, Volume 35, Issue 1, pp 139–155 | Cite as

The Tukey Order and Subsets of ω 1

  • Paul Gartside
  • Ana Mamatelashvili


One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map ϕ : PQ carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients are said to be Tukey equivalent. Let X be a space and denote by \(\mathcal {K}(X)\) the set of compact subsets of X, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of \(\mathcal {K}(S)\) corresponding to various subspaces S of ω 1, their Tukey invariants, and hence the Tukey relations between them. It is shown that ω ω is a strict Tukey quotient of \({\Sigma }(\omega ^{\omega _{1}})\) and thus we distinguish between two Tukey classes out of Isbell’s ten partially ordered sets from (Isbell, J. R.: J. London Math Society 4(2), 394–416, 1972). The relationships between Tukey equivalence classes of \(\mathcal {K}(S)\), where S is a subspace of ω 1, and \(\mathcal {K}(M)\), where M is a separable metrizable space, are revealed. Applications are given to function spaces.


Tukey order Compact set Partial order Subsets of ω1 Stationary Separable metrizable space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arkhangel’skii, A.V. (ed.): General topology III – encyclopaedia of mathematical sciences 51. Springer (1995)Google Scholar
  2. 2.
    Cascales, B., Orihuela, J.: A biased view of topology as a tool in functional analysis. In: Hart, K.P., van Mill, J., Simon, P. (Eds.) Recent Progress in General Topology III, pp. 93–164. Atlantis Press (2014)Google Scholar
  3. 3.
    Cascales, B., Orihuela, J., Tkachuk, V. V.: Domination by second countable spaces and Lindelof Σ property. Topology Appl. 158, 204–214 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Christensen, J.P.R.: Topology and Borel Structure. American Elsevier, New York (1974). North-Holland, Amsterdam-LondonzbMATHGoogle Scholar
  5. 5.
    Dobrinen, N., Todorčević, S.: A new class of Ramsey-classification theorems and their application in the Tukey theory of ultrafilters, Part 1. Trans. Amer. Math. Soc. 366(3), 1659–1684 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dobrinen, N., Todorčević, S.: Tukey types of ultrafilters. Ill. J. Math. 55 (3), 907–951 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Feng, Z., Gartside, P.: Point networks for special subspaces of \(\mathbb {R}^{\kappa }\). Fund. Math. 235(3), 227–255 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Feng, Z., Gartside, P., Morgan, J.: P-paracompact and P-metrizable spaces. Topology Appl. 191, 97–118 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fremlin, D.H.: Measure theory, vol. 5. Torres Fremlin (2000)Google Scholar
  10. 10.
    Fremlin, D.H.: Families of compact sets and Tukey’s ordering. Atti Sem. Mat. Fis. Univ. Modena 39(1), 29–50 (1991)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fremlin, D.H.: The partially ordered sets of measure theory and Tukey’s ordering. Note Mat. 11, 177–214 (1991). Dedicated to the memory of Professor Gottfried KötheMathSciNetzbMATHGoogle Scholar
  12. 12.
    Gartside, P.M., Mamatelashvili, A.: Tukey order, calibres and the rationals. Submitted. Accessed 31 January 2017
  13. 13.
    Gartside, P.M., Mamatelashvili, A.: Tukey order on compact subsets of separable metric spaces. J. Symb. Log. 81(1), 181–200 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gartside, P.M., Morgan, J.: Directed sets with calibre (ω 1,ω) preparationGoogle Scholar
  15. 15.
    Gartside, P.M., Morgan, J.: Calibres, compacta and diagonals. Fund. Math. 232(1), 1–19 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Isbell, J.R.: Seven cofinal types. J. London Math Society 4(2), 394–416 (1972)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Louveau, A., Velickovic, B.: Analytic ideals and cofinal types. Ann. of Pure and Applied Logic 99, 171–195 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mamatelashvili, A.: Tukey Order on Sets of Compact Subsets of Topological Spaces. PhD thesis. Accessed 17 February 2016
  19. 19.
    Milovich, D.: Tukey classes of ultrafilters on ω Spring Topology and Dynamics Conference, Topology Proceedings 32 Spring, pp 351–362 (2008)Google Scholar
  20. 20.
    Moore, E.H., Smith, H.L.: A general theory of limits. Amer. J. of Math. 44, 102–121 (1922)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Moore, J.T., Solecki, S.: A g δ ideal of compact sets strictly above the nowhere dense ideal in the Tukey order. Ann. of Pure and Applied Logic 156, 270–273 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Raghavan, D., Todorčević, S.: Cofinal types of ultrafilters. Ann. of Pure and Applied Logic 163, 185–199 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Solecki, S., Todorčević, S.: Avoiding families and Tukey functions on the nowhere-dense ideal. J. of the Inst. of Math. Jussieu 10(2), 405–435 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Solecki, S., Todorčević, S.: Cofinal types of topological directed orders. Ann. Inst. Fourier, Grenoble 54(6), 1877–1911 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Todorčević, S.: Directed sets and cofinal types. Trans. Amer. Math. Soc. 290 (2), 711–723 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tukey, J.W. Ann. Math Studies: Convergence and Unifomity in Topology, vol. 2. Princeton University Press, Princeton (1940)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA
  2. 2.Department of Mathematics and StatisticsAuburn UniversityAuburnUSA

Personalised recommendations