Israel Journal of Mathematics

, Volume 202, Issue 1, pp 195–225 | Cite as

Noetherian type in topological products

  • Menachem Kojman
  • David Milovich
  • Santi Spadaro


The cardinal invariant Noetherian type Nt(X) of a topological space X was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces.

We prove in Section 2:
  1. (1)

    There are spaces X and Y such that Nt(X×Y)< min{Nt(X), Nt(Y)}.

  2. (2)

    In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace.


The Noetherian type of the Cantor Cube of weight \({\aleph _\omega }\) with the countable box topology, \({({2^{{\aleph _\omega }}})_\delta }\), is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of \({\aleph _\omega }\). We discuss the influence of principles like \({\square _{{\aleph _\omega }}}\) and Chang’s conjecture for \({\aleph _\omega }\) on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms).

Within PCF theory we establish the existence of an (ℵ4, ℵ1)-sparse covering family of countable subsets of \({\aleph _\omega }\) (Theorem 3.20). From this follows an absolute upper bound of ℵ4 on the Noetherian type of \({({2^{{\aleph _\omega }}})_\delta }\). The proof uses a method that was introduced by Shelah in 1993 [33].


Compact Space Topological Product Dense Subspace Countable Subset Regular Cardinal 
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.


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  1. [1]
    U. Abraham and M. Magidor, Cardinal arithmetic, in Handbook of Set Theory, Springer, Dordrecht, 2010, pp. 1149–1227.CrossRefGoogle Scholar
  2. [2]
    A. V. Arhangel’skiĭ, On the metrization of topological spaces, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 8 (1960), 589–595.Google Scholar
  3. [3]
    A. V. Arhangel’skiĭ (Ed.), General Topology III: Paracompactness, Function Spaces, Descriptive Theory, Encyclopedia of the Mathematical Sciences, Vol. 3, Springer, Berlin, 1995.zbMATHGoogle Scholar
  4. [4]
    B. Bailey, δ-OIF spaces, Questions and Answers in General Topology 24 (2006), 79–84.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Z. Balogh and H. Bennett, Total paracompactness of real GO-spaces, Proceedings of the American Mathematical Society 101 (1987), 753–760.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Z. Balogh, H. Bennett, D. Burke, D. Gruenhage, D. Lutzer and J. Mashburn, OIF spaces, Questions and Answers in General Topology 18 (2000), 129–141.MathSciNetzbMATHGoogle Scholar
  7. [7]
    H. Bennett and D. Lutzer, Ordered spaces with special bases, Fundamenta Mathematicae 158 (1998), 289–299.MathSciNetzbMATHGoogle Scholar
  8. [8]
    J. Cummings, Notes on singular cardinal combinatorics, Notre Dame Journal of Formal Logic 46 (2005), 251–282.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J. Cummings, M. Foreman and M. Magidor, Squares, scales, and stationary reflection, Journal of Mathematical Logic 1 (2001), 35–98.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    D. W. Curtis, Total and absolute paracompactness, Fundamenta Mathematicae 77 (1973), 277–283.MathSciNetzbMATHGoogle Scholar
  11. [11]
    A. Dow, An introduction to applications of elementary submodels to topology, Topology Proceedings 13 (1988), 17–72.MathSciNetzbMATHGoogle Scholar
  12. [12]
    M. Foreman and M. Magidor, A very weak square principle, Journal of Symbolic Logic 62 (1997), 175–196.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    M. Foreman, M. Magidor and S. Shelah, Martin’s maximum, saturated ideals and nonregular ultrafilters. Part 1, Annals of Mathematics 127 (1988), 1–47.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S. Geschke and S. Shelah, Some notes concerning the homogeneity of boolean algebras and boolean spaces, Topology and its Applications 133 (2003), 241–253.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M. Gitik, Prikry-type forcings, in Handbook of Set Theory, Springer, Dordrecht, 2010, pp. 1351–1447.CrossRefGoogle Scholar
  16. [16]
    M. Gitik and M. Magidor, The singular cardinal hypothesis revisited, in Set Theory of the Continuum (Berkeley, CA, 1989), Mathematical Sciences Research Institutte Publications, Vol. 26, Springer, New York, 1992, pp. 243–279.CrossRefGoogle Scholar
  17. [17]
    R. W. Heath, Screenability, pointwise paracompactness, and metrization of Moore spaces, Canadian Journal of Mathematics 16 (1964), 763–770.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    I. Juhász, Cardinal Function in Topology — Ten Years Later, Mathematical Centre Tracts, Vol. 123, Mathematisch Centrum, Amsterdam, 1980.Google Scholar
  19. [19]
    I. Juhász, On two problems of A. V. Arhangel’skiĭ, General Topology and its Applications 2 (1972), 151–156.CrossRefzbMATHGoogle Scholar
  20. [20]
    M. Kojman, Exact upper bounds and their uses in set theory, Annals of Pure and Applied Logic 92 (1998), 267–282.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    M. Kojman, A short proof of the PCF theorem, preprint.Google Scholar
  22. [22]
    A. Lelek, Some cover properties of spaces, Fundamenta Mathematicae 64 (1969), 209–218.MathSciNetzbMATHGoogle Scholar
  23. [23]
    J. P. Levinski, M. Magidor and S. Shelah, Chang’s conjecture for \({\aleph _\omega }\), Israel Journal of Mathematics 69 (1990), 161–172.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    M. Magidor and S. Shelah, When does almost free imply free? (For groups, transversals, etc.), Journal of the American Mathematical Society 7 (1994), 769–830.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    V. I. Malykhin, On Noetherian spaces, American Mathematical Society Translations 134 (1987), 83–91.zbMATHGoogle Scholar
  26. [26]
    D. Milovich, Noetherian types of homogeneous compacta and dyadic compacta, Topology and its Applications 156 (2008), 443–464.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    D. Milovich, Splitting families and the Noetherian type of βω \ ω, Journal of Symbolic Logic 73 (2008), 1289–1306.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    S. A. Peregudov, On the Noetherian type of topological spaces, Commentationes Mathematicae Universitatis Carolinae 38 (1997), 581–586.MathSciNetzbMATHGoogle Scholar
  29. [29]
    S. A. Peregudov and B. É. Šhapirovskiĭ, A class of compact spaces, Soviet Math. Dokl. 17 (1976), no. 5, 1296–1300.zbMATHGoogle Scholar
  30. [30]
    A. Sharon and M. Viale, Some consequences of reflection on the approachability ideal, Transactions of the American Mathematical Society 362 (2010), 4201–4212.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    S. Shelah, Cardinal Arithmetic, Oxford University Press, 1994.Google Scholar
  32. [32]
    S. Shelah, More on countably compact, locally countable spaces, Israel Journal of Mathematics 62 (1988), 302–310.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    S. Shelah, Advances in cardinal arithmetic, in Finite and Infinite Combinatorics in Sets and Logic (Banff, AB, 1991), NATO Advanced Science Institutes Series C: Mathematical and Physical Scirnces, Vol. 411, Kluwer Academic, Dordrecht, 1993, 355–383.CrossRefGoogle Scholar
  34. [34]
    S. Shelah, Non-reflection of the bad set for \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}\to {I} _\theta }[\lambda ]\) and pcf, Acta Mathematica Hungarica, in press. Preprint sh:1008 in Shelah’s archive.Google Scholar
  35. [35]
    L. Soukup, A note on Noetherian type of spaces, arXiv:1003.3189.Google Scholar
  36. [36]
    R. Telgárksy, C-scattered and paracompact spaces, Fundamenta Mathematicae 73 (1971/1972), 59–74.MathSciNetGoogle Scholar
  37. [37]
    S. Todorcevic, Directed sets and cofinal types, Transactions of the American Mathematical Society 290 (1985), 711–723.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2014

Authors and Affiliations

  • Menachem Kojman
    • 1
  • David Milovich
    • 2
  • Santi Spadaro
    • 3
  1. 1.Department of MathematicsBen-Gurion University of the NegevBe’er ShevaIsrael
  2. 2.Department of Engineering, Mathematics and PhysicsTexas A&M International UniversityLaredoUSA
  3. 3.Department of Mathematics and Statistics, Faculty of Science and EngineeringYork UniversityTorontoCanada

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