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 $\bs$ dense in $\kk$, the descriptive set-theoretic properties of the set \[ \lbf=\{L\in[\nn]: (f_n)_{n\in L} \text{is pointwise convergent}\} \] are analyzed. It is shown that if $K$ is not first countable, then $\lbf$ is $\PB^1_1$-complete. This can also happen even if $K$ is a pre-metric compactum of degree at most two, in the sense of S. Todorcevic. However, if $K$ is of degree exactly two, then $\lbf$ is always Borel. A deep result of G. Debs implies that $\lbf$ 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 $\PB^1_1$-ranks on $\kk$. We also provide an example of a $\SB^1_1$ Ramsey-null subset $A$ of $[\nn]$ for which there does not exist a Borel set $B\supseteq A$ such that the difference $B\setminus A$ is Ramsey-null.