Archive for Mathematical Logic

, Volume 57, Issue 3–4, pp 195–201 | Cite as

Some remarks on Baire’s grand theorem

Article
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Abstract

We provide a game theoretical proof of the fact that if f is a function from a zero-dimensional Polish space to \( \mathbb N^{\mathbb N}\) that has a point of continuity when restricted to any non-empty compact subset, then f is of Baire class 1. We use this property of the restrictions to compact sets to give a generalisation of Baire’s grand theorem for functions of any Baire class.

Keywords

Baire class \(\xi \) function Wadge game Eraser game Polish zero Dimensional space Compact set 

Mathematics Subject Classification

03E15 

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References

  1. 1.
    Carroy, R.: Playing in the first Baire class. Math. Log. Q. 60, 118–132 (2014)MathSciNetCrossRefMATHGoogle Scholar
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    Duparc, J.: Wadge hierarchy and Veblen hierarchy. I. Borel sets of finite rank. J. Symbol. Log. 66, 56–86 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Kechris, A.S.: Classical Descriptive Set Theory. Springer, New York (1995)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Mathematical Sciences Joseph-Louis LagrangePolitecnico di TorinoTorinoItaly
  2. 2.Département des systèmes d’informationUniversité de LausanneLausanneSwitzerland

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