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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Some consequences of $ (V=L)$ in the theory of analytic sets


Author: R. W. Hansell
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
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0577766-0
Keywords: $ (V = L)$, analytic set, $ \sigma $-discrete, Q-set, $ {Q_A}$-set, completely additive-analytic family, $ \sigma $-discretely decomposable, d-family, metrizable space
Article copyright: © Copyright 1980 American Mathematical Society