If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal

Fleissner, William G.

doi:10.1090/S0002-9947-1982-0664048-8

If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal
HTML articles powered by AMS MathViewer

by William G. Fleissner PDF

Trans. Amer. Math. Soc. 273 (1982), 365-373 Request permission

Abstract:

We formulate an axiom, HYP, and from it construct a normal, metacompact, nonmetrizable Moore space. HYP unifies two better known axioms. The Continuum Hypothesis implies HYP; the nonexistence of an inner model with a measurable cardinal implies HYP. As a consequence, it is impossible to replace Nyikos’ "provisional" solution to the normal Moore space problem with a solution not involving large cardinals. Finally, we discuss how this space continues a process of lowering the character for normal, not collectionwise normal spaces.

References

R. H. Bing, Metrization of topological spaces, Canad. J. Math. 3 (1951), 175–186. MR 43449, DOI 10.4153/cjm-1951-022-3
Keith I. Devlin and R. B. Jensen, Marginalia to a theorem of Silver, $\vDash$ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974) Lecture Notes in Math., Vol. 499, Springer, Berlin, 1975, pp. 115–142. MR 0480036
A. Dodd and R. Jensen, The core model, Ann. Math. Logic 20 (1981), no. 1, 43–75. MR 611394, DOI 10.1016/0003-4843(81)90011-5
Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
William Fleissner, Normal Moore spaces in the constructible universe, Proc. Amer. Math. Soc. 46 (1974), 294–298. MR 362240, DOI 10.1090/S0002-9939-1974-0362240-4
William G. Fleissner, A normal collectionwise Hausdorff, not collectionwise normal space, General Topology and Appl. 6 (1976), no. 1, 57–64. MR 391032
William G. Fleissner, A collectionwise Hausdorff nonnormal Moore space with a $\sigma$-locally countable base, Topology Proc. 4 (1979), no. 1, 83–97 (1980). MR 583690

Normal Moore spaces and large cardinals

Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
F. B. Jones, Concerning normal and completely normal spaces, Bull. Amer. Math. Soc. 43 (1937), no. 10, 671–677. MR 1563615, DOI 10.1090/S0002-9904-1937-06622-5
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
Ernest Michael, Point-finite and locally finite coverings, Canadian J. Math. 7 (1955), 275–279. MR 70147, DOI 10.4153/CJM-1955-029-6
William Mitchell, Hypermeasurable cardinals, Logic Colloquium ’78 (Mons, 1978) Studies in Logic and the Foundations of Mathematics, vol. 97, North-Holland, Amsterdam-New York, 1979, pp. 303–316. MR 567676

ParaLindelöf versus paracompact

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
T. Przymusiński, Collectionwise Hausdorff property in product spaces, Colloq. Math. 36 (1976), no. 1, 49–56. MR 425895, DOI 10.4064/cm-36-1-49-56
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, The normal Moore space problem, Topological structures, II (Proc. Sympos. Topology and Geom., Amsterdam, 1978) Math. Centre Tracts, vol. 116, Math. Centrum, Amsterdam, 1979, pp. 243–261. MR 565845