Codings of separable compact subsets of the first Baire class

Abstract

Let X be a Polish space and K a separable compact subset of the first Baire class on X . For every sequence f = ( f n ) n dense in K , the descriptive set-theoretic properties of the set L f = { L ∈ [ N ] : ( f n ) n ∈ L is pointwise convergent } are analyzed. It is shown that if K is not first countable, then L f is Π 1 1 -complete. This can also happen even if K is a pre-metric compactum of degree at most two, in the sense of S. Todorčević. However, if K is of degree exactly two, then L f is always Borel. A deep result of G. Debs implies that L f contains a Borel cofinal set and this gives a tree-representation of K . We show that classical ordinal assignments of Baire-1 functions are actually Π 1 1 -ranks on K . We also provide an example of a Σ 1 1 Ramsey-null subset A of [ N ] for which there does not exist a Borel set B ⊇ A such that the difference B ∖ A is Ramsey-null.