Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T12:37:06.330Z Has data issue: false hasContentIssue false
  • English
  • Français

Borel's conjecture in topological groups

Published online by Cambridge University Press:  12 March 2014

Fred Galvin  and
Marion Scheepers
Fred Galvin
Affiliation:
Department of Mathematics, The University of Kansas, Lawrence, KS 66045, USA, E-mail: [email protected]
Marion Scheepers
Affiliation:
Department of Mathematics, Boise State University, Boise, ID 83725, USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number κ, let BCκ denote this generalization. Then BCℕ0 is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, ¬BCℕ1 is equivalent to the existence of a Kurepa tree of height ℕ1. Using the connection of BCκ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results:

(1) If it is consistent that there is a 1-inaccessible cardinal then it is consistent that BCℕ1.

(2) If it is consistent that BCℕ1, then it is consistent that there is an inaccessible cardinal.

(3) If it is consistent that there is a 1-inaccessible cardinal with ω inaccessible cardinals above it, then ¬BCℕω + (∀n < ω)BCℕn is consistent.

(4) If it is consistent that there is a 2-huge cardinal, then it is consistent that BCℕω

(5) If it is consistent that there is a 3-huge cardinal, then it is consistent that BCκ for a proper class of cardinals κ of countable cofinality.

Keywords

Rothberger boundedBorel ConjectureKurepa HypothesisChang's Conjecturen-huge cardinal
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 1 , March 2013 , pp. 168 - 184
Copyright
Copyright © Association for Symbolic Logic 2013

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

[1]Barbanel, J. B., Di Prisco, C. A., and Tan, I. B., Many-times huge and superhuge cardinals, this Journal, vol. 49 (1984), no. 1, pp. 112–122.Google Scholar
[2]Bartoszynski, T. and Judah, H., Strong measure zero sets, Israel Mathematical Conference Proceedings, vol. 6 (1993), pp. 13–62.Google Scholar
[3]Baumgartner, J. E., Iterated forcing, Surveys in set theory (Mathias, A. R. D., editor), London Mathematical Society Lecture Notes, vol. 87, Cambridge University Press, 1983, pp. 1–59.Google Scholar
[4]Baumgartner, J. E. and Taylor, A. D., Saturation properties of ideals in generic extensions, II, Transactions of the American Mathematical Society, vol. 271 (1982), no. 2, pp. 587–609.Google Scholar
[5]Borel, E., Sur la classification des ensembles de mesure nulle, Bulletin de la Société Mathématique de France, vol. 47 (1919), pp. 97–125.CrossRefGoogle Scholar
[6]Carlson, T. J., Strong measure zero and strongly meager sets, Proceedings of the American Mathematical Society, vol. 118 (1993), no. 2, pp. 577–586.CrossRefGoogle Scholar
[7]Cummings, J., Foreman, M., and Magidor, M., Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 35–98.CrossRefGoogle Scholar
[8]Devlin, K. J., Constructibility, Springer, 1984.CrossRefGoogle Scholar
[9]Erdős, P., Hajnal, A., and Milner, E. C., On sets of almost disjoint subsets of a set, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 19 (1968), pp. 209–218.CrossRefGoogle Scholar
[10]Guran, I.I., On topological groups close to being Lindelöf, Soviet Mathematics, Doklady, vol. 23 (1981), pp. 173–175.Google Scholar
[11]Halko, A. and Shelah, S., On strong measure zero subsets of κ2, Fundamenta Mathematicae, vol. 170 (2001). no. 3, pp. 219–229.CrossRefGoogle Scholar
[12]Just, W., Miller, A. W., Scheepers, M., and Szeptycki, P. J., The combinatorics of open covers (II), Topology and its Applications, vol. 73 (1996), pp. 241–266.CrossRefGoogle Scholar
[13]Kelley, J. L., General topology, Graduate Texts in Mathematics, vol. 27, Springer, 1975.Google Scholar
[14]Kunen, K., Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, 1980.Google Scholar
[15]Laver, R., On the consistency of Borel's conjecture, Acta Mathematica, vol. 137 (1976), pp. 151–169.CrossRefGoogle Scholar
[16]Levinski, J.-P., Magidor, M., and Shelah, S., Chang's conjecture for ℕω, Israel Journal of Mathematics, vol. 69 (1990), no. 2, pp. 161–172.CrossRefGoogle Scholar
[17]Rothberger, F., Eine Verschärfung der Eigenschaft C, Fundamenta Mathematicae, vol. 30 (1938), pp. 50–55.CrossRefGoogle Scholar
[18]Rowbottom, F., Large cardinals and small constructible sets, Annals of Mathematical Logic, vol. 3 (1971), pp. 1–44.CrossRefGoogle Scholar
[19]Schimmerling, E. and Zeman, M., Square in core models, The Bulletin of Symbolic Logic, vol. 7 (2001), no. 3, pp. 305–314.CrossRefGoogle Scholar
[20]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer, 1982.CrossRefGoogle Scholar
[21]Sierpiński, W., Sur un ensemble non dénombrable, dont toute image continue est de mesure nulle, Fundamenta Mathematicae, vol. 11 (1928), pp. 301–304.CrossRefGoogle Scholar
[22]Silver, J. H., The independence of kurepa's conjecture and two-cardinal conjectures in model theory, American Mathematical Society Proceedings in Symposia in Pure Mathematics, vol. 13 (1971), no. 1, pp. 383–390.Google Scholar
[23]Steel, J. R., PFA implies ADLL(ℝ), this Journal, vol. 70 (2005), no. 4, pp. 1255–1296.Google Scholar
[24]Szpilrajn-Marczewski, E.. La dimension et la mesure, Fundamenta Mathematicae, vol. 28 (1937), pp. 81–89.CrossRefGoogle Scholar
[25]Tkachenko, M., Introduction to topological groups, Topology and its Applications, vol. 86 (1998), pp. 179–231.CrossRefGoogle Scholar
[26]Todorcevic, S., Walks on ordinals and their characteristics, Progress in Mathematics, 263, Birkhäuser Verlag, 2007.CrossRefGoogle Scholar
[27]Tsaban, B. and Weiss, T., Products of special sets of real numbers, Real Analysis Exchange, vol. 30 (2004/2005), pp. 819–836.CrossRefGoogle Scholar