REGULARITY PROPERTIES FOR DOMINATING PROJECTIVE SETS

We show that every dominating analytic set in the Baire space has a do minating closed subset. This improves a theorem of Spinas [15] saying that every dominating analytic set contains the branches of a uniform tree, i.e. a superperfect tree with the property that for every splitn ode all the successor splitnodes have the same length. In [15], a subs et of the Baire space is called u-regular if either it is not dominati ng or it contains the branches of a uniform tree, and it was proved th at Sigma(2)(1)-K-sigma-regularity implies Sigma(2)(1)-u-regularity, He re we show that these properties are in fact equivalent. Since the pro of of analytic u-regularity uses a game argument it was clear that (pr ojective) determinacy implies u-regularity of all (projective) sets. H ere we show that an inaccessible cardinal is enough to construct a mod el for projective u-regularity, namely it holds in Solovay's model. Fi nally we show that forcing with uniform trees is equivalent to Laver f orcing.