DOMINATING PROJECTIVE SETS IN THE BAIRE SPACE

We show that every analytic set in the Baire space which is dominating contains the branches of a uniform tree, i.e. a superperfect tree wit h the property that for every splitnode all the successor splitnodes h ave the same length. We call this property of analytic sets u-regulari ty. However, we show that the concept of uniform tree does not suffice to characterize dominating analytic sets in general. We construct a d ominating closed set with the property that for no uniform tree whose branches are contained in the closed set, the set of these branches is dominating. We also show that from a SIGMA(n+1)1-rapid filter a non-u -regular PI(n)1-set can be constructed. Finally, we prove that SIGMA2( 1)-K(sigma)-regularity implies SIGMA2(1)-u-regularity.