A classification of Baire-1 functions

Author:
P. Kiriakouli

Journal:
Trans. Amer. Math. Soc. **351** (1999), 4599-4609

MSC (1991):
Primary 03E15, 04A15, 46B99, 54C50

DOI:
https://doi.org/10.1090/S0002-9947-99-01907-8

Published electronically:
July 21, 1999

MathSciNet review:
1407705

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Abstract | References | Similar Articles | Additional Information

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 $\xi \ge \omega$.

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Additional Information

**P. Kiriakouli**

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

Received by editor(s):
July 11, 1994

Received by editor(s) in revised form:
December 28, 1995

Published electronically:
July 21, 1999

Article copyright:
© Copyright 1999
American Mathematical Society