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arXiv · 1701.05649

On ordinal ranks of Baire class functions

Denny H. Leung·Hong-Wai Ng·Wee-Kee Tang
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Abstract

The theory of ordinal ranks on Baire class 1 functions developed by Kechris and Loveau was recently extended by Elekes, Kiss and Vidnyánszky to Baire class $ξ$ functions for any countable ordinal $ξ\geq1$. In this paper, we answer two of the questions raised by them in their paper (Ranks on the Baire class $ξ$ functions, Trans. Amer. Math. Soc. 368(2016), 8111-8143). Specifically, we show that for any countable ordinal $ξ\geq1,$ the ranks $β_ξ^{\ast}$ and $γ_ξ^{\ast}$ are essentially equivalent, and that neither of them is essentially multiplicative. Since the rank $β$ is not essentially multiplicative, we investigate further the behavior of this rank with respect to products. We characterize the functions $f$ so that $β(fg)\leq ω^ξ$ whenever $β(g)\leqω^ξ$ for any countable ordinal $ξ.$

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Denny H. Leung, Hong-Wai Ng, Wee-Kee Tang. 2017-01-20. On ordinal ranks of Baire class functions. https://arxiv.org/abs/1701.05649

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