Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T17:33:55.207Z Has data issue: false hasContentIssue false
  • English
  • Français

The Baire category theorem and cardinals of countable cofinality

Published online by Cambridge University Press:  12 March 2014

Arnold W. Miller
Arnold W. Miller*
Affiliation:
University of Wisconsin, Madison, Wisconsin 53706 University of Texas, Austin, Texas 76712
Get access Rights & Permissions [Opens in a new window]

Abstract

Let κB be the least cardinal for which the Baire category theorem fails for the real line R. Thus κB is the least κ such that the real line can be covered by κ many nowhere dense sets. It is shown that κB cannot have countable cofinality. On the other hand it is consistent that the corresponding cardinal for 2ω1 be ℵω. Similar questions are considered for the ideal of measure zero sets, other ω1, saturated ideals, and the ideal of zero-dimensional subsets of Rω1.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 2 , June 1982 , pp. 275 - 288
Copyright
Copyright © Association for Symbolic Logic 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Balcar, B., Pelant, J. and Simon, P., The space of ultrafilters on N covered by nowhere dense sets, Fundamenta Mathematicae, vol. 110 (1980), pp. 11–24.CrossRefGoogle Scholar
Baumgartner, J., Almost-disjoint sets, the dense set problem, and the partition calculus, Annals of Mathematical Logic, vol. 9 (1976), pp. 401–439.CrossRefGoogle Scholar
Bell, M., M. A. for σ-centered posets is equivalent to P(c), preprint, 1979.Google Scholar
Broughan, K., The intersection of a continuum of open dense sets, Bulletin of the Australian Mathematical Society, vol. 16 (1977), pp. 267–272.CrossRefGoogle Scholar
Burgess, J. P., Forcing, Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977.Google Scholar
Comfort, W. W. and Negrepontis, S., On families of large oscillation, Fundamenta Mathematicae, vol. 75 (1972), pp. 275–290.CrossRefGoogle Scholar
Devlin, K., Variations on ⋄, this Journal vol. 44 (1979), pp. 51–58.Google Scholar
Fedorčuk, V. V., A bicompactum whose infinite closed subsets are all n-dimensional, Mathematics of the USSR Sbornik, vol. 25 (1975), pp. 37–57.CrossRefGoogle Scholar
Fedorčuk, V. V., Fully closed mappings and the consistency of some theorems of general topology with the axioms of set theory, Mathematics of the USSR Sbornik, vol. 28 (1976), pp. 1–26.CrossRefGoogle Scholar
Feller, W., An introduction to probability theory and its applications, Wiley, New York, 1950.Google Scholar
Fleissner, W. and Kunen, K., Barely Baire spaces, Fundamenta Mathematicae, vol. 101 (1978), pp. 229–240.CrossRefGoogle Scholar
Fremlin, D. H. and Shelah, S., On partitions of the real line (abstract), Notices of the American Mathematical Society, vol. 25 (1978), p. A–600.Google Scholar
Halmos, P. R., Measure theory, Van Nostrand, Princeton, N. J., 1950.CrossRefGoogle Scholar
Hechler, S., Independence results concerning the number of nowhere dense sets necessary to cover the real line, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 24 (1973), pp. 27–32.CrossRefGoogle Scholar
Hechler, S., A dozen small uncountable cardinals, TOP072-General Topology and Its Applications, Lecture Notes in Mathematics, vol. 378, Springer-Verlag, Berlin and New York, 1974.Google Scholar
Henderson, D. W., An infinite dimensional compactum with no positive dimensional compact subsets—a simpler construction, American Journal of Mathematics, vol. 89 (1967), pp. 105–121.CrossRefGoogle Scholar
Hoffman-Jorgensen, J., and Topsoe, F., Analytic spaces and their application, Analytic sets, Academic press, London, 1980, pp. 317–403.Google Scholar
Hung, H. and Negrepontis, S., Spaces homeomorphic to (2α)α, Bulletin of the American Mathematical Society, vol. 79 (1973), pp. 143–146.CrossRefGoogle Scholar
Hurewicz, W., Uber unendlichdimensionale Punktmengen, Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae (Amsterdam), vol. 31 (1928), p. 916.Google Scholar
Hurewicz, W., Une remarque sur l'hypothèse du continu, Fundamenta Mathematicae, vol. 19 (1932), pp. 8–9.CrossRefGoogle Scholar
Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
Juhasz, I., Cardinal functions in topology, Mathematisch Centrum Amsterdam, 1971.Google Scholar
Kulpa, W. and Szymanski, A., Decompositions into nowhere dense sets, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Physiques et Astronomiques, vol. 25 (1977), pp. 37–39.Google Scholar
Kunen, K., Cohen and random real models, Handbook of set theoretic topology, NorthHolland, Amsterdam (to appear).Google Scholar
Kunen, K., Set theory, North-Holland, Amsterdam, 1980.Google Scholar
Kunen, K. and Tall, F. D., Between Martin's axiom and Souslin's hypothesis, Fundamenta Mathematicae, vol. 102 (1979), pp. 173–181.CrossRefGoogle Scholar
Kuratowski, K., Topology, vol. 1, Academic Press, New York, 1966.Google Scholar
Martin, D. A. and Solovay, R. M., Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), pp. 143–178.CrossRefGoogle Scholar
Miller, A. W., Some properties of measure and category, Transactions of the American Mathematical Society, vol. 266 (1981), pp. 93–114.CrossRefGoogle Scholar
Pincus, D. and Solovay, R. M., Definability of measures and ultrafilters, this Journal, vol. 42 (1977), pp. 179–190.Google Scholar
Pol, R. and Fuzio-Fol, E., Remarks on Cartesian products, fundamenta Mathematicae, vol. 93 (1976), pp. 57–69.CrossRefGoogle Scholar
Ross, K. A. and Stone, A. H., Products of separable spaces, American Mathematical Monthly, vol. 71 (1964), 398–403.CrossRefGoogle Scholar
Rudin, M. E., Martin's axiom, Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, pp. 491–502.CrossRefGoogle Scholar
Shelah, S., Classification theory and the number of nonisomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
Shelah, S., Club is not diamond, preprint, 1979.Google Scholar
Solovay, R. M., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 1–56.CrossRefGoogle Scholar
Solovay, R. M. and Tennenbaum, S., Iterated Cohen extensions and Souslin's problem, Annals of Mathematics, vol. 94 (1971), pp. 201–245.CrossRefGoogle Scholar
Solovay, R. M., 2ℵ0 can be anything it ought to be, The theory of models, North-Holland, Amsterdam, 1972, p. 435.Google Scholar
Stepanek, P. and Vopenka, P., Decompositions of metric spaces into nowhere dense sets, Commentationes Mathematicae Universitatis Carolinae, vol. 8 (1967), pp. 387–404.Google Scholar
Williams, Neil H., Combinatorial set theory, North-Holland, Amsterdam, 1977.Google Scholar