Regularity properties for dominating projective sets Original Research Article

We show that every dominating analytic set in the Baire space has a dominating 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 splitnode all the successor splitnodes have the same length. In [15], a subset of the Baire space is called u-regular if either it is not dominating or it contains the branches of a uniform tree, and it was proved that Σ21-Kσ-regularity implies Σ21-u-regularity. Here we show that these properties are in fact equivalent. Since the proof of analytic u-regularity uses a game argument it was clear that (projective) determinacy implies u-regularity of all (projective) sets. Here we show that an inaccessible cardinal is enough to construct a model for projective u-regularity, namely it holds in Solovayʹs model. Finally we show that forcing with uniform trees is equivalent to Laver forcing.