Skip to main content
SpringerLink
Log in

Games characterizing certain families of functions

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

We obtain several game characterizations of Baire 1 functions between Polish spaces X, Y which extends the recent result of V. Kiss. Then we propose similar characterizations for equi-Bare 1 families of functions. Also, using related ideas, we give game characterizations of Baire measurable and Lebesgue measurable functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We would like to thank the reviewer for noting that this result was announced by Louveau and Semmes at a conference in 2010.

References

  1. Alikhani-Koopaei, A.: Equi-Baire one family on metric spaces: a generalization of equi-continuity; and some applications. Topol. Appl. 277, 107170 (2020)

    Article  MathSciNet  Google Scholar 

  2. Andretta, A.: More on Wadge determinacy. Ann. Pure Appl. Logic 144, 2–32 (2006)

    Article  MathSciNet  Google Scholar 

  3. Aurichi, L.F., Dias, R.R.: A minicourse on topological games. Topol. Appl. 258, 305–335 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bouziad, A.: The point of continuity property, neighbourhood assignments and filter convergences. Fund. Math. 218, 225–242 (2012)

    Article  MathSciNet  Google Scholar 

  5. Bukovský, L.: The Structure of the Real Line. Birkhäuser, Basel (2011)

    Book  Google Scholar 

  6. Camerlo, R., Duparc, J.: Some remarks on Baire’s grand theorem. Arch. Math. Logic 57, 195–201 (2018)

    Article  MathSciNet  Google Scholar 

  7. Cao, J., Moors, W.B.: A survey on topological games and their applications in analysis, RACSAM Rev. R. Acad. Cien. Serie A. Mat. 100, 39–49 (2006)

    Google Scholar 

  8. Carroy, R.: Playing in the first Baire class. Math. Log. Q. 60, 118–132 (2014)

    Article  MathSciNet  Google Scholar 

  9. Duparc, J.: Wadge hierarchy and Veblen hierarchy I. Borel sets of finite rank. J. Symb. Logic 66, 56–86 (2001)

    Article  MathSciNet  Google Scholar 

  10. Elekes, M., Flesch, J., Kiss, V., Nagy, D., Poór, M., Predtetchinski, A.: Games characterizing limsup functions and Baire class 1 functions. J. Symb. Log. 87(4), 1459–1473 (2022)

    Article  MathSciNet  Google Scholar 

  11. Glasner, E., Megrelishvili, M.: Hereditarily non-sensitive dynamical systems and linear representations. Colloq. Math. 104, 223–283 (2006)

    Article  MathSciNet  Google Scholar 

  12. Gruenhage, G.: Infinite games and generalizations of first-countable spaces. Gen. Topol. Appl. 6, 339–352 (1976)

    Article  MathSciNet  Google Scholar 

  13. Gruenhage, G.: The story of a topological game. Rocky Mt. Math. J. 36, 1885–1914 (2006)

    Article  MathSciNet  Google Scholar 

  14. Kechris, A.S.: Classical Descriptive Set Theory. Springer, New York (1995)

    Book  Google Scholar 

  15. Kiss, V.: A game characterizing Baire class 1 functions. J. Symb. Logic 85, 456–466 (2020)

    Article  MathSciNet  Google Scholar 

  16. Lecomte, D.: How can we recover Baire class one functions? Mathematika 50, 171–198 (2003)

    Article  MathSciNet  Google Scholar 

  17. Lee, P.Y., Tang, W.-K., Zhao, D.: An equivalent defnition of functions of the first Baire class. Proc. Amer. Math. Soc. 129, 2273–2275 (2001)

    Article  MathSciNet  Google Scholar 

  18. Motto Ros, L.: Game representations of classes of piecewise definable functions, MLQ Math. Log. Q. 57, 95–112 (2011)

    Article  MathSciNet  Google Scholar 

  19. Nobrega, H.: Games for Functions Baire classes, Weihrauch Degrees, Transfinite Computations, and Ranks, Ph.D. thesis, University of Amsterdam (2018)

  20. Notaro, L.: A game for Baire’s grand theorem, preprint, arXiv:2301.11805

  21. Oxtoby, J.C.: Measure and Category. Springer, New York (1980)

    Book  Google Scholar 

  22. Semmes, B.: Games, Trees, and Borel Functions, Ph.D. thesis, University of Amsterdam, (2009)

  23. Stone, A.H.: Lusin’s theorem. Atti Sem. Mat. Fis. Univ. Modena 44, 201–207 (1996)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the referee for several useful remarks that have helped us to improve the former version of the paper.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Lodz University of Technology, al. Politechniki 8, 93-590, Lodz, Poland

    Marek Balcerzak

  2. Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308, Gdańsk, Poland

    Tomasz Natkaniec & Piotr Szuca

Authors
  1. Marek Balcerzak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tomasz Natkaniec
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Piotr Szuca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomasz Natkaniec.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest, and add that there is no data associated with this work.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Balcerzak, M., Natkaniec, T. & Szuca, P. Games characterizing certain families of functions. Arch. Math. Logic (2024). https://doi.org/10.1007/s00153-024-00922-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00153-024-00922-9

Keywords

Mathematics Subject Classification

Search

Navigation