TY - JOUR

T1 - THREE TOPOLOGICAL REDUCIBILITIES FOR DISCONTINUOUS FUNCTIONS

AU - Day, Adam R.

AU - Downey, Rod

AU - Westrick, Linda

N1 - Funding Information:
Received by the editors September 21, 2020, and, in revised form, August 2, 2021, and March 8, 2022. 2020 Mathematics Subject Classification. Primary 03D30, 03D78. This work was supported in part by the Marsden Fund of New Zealand. The third author was supported in part by Noam Greenberg’s Rutherford Discovery Fellowship as a postdoctoral fellow.
Publisher Copyright:
© 2022 by the author(s).

PY - 2022/10/17

Y1 - 2022/10/17

N2 - We define a family of three related reducibilities, ≤T, ≤tt and ≤m, for arbitrary functions f, g: X → R, where X is a compact separable metric space. The ≡T-equivalence classes mostly coincide with the proper Baire classes. We show that certain α-jump functions jα: 2ω → R are ≤m-minimal in their Baire class. Within the Baire 1 functions, we completely characterize the degree structure associated to ≤tt and ≤m, finding an exact match to the α hierarchy introduced by Bourgain [Bull. Soc. Math. Belg. Sér. B 32 (1980), pp. 235–249] and analyzed in Kechris & Louveau [Trans. Amer. Math. Soc. 318 (1990), pp. 209–236].

AB - We define a family of three related reducibilities, ≤T, ≤tt and ≤m, for arbitrary functions f, g: X → R, where X is a compact separable metric space. The ≡T-equivalence classes mostly coincide with the proper Baire classes. We show that certain α-jump functions jα: 2ω → R are ≤m-minimal in their Baire class. Within the Baire 1 functions, we completely characterize the degree structure associated to ≤tt and ≤m, finding an exact match to the α hierarchy introduced by Bourgain [Bull. Soc. Math. Belg. Sér. B 32 (1980), pp. 235–249] and analyzed in Kechris & Louveau [Trans. Amer. Math. Soc. 318 (1990), pp. 209–236].

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U2 - 10.1090/btran/115

DO - 10.1090/btran/115

M3 - Article

AN - SCOPUS:85147868410

SN - 2330-0000

VL - 9

SP - 859

EP - 895

JO - Transactions of the American Mathematical Society Series B

JF - Transactions of the American Mathematical Society Series B

IS - 28

ER -