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On Weakly Tight Families

  • Dilip Raghavan (a1) and Juris Steprāns (a2)

Abstract

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $c<{{\aleph }_{\omega }}$ , we construct a weakly tight family under the hypothesis $\mathfrak{s}\le \mathfrak{b}<{{\aleph }_{\omega }}$ . The case when $\mathfrak{s}<\mathfrak{b}$ is handled in ZFC and does not require $\mathfrak{b}<{{\aleph }_{\omega }}$ , while an additional PCF type hypothesis, which holds when $\mathfrak{b}<{{\aleph }_{\omega }}$ is used to treat the case $\mathfrak{s}=\mathfrak{b}$ . The notion of a weakly tight family is a natural weakening of the well-studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hrušák and García Ferreira [8], who applied it to the Katétov order on almost disjoint families.

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References

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[1] Balcar, B., J. Dočkálková, and Simon, P., Almost disjoint families of countable sets. In: Finite and infinite sets, Vols. I, II (Eger, 1981), Colloq. Math. Soc. János Bolyai 37, North-Holland, Amsterdam, 1984, 5988.
[2] Balcar, B. and Simon, P., Disjoint refinement. In: Handbook of Boolean algebras, Vol. 2, North-Holland, Amsterdam, 1989, 333388.
[3] Brendle, J., Mob families and mad families. Arch. Math. Logic 37(1997), 183197. http://dx.doi.org/10.1007/s001530050091
[4] Brendle, J. and Yatabe, S., Forcing indestructibility of MAD families. Ann. Pure Appl. Logic 132(2005), 271312. http://dx.doi.org/10.1016/j.apal.2004.09.002
[5] Erdʺos, P. and Shelah, S., Separability properties of almost-disjoint families of sets. Israel J. Math. 12(1972), 207214. http://dx.doi.org/10.1007/BF02764666
[6] Fuchino, S., Koppelberg, S., and Shelah, S., Partial orderings with the weak Freese–Nation property. Ann. Pure Appl. Logic 80(1996), 3554. http://dx.doi.org/10.1016/0168-0072(95)00047-X
[7] Hrušák, M., MAD families and the rationals. Comment. Math. Univ. Carolin. 42(2001), 345352.
[8] Hrušák, M. and S. Garcıa Ferreira, Ordering MAD families a la Katétov. J. Symbolic Logic 68(2003), 13371353. http://dx.doi.org/10.2178/jsl/1067620190
[9] Kurilic, M. S., Cohen-stable families of subsets of integers. J. Symbolic Logic 66(2001), 257270. http://dx.doi.org/10.2307/2694920
[10] Larson, P. B., Almost-disjoint coding and strongly saturated ideals. Proc. Amer. Math. Soc. 133(2005), 27372739. http://dx.doi.org/10.1090/S0002-9939-05-07824-X
[11] Malykhin, V. I., Topological properties of Cohen generic extensions. Trudy Moskov. Mat. Obshch. 52(1989), 333, 247.
[12] Miller, A. W., Arnie Miller's problem list. In: Set theory of the reals (Ramat Gan, 1991), Israel Math. Conf. Proc. 6, Bar-Ilan Univ., Ramat Gan, 1993, 645654.
[13] Mrówka, S., On completely regular spaces. Fund. Math. 41(1954), 105106.
[14] Mró;wka, S., Some set-theoretic constructions in topology. Fund. Math. 94(1977), 8392.
[15] Raghavan, D., Maximal almost disjoint families of functions. Fund. Math. 204(2009), 241282. http://dx.doi.org/10.4064/fm204-3-3
[16] Raghavan, D., Almost disjoint families and diagonalizations of length continuum. Bull. Symbolic Logic 16(2010), 240260. http://dx.doi.org/10.2178/bsl/1286889125
[17] Raghavan, D., There is a Van Douwen MAD family. Trans. Amer. Math. Soc. 362(2010), 58795891. http://dx.doi.org/10.1090/S0002-9947-2010-04975-X
[18] Shelah, S., Mad families and sane player. Canad. J. Math. 63(2011), 1416-1436. http://dx.doi.org/10.4153/CJM-2011-057-1
[19] Shelah, S., On cardinal invariants of the continuum. In: Axiomatic set theory (Boulder, Colo., 1983), Contemp. Math. 31, Amer. Math. Soc., Providence, RI, 1984, 183207.
[20] Simon, P., A note on almost disjoint refinement. Acta Univ. Carolin. Math. Phys. 37(1996), 8999.
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On Weakly Tight Families

  • Dilip Raghavan (a1) and Juris Steprāns (a2)

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