Abstract
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
to mean the following statement.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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]
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]
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]
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) 12 ]
Friedman, H. (1975): One hundred and two problems in mathematical logic. J. Symb. Logic 40, 113–129 [ω2→ (top ω1) 12 is problem number 72]
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]
Malyhin, V.I. (1979): On Ramsey spaces. Sov. Math., Dokl. 20, 894–898 [This has an independent proof of Corollary 2.16]
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]
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]
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) 12 ]
Shelah, S. (1982): Proper forcing. Springer Verlag, Berlin, Heidelberg (Lect. Notes Math., Vol. 940) [This contains the proof of Theorem 2.12, Part (i)]
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]
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]
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]
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Weiss, W. (1990). Partitioning Topological Spaces. In: Nešetřil, J., Rödl, V. (eds) Mathematics of Ramsey Theory. Algorithms and Combinatorics, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-72905-8_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-72905-8_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-72907-2
Online ISBN: 978-3-642-72905-8
eBook Packages: Springer Book Archive