An axiom for nonseparable Borel theory

Fleissner, William G.

doi:10.1090/S0002-9947-1979-0531982-9

An axiom for nonseparable Borel theory
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by William G. Fleissner PDF

Trans. Amer. Math. Soc. 251 (1979), 309-328 Request permission

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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