Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T03:41:33.675Z Has data issue: false hasContentIssue false
  • English
  • Français

Ordering MAD families a la Katětov

Published online by Cambridge University Press:  12 March 2014

Michael Hrušák  and
Salvador García Ferreira
Michael Hrušák
Affiliation:
Instituto De Matematicas, UNAM A. P. 61-3, Xangari C. P. 58089, Morelia, Mich., Mexico, E-mail: [email protected]
Salvador García Ferreira
Affiliation:
Instituto De Matematicas, UNAM A. P. 61-3, Xangari C. P. 58089, Morelia, Mich., Mexico, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

An ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size c and decreasing chains of length c+ bellow every element. Assuming t = c a MAD family equivalent to all of its restrictions is constructed. It is also shown here that the Continuum Hypothesis implies that for every ωω-bounding forcing ℙ of size c there is a Cohen-destructible, ℙ-indestructible MAD family. Finally, two other orderings on MAD families are suggested and an old construction of Mrówka is revisited.

Keywords

. Maximal almost disjoint familyRudin-Keisler ordering of filtersKatětov ordercardinal invariants of the continuumindestructibility of MAD families
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 4 , December 2003 , pp. 1337 - 1353
Copyright
Copyright © Association for Symbolic Logic 2003

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] Balcar, B. and Simon, P., Disjoint refinement, Handbook of Boolean Algebras (Monk, J. D. and Bonnet, R., editors), vol. 2, 1989, pp. 333–386.Google Scholar
[2] Bartoszyński, T. and Judah, H., Set Theory, On the structure of the real line, A. K. Peters, 1995.Google Scholar
[3] Bashkirov, A. I., On continuous maps of Isbell spaces and strong O-dimensionality, Bulletin of the Polish Academy of Sciences, vol. 27 (1979), pp. 605–611.Google Scholar
[4] Bashkirov, A. I., On Stone-čech compactifications of Isbell spaces. Bulletin of the Polish Academy of Sciences, vol. 27 (1979), pp. 613–619.Google Scholar
[5] Baumgartner, J. E. and Weese, M., Partition algebras for almost disjoint families, Transactions of the American Mathematical Society, vol. 274 (1982), pp. 619–630.CrossRefGoogle Scholar
[6] Brendle, J., Mob and mad families, Archive for Mathematical Logic, vol. 37 (1998), pp. 183–197.Google Scholar
[7] Brendle, J. and Yatabe, S., Forcing indestructibility of MAD families, 2003, preprint.Google Scholar
[8] Dow, A. and Frankiewicz, R., Remarks on partitioner algebras, Proceedings of the American Mathematical Society, vol. 114 (1991), no. 4, pp. 1067–1070.Google Scholar
[9] Dow, A. and Nyikos, P., Representing free Boolean algebra, Fundamenta Mathematicae, vol. 141 (1992), pp. 21–30.CrossRefGoogle Scholar
[10] Erdös, P. and Shelah, S., Separability properties of almost-disjoint families of sets, Israel Journal of Mathematics, vol, 12 (1972), pp. 207–214.Google Scholar
[11] Farah, I., Analytic quotients: Theory of liftings for quotients over analytic ideals on the integers, Memoirs of the American Mathematical Society, vol. 148 (2000), no. 702.Google Scholar
[12] Ferreira, F. Garcia, Continuous functions between Isbell-Mrówka spaces, Commentationes Mathematicae Universitatis Carolinae, vol. 39 (1998), no. 1, pp. 185–195.Google Scholar
[13] Hrušák, M., Selectivity of almost disjoint families, Acta Universitatis Carolinae, vol. 41 (2000), no. 2, pp. 13–21.Google Scholar
[14] Hrušák, M., Another ⋄-like principle, Fundamenta Mathematicae, vol. 167 (2001), pp. 277–289.CrossRefGoogle Scholar
[15] Hrušák, M., MAD families and the rationals, Commentationes Mathematicae Universitatis Carolinae, vol. 42 (2001), pp. 245–352.Google Scholar
[16] Katětov, M., Products of filters, Commentationes Mathematicae Universitatis Carolinae, vol. 9 (1968), pp. 173–189.Google Scholar
[17] Kunen, K., Set Theory. An Introduction to Independence Proofs, North Holland, Amsterdam, 1980.Google Scholar
[18] Kurilić, M., Cohen-stable families of subsets of integers, this Journal, vol. 66 (2001), no. 1, pp. 257–270.Google Scholar
[19] Laflamme, G., Zapping small filters, Proceedings of the American Mathematical Society, vol. 114 (1992), pp. 535–544.Google Scholar
[20] Malykhin, V. I., Topological properties of Cohen generic extensions, Transactions of the Moscow Mathematical Society, vol. 52 (1990), pp. 1–32.Google Scholar
[21] Malykhin, V. I. and Tamariz-Mascarua, A., Extensions of functions in Mrówka-Isbell spaces, Topology and its Applications, vol. 81 (1997), pp. 85–102.CrossRefGoogle Scholar
[22] Mathias, A. R. D., Happy families, Annals of Mathematical Logic, vol. 12 (1977), pp. 59–111.CrossRefGoogle Scholar
[23] Moore, J. T., Hrušák, M., and Džamonja, M., Parametrized ⋄ principles, Transactions of the American Mathematical Society, to appear.Google Scholar
[24] Mrówka, S., Some set-theoretic constructions in topology, Fundamenta Mathematicae, vol. 94 (1977), pp. 83–92.Google Scholar
[25] Shelah, S., On Cardinal invariants of the continuum, Contemporary Mathematics, vol. 31 (1984), pp. 183–207, Also in Axiomatic Set Theory (J. Baumgartner, D. Martin and S. Shelah, editors).CrossRefGoogle Scholar
[26] Sierpiński, W., Cardinal and ordinal numbers, Panstwowe wydawn naukowe, Warsaw, 1958.Google Scholar
[27] Solomon, R. S., A scattered space that is not zero-dimensional, Bulletin of the London Mathematical Society, vol. 8 (1976), pp. 239–240.Google Scholar
[28] Steprāns, J., Combinatorial consequences of adding Cohen reals, Set theory of the reals, Proceedings of the Israel Conference on Mathematics (Judah, H., editor), vol. 6, 1993, pp. 583–617.Google Scholar
[29] Teresawa, J., Spaces N ⋃ R need not be strongly 0-dimensional, Bulletin of the Polish Academy of Sciences, vol. 25 (1977), pp. 279–281.Google Scholar
[30] van Douwen, E., The integers and topology, Handbook of Set Theoretic Topology (Kunen, K. and Vaughan, J., editors), North-Holland, Amsterdam, 1984, pp. 111–167.Google Scholar