Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: July 21, 1999
MathSciNet review: 1407705
Full-text PDF Free Access

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 $\xi\ge \omega$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 03E15, 04A15, 46B99, 54C50

Retrieve articles in all journals with MSC (1991): 03E15, 04A15, 46B99, 54C50


Additional Information

P. Kiriakouli
Affiliation: Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece

DOI: http://dx.doi.org/10.1090/S0002-9947-99-01907-8
PII: S 0002-9947(99)01907-8
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