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arxiv: 2604.04492 · v1 · submitted 2026-04-06 · 🧮 math.LO

An effective version of the Stone duality

Nikolay A. Bazhenov , Iskander Sh. Kalimullin , Marina V. Schwidefsky This is my paper

Pith reviewed 2026-05-10 20:09 UTC · model grok-4.3

classification 🧮 math.LO
keywords effective Stone dualitycomputable topologyZ-computably enumerable presentationsdegree spectraalmost semispectral spacesdistributive c-posetstopological spaces with base
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The pith

The Stone duality between almost semispectral spaces and distributive c-posets extends effectively to objects with Z-computably enumerable presentations for any set Z.

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

The paper introduces Z-computably enumerable presentations for countable c-posets and second-countable T0-spaces with bases. It proves that the classical Stone-type duality between the category of almost semispectral spaces (with spectral mappings) and the category of distributive c-posets (with strict mappings) remains valid when both sides are restricted to objects possessing only Z-c.e. presentations. This effective duality transfers degree spectra of countable algebraic structures to topological spaces with bases and allows construction of computable spaces having exactly any chosen finite number of computable copies up to effective spectral homeomorphisms. A reader would care because the result shows how computability constraints can be imposed on a classical duality without breaking the correspondence, thereby importing algebraic computability phenomena into topology in a controlled way.

Core claim

We introduce effective versions of countable c-posets and topological spaces with base using Z-computably enumerable presentations for an arbitrary set Z. We prove that the Stone duality between the category AS of almost semispectral spaces with base (morphisms are spectral mappings) and the category DP of distributive c-posets (morphisms are strict mappings) is preserved under restriction to objects having Z-c.e. presentations only. As consequences, every degree spectrum of a countable algebraic structure is realized as the degree spectrum of some topological space with base, and for each positive integer N there exists a computable topological space with base possessing precisely N computa

What carries the argument

Z-computably enumerable presentations of objects in the categories AS (almost semispectral spaces with base) and DP (distributive c-posets), which carry the effective Stone duality.

If this is right

  • Every degree spectrum arising from a countable algebraic structure can be realized exactly as the degree spectrum of a topological space with base.
  • For any positive integer N there exists a computable topological space with base having precisely N computable copies up to effective spectral homeomorphisms.
  • The correspondence between spectral mappings and strict mappings remains effective when both categories are restricted to Z-c.e. objects.

Where Pith is reading between the lines

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

  • Many existing computability results about algebraic structures can be translated into statements about topological spaces via the preserved duality.
  • The same restriction-to-Z-c.e. technique might apply to other classical dualities between posets and spaces.
  • One could test whether the number of computable copies or the degree spectrum behaves similarly under other notions of effective homeomorphism.

Load-bearing premise

The newly defined notions of Z-c.e. presentation for c-posets and spaces with base, along with the morphisms in the categories AS and DP, correctly capture the intended effective counterparts so that the classical duality lifts directly.

What would settle it

A specific Z and a Z-c.e. almost semispectral space whose corresponding distributive c-poset under the duality fails to admit a Z-c.e. presentation, or an algebraic degree spectrum that cannot be matched by any space with base.

read the original abstract

The paper studies computability-theoretic aspects of topological $T_0$-spaces. We introduce effective versions of the notions of a countable $c$-poset and a (second-countable) topological space with base. Based on this, we prove an effective version of the known Stone-type duality between the category $\mathbf{AS}$ (whose objects are almost semispectral spaces with base and whose morphisms are spectral mappings) and the category $\mathbf{DP}$ (whose objects are distributive $c$-posets and whose morphisms are strict mappings). Namely, we show that for an arbitrary set $Z\subseteq \omega$, this duality is preserved when one restricts to objects which have $Z$-computably enumerable presentations only. Following this approach, we establish several results in computable topology. We prove that every degree spectrum of a countable algebraic structure can be realized as the degree spectrum of a topological space with base. We show that for any non-zero natural number $N$, there is a computable topological space with base that has precisely $N$-many computable copies, up to effective spectral homeomorphisms.

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

0 major / 3 minor

Summary. The paper introduces Z-c.e. presentations for countable c-posets (via enumerations of the order and distinguished elements) and for second-countable T0-spaces with base (via enumerations of a countable base and specialization preorder). It proves that the classical Stone duality between the category AS of almost semispectral spaces with base (morphisms: spectral maps) and the category DP of distributive c-posets (morphisms: strict maps) lifts to the relativized categories AS_Z and DP_Z for arbitrary Z ⊆ ω, with the effective functors defined by the same formulas but relativized to Z and shown to be mutually inverse on presentations while preserving the relevant morphisms. Two applications follow: every degree spectrum of a countable algebraic structure is realizable as the degree spectrum of a space with base, and for each N ≥ 1 there exists a computable space with base having exactly N computable copies up to effective spectral homeomorphisms.

Significance. If the effective duality and its preservation under Z-c.e. presentations hold, the work supplies a computability-theoretic transfer principle between algebraic structures and topological spaces that is uniform in the oracle Z. The explicit inductive definitions of the presentations and the direct relativization of the classical functors (without extra hidden assumptions) constitute a clear technical advance over prior effective dualities that were often restricted to computable or low-degree cases. The two applications demonstrate that degree spectra and computable-copy counts transfer non-trivially, providing concrete new results in computable topology.

minor comments (3)
  1. §2.3: the definition of 'almost semispectral' space is stated only by reference to the classical literature; a self-contained characterization in terms of the base and specialization preorder would improve readability for readers focused on the effective setting.
  2. §4.2, Definition 4.5: the notion of 'effective spectral homeomorphism' is introduced without an explicit clause confirming that it preserves the Z-c.e. presentations of the bases; adding one sentence would clarify that the application in Theorem 5.3 is well-defined.
  3. The paper would benefit from a short table comparing the classical and effective categories (objects, morphisms, functors) to make the lifting argument easier to follow at a glance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the report and the recommendation of minor revision. No major comments were provided.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from new definitions

full rationale

The paper introduces explicit inductive definitions of Z-c.e. presentations for c-posets (via enumerations of the order and distinguished elements) and for second-countable spaces with base (via enumerations of a countable base and specialization preorder). The effective Stone functors are obtained by direct relativization of the classical formulas to the oracle Z; the proofs then verify mutual inversion on the level of presentations and preservation of strict/spectral morphisms. Degree-spectrum transfer results are obtained by composing these functors with known algebraic constructions. No equation or central claim reduces by construction to a fitted parameter, a self-definition, or a load-bearing self-citation; the entire chain is built from the newly supplied effective notions and standard relativization, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard axioms of topology (T0 separation, second-countability via bases) and order theory (distributivity of c-posets) plus the new definitions of effective presentations; no free parameters or invented entities beyond the effective notions themselves are apparent from the abstract.

axioms (2)
  • standard math T0 topological spaces admit a base of open sets closed under finite intersections in the usual way.
    Invoked when defining spaces with base and spectral mappings.
  • standard math Distributive c-posets satisfy the standard order-theoretic distributivity laws used in classical Stone duality.
    Required for the objects of category DP.

pith-pipeline@v0.9.0 · 5500 in / 1456 out tokens · 43330 ms · 2026-05-10T20:09:26.945352+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean Recovery theorem (LogicNat ≃ Nat) unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We introduce effective versions of the notions of a countable c-poset and a (second-countable) topological space with base... Theorem 21 and Corollary 22 establish an effective version of Theorem 15: a c-poset P from DP has a Z-c.e. presentation if and only if its dual topological space with base T(P) from AS has a Z-c.e. presentation.

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extends
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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