Classifying Projective-like Hierarchies

The starting point for the investigations reported in this paper is the following surprising result of Steel [St]: Assuming AD + DC let Г be a collection of pointsets containing all open sets and closed under continuous preimages and countable intersections and unions. Then either Г or Г ( = the dual of Г) has the reduction property. This result raises naturally the question if a similar phenomenon occurs with the stronger prewellordering property, perhaps under stronger closure assumptions on Г.