# Article

 Title: Classifications and characterizations of Baire-1 functions (English) Author: Kiriakouli, P. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 39 Issue: 4 Year: 1998 Pages: 733-748 . Category: math . Summary: Kechris and Louveau in [5] classified the bounded Baire-1 functions, which are defined on a compact metric space $K$, to the subclasses $\Cal B_{1}^{\xi }(K)$, $\xi < \omega_1$. In [8], for every ordinal $\xi < \omega_{1}$ we define a new type of convergence for sequences of real-valued functions ($\xi$-uniformly pointwise) which is between uniform and pointwise convergence. In this paper using this type of convergence we obtain a classification of pointwise convergent sequences of continuous real-valued functions defined on a compact metric space $K$, and also we give a characterization of the classes $\Cal B_{1}^{\xi }(K)$, $1 \leq \xi < \omega_{1}$. (English) Keyword: Baire-1 functions Keyword: convergence index Keyword: oscillation index Keyword: trees MSC: 46E99 MSC: 54C30 MSC: 54C35 MSC: 54C50 idZBL: Zbl 1060.54506 idMR: MR1715462 . Date available: 2009-01-08T18:48:02Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119048 . Reference: [1] Alspach D., Argyros S.: Complexity of weakly null sequences.Dissertationes Math. CCCXXI (1992), 1-44. Zbl 0787.46009, MR 1191024 Reference: [2] Alspach D., Odell E.: Averaging null sequences.Lectures Notes in Math. 1332, Springer, Berlin, 1988. MR 0967092 Reference: [3] Bourgain J.: On convergent sequences of continuous functions.Bull. Soc. Math. Belg. Ser. B 32 (1980), 235-249. Zbl 0474.54008, MR 0682645 Reference: [4] Haydon R., Odell E., Rosenthal H.: On certain classes of Baire-1 functions with applications to Banach space theory.Longhorn Notes, The University of Texas at Austin, Functional Analysis Seminar 1987-89. Zbl 0762.46006 Reference: [5] Kechris A.S., Louveau A.: A classification of Baire class 1 functions.Trans. Amer. Math. Soc. 318 (1990), 209-236. Zbl 0692.03031, MR 0946424 Reference: [6] Kiriakouli P.: Namioka spaces, Baire-1 functions, Combinatorial principles of the type of Ramsey and their applications in Banach spaces theory (in Greek).Doctoral Dissertation, Athens Univ., 1994. Reference: [7] Kiriakouli P.: A classification of Baire-1 functions.Trans. Amer. Math. Soc., to appear. Zbl 0926.03056, MR 1407705 Reference: [8] Kiriakouli P.: On pointwise convergent sequences of continuous functions with continuous limits.preprint. Reference: [9] Kiriakouli P., Papanastassiou N.: Convergence for sequences of functions and an Egorov type theorem.preprint. Zbl 1034.28001, MR 2018591 Reference: [10] Mercourakis S.: On Cesaro summable sequences of continuous functions.Mathematika 42 (1995), 87-104. Zbl 0826.46001, MR 1346674 Reference: [11] Mercourakis S.: On some generalizations of the classical Banach-Saks properties.preprint, 1994. Reference: [12] Mercourakis S., Negrepontis S.: Banach spaces and topology II.Recent Progress in General Topology, M. Hušek and J. van Mill, eds., Elsevier Science Publishers B.V., 1992, pp.495-536. Zbl 0832.46005 .

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