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$ \sigma$-centred forcing and reflection of (sub)metrizability


Author: Franklin D. Tall
Journal: Proc. Amer. Math. Soc. 121 (1994), 299-306
MSC: Primary 54A35; Secondary 03E35, 03E55, 54D15, 54E30, 54E35
DOI: https://doi.org/10.1090/S0002-9939-1994-1179593-2
MathSciNet review: 1179593
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Abstract: By using supercompact reflection and preservation lemmas for random real forcing and $ \sigma $-centred forcing, we obtain a model in which every normal Moore space is submetrizable, but not every normal Moore space is metrizable.


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  • [A] C. E. Aull, Some base axioms for topology involving enumerability, General Topology and Its Relations to Modern Analysis and Algebra (Proc. Kanpur Topology Conf., 1968), Academia, Prague, 1971, pp. 54-61. MR 0287508 (44:4712)
  • [B] M. Bell, On the combinatorial principle $ P(c)$, Fund. Math. 114 (1981), 149-157. MR 643555 (83e:03077)
  • [Bu] D. K. Burke, PMEA and first countable, countably paracompact spaces, Proc. Amer. Math. Soc. 92 (1984), 455-460. MR 759673 (85h:54032)
  • [vD] E. K. van Douwen, The integers and topology, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. 111-167. MR 776619 (85k:54001)
  • [D] A. Dow, An empty class of nonmetric spaces, Proc. Amer. Math. Soc. 104 (1988), 999-1001. MR 964886 (89j:54032)
  • [DTW1] A. Dow, F. D. Tall, and W. Weiss, New proofs of the consistency of the normal Moore space conjecture. I, Topology Appl. 37 (1990), 33-51. MR 1075372 (92b:54008a)
  • [DTW2] -, New proofs of the consistency of the normal Moore space conjecture. II, Topology Appl. 37 (1990), 115-130. MR 1080345 (92b:54008b)
  • [E] R. Engelking, General topology, Heldermann-Verlag, Berlin, 1989. MR 1039321 (91c:54001)
  • [F] W. Fleissner, Forcing and topological properties, preprint. MR 1206451 (93m:54009)
  • [G] G. Gruenhage, Generalized metric spaces , Handbook of Set-Theoretic Topology (K Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. 423-501. MR 776629 (86h:54038)
  • [J] T. Jech, Set theory, Academic Press, New York, 1978. MR 506523 (80a:03062)
  • [N] P. J. Nyikos, A provisional solution to the normal Moore space problem, Proc. Amer. Math. Soc. 78 (1980), 429-435. MR 553389 (81k:54044)
  • [R1] G. M. Reed, On normality and countable paracompactness, Fund. Math. 110 (1980), 145-152. MR 600588 (82d:54033)
  • [R2] -, Set-theoretic problems in Moore spaces, Open Problems in Topology (J. van Mill and G. M. Reed, eds.), North-Holland, Amsterdam, 1990, pp. 163-181. MR 1078645
  • [RZ] G. M. Reed and P. L. Zenor, Metrization of Moore spaces and generalized manifolds, Fund. Math. 91 (1976), 213-220. MR 0425918 (54:13868)
  • [STW] D. Shakhmatov, F. D. Tall, and S. Watson, A normal Moore space which is not submetrizable (in preparation).
  • [T1] F. D. Tall, Set-theoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problems, Thesis, Univ. of Wisconsin, Madison, 1969, Dissertationes Math. (Rozprawy Mat.) 148 (1977), 1-53. MR 0454913 (56:13156)
  • [T2] -, Countably paracompact Moore spaces are metrizable in the Cohen model, Topology Proc. 9 (1984), 145-148. MR 781557 (86k:54011)
  • [T3] -, Normality versus collectionwise normality, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. 685-732. MR 776634 (86m:54022)
  • [T4] -, Topological applications of generic huge embeddings, Trans. Amer. Math. Soc. 341 (1994), 45-68. MR 1223302 (94j:03108)
  • [WFR] M. L. Wage, W. G. Fleissner, and G. M. Reed, Countable paracompactness in perfect spaces, Bull. Amer. Math. Soc. 82 (1976), 635-639. MR 0410665 (53:14413)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1179593-2
Keywords: Supercompact, reflection, random reals, $ \sigma $-centred forcing, normal Moore space, submetrizable, nonmetrizable, countably paracompact
Article copyright: © Copyright 1994 American Mathematical Society

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