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ABsTRAcr. Following G. M. Reed's definition of a Q-set, we define a QA-set to be any non-a-discrete topological space with the property that each subset is (relatively) analytic (= Souslin-'F set). Clearly every Q-set is a QA-set. The discrete irrational extension of the space of real numbers is an example of a first countable hereditarily paracompact (Hausdorff) QA -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 a-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.

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