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On ordinal ranks of Baire class functions

Volume 247 / 2019

Denny H. Leung, Hong-Wai Ng, Wee-Kee Tang Fundamenta Mathematicae 247 (2019), 109-130 MSC: Primary 26A21; Secondary 03E15, 54H05. DOI: 10.4064/fm616-3-2019 Published online: 5 September 2019

Abstract

The theory of ordinal ranks on Baire class 1 functions developed by Kechris and Loveau was extended by Elekes, Kiss and Vidnyánszky (2016) to Baire class $\xi $ functions for any countable ordinal $\xi \geq 1$. We answer two of the questions raised by them. Specifically, we show that for any countable ordinal $\xi \geq 1,$ the ranks $\beta _{\xi }^{\ast }$ and $\gamma _{\xi }^{\ast }$ are essentially equivalent, and that neither of them is essentially multiplicative. Since the rank $\beta $ is not essentially multiplicative, we investigate further its behavior with respect to products. We characterize the functions $f$ such that $\beta (fg)\leq \omega ^{\xi }$ whenever $\beta (g)\leq \omega ^{\xi }$ for any countable ordinal $\xi .$

Authors

  • Denny H. LeungDepartment of Mathematics
    National University of Singapore
    Singapore 119076
    e-mail
  • Hong-Wai NgSchool of Physical and Mathematical Sciences
    Division of Mathematical Sciences
    Nanyang Technological University
    Singapore 637371
    e-mail
  • Wee-Kee TangSchool of Physical and Mathematical Sciences
    Division of Mathematical Sciences
    Nanyang Technological University
    Singapore 637371
    e-mail

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