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Cohen-stable families of subsets of integers

  • Miloš S. Kurilić (a1)

Abstract

A maximal almost disjoint (mad) family ⊆ [ω]ω is Cohen-stable if and only if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-unstable. It is shown that a mad family. .is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the sets G[A]. A are nowhere dense. An ℵ0-mad family, . is a mad family with the property that given any countable family ℬ ⊂ [ω]ω such that each element of ℬ meets infinitely many elements of in an infinite set there is an element of meeting each element of ℬ in an infinite set. It is shown that Cohen-stable mad families exist if and only if there exist ℵ0-mad families. Either of the conditions b = c or a < cov() implies that there exist Cohen-stable mad families. Similar results are obtained for splitting families. For example, a splitting family. . is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the boundaries of the sets G[S], S are nowhere dense. Also. Cohen-stable splitting families of cardinality ≤ κ exist if and only if ℵ0-splitting families of cardinality ≤ κ exist.

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[1]Hechler, S. H., Short complete nested sequences in βℕ ∖ ℕ and small almost-disjoint families, General Topology and Applications, vol. 2 (1972), no. 3, pp. 139149.
[2]Kamburelis, A. and Wȩglorz, B., Splittings, Archive for Mathematical Logic, vol. 35 (1996), no. 4. pp. 263277.
[3]Kunen, K., Set theory, an introduction to independence proofs. North-Holland, Amsterdam, 1980.
[4]Kunen, K., Random and Cohen reals, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors). North-Holland, Amsterdam, 1984, pp. 887911.
[5]Malyhin, V. I., Topological properties of Cohen generic extensions, Transactions of the Moscow Mathematical Society, vol. 52 (1990), pp. 132.
[6]Solovay, R. M., A model of set-theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970). pp. 156.
[7]van Douwen, E. K., The integers and topology, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors). North-Holland, Amsterdam, 1984, pp. 111167.
[8]Vaughan, J. E., Small uncountable cardinals and topology, Open problems in topology (van Mill, J. and Reed, G. M., editors). North-Holland, Amsterdam, 1990, pp. 195218.

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