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

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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 for every (where is a compact metric space). The first basic result of this paper is that for , iff there exists a sequence of differences of bounded semicontinuous functions on with pointwise and (where ``'' denotes the convergence rank). This extends the work of Kechris and Louveau who obtained this result for . We also show that the result fails for . The second basic result of the paper involves the introduction of a new ordinal-rank on sequences , called the -rank, which is smaller than the convergence rank . This result yields the following characterization of iff there exists a sequence of continuous functions with pointwise and if , resp. if .

**[G-H]**D. C. Gillespie and H. A. Hurwicz,*On sequences of continuous functions having continuous limits*, Trans. Amer. Math. Soc.**32**(1930), 527-543.**[H-O-R]**R. Haydon, E. Odell and H. P. Rosenthal,*Certain subclasses of Baire- functions with Banach space applications*, Longhorn Notes, University of Texas at Austin Functional Analysis Seminar 1987-89.**[K-L]**A. S. Kechris and A. Louveau,*A classification of Baire class 1 functions*, Trans. Amer. Math. Soc.**318**(1990), no. 1, 209–236. MR**946424**, 10.1090/S0002-9947-1990-0946424-3**[K-N]**P. Kiriakouli and S. Negrepontis,*A classification of Baire- functions*, unpublished.**[Z]**Z. Zalcwasser,*Sur une propriete du champ des fonctions continues*, Studia Math.**2**(1930), 63-67.

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Additional Information

**P. Kiriakouli**

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

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

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