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A classification of Baire-1 functions

Authors:	P Kiriakouli

Institution:	Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece

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 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 of differences of bounded semicontinuous functions on 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 , 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 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 $\xi\ge \omega$ .

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