A Python Framework Enhancement for Neutrosophic Topologies

Giorgio Nordo; Maikel Yelandi Leyva Vazquez; Saeid Jafari

REVIEW 3 major objections 7 minor 30 references

A Python Framework Enhancement for Neutrosophic Topologies

T0 review · 3 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper adds an NSfamily class to the PYNS framework that generates a neutrosophic topology from any finite sub-basis and verifies the topology axioms.

desk verdict A useful but buggy tool for computational neutrosophic topology; the code artifact is new, but the printed code doesn't run and getNSBase doesn't match the paper's own definition. read the letter →

arxiv 2412.00047 v1 pith:EL4FHLNL submitted 2024-11-23 math.GM

Giorgio Nordo , Saeid Jafari , Maikel Yelandi Leyva Vazquez This is my paper · ORCID

classification math.GM MSC 03E7254A4054D9968W99

keywords neutrosophicsettopologysingle-valuedtopologicalbasissub-basisPythonframeworkNSfamilyclosureproperties

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that a single new software class can carry the full machinery of neutrosophic topology: given any finite family of single-valued neutrosophic sets over a common universe, the class treats that family as a sub-basis, computes the topology it generates, and then verifies that the result obeys the four topology axioms. The point is practical. Neutrosophic topology is a recent extension of ordinary topology that assigns each element separate degrees of truth, indeterminacy, and falsity, and hand-checking closure under union and intersection quickly becomes tedious even for small universes. If the implementation is correct, a researcher can generate complete topologies from a few starting sets and check arbitrary candidate families in a fraction of a second. The paper demonstrates the workflow on two- and three-element universes where the generated topology has the expected six members.

What carries the argument

The load-bearing object is the class NSfamily, a list of NSset objects tied to one NSuniverse. Its generative core is the two-step pipeline getNSBase() then getNSTopologyByBase(): the first enumerates every combination of the stored sets and reduces it by the neutrosophic intersection operation, adding the absolute set; the second enumerates every combination of the resulting basis and reduces it by the neutrosophic union operation. A shared private method __checkClosure takes an operation (union or intersection) and returns False if any combination's result is missing from the family, which is what lets isNeutrosophicTopology() verify the axioms. The mathematical warrant for the pipeline is Proposition 2.16, T(S)=T(B(S)), with B(S) the finite intersections of S together with the absolute set.

What would settle it

On a finite universe, say $\{1,2\}$, take the family $S=\{A,B\}$ with $A$ and $B$ chosen so that their neutrosophic union and intersection are not already in $S$. Run getNSTopologyBySubBase() and compare its output to the family obtained by repeatedly adjoining all finite intersections and all finite unions until closure; also check whether isNeutrosophicTopology() returns True for the output. Any mismatch between the returned family and the closure, or any output on which the four axioms fail, would refute the claim that the class builds and verifies the generated topology.

Watch

Extended reading notes

Core claim

The central claim is that the NSfamily class makes the construction of neutrosophic topologies algorithmic. A single-valued neutrosophic set assigns each element of a universe three numbers in [0,1]—membership, indeterminacy, and non-membership—and the class stores families of such sets sharing one universe. From any such family S, getNSBase() forms all finite neutrosophic intersections and adds the absolute set, thereby forming the basis B(S); getNSTopologyByBase() then forms all finite neutrosophic unions of that basis, and getNSTopologyBySubBase() chains the two steps to produce the generated topology T(S). The companion method isNeutrosophicTopology() checks the four axioms—presence of the empty and absolute sets, closure under union, and closure under finite intersection—by enumerating combinations and applying the corresponding operations. For the worked examples, the generated family is indeed a neutrosophic topology containing the starting family.

Load-bearing premise

The generator depends on Proposition 2.16, which the paper states without proof: the topology generated by a family S is exactly the topology generated by all finite intersections of S plus the absolute set; if that equality fails, getNSTopologyBySubBase may not return the smallest topology containing S.

Editorial extensions

If this is right

If the central claim holds, a researcher can generate the full neutrosophic topology from a finite sub-basis with one method call, so exhaustively listing the open sets of a finite-space topology becomes automatic.
The same class can check whether any candidate family is a neutrosophic topology, making verification of examples in papers or exercises a single boolean call.
Because the generator and checker share the same union and intersection operations, the framework gives a consistent reference implementation of the axioms across different neutrosophic set constructions.
The worked examples show that the generated topology from a two-set sub-basis on a three-element universe has exactly six members, matching the theoretical expectation for the minimal topology containing the sub-basis.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension would be to expose the private closure checker so users can test other binary operations, such as neutrosophic difference or symmetric difference, for closure.
Because the enumeration is combinatorial, the practical range of the tool is limited to small finite universes; a generalization to infinite or continuous universes would need symbolic or lazy representations instead of listing all combinations.
The same generation scheme could be used to search for small counterexamples to proposed theorems about neutrosophic bases, since any finite search space of sets can be filtered through isNeutrosophicTopology().

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague, this is a software paper that extends the authors' own PYNS framework with an NSfamily class for building and checking neutrosophic topologies. The genuinely new bit is the code artifact: methods like getNSTopologyByBase() and isNeutrosophicTopology() automate finite enumeration of unions and intersections. That's real and potentially useful for the small group of researchers doing computational neutrosophic topology. The math is a direct analogy to classical topology, and the paper says so itself, so don't expect new theory.

The paper does several things right: the examples are concrete, the definitions are mostly standard, and the idea of generating a topology by all finite intersections then all finite unions is correct for finite families. The GitHub repo is a plus; GPL licensing and a cross-checkable artifact count for something.

But the soft spots are substantial. The printed code cannot run: the constructor is named init rather than __init__, __str__ and __format__ appear as str and format, and NSintersectionClosed has a missing parenthesis. That's not a trivial typo; it means every example in the paper would fail if copied. More importantly, getNSBase() does not append the absolute set that Definition 2.14 puts into B(S). So the object it returns is not a neutrosophic basis by the paper's own Definition 2.13. The stress-test note gets this right: the topology generated by getNSTopologyByBase() still includes the absolute set, so getNSTopologyBySubBase() may give the correct topology in the finite case, but the claim that getNSBase() returns a neutrosophic basis is false as implemented. The fix is a one-liner, but as printed the framework does not implement Proposition 2.16.

There's also a symbol inconsistency: the paper defines ⋒ as union and ⋓ as intersection, but the printed output uses them in the opposite way. Not load-bearing, but confusing.

Proposition 2.16 is stated without proof and is load-bearing. For the finite case it's fine, but the paper doesn't say it's restricted to finite families, and the algorithms are finite enumerations anyway.

Bottom line: the conceptual framework holds up for the finite case, and the intended behavior is clear. The code needs a careful rewrite and the basis method needs to match the definitions. This deserves a serious referee, but it needs major revision before it's usable.

Referee Report

3 major / 7 minor

Summary. The paper proposes an extension of the PYNS Python framework with a new NSfamily class for representing families of single-valued neutrosophic sets. The class is intended to build neutrosophic bases and sub-bases, generate the neutrosophic topology generated by a finite family, verify the four topology axioms, and support standard set-theoretic operations on families. The theoretical part recalls definitions of neutrosophic sets, neutrosophic topology, bases, and sub-bases, and states Proposition 2.16 that the topology generated by a family S coincides with the topology generated by the family B(S) of finite intersections plus the absolute set. The implementation part gives pseudo-code, Python listings, and interactive examples that demonstrate the advertised functionality.

Significance. If the implementation were corrected, the paper would provide a practical, open-source tool for finite neutrosophic topology generation and verification, with concrete examples and a public GitHub repository. The conceptual method is sound: enumerating all finite intersections and then all finite unions does generate the topology for finite families in the pointwise min/max semantics, and isNeutrosophicTopology directly checks the four axioms. The paper also deserves credit for shipping complete source listings and explicit numeric examples rather than only describing algorithms. However, as printed the code cannot run, and one of the central methods, getNSBase, does not implement the paper's own definition of a neutrosophic basis. These are load-bearing defects for a software paper, so the contribution is not yet usable in its current form.

major comments (3)

[Section 3, constructor (pseudo-code and Python listing)] The constructor is defined as `def init(self, *args)` instead of `def __init__(self, *args)`, so the call `NSfamily(A1, A2, A3)` used in every example will not invoke this method and will raise a TypeError. Moreover, the constructor stores `self.universe`, `self.neutrosophicfamily`, and `self.name`, while later methods access `self.__universe`, `self.__neutrosophicfamily`, and `self.__name`; because of Python name mangling these are different attributes, so even after renaming `init` to `__init__` the object would raise AttributeError on the first access. This invalidates all interactive examples as they stand.
[Section 3, getNSBase listing vs. Notation 2.14 and Definition 2.13] The method `getNSBase()` enumerates only non-empty combinations of the current family and never adds `NSset.ABSOLUTE(universe)`, although Notation 2.14 defines B(S) as the family of all finite intersections together with the absolute set. Therefore the returned object is not B(S), and it is not guaranteed to be a neutrosophic basis for T(S) in the sense of Definition 2.13. The paper's own example confirms the mismatch: the printed basis is {B1, B2, B1 ⋓ B2}, while the generated topology also contains ~U, and ~U is not a neutrosophic union of those three sets. The method should append the absolute set before constructing the NSfamily; without this correction the advertised basis/sub-basis functionality is not implemented.
[Section 3, NSintersectionClosed listing] The definition line reads `def NSintersectionClosed(self:` with no closing parenthesis, which is a Python syntax error. Because `isNeutrosophicTopology()` calls this method, the topology verification cannot run even after the constructor is repaired. This is not a cosmetic typo: the module as printed cannot be imported.

minor comments (7)

[Title/header] The running title contains the typo 'Neutrosophic Topologie s'; this should be corrected.
[Section 2.1] The text repeatedly uses the placeholder `/CD` instead of a typeset universe symbol, e.g., in Definitions 2.1, 2.2, and 2.4; the LaTeX macros need to be repaired throughout.
[Definition 2.8] The phrase 'two nutrosophic topologies T1 and τ2' mixes a typo and inconsistent notation; it should read 'two neutrosophic topologies T1 and T2'.
[Section 3, getNSTopologyByBase] The listing uses helper functions `nameToBB` and `isBB` without defining or importing them; these should either be documented and included or removed from the presented code.
[Section 3, general] The paper never states explicitly that the implementation assumes finite universes and finite families; since `combinations` and `reduce` enumerate all subsets, this assumption should be stated up front and its consequences for scalability mentioned.
[Section 3, storeName] The `storeName()` method relies on `inspect.currentframe()` and the caller's local variables, which is fragile outside interactive sessions; this should be documented as an optional convenience rather than part of the mathematical functionality.
[Section 2.2, Proposition 2.16] Proposition 2.16 is stated without proof or reference; a short proof or a pointer to a standard text would make the paper more self-contained, since the correctness of getNSTopologyBySubBase depends on it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the framework's topology generator and verifier directly execute the paper's own definitions; the flagged absolute-set omission and unproved Proposition 2.16 are correctness issues, not circular reasoning.

full rationale

The paper does not fit parameters, rename known results, or import a uniqueness theorem to force its conclusions. Its central claims are algorithmic: Definition 2.7 states the topology axioms, Definition 2.13 defines a basis, Notation 2.14 defines B(S), and the Section 3 code implements these definitions directly. getNSTopologyBySubBase constructs all finite nonempty intersections and then all finite nonempty unions, while getNSTopologyByBase appends the empty and absolute sets; isNeutrosophicTopology checks the axioms by exhaustive enumeration. Thus the verification is not equivalent by construction to the claim being verified. The dependence on the authors' earlier PYNS framework [13] is a software dependency rather than a circular justification: the relevant NSset operations are imported and the new methods are displayed in full. Self-citations [9-11,13,23] appear, but they do not supply the load-bearing mathematical content of the topology-generation claim. Two non-circular caveats should be weighed separately. First, Proposition 2.16, stating T(S)=T(B(S)), is given without proof, so the generator's correctness rests on an unproved though standard equivalence. Second, getNSBase enumerates only nonempty finite intersections and never appends NSset.ABSOLUTE(universe), despite Notation 2.14 defining B(S) with the absolute set added; this makes the returned object not exactly the B(S) of Proposition 2.16. Since getNSTopologyByBase appends the absolute set, the full topology may still be correct, but the basis method does not match the paper's own definition as printed. These are correctness and documentation gaps, not circular reductions.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new mathematical entities. It relies on standard neutrosophic set operations from prior literature, on an unproved analogue of the classical subbase-basis-topology theorem (Prop 2.16), and on a silent finiteness assumption required by the enumeration-based algorithms.

assumptions (3)

domain assumption Neutrosophic union, intersection, and complement are defined pointwise via max, min, and the complement map on [0,1] as in Definitions 2.5 and 2.6, and these operations satisfy the algebraic properties of a De Morgan algebra.
Taken from cited literature (Smarandache [25], Wang et al. [28], Salama [19]) and used throughout the topology algorithms; the paper does not prove these properties.
standard math Proposition 2.16: for any family S, the topology generated by S equals the topology generated by the family of finite intersections B(S), and B(S) is a basis.
Stated as Proposition 2.16 and used to justify getNSTopologyBySubBase(); it is an analogue of the classical topology theorem but no proof is given in the paper.
domain assumption The universe and all families are finite, so that combinations() enumerations terminate.
Section 3 methods (getNSBase, getNSTopologyByBase, __checkClosure) iterate over itertools.combinations; no finiteness guard or infinite-family handling is provided.

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Cite this review

Pith. "Pith review of A Python Framework Enhancement for Neutrosophic Topologies." pith.science (2026). https://pith.science/paper/EL4FHLNL

@misc{pith2026241200047,
  author       = {Pith},
  title        = {Pith review of: A Python Framework Enhancement for Neutrosophic Topologies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EL4FHLNL}},
  note         = {Machine review of arXiv:2412.00047}
}

read the original abstract

This paper introduces an extension to the Python Neutrosophic Sets (PYNS) framework, originally detailed in Nordo et al. 2024, with the addition of the NSfamily class for constructing and manipulating neutrosophic topologies. Building on existing classes like NSuniverse and NSset, the NSfamily class enables the definition and testing of neutrosophic families as basis and sub-basis for neutrosophic topological spaces. This extension provides tools for verifying closure properties under union and intersection, and for determining whether a given family constitutes a neutrosophic topology. Through implemented algorithms, the framework automates the generation of topologies from families of neutrosophic sets, offering an efficient tool for advancing research in neutrosophic topology. Practical applications are demonstrated with detailed examples, showcasing how this class enhances the scope and flexibility of neutrosophic modeling within the PYNS framework.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Intuitionistic Fuzzy Sets

Atanassov K.T. Intuitionistic Fuzzy Sets. Fuzzy Sets and Systems 20 (1), pp. 87-96, 1986

work page 1986

[2] [2]

Single val ued neutrosophic graphs

Broumi S., Talea M., Bakali A., Smarandache F. Single val ued neutrosophic graphs. Journal of New theory 10, pp. 86-101, 2016

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Broumi S., Smarandache F., Bakali A. Connectedness and S tratiﬁcation of Single-V alued Neutrosophic Topological Spaces. Neutrosophic Sets and Systems , 36, pp. 77-84, 2020

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