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A Classification of Baire-1 Functions

P. Kiriakouli
Transactions of the American Mathematical Society
Vol. 351, No. 11 (Nov., 1999), pp. 4599-4609
Stable URL: http://www.jstor.org/stable/117963
Page Count: 11
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A Classification of Baire-1 Functions
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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 B1 ξ(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∈ B1 ξ +1(K) iff there exists a sequence (fn) of differences of bounded semicontinuous functions on K with fn→ f pointwise and γ ((fn))≤ ω ξ (where ``γ '' denotes the convergence rank). This extends the work of Kechris and Louveau who obtained this result for ξ =1. We also show that the result fails for ξ ≥ ω . The second basic result of the paper involves the introduction of a new ordinal-rank on sequences (fn), called the δ -rank, which is smaller than the convergence rank γ . This result yields the following characterization of B1 ξ(K): f∈ B1 ξ(K) iff there exists a sequence (fn) of continuous functions with fn→ f pointwise and δ ((fn))≤ ω ξ -1 if $1\leq \xi <\omega $, resp. δ ((fn))≤ ω ξ if ξ ≥ ω .

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