201. Functions of Baire class one
- Author
-
Wee-Kee Tang and Denny H. Leung
- Subjects
Pointwise ,Sequence ,Class (set theory) ,Algebra and Number Theory ,Mathematical analysis ,Mathematics::General Topology ,Function (mathematics) ,Functional Analysis (math.FA) ,Combinatorics ,Mathematics - Functional Analysis ,Mathematics::Logic ,Compact space ,26A21, 03E15, 54C30 ,Mathematics::Probability ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Limit of a sequence ,Tietze extension theorem ,Mathematics - Abstract
Let $K$ be a compact metric space. A real-valued function on $K$ is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. In this paper, we study two well known ordinal indices of Baire-1 functions, the oscillation index $\beta$ and the convergence index $\gamma$. It is shown that these two indices are fully compatible in the following sense : a Baire-1 function $f$ satisfies $\beta(f) \leq \omega^{\xi_1} \cdot \omega^{\xi_2}$ for some countable ordinals $\xi_1$ and $\xi_2$ if and only if there exists a sequence of Baire-1 functions $(f_n)$ converging to $f$ pointwise such that $\sup_n\beta(f_n) \leq \omega^{\xi_1}$ and $\gamma((f_n)) \leq \omega^{\xi_2}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $\beta(f) \leq \omega^{\xi_1}$ and $\beta(g) \leq \omega^{\xi_2},$ then $\beta(fg) \leq \omega^{\xi},$ where $\xi=\max\{\xi_1+\xi_2, \xi_2+\xi_1}\}.$ These results do not assume the boundedness of the functions involved.
- Published
- 2000