A game for Baire’s grand theorem

Notaro, Lorenzo

doi:10.1090/proc/17025

A game for Baire’s grand theorem
HTML articles powered by AMS MathViewer

by Lorenzo Notaro;

Proc. Amer. Math. Soc. 152 (2024), 4903-4914

DOI: https://doi.org/10.1090/proc/17025

Published electronically: September 20, 2024

HTML | PDF | Request permission

Abstract:

Generalizing a result of Kiss, we provide a game that characterizes Baire class $1$ functions between arbitrary separable metrizable spaces. We show that the determinacy of our game is equivalent to a generalization of Baire’s grand theorem, and that both these statements hold under $\mathsf {AD}$ and in Solovay’s model.

References

Viktor Kiss, A game characterizing Baire class 1 functions, J. Symb. Log. 85 (2020), no. 1, 456–466. MR 4085069, DOI 10.1017/jsl.2019.38
Raphaël Carroy, Playing in the first Baire class, MLQ Math. Log. Q. 60 (2014), no. 1-2, 118–132. MR 3171615, DOI 10.1002/malq.201200064
Raphaël Carroy, Benjamin D. Miller, and Dániel T. Soukup, The open dihypergraph dichotomy and the second level of the Borel hierarchy, Trends in set theory, Contemp. Math., vol. 752, Amer. Math. Soc., [Providence], RI, [2020] ©2020, pp. 1–19. MR 4132098, DOI 10.1090/conm/752/15127
J. Gerlits and M. Laczkovich, On barely continuous and Baire $1$ functions, Topology, Vol. I, II (Proc. Fourth Colloq., Budapest, 1978) Colloq. Math. Soc. János Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 493–499. MR 588800
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
William Wilfred Wadge, Reducibility and determinateness on the Baire space, ProQuest LLC, Ann Arbor, MI, 1983. Thesis (Ph.D.)–University of California, Berkeley. MR 2633374
J. Duparc, Wadge hierarchy and Veblen hierarchy. I. Borel sets of finite rank, J. Symbolic Logic 66 (2001), no. 1, 56–86. MR 1825174, DOI 10.2307/2694911
Carlos Augusto Di Prisco and Stevo Todorcevic, Perfect-set properties in $L(\textbf {R})[U]$, Adv. Math. 139 (1998), no. 2, 240–259. MR 1654181, DOI 10.1006/aima.1998.1752
R. Engelking, On closed images of the space of irrationals, Proc. Amer. Math. Soc. 21 (1969), 583–586. MR 239571, DOI 10.1090/S0002-9939-1969-0239571-X
Akihiro Kanamori, The higher infinite, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Large cardinals in set theory from their beginnings. MR 1994835
Luca Motto Ros, Game representations of classes of piecewise definable functions, MLQ Math. Log. Q. 57 (2011), no. 1, 95–112. MR 2779711, DOI 10.1002/malq.200910124
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 217751
Riccardo Camerlo and Jacques Duparc, Some remarks on Baire’s grand theorem, Arch. Math. Logic 57 (2018), no. 3-4, 195–201. MR 3778956, DOI 10.1007/s00153-017-0563-6
René Baire, Leçons sur les fonctions discontinues, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1995 (French). Reprint of the 1905 original. MR 1400223
Olena Karlova, A generalization of a Baire theorem concerning barely continuous functions, Topology Appl. 258 (2019), 433–438. MR 3927904, DOI 10.1016/j.topol.2019.03.016
Olena Karlova, On Baire-one mappings with zero-dimensional domains, Colloq. Math. 146 (2017), no. 1, 129–141. MR 3570207, DOI 10.4064/cm6530-1-2016
Petr Pošta, Dirichlet problem and subclasses of Baire-one functions, Israel J. Math. 226 (2018), no. 1, 177–188. MR 3819690, DOI 10.1007/s11856-018-1707-z
E. Odell and H. P. Rosenthal, A double-dual characterization of separable Banach spaces containing $l^{1}$, Israel J. Math. 20 (1975), no. 3-4, 375–384. MR 377482, DOI 10.1007/BF02760341
Felix Hausdorff, Set theory, 2nd ed., Chelsea Publishing Co., New York, 1962. Translated from the German by John R. Aumann et al. MR 141601
Alessandro Andretta, More on Wadge determinacy, Ann. Pure Appl. Logic 144 (2006), no. 1-3, 2–32. MR 2279651, DOI 10.1016/j.apal.2006.05.008
Márton Elekes, János Flesch, Viktor Kiss, Donát Nagy, Márk Poór, and Arkadi Predtetchinski, Games characterizing limsup functions and Baire class 1 functions, J. Symb. Log. 87 (2022), no. 4, 1459–1473. MR 4510828, DOI 10.1017/jsl.2022.29
Marek Balcerzak, Tomasz Natkaniec, and Piotr Szuca, Games characterizing certain families of functions, Arch. Math. Logic 63 (2024), no. 7, 759–772., DOI 10.1007/s00153-024-00922-9