Some consequences of $(V=L)$ in the theory of analytic sets
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- by R. W. Hansell
- Proc. Amer. Math. Soc. 80 (1980), 311-319
- DOI: https://doi.org/10.1090/S0002-9939-1980-0577766-0
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Abstract:
Following G. M. Reed’s definition of a Q-set, we define a ${Q_A}$-set to be any non-$\sigma$-discrete topological space with the property that each subset is (relatively) analytic (= Souslin-$\mathcal {F}$ set). Clearly every Q-set is a ${Q_A}$-set. The discrete irrational extension of the space of real numbers is an example of a first countable hereditarily paracompact (Hausdorff) ${Q_A}$-set which is not a Q-set. Theorem. $(V = L)$ Let X be a first countable normal space all of whose subsets are analytic. Then X is $\sigma$-discrete if and only if the product of X with the space of irrational numbers is normal. A new structural property of analytic sets is developed in order to utilize a proof technique due to Reed. Several corollaries are obtained on properties of completely additive-analytic families of sets in general metric spaces.References
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Bibliographic Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 80 (1980), 311-319
- MSC: Primary 54H05; Secondary 03E15, 03E35, 54E35
- DOI: https://doi.org/10.1090/S0002-9939-1980-0577766-0
- MathSciNet review: 577766