contains the proof of Theorem 4.7, Part (i) explicitly and Part (ii) implicitly]zbMATHGoogle Scholar
  13. Weiss, W. (1980): Partitioning topological spaces. In: Csaśzár, A. (ed.): Topology II. North Holland, Amsterdam, pp. 1249–1255 (Colloq. Math. Soc. János Bolyai, Vol. 23) [This contains an incorrect proof of Theorem 2.15; unfortunately the correction didn’t reach the editors in time. I would like to thank V.I. Malyhin for pointing out the gap in that proof]Google Scholar
  14. Wolfsdorf, K. (1983): Färbungen großer Würfel mit bunten Wegen. Arch. Math, 40, 569–576 [Here it is pointed out that the proof of Theorem 2.15 can be slightly modified to partition into continuum-many pieces]MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • William Weiss

There are no affiliations available

Personalised recommendations