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arxiv: 1906.10573 · v1 · pith:CUKBVCIGnew · submitted 2019-06-25 · 🧮 math.LO · math.GN

Topological reducibilities for discontinuous functions and their structures

Takayuki Kihara This is my paper

Pith reviewed 2026-05-25 15:44 UTC · model grok-4.3

classification 🧮 math.LO math.GN
keywords topological many-one reducibilityWeihrauch degreesBaire-one functionsMartin conjectureaxiom of determinacyPolish spacesdiscontinuous functionsdegree structures
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The pith

The topological many-one degrees of real-valued functions admit a complete structural description.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes a complete classification of the topological many-one reducibility degrees among real-valued functions. It extends an existing characterization of the Bourgain rank for Baire-one functions from compact to noncompact Polish domains. Under the axiom of determinacy it further shows that the continuous Weihrauch degrees of parallelizable single-valued functions form a well-ordering in which the jump operation increases rank by exactly one. A reader would care because these results organize the hierarchy of discontinuous functions according to continuous reductions and tie that hierarchy to the Martin conjecture.

Core claim

We give a full description of a topological many-one degree structure of real-valued functions. We also point out that their characterization of the Bourgain rank of a Baire-one function of compact Polish domain can be extended to noncompact Polish domain. Finally, we 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 continuousWe

What carries the argument

topological many-one reducibility, which assigns degrees to real-valued functions according to the existence of continuous pre- and post-processors that reduce one function to another

If this is right

  • The topological many-one degrees of all real-valued functions receive a complete classification.
  • The Bourgain rank characterization for Baire-one functions holds on every Polish domain.
  • Continuous Weihrauch degrees of parallelizable single-valued functions form a well-ordering under the axiom of determinacy.
  • The jump operation on such a function raises its continuous Weihrauch rank by precisely one.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The classification may supply a concrete model in which to test further instances of the Martin conjecture for other reducibility notions.
  • Specific computable or continuous examples could be checked directly to verify the rank-increase claim without invoking the full strength of the axiom of determinacy.
  • The same structural description might be adapted to compare other classes of discontinuous functions such as Baire-two or higher.

Load-bearing premise

The axiom of determinacy is needed to prove that the continuous Weihrauch degrees are well-ordered.

What would settle it

An explicit pair of parallelizable single-valued real functions whose continuous Weihrauch degrees are incomparable in any model satisfying the axiom of determinacy.

Figures

Figures reproduced from arXiv: 1906.10573 by Takayuki Kihara.

Figure 1
Figure 1. Figure 1: The structure of m-Wadge degrees of subsets of ω ω (The structure of non-proper m-Wadge degrees of 2⊥-valued functions of ω ω ) which implies that B ≤mw A. By a symmetric argument, we also have A ≤mw B, and conclude that A ≡mw B. Lemmas 2.2 and 2.3 give the complete description of the non-proper 2⊥-m￾Wadge degree structure, hence the m-Wadge degree structure of subsets of ω ω. Each selfdual Wadge degree sp… view at source ↗
Figure 2
Figure 2. Figure 2: The structure of m-degrees of real-valued functions on 2ω We will show that the map f 7→ Levf induces an isomorphism between the m-degrees on real-valued functions on ω ω and the m-Wadge degrees on nonempty proper subsets of ω ω. Theorem 3.3. The map f 7→ Levf induces an isomorphism between the quotients of (F(ω ω, R), ≤m) and (P(ω ω) \ {∅, ωω}, ≤mw). The proof of Theorem 3.3 will be given in the rest of t… view at source ↗
read the original abstract

In this article, we give a full description of a topological many-one degree structure of real-valued functions, recently introduced by Day-Downey-Westrick. We also point out that their characterization of the Bourgain rank of a Baire-one function of compact Polish domain can be extended to noncompact Polish domain. Finally, we 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 $f$ is has continuous Weihrauch rank $\alpha$, then $f'$ has continuous Weihrauch rank $\alpha+1$, where $f'(x)$ is defined as the Turing jump of $f(x)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript gives a complete description of the topological many-one degree structure on real-valued functions introduced by Day-Downey-Westrick. It extends the Bourgain-rank characterization of Baire-one functions from compact to arbitrary Polish domains. Under the axiom of determinacy it proves that the continuous Weihrauch degrees of parallelizable single-valued functions are well-ordered and that the continuous Weihrauch rank of the jump f' is exactly one greater than the rank of f.

Significance. If the claims hold, the unconditional results on many-one degrees and the Bourgain-rank extension supply concrete structural information that can be used in ZFC. The AD-based well-ordering result supplies a precise clarification of the relationship between the Martin conjecture and parallelized continuous strong Weihrauch reducibility for single-valued functions, a contribution that is valuable within the descriptive-set-theoretic literature that routinely employs AD for linearity and well-foundedness statements.

major comments (1)
  1. [final section / abstract claim on continuous Weihrauch degrees] The well-ordering and successor-rank statements are proved only under AD (as stated in the abstract and the final section). The manuscript should explicitly record whether any fragment of the well-ordering (e.g., linearity or absence of infinite descending chains) is provable in ZFC alone, or whether the authors know of ZF models in which the structure fails to be well-ordered; without this discussion the scope of the Martin-conjecture clarification remains imprecise.
minor comments (2)
  1. [introduction / preliminaries] Notation for the jump operation f' is introduced only in the abstract; a formal definition should appear in the body before the rank-increase theorem is stated.
  2. [section on Bourgain rank] The extension of the Bourgain-rank result to non-compact domains is asserted without a displayed statement of the original compact-domain theorem; a brief recall of the cited result would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment. We address the single major comment below.

read point-by-point responses

  1. Referee: [final section / abstract claim on continuous Weihrauch degrees] The well-ordering and successor-rank statements are proved only under AD (as stated in the abstract and the final section). The manuscript should explicitly record whether any fragment of the well-ordering (e.g., linearity or absence of infinite descending chains) is provable in ZFC alone, or whether the authors know of ZF models in which the structure fails to be well-ordered; without this discussion the scope of the Martin-conjecture clarification remains imprecise.

    Authors: We agree that the well-ordering and successor-rank results are proved under AD, as already indicated in the abstract and final section. The manuscript does not contain any ZFC proofs of linearity or well-foundedness for this structure, nor do the authors know of ZF models in which the structure fails to be well-ordered. In the revised manuscript we will add an explicit remark in the final section stating that these properties rely on AD and that their status in ZFC is open; this will make the precise scope of the Martin-conjecture clarification clear. revision: yes

Circularity Check

0 steps flagged

No circularity; new structural results and AD-based ordering stand independently of inputs

full rationale

The paper describes the topological many-one degree structure by building on the external prior work of Day-Downey-Westrick (distinct authors), extends the Bourgain-rank characterization to non-compact domains without any fitted parameters or self-referential definitions, and establishes the well-ordering plus rank-increase claim explicitly under the axiom of determinacy as a fresh theorem. No equations reduce a claimed prediction to a fitted input by construction, no uniqueness theorems are imported from the author's own prior work, and no ansatz is smuggled via self-citation. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background from set theory and computability plus the axiom of determinacy for the ordering result; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption Axiom of Determinacy (AD)
    Invoked to prove that continuous Weihrauch degrees of parallelizable single-valued functions are well-ordered and that the jump increases rank by exactly one.

pith-pipeline@v0.9.0 · 5681 in / 1307 out tokens · 25617 ms · 2026-05-25T15:44:14.253045+00:00 · methodology

discussion (0)

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Reference graph

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