A classification of Baire-1 functions

Abstract:

In this paper we give some topological characterizations of bounded Baire-1 functions using some ranks. Kechris and Louveau classified the Baire-1 functions to the subclasses $\mathbb {B}^\xi _1(K)$ for every $\xi <\omega _1$ (where $K$ is a compact metric space). The first basic result of this paper is that for $\xi <\omega$, $f\in \mathbb {B}^{\xi +1}_1(K)$ iff there exists a sequence $(f_n)$ of differences of bounded semicontinuous functions on $K$ with $f_n\to f$ pointwise and $\gamma ((f_n))\le \omega ^\xi$ (where “$\gamma$” denotes the convergence rank). This extends the work of Kechris and Louveau who obtained this result for $\xi =1$. We also show that the result fails for $\xi \ge \omega$. The second basic result of the paper involves the introduction of a new ordinal-rank on sequences $(f_n)$, called the $\delta$-rank, which is smaller than the convergence rank $\gamma$. This result yields the following characterization of $\mathbb {B}^\xi _1(K): f\in \mathbb {B}^\xi _1(K)$ iff there exists a sequence $(f_n)$ of continuous functions with $f_n\to f$ pointwise and $\delta ((f_n))\le \omega ^{\xi -1}$ if $1\le \xi <\omega$, resp. $\delta ((f_n))\le \omega ^\xi$ if $\xi \ge \omega$.