Codings of separable compact subsets of the first Baire class

Let X be a Polish space and K a separable compact subset of the first Baire class on X. For every sequence f = (fn)n dense in K, the descriptive set-theoretic properties of the set Lf = {L ∈ [N] : (fn)n∈L is pointwise convergent} are analyzed. It is shown that if K is not first countable, then Lf 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 Lf is always Borel. A deep result of G. Debs implies that Lf 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 -ranks on K. We also provide an example of a Σ 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.