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The Arkhangel'skiĭ–Tall problem under Martin’s Axiom

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We show that MA$_{σ-centered}(ω_1)$ implies that normal locally compact metacompact spaces are paracompact, and that MA($ω_1$) implies normal locally compact metalindelöf spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is a normal locally compact metalindelöf space which is not paracompact.
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  • Department of Mathematics, Auburn University, Auburn, Alabama 36849, U.S.A.
  • Department of Mathematics, Auburn University, Auburn, Alabama 36849, U.S.A.
  • [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).
  • [B1] Z. Balogh, On collectionwise normality of locally compact spaces, Trans. Amer. Math. Soc. 323 (1991), 389-411.
  • [B2] Z. Balogh, Paracompactness in locally Lindelöf spaces, Canad. J. Math. 38 (1986), 719-727.
  • [E] R. Engelking, General Topology, Heldermann, 1989.
  • [G] G. Gruenhage, Applications of a set-theoretic lemma, Proc. Amer. Math. Soc. 92 (1984), 133-140.
  • [GK] G. Gruenhage and P. Koszmider, The Arkhangel'skiĭ-Tall problem: a consistent counterexample, Fund. Math. 149 (1996), 143-166.
  • [GM] G. Gruenhage and E. Michael, A result on shrinkable open covers, Topology Proc. 8 (1983), 37-43.
  • [K] K. Kunen, Set Theory, North-Holland, 1980.
  • [T] F. D. Tall, On the existence of normal metacompact Moore spaces which are not metrizable, Canad. J. Math. 26 (1974), 1-6.
  • [W1] S. Watson, Locally compact normal spaces in the constructible universe, ibid. 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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