Dominating projective sets in the Baire space (Q1331915)

The Cantor-Bendixson theorem says that a closed set in a Polish space is either countable or contains a perfect subset. Souslin proved this even for analytic sets. In this paper a similar regularity property is studied for the Baire space. It is shown by the game-theoretic method that every analytic set in the Baire space which is dominating contains the branches of a uniform tree. This property of analytic sets is called \(u\)- regularity. However, the concept of \(u\)-regularity does not suffice to characterize dominating analytic sets in general. It is also shown that from a \(\Sigma_{n+1}^ 1\)-rapid filter a non-\(u\)-regular \(\Pi_ n^ 1\)-set can be constructed. Finally, it is proved that \(\Sigma_ 2^ 1\)- \(K_ \sigma\)-regularity implies \(\Sigma_ 2^ 1\)-\(u\)-regularity.