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## Fundamenta Mathematicae

2003 | 179 | 3 | 225-247
Tytuł artykułu

### Functions of Baire class one

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EN
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. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies $β(f) ≤ ω^{ξ₁} · ω^{ξ₂}$ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1 functions converging to f pointwise such that $supₙβ(fₙ) ≤ ω^{ξ₁}$ and $γ((fₙ)) ≤ ω^{ξ₂}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $β(f) ≤ ω^{ξ₁}$ and $β(g) ≤ ω^{ξ₂}$, then $β(fg) ≤ ω^{ξ}$, where ξ = max{ξ₁+ξ₂,ξ₂+ξ₁}. These results do not assume the boundedness of the functions involved.
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Kategorie tematyczne
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Rocznik
Tom
Numer
Strony
225-247
Opis fizyczny
Daty
wydano
2003
Twórcy
autor
• Denny H. Leung, Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
autor
• Wee-Kee Tang, Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616
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