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arxiv: 2604.24522 · v1 · submitted 2026-04-27 · 🧮 math.LO

Recognition: unknown

NTP2 topological structures

Pablo and\'ujar Guerrero

Pith reviewed 2026-05-07 17:08 UTC · model grok-4.3

classification 🧮 math.LO
keywords NTP2constructible setsdefinable setsordered groupsp-adic fieldso-minimalitytopological dimensiondefinable choice
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The pith

NTP2 expansions of the reals by constructible sets define only constructible sets.

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

The paper establishes that adding constructible predicates to the structure of the real numbers with order and addition, under the NTP2 tameness condition, results in a structure where all definable sets are still constructible. This means the definable sets are finite Boolean combinations of closed sets, preserving a form of topological simplicity. The same preservation holds for the p-adic numbers with addition and multiplication, and for definably complete expansions of ordered groups. In the group case, this implies additional properties like generic local o-minimality and a well-behaved dimension. A sympathetic reader would care because this shows how model-theoretic tameness controls topological complexity in expansions.

Core claim

Every NTP2 expansion of (ℝ, <, +) by constructible sets defines only constructible sets, and definable functions are generically piecewise continuous. This extends to NTP2 expansions of (ℚ_p, +, ·) and NTP2 definably complete expansions of ordered groups, where the structure is generically locally o-minimal, has definable choice, and carries a well-behaved notion of naive topological dimension. For NIP uniform topological structures, constructibility is preserved in the Shelah expansion.

What carries the argument

The NTP2 condition on expansions of topological structures by constructible sets, which forces definable sets to remain finite Boolean combinations of closed sets.

If this is right

  • Definable functions in such expansions are generically piecewise continuous.
  • The result applies to all NTP2 expansions of the p-adics with addition and multiplication.
  • NTP2 definably complete expansions of ordered groups are generically locally o-minimal and have definable choice.
  • Constructibility of definable sets is preserved in the Shelah expansion for NIP uniform topological structures.
  • Strong expansions of (ℝ, <, +) by constructible sets can be classified.

Where Pith is reading between the lines

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

  • This tameness suggests that NTP2 structures limit the complexity of definable sets in topological settings.
  • Similar preservation might hold for other ordered fields or groups under NTP2 assumptions.
  • These results could inform classifications of tame expansions in model theory beyond the reals.
  • The generic local o-minimality provides a way to study dimension without full o-minimality.

Load-bearing premise

The expansions must satisfy the NTP2 property and the added sets must be constructible in the original topology.

What would settle it

Constructing an NTP2 expansion of (ℝ, <, +) by a constructible predicate that defines a set which is not a finite Boolean combination of closed sets would disprove the main claim.

read the original abstract

A subset of a topological space is constructible if it is a finite Boolean combination of closed sets. We prove that every NTP$_2$ expansion of $(\mathbb{R},<,+)$ by constructible sets defines only constructible sets, and that definable functions are generically piecewise continuous. The result also holds for all NTP$_2$ expansions of $(\mathbb{Q}_p,+,\cdot)$, and all NTP$_2$ definably complete expansions of ordered groups. In the latter case, the structure is generically locally o-minimal, has definable choice, and carries a well-behaved notion of naive topological dimension. For NIP uniform topological structures, constructibility of definable sets is preserved in the Shelah expansion. We classify strong expansions of $(\mathbb{R},<,+)$ by constructible sets, and obtain results on NTP$_2$ d-minimal structures.

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 / 2 minor

Summary. The manuscript proves that every NTP₂ expansion of (ℝ, <, +) by constructible sets defines only constructible sets, with definable functions generically piecewise continuous. Analogous preservation of constructibility holds for NTP₂ expansions of (ℚ_p, +, ·) and for all NTP₂ definably complete expansions of ordered groups; in the latter case the structures are generically locally o-minimal, admit definable choice, and carry a well-behaved naive topological dimension. Additional results include preservation of constructibility under the Shelah expansion for NIP uniform topological structures, a classification of strong expansions of (ℝ, <, +) by constructible sets, and theorems on NTP₂ d-minimal structures.

Significance. If the central claims hold, the work advances the model theory of topological tameness by demonstrating that NTP₂ controls definability so that expansions by constructible sets remain within the Boolean algebra generated by closed sets. The generic piecewise continuity, local o-minimality, and dimension results extend known preservation phenomena from o-minimal and NIP settings. The case-by-case analysis on the three base structures, combined with standard NTP₂ forking control, constitutes a coherent framework whose strength lies in the explicit topological hypotheses rather than ad-hoc parameters.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'naive topological dimension' appears without a one-sentence gloss or forward reference, which may hinder readers who encounter the term for the first time.
  2. [Introduction] The manuscript invokes 'standard facts about NTP₂' at several points; a brief parenthetical reminder of the precise property used (e.g., absence of 2-trees or forking calculus) would improve readability in the introductory sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on NTP₂ topological structures and for recommending minor revision. The summary and significance evaluation are appreciated as they accurately capture the main contributions regarding preservation of constructibility, generic piecewise continuity, and related tameness properties in the specified base structures.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external NTP2 facts

full rationale

The paper's central results on preservation of constructibility and generic piecewise continuity in NTP2 expansions are derived via case analysis on base structures (reals, p-adics, ordered groups) combined with standard external properties of NTP2 (no 2-tree property, forking control) and topological assumptions on constructible sets. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear; the argument flow invokes independent mathematical facts about NTP2 and definable completeness without reducing the target claims to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on the domain assumption that the structures are NTP2 and on standard definitions of constructible sets and definable completeness; no free parameters or new entities are introduced.

axioms (3)
  • domain assumption The expansion satisfies the NTP2 property
    Invoked to conclude that definable sets remain constructible.
  • domain assumption The added sets are constructible
    The expansions are restricted to constructible predicates.
  • domain assumption Definable completeness for ordered-group expansions
    Used to obtain generic local o-minimality and definable choice.

pith-pipeline@v0.9.0 · 5435 in / 1423 out tokens · 65409 ms · 2026-05-07T17:08:23.267736+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

3 extracted references · 2 canonical work pages

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    [FS10] Antongiulio Fornasiero and Tamara Servi

    arXiv:2107.04293. [FS10] Antongiulio Fornasiero and Tamara Servi. Definably complete Baire structures. Fund. Math., 209(3):215–241,

    work page arXiv

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    A note on topological model theory.Fund

    [Rob74] Abraham Robinson. A note on topological model theory.Fund. Math., 81(2):159– 171, 1973/74. [Sim11] Pierre Simon. On dp-minimal ordered structures.J. Symbolic Logic, 76(2):448– 460,

    1973

  3. [3]

    [Wal22] Erik Walsberg

    arXiv:1910.10572. [Wal22] Erik Walsberg. Externally definable quotients and NIP expansions of the real ordered additive group.Trans. Amer. Math. Soc., 375(3):1551–1578,

    work page arXiv 1910