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arxiv: 1907.00798 · v1 · pith:IDYPR5YTnew · submitted 2019-06-28 · 🧮 math.GM

Neutrosophic metric Spaces

Murat Kiri\c{s}ci , Necip \c{S}im\c{s}ek This is my paper

Pith reviewed 2026-05-25 12:40 UTC · model grok-4.3

classification 🧮 math.GM
keywords neutrosophic metric spaceBaire category theoremuniform convergence theoremneutrosophic numbersmetric axiomstopological properties
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The pith

A distance function on neutrosophic numbers creates a metric space in which the Baire category theorem and the uniform convergence theorem have direct analogues.

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

The paper defines a new metric space whose distances take values in neutrosophic numbers. It verifies that this distance satisfies the metric axioms and studies the resulting topological and structural properties. Analogues of the classical Baire category theorem and the uniform convergence theorem are proved inside this setting. A reader would care because the construction supplies a single framework in which indeterminacy is built into the distances while standard completeness and convergence results remain available.

Core claim

The authors introduce a distance function d that maps pairs of neutrosophic numbers to neutrosophic numbers, verify that d obeys non-negativity, symmetry and the triangle inequality, and then show that the induced topology supports the statement and proof of the Baire category theorem together with the statement and proof of the uniform convergence theorem.

What carries the argument

The neutrosophic metric: a distance function from pairs of neutrosophic numbers to neutrosophic numbers that satisfies the three metric axioms and induces a topology on the space.

If this is right

  • In a complete neutrosophic metric space the intersection of countably many dense open sets is dense.
  • The uniform limit of a sequence of continuous functions between neutrosophic metric spaces is itself continuous.
  • Cauchy sequences with respect to the neutrosophic distance converge when the space is complete.
  • Open and closed sets can be defined directly from the neutrosophic distance values.

Where Pith is reading between the lines

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

  • The same distance construction could be tested on other generalized number systems to see which classical theorems survive.
  • Fixed-point results that rely on completeness might be examined next in neutrosophic metric spaces.
  • The topology induced by the neutrosophic metric supplies a concrete setting in which to compare ordinary convergence with convergence that tracks indeterminacy.

Load-bearing premise

The proposed distance function on neutrosophic numbers satisfies the metric axioms and induces a topology in which the classical proofs of the Baire category theorem and the uniform convergence theorem remain valid.

What would settle it

An explicit triple of neutrosophic numbers for which the proposed distance violates the triangle inequality, or a concrete sequence of functions that converges uniformly in the neutrosophic metric yet whose pointwise limit fails to be continuous.

read the original abstract

In present paper, the definition of new metric space with neutrosophic numbers is given. Several topological and structural properties have been investigated. The analogues of Baire Category Theorem and Uniform Convergence Theorem are given for Neutrosophic metric spaces.

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 introduces a definition of a metric space whose underlying set consists of neutrosophic numbers, investigates several topological and structural properties of this space, and states analogues of the Baire Category Theorem and the Uniform Convergence Theorem that are claimed to hold in the new setting.

Significance. A correctly verified metric on neutrosophic numbers would supply a concrete topological framework for handling truth-indeterminacy-falsity triples, allowing classical results such as Baire category and uniform convergence to be transferred. The paper supplies no machine-checked proofs, no reproducible code, and no parameter-free derivations, so its contribution remains conditional on the metric axioms being satisfied.

major comments (1)
  1. [Definition section / main construction] Definition of the distance function on neutrosophic numbers: the manuscript asserts that the proposed distance satisfies the metric axioms (non-negativity, symmetry, triangle inequality) but supplies neither an explicit verification of the triangle inequality for arbitrary triples of neutrosophic numbers nor a derivation from the component-wise operations on the truth, indeterminacy and falsity components. This verification is load-bearing for the claim that the structure is a metric space and therefore for the validity of the stated analogues of the Baire Category Theorem and Uniform Convergence Theorem.
minor comments (2)
  1. [Abstract] Abstract: the sentence 'In present paper' should read 'In the present paper'.
  2. [Definition section] The manuscript does not indicate which specific neutrosophic operations (e.g., component-wise addition or a custom combination) are used to define the distance; this notation should be made explicit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and for identifying the need for explicit verification of the metric axioms. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Definition section / main construction] Definition of the distance function on neutrosophic numbers: the manuscript asserts that the proposed distance satisfies the metric axioms (non-negativity, symmetry, triangle inequality) but supplies neither an explicit verification of the triangle inequality for arbitrary triples of neutrosophic numbers nor a derivation from the component-wise operations on the truth, indeterminacy and falsity components. This verification is load-bearing for the claim that the structure is a metric space and therefore for the validity of the stated analogues of the Baire Category Theorem and Uniform Convergence Theorem.

    Authors: We agree that the triangle inequality requires explicit verification for arbitrary triples and that a derivation from the component-wise operations on the truth, indeterminacy, and falsity components should be supplied. In the revised manuscript we will insert a complete, self-contained proof of all metric axioms, with the triangle inequality proved directly from the definition of the distance function. This addition will also clarify the foundation for the Baire Category and Uniform Convergence analogues. revision: yes

Circularity Check

0 steps flagged

No circularity; definition and theorems are self-contained.

full rationale

The paper defines a neutrosophic metric space and states analogues of the Baire Category Theorem and Uniform Convergence Theorem for it. No quoted equations, self-citations, or constructions reduce any claimed result to its inputs by construction, rename a fitted parameter as a prediction, or import uniqueness via author overlap. The load-bearing premise (that the proposed distance satisfies metric axioms) is a standard definitional step whose verification is external to any circular loop; the derivation chain therefore remains independent of the patterns that trigger circularity scores.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.0 · 5554 in / 1045 out tokens · 48026 ms · 2026-05-25T12:40:50.625569+00:00 · methodology

discussion (0)

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Reference graph

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