Codings of separable compact subsets of the first Baire class

Let XX be a Polish space and KK a separable compact subset of the first Baire class on XX. For every sequence f=(fn)n dense in KK, the descriptive set-theoretic properties of the set Lf={L∈[N]:(fn)n∈L is pointwise convergent} are analyzed. It is shown that if KK is not first countable, then Lf is Π11-complete. This can also happen even if KK is a pre-metric compactum of degree at most two, in the sense of S. Todorčević. However, if KK 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 KK. We show that classical ordinal assignments of Baire-1 functions are actually Π11-ranks on KK. We also provide an example of a Σ11 Ramsey-null subset AA of [N][N] for which there does not exist a Borel set B⊇AB⊇A such that the difference B∖AB∖A is Ramsey-null.