𝜎-centred forcing and reflection of (sub)metrizability

Tall, Franklin D.

doi:10.1090/S0002-9939-1994-1179593-2

$\sigma$-centred forcing and reflection of (sub)metrizability
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Proc. Amer. Math. Soc. 121 (1994), 299-306 Request permission

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.

References

C. E. Aull, Some base axioms for topology involving enumerability, General Topology and its Relations to Modern Analysis and Algebra, III (Proc. Conf., Kanpur, 1968) Academia, Prague, 1971, pp. 55–61. MR 0287508
Murray G. Bell, On the combinatorial principle $P({\mathfrak {c}})$, Fund. Math. 114 (1981), no. 2, 149–157. MR 643555, DOI 10.4064/fm-114-2-149-157
Dennis K. Burke, PMEA and first countable, countably paracompact spaces, Proc. Amer. Math. Soc. 92 (1984), no. 3, 455–460. MR 759673, DOI 10.1090/S0002-9939-1984-0759673-7
Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of set-theoretic topology, North-Holland Publishing Co., Amsterdam, 1984. MR 776619
Alan Dow, An empty class of nonmetric spaces, Proc. Amer. Math. Soc. 104 (1988), no. 3, 999–1001. MR 964886, DOI 10.1090/S0002-9939-1988-0964886-9
Alan Dow, Franklin D. Tall, and William A. R. Weiss, New proofs of the consistency of the normal Moore space conjecture. I, Topology Appl. 37 (1990), no. 1, 33–51. MR 1075372, DOI 10.1016/0166-8641(90)90013-R
Alan Dow, Franklin D. Tall, and William A. R. Weiss, New proofs of the consistency of the normal Moore space conjecture. I, Topology Appl. 37 (1990), no. 1, 33–51. MR 1075372, DOI 10.1016/0166-8641(90)90013-R
Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
William Fleissner, Forcing and topological properties, Topology Proc. 16 (1991), 41–43. MR 1206451
Gary Gruenhage, Generalized metric spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 423–501. MR 776629
Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
Peter J. Nyikos, A provisional solution to the normal Moore space problem, Proc. Amer. Math. Soc. 78 (1980), no. 3, 429–435. MR 553389, DOI 10.1090/S0002-9939-1980-0553389-4
G. M. Reed, On normality and countable paracompactness, Fund. Math. 110 (1980), no. 2, 145–152. MR 600588, DOI 10.4064/fm-110-2-145-152
G. M. Reed, Set-theoretic problems in Moore spaces, Open problems in topology, North-Holland, Amsterdam, 1990, pp. 163–181. MR 1078645, DOI 10.1080/07468342.1989.11973227
G. M. Reed and P. L. Zenor, Metrization of Moore spaces and generalized manifolds, Fund. Math. 91 (1976), no. 3, 203–210. MR 425918, DOI 10.4064/fm-91-3-203-210

A normal Moore space which is not submetrizable

F. D. Tall, Set-theoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problems, Dissertationes Math. (Rozprawy Mat.) 148 (1977), 53. MR 454913
Franklin D. Tall, Countably paracompact Moore spaces are metrizable in the Cohen model, Proceedings of the 1984 topology conference (Auburn, Ala., 1984), 1984, pp. 145–148. MR 781557
Franklin D. Tall, Normality versus collectionwise normality, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 685–732. MR 776634
Franklin D. Tall, Topological applications of generic huge embeddings, Trans. Amer. Math. Soc. 341 (1994), no. 1, 45–68. MR 1223302, DOI 10.1090/S0002-9947-1994-1223302-X
M. L. Wage, W. G. Fleissner, and G. M. Reed, Normality versus countable paracompactness in perfect spaces, Bull. Amer. Math. Soc. 82 (1976), no. 4, 635–639. MR 410665, DOI 10.1090/S0002-9904-1976-14150-X