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Large cardinals and small Dowker spaces


Author: Chris Good
Journal: Proc. Amer. Math. Soc. 123 (1995), 263-272
MSC: Primary 03E35; Secondary 03E55, 54D15, 54D20, 54G15
DOI: https://doi.org/10.1090/S0002-9939-1995-1216813-0
MathSciNet review: 1216813
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Abstract: We prove that, if there is a model of set-theory which contains no first countable, locally compact, scattered Dowker spaces, then there is an inner model which contains a measurable cardinal.


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  • [Bl] Z. Balogh, Locally nice spaces under Martin's axiom, Comment. Math. Univ. Carolin. 24 (1983), 63-87. MR 703926 (85b:54005)
  • [Be] M. G. Bell, On the combinatorial principal $ P(\mathfrak{c})$, Fund. Math. 114 (1981), 149-157. MR 643555 (83e:03077)
  • [dC] P. de Caux, A collectionwise normal weakly $ \theta $-refinable Dowker space which is neither irreducible nor realcompact, Topology Proc. 1 (1976), 67-77. MR 0448322 (56:6629)
  • [De] K. Devlin, Constructability, Springer-Verlag, Berlin, 1984. MR 750828 (85k:03001)
  • [Dd] A. J. Dodd, Core models, J. Symbolic Logic 48 (1983), 78-90. MR 693251 (84g:03081)
  • [DJ] A. J. Dodd and R. Jensen, The covering lemma for K, Ann. Math. Logic 22 (1982), 1-30. MR 661475 (83i:03082a)
  • [Do] C. H. Dowker, On countably paracompact spaces, Canad. J. Math. 3 (1951), 219-224. MR 0043446 (13:264c)
  • [E] R. Engelking, General topology, Heldermann Verlag, Berlin, 1989. MR 1039321 (91c:54001)
  • [F1] W. G. Fleissner, If all normal Moore spaces are metrizable then there is an inner model with a measurable cardinal, Trans. Amer. Math. Soc. 273 (1982), 365-373. MR 664048 (84h:03118)
  • [F2] -, The normal Moore space conjecture and large cardinals, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, 1984. MR 776635 (86m:54023)
  • [G] G. Gruenhage, Generalized metric spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, 1984. MR 776629 (86h:54038)
  • [JKR] I. Juhász, K. Kunen, and M. E. Rudin, Two more hereditarily separable non-Lindelöf spaces, Canad. J. Math. 28 (1976), 998-1005. MR 0428245 (55:1270)
  • [Ka] M. Katětov, Complete normality of Cartesian products, Fund. Math. 38 (1948), 271-274. MR 0027501 (10:315h)
  • [Ku] K. Kunen, Set theory, an introduction to independence proofs, North-Holland, Amsterdam, 1984. MR 756630 (85e:03003)
  • [KT] K. Kunen and F. D. Tall, Between Martin's axiom and Souslin's hypothesis, Fund. Math. 102 (1979), 173-181. MR 532951 (83e:03078)
  • [KV] K. Kunen and J. E. Vaughan, eds., Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984. MR 776619 (85k:54001)
  • [M] J. Mack, Countable paracompactness and weak normality properties, Trans. Amer. Math. Soc. 148 (1970), 265-272. MR 0259856 (41:4485)
  • [vMR] J. van Mill and G. M. Reed, eds., Open problems in topology, North-Holland, Amsterdam, 1990. MR 1078636 (92c:54001)
  • [N1] P. J. Nyikos, Covering properties on $ \sigma $-scattered spaces, Topology Proc. 2 (1977), 509-541. MR 540626 (80k:54045)
  • [N2] -, A provisional solution to the normal Moore space problem, Proc. Amer. Math. Soc. 78 (1980), 429-435. MR 553389 (81k:54044)
  • [NP] P. J. Nyikos and S. Purisch, Monotone normality and paracompactness in scattered spaces, Papers on General Topology and Related Category Theory and Topological Algebra, Ann. New York Acad. Sci., vol. 552, New York Acad. Sci., New York, 1989, pp. 124-137. MR 1020780 (91b:54044)
  • [O] A. J. Ostaszewski, On countably compact, perfectly normal spaces, J. London Math. Soc. 14 (1976), 505-516. MR 0438292 (55:11210)
  • [P] T. C. Przymusiński, Products of normal spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, 1984. MR 776637 (86c:54007)
  • [Ro] J. Roitman, Basic S and L, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, 1984. MR 776619 (85k:54001)
  • [Ru1] M. E. Rudin, Countable paracompactness and Souslin's problem, Canad. J. Math. 7 (1955), 543-547. MR 0073155 (17:391e)
  • [Ru2] -, A normal space for which $ X \times I$ is not normal, Fund. Math. 72 (1971), 179-186.
  • [Ru3] -, Dowker spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, 1984. MR 776619 (85k:54001)
  • [S] B. M. Scott, Toward a product theory for orthocompactness, Studies in Topology, Academic Press, New York, 1975, pp. 517-537. MR 0372820 (51:9024)
  • [Ta] F. D. Tall, Normality versus collectionwise normality, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, 1984. MR 776634 (86m:54022)
  • [Tr] I. J. Tree, Pseudocompactness and chain conditions, Ph.D., University of Oxford, London, 1991.
  • [Wt] W. S. Watson, A construction of a Dowker space, Proc. Amer. Math. Soc. 109 (1990), 835-841. MR 1019285 (91b:54045)
  • [Ws] W. Weiss, Small Dowker spaces, Pacific J. Math. 91 (1981), 485-492. MR 628595 (83d:54036)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1216813-0
Keywords: Small Dowker space, normality, countable paracompactness, measurable cardinals, Covering Lemma
Article copyright: © Copyright 1995 American Mathematical Society

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