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Transactions of the American Mathematical Society

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The almost-disjointness number may have countable cofinality


Author: Jörg Brendle
Journal: Trans. Amer. Math. Soc. 355 (2003), 2633-2649
MSC (2000): Primary 03E17; Secondary 03E35
DOI: https://doi.org/10.1090/S0002-9947-03-03271-9
Published electronically: February 27, 2003
MathSciNet review: 1975392
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Abstract: We show that it is consistent for the almost-disjointness number $\mathfrak{a}$ to have countable cofinality. For example, it may be equal to $\aleph_\omega$.


References [Enhancements On Off] (What's this?)

  • [BJ] T. Bartoszynski and H. Judah, Set Theory. On the structure of the real line, A K Peters, Wellesley, MA, 1995. MR 96k:03002
  • [Bl] A. Blass, Combinatorial cardinal characteristics of the continuum, in: Handbook of Set Theory (A. Kanamori et al., eds.), to appear.
  • [Br] J. Brendle, Mad families and iteration theory, in: Logic and Algebra (Y. Zhang, ed.), Contemp. Math. 302 (2002), Amer. Math. Soc., Providence, RI, 1-31.
  • [H1] S. Hechler, Short complete nested sequences in $\beta \mathbb{N}\setminus \mathbb{N} $ and small maximal almost-disjoint families, General Topology and Appl. 2 (1972), 139-149. MR 46:7028
  • [H2] S. Hechler, On the existence of certain cofinal subsets of $\omega^\omega$, in: Axiomatic Set Theory (T. Jech, ed.), Proc. Sympos. Pure Math. 13 (1974), 155-173. MR 50:12716
  • [M] A. Miller, Arnie Miller's problem list, in: Set Theory of the Reals (H. Judah, ed.), Israel Math. Conf. Proc. 6 (1993), 645-654. MR 94m:03073
  • [S1] S. Shelah, Covering of the null ideal may have countable cofinality, Fund. Math. 166 (2000), 109-136. (publication number 592) MR 2001m:03101
  • [S2] S. Shelah, Are $\mathfrak{a}$ and $\mathbb{D} $ your cup of tea? Acta Math., to appear. (publication number 700)
  • [vD] E. van Douwen, The integers and topology, in: Handbook of Set-theoretic Topology (K. Kunen and J. Vaughan, eds.), North-Holland, Amsterdam (1984), 111-167. MR 87f:54008
  • [V] J. E. Vaughan, Small uncountable cardinals and topology, in: Open Problems in Topology (J. van Mill and G. M. Reed, eds.), North-Holland (1990), 195-218.

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Additional Information

Jörg Brendle
Affiliation: The Graduate School of Science and Technology, Kobe University, Rokko–dai 1–1, Nada–ku, Kobe 657–8501, Japan
Email: brendle@kurt.scitec.kobe-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-03-03271-9
Keywords: Maximal almost-disjoint families, almost-disjointness number, iterated forcing.
Received by editor(s): October 3, 2001
Published electronically: February 27, 2003
Additional Notes: Supported by Grant–in–Aid for Scientific Research (C)(2)12640124, Japan Society for the Promotion of Science
Article copyright: © Copyright 2003 American Mathematical Society

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