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Transactions of the American Mathematical Society

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An axiom for nonseparable Borel theory


Author: William G. Fleissner
Journal: Trans. Amer. Math. Soc. 251 (1979), 309-328
MSC: Primary 03E15; Secondary 03E35, 03E55, 04A15, 26A21
DOI: https://doi.org/10.1090/S0002-9947-1979-0531982-9
MathSciNet review: 531982
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Abstract: Kuratowski asked whether the Lebesgue-Hausdorff theorem held for metrizable spaces. A. Stone asked whether a Borel isomorphism between metrizable spaces must be a generalized homeomorphism. The existence of a Q set refutes the generalized Lebesgue-Hausdorff theorem. In this paper we discuss the consequences of the axiom of the title, among which are ``yes'' answers to both Kuratowski's and Stone's questions. The axiom states that a point finite analytic additive family is $ \sigma $ discretely decomposable. We show that this axiom is valid in the model constructed by collapsing a supercompact cardinal to $ {\omega _2}$ using Lévy forcing. Our proof displays relationships between $ \sigma $ discretely decomposable families, analytic additive families and d families.


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  • [1] R. Baire, Leçons sur les fonctions discontinues, Gauthier-Villars, Paris, 1905.
  • [2] S. Banach, Über analytisch darstellbare Operationen in abstrakten Raumen, Fund. Math. 17 (1931), 283-295.
  • [3] J. Baumgartner, Some new order types, Ann. Math. Logic 9 (1976), 187-222. MR 0416925 (54:4988)
  • [4] W. Fleissner, Separation properties in Moore spaces, Fund. Math. 98 (1978), 279-286. MR 0478111 (57:17600)
  • [5] R. Hansell, Borel measurable mappings for nonseparable metric spaces, Trans. Amer. Math. Soc. 161 (1971), 145-169. MR 44 #5426. MR 0288228 (44:5426)
  • [6] -, On Borel mappings and Baire functions, Trans. Amer. Math. Soc. 194 (1974), 195-211. MR 0362270 (50:14712)
  • [7] F. Hausdorff, Mengenlehre, de Gruyter, Berlin, 1937; English translation, Chelsea, New York, 1957. MR 19, 11.
  • [8] M. Lavrentiev, Sur la recherche des ensembles homéomorphes, C. R. Acad. Sci. Paris 178 (1924), 187.
  • [9] H. Lebesgue, Sur les fonctions représentables anatytiquement, J. Math. Pures Appl. 1 (1905), 139-216.
  • [10] J. Kaniewski and R. Pol, Borel-measurable selectors for compact-valued mappings in the nonseparable case, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 1043-1050. MR 0410657 (53:14405)
  • [11] K. Kuratowski, Quelques problèmes concernant espaces métriques non-séparables, Fund. Math. 25 (1935), 532-545.
  • [12] -, Topology. Vol. 1, Academic Press, New York; PWN, Warsaw, 1966. MR 19 #873, MR 36 #840. MR 0193605 (33:1823)
  • [13] A. Martin and R. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), 143-178. MR 0270904 (42:5787)
  • [14] E. Michael and I. Namioka, Barely continuous functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 889-892. MR 0431092 (55:4094)
  • [15] A. Miller, On the length of Borel hierarchies (to appear). MR 548475 (80m:04003)
  • [16] E. Milnor and S. Shelah, Some theorems on transversals, Colloq. Math. Soc. János Bolyai 10 (1973), 1115-1126. MR 0376358 (51:12534)
  • [17] J. Nagata, Modern general topology, North Holland, Amsterdam, 1968.
  • [18] R. Pol, Note on decompositions of metrizable spaces, Fund. Math. 95 (1977), 95-103. MR 0433371 (55:6347)
  • [19] -, Note on decompositions of metrizable spaces. II, Fund. Math. 100 (1978), 129-143. MR 0494011 (58:12953)
  • [20] D. Preiss, Completely additive disjoint systems of Baire sets is of bounded class, Comment. Math. Univ. Carolinae 15 (1972), 341-344. MR 0346116 (49:10842)
  • [21] A. Kanamori, W. Reinhardt and R. Solovay, Strong axioms of infinity and elementary embeddings, Ann. Math. Logic 13 (1978), 73-116. MR 482431 (80h:03072)
  • [22] S. Shelah, A compactness theorem for singular cardinals, free algebras, Whitehead problem, and transversals, Israel J. Math. 21 (1975), 319-349. MR 0389579 (52:10410)
  • [23] R. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin's problem, Ann. of Math. (2) 94 (1971), 201-245. MR 0294139 (45:3212)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0531982-9
Keywords: Borel mapping, Baire function, $ \sigma $ discretely decomposable, analytically additive, d family, decompositions of metrizable spaces, Q sets, Lévy forcing, $ \diamondsuit $ $ E(\kappa )$ supercompact cardinals, Borel isomorphism, generalized homeomorphism
Article copyright: © Copyright 1979 American Mathematical Society

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