Codings of separable compact subsets of the first Baire class

TL;DR

This paper investigates the descriptive set-theoretic complexity of pointwise convergence sets within separable compact subsets of the first Baire class on Polish spaces, revealing conditions under which these sets are Borel or more complex.

Contribution

It characterizes the complexity of convergence sets for Baire-1 functions, showing they are Borel when the degree is exactly two and analyzing their structure in more complex cases.

Findings

If $K$ is not first countable, the set $ ext{L}_f$ is $oldsymbol{ ext{Pi}}^1_1$-complete.

When $K$ has degree exactly two, $ ext{L}_f$ is always Borel.

Existence of a $oldsymbol{ ext{Sigma}}^1_1$ Ramsey-null set with no Borel superset differing by a Ramsey-null set.

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.