A Strong Boundedness Result for Separable Rosenthal Compacta

Our main result is a strong boundedness result for the class of separable Rosenthal compacta (that is, separable compact subsets of the first Baire class – see [ADK] and [Ro2]) on the Cantor set having a uniformly bounded dense sequence of continuous functions. We shall denote this class by SRC. The phenomenon of strong boundedness, which was first touched by A. S. Kechris and W. H. Woodin in [KW], is a strengthening of the classical property of boundedness of Π1-ranks. Abstractly, one has a Π1 set B, a natural notion of embedding between elements of B and a canonical Π1-rank φ on B which is coherent with the embedding, in the sense that if x, y ∈ B and x embeds into y, then φ(x) ≤ φ(y). The strong boundedness of B is the fact that for every analytic subset A of B there exists y ∈ B such that x embeds into y for every x ∈ A. Basic examples of strongly bounded classes are the well-orderings WO and the well-founded trees WF (although, in these cases strong boundedness is easily seen to be equivalent to boundedness). Recently, it was shown (see [AD] and [DF]) that several classes of separable Banach spaces are strongly bounded, where the corresponding notion of embedding is that of (linear) isomorphic embedding. These results have, in turn, important consequences in the study of universality problems in Banach space Theory. We will add another example to the list of strongly bounded classes, namely the class SRC. We notice that every K in SRC can be naturally coded by its dense sequence of continuous functions. Hence, we identify SRC with the set