The Arkhangel’skiĭ–Tall problem: a consistent counterexample

Gary Gruenhage; Piotr Koszmider

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 2, page 143-166
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Gruenhage, Gary, and Koszmider, Piotr. "The Arkhangel’skiĭ–Tall problem: a consistent counterexample." Fundamenta Mathematicae 149.2 (1996): 143-166. <http://eudml.org/doc/212113>.

@article{Gruenhage1996,
abstract = {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.},
author = {Gruenhage, Gary, Koszmider, Piotr},
journal = {Fundamenta Mathematicae},
keywords = {normal space; locally compact space; paracompact space; metacompact space; metacompactness; meta-Lindelöfness; forcing},
language = {eng},
number = {2},
pages = {143-166},
title = {The Arkhangel’skiĭ–Tall problem: a consistent counterexample},
url = {http://eudml.org/doc/212113},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Gruenhage, Gary
AU - Koszmider, Piotr
TI - The Arkhangel’skiĭ–Tall problem: a consistent counterexample
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 2
SP - 143
EP - 166
AB - 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.
LA - eng
KW - normal space; locally compact space; paracompact space; metacompact space; metacompactness; meta-Lindelöfness; forcing
UR - http://eudml.org/doc/212113
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. [B] Z. Balogh, On collectionwise normality of locally compact spaces, Trans. Amer. Math. Soc. 323 (1991), 389-411. Zbl0736.54017
  4. [BL] J. Baumgartner and R. Laver, Iterated perfect set forcing, Ann. Math. Logic 17 (1979), 271-288. Zbl0427.03043
  5. [Bi] R. H. Bing, Metrization of topological spaces, Canad. J. Math. 3 (1951), 175-186. Zbl0042.41301
  6. [D] P. Daniels, Normal locally compact boundedly metacompact spaces are paracompact: an application of Pixley-Roy spaces, ibid. 35 (1983), 807-823. Zbl0526.54009
  7. [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. 
  8. [GK] G. Gruenhage and P. Koszmider, The Arkhangel'skiĭ-Tall problem under Martin's axiom, Fund. Math., to appear. Zbl0855.54006
  9. [J] T. Jech, Multiple Forcing, Cambridge University Press, New York, 1986. Zbl0601.03019
  10. [JS] H. Judah and S. Shelah, Q-sets, Sierpinski sets, and rapid filters, Proc. Amer. Math. Soc. 111 (1991), 821-832. Zbl0751.03023
  11. [Mi] E. A. Michael, Point-finite and locally finite coverings, Canad. J. Math. 7 (1955), 275-279. 
  12. [T] F. D. Tall, On the existence of normal metacompact Moore spaces which are not metrizable, Canad. J. Math. 26 (1974), 1-6. 
  13. [To] S. Todorčević, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), 261-294. Zbl0658.03028
  14. [V] D. Velleman, Simplified morasses, J. Symbolic Logic 49 (1984), 257-271. Zbl0575.03035
  15. [W1] S. Watson, Locally compact normal spaces in the constructible universe, Canad. J. Math. 34 (1982), 1091-1095. 
  16. [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
  17. [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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