Skip to main content
SpringerLink
Log in

On the maximal resolvability of monotonically normal spaces

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We continue the work started in [6] and show that all monotonically normal (in short: MN) spaces are maximally resolvable if and only if all uniform ultrafilters are maximally decomposable. As a consequence we get that the existence of an MN space which is not maximally resolvable is equi-consistent with the existence of a measurable cardinal. We also show that it is consistent (modulo the consistency of a measurable cardinal) that there is an MN space X with |X| = Δ(X) = ℵ ω which is not ω 1-resolvable. It follows from the results of [6] that this is best possible.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Ben-David and M. Magidor, The weak □* is really weaker than the full □, The Journal of Symbolic Logic 51 (1986), 1029–1033.

    Article  MathSciNet  MATH  Google Scholar 

  2. H.-D. Donder, Regularity of ultrafilters and the core model, Israel Journal of Mathematics 63 (1988), 289–322.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Dow, A. V. Gubbi and A. Szymanski, Rigid Stone Spaces within ZFC, Proceedings of the American Mathematical Society 102 (1988), 745–748.

    MathSciNet  MATH  Google Scholar 

  4. A. Dow, M. G. Tkachenko, V. V. Tkachuk and R. G. Wilson, Topologies generated by discrete subspaces, Glasnik Matematički, Serija III 37 (2002), 187–210.

    MathSciNet  MATH  Google Scholar 

  5. A. G. Elkin, Resolvable spaces which are not maximally resolvable, Vestnik Moskovskogo Universiteta, Seriya I. Matematika, Mekhanika 24 (1969), 66–70.

    MathSciNet  Google Scholar 

  6. I. Juhász, L. Soukup and Z. Szentmiklóssy, Resolvability and monotone normality, Israel Journal of Mathematics 166 (2008), 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  7. K. Kunen and K. Prikry, On descendingly incomplete ultrafilters, The Journal of Symbolic Logic 36 (1971), 650–652.

    Article  MathSciNet  Google Scholar 

  8. M. Magidor, On the singular cardinals problem I, Israel Journal of Mathematics 28 (1977), 1–31.

    Article  MathSciNet  MATH  Google Scholar 

  9. K. L. Prikry, Changing measurable into accessible cardinals, Dissertationes Mathematicae (Rozprawy Matematyczne) 68 (1970), 55 pp.

  10. K. Prikry, On descendingly complete ultrafilters, in Cambridge Summer School in Mathematical Logic (1971), Lecture Notes in Mathematics, Vol. 337, Springer, Berlin, 1973, pp. 459–488.

    Chapter  Google Scholar 

  11. E. G. Pytkeev, Maximally resolvable spaces, Trudy Matematicheskogo Instituta Imeni V. A. Steklova 154(B) (1983), 209–213.

    MathSciNet  Google Scholar 

  12. E. Schimmerling and M. Zeman, Square in core models, The Bulletin of Symbolic Logic 7 (2001), 305–314.

    Article  MathSciNet  MATH  Google Scholar 

  13. W. H. Woodin, Descendingly complete ultrafilter on N ω, Personal communication.

Download references

Author information

Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, 1053, Budapest, Hungary

    Istvan Juhász

  2. Einstein Institute of Mathematics, Hebrew University of Jerusalem Givat Ram, Jerusalem, 91904, Israel

    Menachem Magidor

Authors
  1. Istvan Juhász
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Menachem Magidor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Istvan Juhász.

Additional information

The first author was partially supported by OTKA grant no. 68262.

Both authors would like to thank the Mittag-Leffler Institute where the research on this paper was started in the fall semester of 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Juhász, I., Magidor, M. On the maximal resolvability of monotonically normal spaces. Isr. J. Math. 192, 637–666 (2012). https://doi.org/10.1007/s11856-012-0042-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-012-0042-z

Keywords

Search

Navigation