  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.
