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.
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S. Ben-David and M. Magidor, The weak □* is really weaker than the full □, The Journal of Symbolic Logic 51 (1986), 1029–1033.
H.-D. Donder, Regularity of ultrafilters and the core model, Israel Journal of Mathematics 63 (1988), 289–322.
A. Dow, A. V. Gubbi and A. Szymanski, Rigid Stone Spaces within ZFC, Proceedings of the American Mathematical Society 102 (1988), 745–748.
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.
A. G. Elkin, Resolvable spaces which are not maximally resolvable, Vestnik Moskovskogo Universiteta, Seriya I. Matematika, Mekhanika 24 (1969), 66–70.
I. Juhász, L. Soukup and Z. Szentmiklóssy, Resolvability and monotone normality, Israel Journal of Mathematics 166 (2008), 1–16.
K. Kunen and K. Prikry, On descendingly incomplete ultrafilters, The Journal of Symbolic Logic 36 (1971), 650–652.
M. Magidor, On the singular cardinals problem I, Israel Journal of Mathematics 28 (1977), 1–31.
K. L. Prikry, Changing measurable into accessible cardinals, Dissertationes Mathematicae (Rozprawy Matematyczne) 68 (1970), 55 pp.
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.
E. G. Pytkeev, Maximally resolvable spaces, Trudy Matematicheskogo Instituta Imeni V. A. Steklova 154(B) (1983), 209–213.
E. Schimmerling and M. Zeman, Square in core models, The Bulletin of Symbolic Logic 7 (2001), 305–314.
W. H. Woodin, Descendingly complete ultrafilter on N ω, Personal communication.
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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.
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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
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DOI: https://doi.org/10.1007/s11856-012-0042-z