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Ordering MAD families a la Katětov

  • Michael Hrušák (a1) and Salvador García Ferreira (a2)

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 ω ω -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.

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[1] Balcar, B. and Simon, P., Disjoint refinement, Handbook of Boolean Algebras (Monk, J. D. and Bonnet, R., editors), vol. 2, 1989, pp. 333386.
[2] Bartoszyński, T. and Judah, H., Set Theory, On the structure of the real line, A. K. Peters, 1995.
[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. 605611.
[4] Bashkirov, A. I., On Stone-čech compactifications of Isbell spaces. Bulletin of the Polish Academy of Sciences, vol. 27 (1979), pp. 613619.
[5] Baumgartner, J. E. and Weese, M., Partition algebras for almost disjoint families, Transactions of the American Mathematical Society, vol. 274 (1982), pp. 619630.
[6] Brendle, J., Mob and mad families, Archive for Mathematical Logic, vol. 37 (1998), pp. 183197.
[7] Brendle, J. and Yatabe, S., Forcing indestructibility of MAD families, 2003, preprint.
[8] Dow, A. and Frankiewicz, R., Remarks on partitioner algebras, Proceedings of the American Mathematical Society, vol. 114 (1991), no. 4, pp. 10671070.
[9] Dow, A. and Nyikos, P., Representing free Boolean algebra, Fundamenta Mathematicae, vol. 141 (1992), pp. 2130.
[10] Erdös, P. and Shelah, S., Separability properties of almost-disjoint families of sets, Israel Journal of Mathematics, vol, 12 (1972), pp. 207214.
[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.
[12] Ferreira, F. Garcia, Continuous functions between Isbell-Mrówka spaces, Commentationes Mathematicae Universitatis Carolinae, vol. 39 (1998), no. 1, pp. 185195.
[13] Hrušák, M., Selectivity of almost disjoint families, Acta Universitatis Carolinae, vol. 41 (2000), no. 2, pp. 1321.
[14] Hrušák, M., Another ⋄-like principle, Fundamenta Mathematicae, vol. 167 (2001), pp. 277289.
[15] Hrušák, M., MAD families and the rationals, Commentationes Mathematicae Universitatis Carolinae, vol. 42 (2001), pp. 245352.
[16] Katětov, M., Products of filters, Commentationes Mathematicae Universitatis Carolinae, vol. 9 (1968), pp. 173189.
[17] Kunen, K., Set Theory. An Introduction to Independence Proofs, North Holland, Amsterdam, 1980.
[18] Kurilić, M., Cohen-stable families of subsets of integers, this Journal, vol. 66 (2001), no. 1, pp. 257270.
[19] Laflamme, G., Zapping small filters, Proceedings of the American Mathematical Society, vol. 114 (1992), pp. 535544.
[20] Malykhin, V. I., Topological properties of Cohen generic extensions, Transactions of the Moscow Mathematical Society, vol. 52 (1990), pp. 132.
[21] Malykhin, V. I. and Tamariz-Mascarua, A., Extensions of functions in Mrówka-Isbell spaces, Topology and its Applications, vol. 81 (1997), pp. 85102.
[22] Mathias, A. R. D., Happy families, Annals of Mathematical Logic, vol. 12 (1977), pp. 59111.
[23] Moore, J. T., Hrušák, M., and Džamonja, M., Parametrized ⋄ principles, Transactions of the American Mathematical Society, to appear.
[24] Mrówka, S., Some set-theoretic constructions in topology, Fundamenta Mathematicae, vol. 94 (1977), pp. 8392.
[25] Shelah, S., On Cardinal invariants of the continuum, Contemporary Mathematics, vol. 31 (1984), pp. 183207, Also in Axiomatic Set Theory (J. Baumgartner, D. Martin and S. Shelah, editors).
[26] Sierpiński, W., Cardinal and ordinal numbers, Panstwowe wydawn naukowe, Warsaw, 1958.
[27] Solomon, R. S., A scattered space that is not zero-dimensional, Bulletin of the London Mathematical Society, vol. 8 (1976), pp. 239240.
[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. 583617.
[29] Teresawa, J., Spaces N ⋃ R need not be strongly 0-dimensional, Bulletin of the Polish Academy of Sciences, vol. 25 (1977), pp. 279281.
[30] van Douwen, E., The integers and topology, Handbook of Set Theoretic Topology (Kunen, K. and Vaughan, J., editors), North-Holland, Amsterdam, 1984, pp. 111167.

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Ordering MAD families a la Katětov

  • Michael Hrušák (a1) and Salvador García Ferreira (a2)

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