Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
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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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