Israel Journal of Mathematics

, Volume 166, Issue 1, pp 1–16 | Cite as

Resolvability and monotone normality

  • István Juhász
  • Lajos Soukup
  • Zoltán Szentmiklóssy


A space X is said to be κ-resolvable (resp., almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp., almost disjoint over the ideal of nowhere dense subsets). X is maximally resolvable if and only if it is Δ(X)-resolvable, where Δ(X) = min{|G| : G\(\not 0\) open}.

We show that every crowded monotonically normal (in short: MN) space is ω-resolvable and almost μ-resolvable, where μ = min{2ω, ω2}. On the other hand, if κ is a measurable cardinal then there is a MN space X with Δ(X) = κ such that no subspace of X is ω1-resolvable.

Any MN space of cardinality < ℵω is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space X with |X| = Δ(X) = ℵω such that no subspace of X is ω2-resolvable.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Ben-David and M. Magidor, The weak □* is really weaker than the full □, Journal of Symbolic Logic 51 (1986), 1029–1033.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    J. Ceder and T. Pearson, On products of maximally resolvable spaces, Pacific Journal of Mathematics 22 (1967), 31–45.MATHMathSciNetGoogle Scholar
  3. [3]
    A. Dow, M. G. Tkachenko, V. V. Tkachuk, and R. G. Wilson, Topologies generated by discrete subspaces, Glasnik Matematički. Serija III 37(57) (2002), 187–210.MATHMathSciNetGoogle Scholar
  4. [4]
    F. W. Eckertson, Resolvable, not maximally resolvable spaces, Topology and its Applications 79 (1997), 1–11.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    A. G. El’kin, Resolvable spaces which are not maximally resolvable, Vestnik Moskovskogo Universiteta. Seriya I. Matematika. Mekhanika 24 (1969), 66–70.MATHMathSciNetGoogle Scholar
  6. [6]
    L. Hegedűs, Szűrők és Lyukak, Master’s Thesis, in Hungarian, Eötvös Loránd University, Budapest, 1998.Google Scholar
  7. [7]
    E. Hewitt, A problem of set theoretic topology, Duke Mathematical Journal 10 (1943), 309–333.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    I. Juhász, L. Soukup, and Z. Szentmiklóssy, \(\mathcal{D}\)-forced spaces: a new approach to resolvability, Topology and its Applications 153 (2006), 1800–1824.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    I. Juhász, L. Soukup, and Z. Szentmiklóssy, Resolvability of spaces having small spread or extent, Topology and its Applications 154 (2007), 144–154.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    K. Kunen and K. Prikry, On descendingly incomplete ultrafilters, Journal of Symbolic Logic 36 (1971), 650–652.CrossRefMathSciNetGoogle Scholar
  11. [11]
    M. Magidor, private communication.Google Scholar
  12. [12]
    O. Pavlov, On resolvability of topological spaces, Topology and its Applications 126 (2002), 37–47.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    S. W. Williams and H. X. Zhou, Strong versions of normality, in General topology and applications (Staten Island, NY, 1989), Lecture Notes in Pure and Applied Mathematics, 134, Dekker, New York, 1991, pp. 379–389.Google Scholar

Copyright information

© Hebrew University Magnes Press 2008

Authors and Affiliations

  • István Juhász
    • 1
  • Lajos Soukup
    • 1
  • Zoltán Szentmiklóssy
    • 2
  1. 1.Alfréd Rényi Institute of MathematicsBudapestHungary
  2. 2.Department of AnalysisEötvös University of BudapestBudapestHungary

Personalised recommendations