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

Hansell, R. W.

doi:10.1090/S0002-9939-1980-0577766-0

Some consequences of $(V=L)$ in the theory of analytic sets
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Proc. Amer. Math. Soc. 80 (1980), 311-319 Request permission

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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