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

Gary Gruenhage; Piotr Koszmider

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 3, page 275-285
  • ISSN: 0016-2736

Abstract

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We show that MA σ - c e n t e r e d ( 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.

How to cite

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Gruenhage, Gary, and Koszmider, Piotr. "The Arkhangel'skiĭ–Tall problem under Martin’s Axiom." Fundamenta Mathematicae 149.3 (1996): 275-285. <http://eudml.org/doc/212124>.

@article{Gruenhage1996,
abstract = {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.},
author = {Gruenhage, Gary, Koszmider, Piotr},
journal = {Fundamenta Mathematicae},
keywords = {normal locally compact metacompact space; Martin's axiom},
language = {eng},
number = {3},
pages = {275-285},
title = {The Arkhangel'skiĭ–Tall problem under Martin’s Axiom},
url = {http://eudml.org/doc/212124},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Gruenhage, Gary
AU - Koszmider, Piotr
TI - The Arkhangel'skiĭ–Tall problem under Martin’s Axiom
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 3
SP - 275
EP - 285
AB - 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.
LA - eng
KW - normal locally compact metacompact space; Martin's axiom
UR - http://eudml.org/doc/212124
ER -

References

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  1. [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. 
  2. [AP] A. V. Arkhangel'skiĭ and V. I. Ponomarev, General Topology in Problems and Exercises, Nauka, Moscow, 1974 (in Russian). 
  3. [B1] Z. Balogh, On collectionwise normality of locally compact spaces, Trans. Amer. Math. Soc. 323 (1991), 389-411. Zbl0736.54017
  4. [B2] Z. Balogh, Paracompactness in locally Lindelöf spaces, Canad. J. Math. 38 (1986), 719-727. Zbl0599.54027
  5. [E] R. Engelking, General Topology, Heldermann, 1989. 
  6. [G] G. Gruenhage, Applications of a set-theoretic lemma, Proc. Amer. Math. Soc. 92 (1984), 133-140. Zbl0547.54015
  7. [GK] G. Gruenhage and P. Koszmider, The Arkhangel'skiĭ-Tall problem: a consistent counterexample, Fund. Math. 149 (1996), 143-166. Zbl0862.54020
  8. [GM] G. Gruenhage and E. Michael, A result on shrinkable open covers, Topology Proc. 8 (1983), 37-43. Zbl0547.54017
  9. [K] K. Kunen, Set Theory, North-Holland, 1980. 
  10. [T] F. D. Tall, On the existence of normal metacompact Moore spaces which are not metrizable, Canad. J. Math. 26 (1974), 1-6. 
  11. [W1] S. Watson, Locally compact normal spaces in the constructible universe, ibid. 34 (1982), 1091-1095. 
  12. [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. Zbl0604.54021
  13. [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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