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1996 | 149 | 2 | 143-166
Tytuł artykułu

The Arkhangel’skiĭ–Tall problem: a consistent counterexample

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We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel'skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in $[ω]^ω$, and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.
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Bibliografia
  • [A] A. V. Arkhangel'skiĭ, The property of paracompactness in the class of perfectly normal locally bicompact spaces, Soviet Math. Dokl. 12 (1971), 1253-1257.
  • [AP] A. V. Arkhangel'skiĭ and V. I. Ponomarev, General Topology in Problems and Exercises, Nauka, Moscow, 1974 (in Russian).
  • [B] Z. Balogh, On collectionwise normality of locally compact spaces, Trans. Amer. Math. Soc. 323 (1991), 389-411.
  • [BL] J. Baumgartner and R. Laver, Iterated perfect set forcing, Ann. Math. Logic 17 (1979), 271-288.
  • [Bi] R. H. Bing, Metrization of topological spaces, Canad. J. Math. 3 (1951), 175-186.
  • [D] P. Daniels, Normal locally compact boundedly metacompact spaces are paracompact: an application of Pixley-Roy spaces, ibid. 35 (1983), 807-823.
  • [vD] E. K. van Douwen, The integers and topology, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, Amsterdam, 1984, 111-167.
  • [GK] G. Gruenhage and P. Koszmider, The Arkhangel'skiĭ-Tall problem under Martin's axiom, Fund. Math., to appear.
  • [J] T. Jech, Multiple Forcing, Cambridge University Press, New York, 1986.
  • [JS] H. Judah and S. Shelah, Q-sets, Sierpinski sets, and rapid filters, Proc. Amer. Math. Soc. 111 (1991), 821-832.
  • [Mi] E. A. Michael, Point-finite and locally finite coverings, Canad. J. Math. 7 (1955), 275-279.
  • [T] F. D. Tall, On the existence of normal metacompact Moore spaces which are not metrizable, Canad. J. Math. 26 (1974), 1-6.
  • [To] S. Todorčević, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), 261-294.
  • [V] D. Velleman, Simplified morasses, J. Symbolic Logic 49 (1984), 257-271.
  • [W1] S. Watson, Locally compact normal spaces in the constructible universe, Canad. J. Math. 34 (1982), 1091-1095.
  • [W2] S. Watson, Locally compact normal metalindelöf spaces may not be paracompact: an application of uniformization and Suslin lines, Proc. Amer. Math. Soc. 98 (1986), 676-680.
  • [W3] S. Watson, Problems I wish I could solve, in: Open Problems in Topology, J. van Mill and G. M. Reed (eds.), North-Holland, Amsterdam, 1990, 37-76.
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bwmeta1.element.bwnjournal-article-fmv149i2p143bwm
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