Abstract
In this article, we give a full description of the topological many-one degree structure of real-valued functions, recently introduced by Day—Downey—Westrick. We also clarify the relationship between the Martin conjecture and Day—Downey—Westrick’s topological Turing-like reducibility, also known as parallelized continuous strong Weihrauch reducibility, for single-valued functions: Under the axiom of determinacy, we show that the continuous Weihrauch degrees of parallelizable single-valued functions are well-ordered; and moreover, if f has continuous Weihrauch rank α, then f′ has continuous Weihrauch rank α + 1, where f′(x) is defined as the Turing jump of f(x).
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The author would like to thank Adam Day, Rod Downey, Antonio Montalbán, and Linda Brown Westrick for valuable discussions.
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The author was partially supported by JSPS KAKENHI Grant 19K03602, 21H03392, the JSPS Core-to-Core Program (A. Advanced Research Networks), and the JSPS-RFBR Bilateral Joint Research Project JPJSBP120204809.
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Kihara, T. Topological reducibilities for discontinuous functions and their structures. Isr. J. Math. 252, 461–500 (2022). https://doi.org/10.1007/s11856-022-2367-6
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DOI: https://doi.org/10.1007/s11856-022-2367-6