

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5839
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Key Characteristics of Quadri-Partitioned Neutrosophic
Riemann integrals and Quadri-Partitioned Neutrosophic

Soft Topological Spaces

Abdallah Shihadeh1, Mayada Abualhomos2, Alaa M. Abd El-latif3,∗,
Abdallah Alhusban4, Shaaban M. Shaaban5, Muhammad Arslan6, Arif Mehmood6

1 Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa 13133,
PO box 330127, Jordan
2 Applied Science Private University Amman, 11931, Jordan
3 Mathematics Department, College of Science, Northern Border University, Arar 91431,
Saudi Arabia
4 Department of Mathematics, Faculty of Science and Technology, Irbid National University,
P.O. Box: 2600, Irbid, Jordan
Jadara University Research Center, Jadara University, Jordan
5 Center for Scientific Research and Entrepreneurship, Northern Border University, Arar 73213,
Saudi Arabia
6 Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail
Khan 29050, KPK, Pakistan

Abstract. Neutrosophic Set Theory (NST) is an extension of Intuitionistic Fuzzy Set Theory (IFST).
While IFST relies on two possibilities for the complete depiction of a set, neutrosophic set theory fa-
miliarizes an additional third possibility, thus providing a more delicate representation. Our research
builds upon a further extension of neutrosophic set theory, known as quadri-partitioned neutrosophic set
theory (QPNST), which brings in a fourth possibility for a more detailed and complete description of
sets. In this study, we define the Riemann Integral Theory (RIT) within the framework of QPNST. This
opens new doors for probing the properties and characteristics of the Riemann integral in this extended
context. One strategic concept that arises in this work is the level cut. In QPNST, the level cut is defined
as a four-tuple (i, j, k, l), which represents the different possibilities inherent in the theory. The notion
of the Quadri-Partitioned Neutrosophic Riemann Integral Theory (QPNRIT) is explored numerically in
this study, and the results are systematically presented in tabular form. This numerical approach sheds
light on the integral’s properties and facilitates the understanding of its behavior within the QPNST
framework. This study explores quadripartitioned neutrosophic soft topological spaces, extending neu-
trosophic set theory (NST), which incorporates three membership values: true, false, and indeterminacy.
The study introduces new concepts such as QPNS semi-open, QPNS pre-open, and QPNS ∗b open sets,
and builds on these to define QPNS closure, exterior, boundary, and interior. A key development is the
definition of a quadripartitioned neutrosophic soft base, which plays a central role in these topological
structures. The paper also explores the concept of a quadripartitioned neutrosophic soft sub-base and
discusses local bases, as well as the first- and second-countability axioms. The study further examines
hereditary properties of these spaces, distinguishing between inherited and non-inherited properties. Key
results include that a quadripartitioned neutrosophic soft subspace of a first-countable space is also first-
countable, and a second-countable subspace of a second-countable space remains second-countable. It
also highlights the relationship between second countability and separability in these spaces, asserting
that a second-countable quadripartitioned neutrosophic soft space is separable, though the converse is
not always true. This work lays the foundation for further research in neutrosophic soft topologies.
2020 Mathematics Subject Classifications: 54A05
Key Words and Phrases: Closed quadri-neutrosophic number, bounded quadri-neutrosophic neutro-
sophic number, quadri- neutrosophic Riemann integration. quadri-neutrosophic neutrosophic soft set

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5839

Email addresses: alaa.ali@nbu.edu.sa (A. M. Abd El-latif), mehdaniyal@gmail.com (A. Mehmood)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 2 of 54

1. Introduction

Zadeh [1] introduced the concept of fuzzy set theory (FST), where a set is characterized
by a membership function. The foundational operations—union, intersection, complement,
and convexity—were established. Additionally, the separation theorem for convex fuzzy sets
was formulated. Zadeh [2] introduced the concept of linguistic variables and their applications
within the context of FST. These variables have found widespread use in various fields, including
medicine, law, psychology, economics, and others. Zadeh [3] generalized the ideas presented
in [2] and introduced the notion of fuzzy variables. Zadeh [4] applied linguistic variables to
approximate reasoning. Zadeh [5] proposed the generalized theory of uncertainty, which enables
a broader perspective on uncertainty.

Ye [6] discussed the concept of simplified neutrosophic sets (SNSs), which are a sub-class of
neutrosophic sets, and explored several aggregation operators. Furthermore, decision-making
techniques were developed. Liu and Luo [7] introduced the notion of multi-attribute group
decision-making (MAGDM). Atanassov [8] introduced intuitionistic fuzzy sets (IFS), an exten-
sion of fuzzy sets, and described the basic operations associated with this theory, as well as
the development of topological operators. Atanassov and Gargov [9] advanced the concept of
interval-valued IFSs, utilizing intervals in their formulation. Smarandache [10, 11] explored
various techniques, including neutrosophic probability and neutrosophic logic. Ye [12] exam-
ined applications of neutrosophic sets in multi-criteria decision-making problems. Techniques
based on weighted distance measures and generalized hybrid weighted average operators, em-
ploying neutrosophic hesitant sets, were discussed in [13, 14], along with their applications in
multiple-attribute decision-making.

Li and Luo [15] introduced the super strong theory, known as soft set theory (SST), and
defined its fundamental operations. Maji et al. [16] applied SST to decision-making problems,
redefining key operations and illustrating their validity with examples. Molodtsov [17] estab-
lished a connection between soft sets and fuzzy sets, leading to the development of a hybrid
theory known as fuzzy soft set theory (FSST). Wang et al. [18] introduced the concept of HFSS
and effectively applied it in multi-criteria decision-making problems. Pei and Miao [19] bridged
soft sets with information theory, discussing their practical applications. John [20–22] explored
various structures based on SST, offering examples and applications across different areas of
mathematics.

Al-Shami et al. [23] proposed a new structure known as Menger space. Al-Shami et al. [24]
investigated the structure of infra soft topological spaces (ISTS) with respect to crisp points.
Al-Shami et al. [25] discussed weak forms of soft separation axioms and fixed points. Al-Shami
et al. [26] defined concepts of connectedness and local connectedness within the context of infra
soft topological spaces, presenting several results related to this strong structure. Al-Shami
[27] discussed quantum mechanics (QM) in the framework of ISTS. Further results on ISTS
and soft topological spaces (STS) were examined in [28, 29]. Ozturk et al. [30] introduced
new operators in Neutrosophic Soft Topological Spaces (NSTS), leading to novel approaches
to several existing results, with numerous examples provided for clarification. Ahmad et al.
[31] discussed Irreversible k-Threshold conversion number for some graph products and neutro-
sophic graphs. Hatamleh et al. [32] studied complex tangent trigonometric approach applied
to q-rung fuzzy set using weighted averaging, geometric operators and its extension. Hatamleh
et al. [33] studied different weighted operators such as generalized averaging and generalized
geometric based on trigonometric P-rung interval-valued approach and in addition to this some
examples were given for clear understanding. Shihadeh et al. [34] discussed algebraic struc-
tures towards different intuitionistic fuzzy ideals and its characterization of an ordered ternary
semigroup. Hatamleh et al. [35, 36] studied operators via weighted averaging and geometric
approach using trigonometric neutrosophic interval-valued set and its extension and character-
ization of interaction aggregating operators setting interval-valued Pythagorean neutrosophic
set. Hatamleh et al. [37] discussed applications of complex interval-valued picture fuzzy soft


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 3 of 54

relations. El-Sheikh and Abd El-Latif [38] discussed decompositions of some types of supra
soft sets and soft continuity and cited some excellent examples for clear understanding of the
concept. Abd El-Latif [39] discussed soft supra compactness in supra soft topological spaces.
Abd El-Latif and Hosny [40] discussed the eye-catching concept of soft separation axioms and
provided examples. Abd El-Latif and Hosny discussed some more structures in [41, 42].

1.1. Research Gap

While neutrosophic set theory (NST) extends intuitionistic fuzzy set theory (IFST) by in-
troducing a third possibility for uncertainty representation, the extension to quadri-partitioned
neutrosophic set theory (QPNST), which introduces a fourth possibility, remains underexplored.
Additionally, the application of QPNST to classical mathematical theories, such as the Riemann
Integral, has not been rigorously studied. There is a lack of formal mathematical analysis and
numerical exploration of the properties of the Riemann Integral within the QPNST framework,
particularly with respect to the behavior of the quadri-partitioned neutrosophic Riemann inte-
gral theory (QPNRIT). This research gap hinders a deeper understanding of how higher-order
uncertainty models can be integrated into classical mathematical analysis.

1.2. Motivation

The research on neutrosophic Riemann integration and its properties [43], which delves
into neutrosophic Riemann set theory, provided a foundational understanding of integrating
neutrosophic functions. This exploration highlighted the limitations and potential for further
development in the field. As a result, it motivated the development of the theory of quadri-
partitioned neutrosophic Riemann integrals, which extends the concept by introducing a fourth
partition. This new extension allows for a more comprehensive representation of uncertainty,
offering greater flexibility and accuracy in modeling complex systems with multiple layers of
uncertainty.

1.3. Novelty

The novelty of neutrosophic set theory (NST) lies in its extension of intuitionistic fuzzy
set theory (IFST). While IFST provides two possibilities for a set’s complete representation,
neutrosophic set theory introduces an additional third possibility, allowing for a more refined and
nuanced depiction of sets. Building upon this, our research delves into an even further extension
of NST, called quadri-partitioned neutrosophic set theory (QPNST), which incorporates a fourth
possibility. This additional possibility enhances the level of detail and completeness in the
representation of sets.

In this study, we define the Riemann integral theory (RIT) within the context of QPNST,
offering a novel way to explore the properties and characteristics of the Riemann integral in
this expanded framework. A key concept that emerges in this work is the level cut, which in
the context of QPNST is represented as a four-tuple (i, j, k, l), encapsulating the various possi-
bilities inherent in the theory. We also explore the quadri-partitioned neutrosophic Riemann
integral theory (QPNRIT), applying it numerically and presenting the results in tabular form.
This numerical exploration allows us to investigate the behavior of the integral in the QPNST
framework, providing a deeper understanding of its properties and showcasing the potential of
this extended theory for future mathematical and practical applications.

1.4. Importance of The Study

A significant development in our study is the application of the Riemann Integral Theory
(RIT) within the context of QPNST. This extension paves the way for exploring the properties
and characteristics of the Riemann integral in a richer and more nuanced mathematical setting.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 4 of 54

The concept of the level cut, defined as a four-tuple (i, j, k, l), is critical in this work as it
captures the different possibilities within the QPNST framework, offering a unique perspective
for analyzing integral properties.

Through the introduction of the quadri-partitioned neutrosophic Riemann integral theory
(QPNRIT), our study provides numerical insights into the behavior of integrals within this
extended framework. The results are presented systematically in tabular form, enhancing the
understanding of the integrals properties and illustrating how it behaves under different condi-
tions of uncertainty. This numerical approach not only enriches the theoretical foundations of
QPNST but also holds promise for practical applications in fields such as engineering, decision-
making, and data analysis, where complex uncertainties and ambiguities need to be addressed.

1.5. Literature Review

Ozturk et al. [44] explored soft continuous mappings. Gunduz et al. [45] focused on criti-
cal structures within STS, particularly separation axioms. Ozturk [46] expanded the analysis
of additional structures within STS. Mehmood et al. [47] made significant contributions to
Neutrosophic Soft Bounded Topological Spaces (NSBTS) and discussed a comprehensive set of
results related to crisp points. Mehmood et al. [48, 49] presented results on Neutrosophic Soft
Open Sets (NSOSs). Mehmood et al.[50] introduced the concept of Neutrosophic Soft Quasi-
Sets (NSQS), with respect to neutrosophic soft points (NSP). Mehmood et al. [51] provided an
in-depth analysis of NSQS, specifically focusing on soft p-open sets. Kim et al. [52] addressed
several differential problems and illustrated their solutions with examples. Moi et al. [53] in-
vestigated second-order problems within the context of NST, enhancing applicability through
well-chosen examples. Shami et al. [54] introduced supra-soft topologically ordered spaces as an
extension of soft topologically ordered spaces. They discussed key notions like monotone interior
and closure operators, formulated supra-soft separation axioms, and explored the relationships
between these concepts and their parametric supra topologies. They also characterized supra
p-soft Ti-ordered spaces, supra p-soft regularly ordered spaces, and supra-soft normally ordered
spaces. T. M. Al-Shami and Shafei [55] discussed two types of separation axioms in supra-soft
topological spaces and provided the best examples for better understanding of the results.

2. Preliminaries

Definition 1. [43] Let U be the universal set. Then, a single-valued neutrosophic set (SVNS) N
over the set U is a neutrosophic set over U , but the truth, indeterminacy, and falsity membership
functions are defined as

TN : U → [0, 1], IN : U → [0, 1], FN : U → [0, 1],

respectively.

Definition 2. [43] A neutrosophic set N over the universal set of real numbers R is said to be
a neutrosophic number if it satisfies the following conditions:

1. N is normal, i.e., there exists X0 ∈ R such that:

TN (X0) = 1, IN (X0) = 0, FN (X0) = 0.

2. N is convex for the truth membership function TN (X), i.e.,

TN (µX1 + (1− µ)X2) ≥ min
(
TN (X1), TN (X2)

)
, ∀X1,X2 ∈ R, µ ∈ [0, 1].

3. N is concave for the indeterminacy and falsity membership functions, IN (X) and FN (X),
respectively:

IN (µX1 + (1− µ)X2) ≥ max
(
IN (X1), IN (X2)

)
,


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FN (µX1 + (1− µ)X2) ≥ max
(
FN (X1), FN (X2)

)
,

for all X1,X2 ∈ R and µ ∈ [0, 1].

Definition 3. [43] A truth, indeterminacy, and falsity membership function describes an in-
terval neutrosophic set N over the universal set U and is given as TN (X), IN (X), and FN (X),
respectively. For all X ∈ U , we have:

TN (X) = [inf TN (X), supTN (X)],

IN (X) = [inf IN (X), sup IN (X)],

FN (X) = [inf FN (X), supFN (X)] ⊆ [0, 1], ∀X ∈ U.

Here, we focus on the sub-unitary range [0, 1]. Let

ñ = ⟨[TL
ñ , T

U
ñ ], [ILñ , I

U
ñ ], [FL

ñ , F
U
ñ ]⟩

indicate a neutrosophic interval number (INN), where TL
ñ , T

U
ñ , ILñ , I

U
ñ , FL

ñ , and FU
ñ denote:

inf TÑ (X), supTÑ (X), inf IÑ (X), sup IÑ (X), inf FÑ (X), supFÑ (X),

respectively.

3. Single Valued Quadri-partitioned Neutrosophic Set

This section introduces the Single Valued Quadri-Partitioned Neutrosophic Set (SVQNS),
characterized by membership functions representing absolute truth, relative truth, absolute
falsehood, and relative falsehood. We define the inclusion of one SVQNS within another based
on membership values. The union and intersections are also defined in this study. The Quadri
Single Valued Neutrosophic Number (QSVNN) is also proposed, a unique neutrosophic set on
the real number line R, incorporating truth, indeterminacy, hesitation, and falsity membership
functions. Additionally, we define the cut of a neutrosophic set, providing a framework for
analyzing subsets.

Definition 4. A Single Valued Quadri-Partitioned Neutrosophic Set (SVQNS) A on the uni-
verse of discourse X is characterized by the following membership functions:

• Absolute true membership function: TA(x),

• Relative true membership function: ReTA(x),

• Absolute false membership function: FA(x),

• Relative false membership function: ReFA(x).

These functions are subsets of ]0, 1[, i.e.,

TA(x) : X →]0, 1[, ReTA(x) : X →]0, 1[, FA(x) : X →]0, 1[, ReFA(x) : X →]0, 1[

with the condition:

0 ≤ supTA(x) + supReTA(x) + supFA(x) + supReFA(x) ≤ 4.

Thus, the SVQNS can be represented as:

A = {⟨x, TA(x), ReTA(x), ReFA(x), FA(x)⟩ : x ∈ X} .


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 6 of 54

Definition 5. A single valued quadri-partitioned neutrosophic set A is contained in another
single valued quadri-partitioned neutrosophic set B if the following conditions hold for each
X ∈ X:

TA(X) ≤ TB(X), ReTA(X) ≤ ReTB(X), FA(X) ≥ FB(X), ReFA(X) ≥ ReFB(X).

Equality of two SVQNSs is defined as:

A = B ⇐⇒ A ⊆ B and B ⊆ A.

The complement of A, denoted as Ac, is given by:

Ac = {⟨X, FA(X), ReFA(X), TA(X), ReTA(X)⟩ : X ∈ X} .

Definition 6. Let A and B be two single valued quadri-partitioned neutrosophic sets on the
universe of discourse X, then:

(i) The union of A and B is defined as:

A ∪B =

{
X,
[
max(TA(X), TB(X)),max(ReTA(X), ReTB(X)),

min(ReFA(X), ReFB(X)),min(FA(X), FB(X))
]
: X ∈ X

}
.

(ii) The intersection of A and B is defined as:

A ∩B =

{
X,
[
min(TA(X), TB(X)),min(ReTA(X), ReTB(X)),

max(ReFA(X), ReFB(X)),max(FA(X), FB(X))
]
: X ∈ X

}
.

Definition 7. A Quadri-Single-Valued-Neutrosophic-Number (Q-SVNN) is denoted as:

N = ⟨(p, q, r, s); ρN , υN , κN , τN ⟩

where N is a unique neutrosophic set on R, with its truth-membership function TN (X) defined
as:

TN (X) =


X−p
q−p ρN , for p ⪯ X ⪯ q,
r−X
r−q ρN , for q ⪯ X ⪯ r,

0, otherwise.

The definitions of the indeterminacy-membership, hesitation, and falsity-membership functions
follow a similar structure.

IN (X) =


q−X+υN (X−p)

q−p , for p ⪯ X ⪯ q,
q−X+υN (r−X)

r−q , for q ⪯ X ⪯ r,

0, otherwise.

HN (X) =


q−X+κN (X−p)

q−p , for p ⪯ X ⪯ q,
q−X+κN (r−X)

r−q , for q ⪯ X ⪯ r,

0, otherwise.


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FN (X) =


q−X+τN (X−p)

q−p , for p ⪯ X ⪯ q,
q−X+τN (r−X)

r−q , for q ⪯ X ⪯ r,

0, otherwise.

Respectively, these represent the indeterminacy-membership function IN (X), hesitation-membership
function HN (X), and falsity-membership function FN (X).

Definition 8. A Quadri-Single-Valued-Neutrosophic Number (Q-SVN number) is defined as:

N = ⟨(p, q, r, s); ρN , υN , κN , τN ⟩

where N represents a unique neutrosophic set on R. The corresponding membership functions
for truth, indeterminacy, hesitation, and falsity are defined as follows.

N(X) =


X−p
q−p ρN , for p ⪯ X ⪯ q,
s−X
s−q ρN , for q ⪯ X ⪯ r,

τN , for r ⪯ X ⪯ s,

0, otherwise.

TN (X) =


q−X+υN (X−p)

q−p , for p ⪯ X ⪯ q,
X−r+υN (s−X)

s−r , for q ⪯ X ⪯ r,

τN , for r ⪯ X ⪯ s,

0, otherwise.

FN (X) =


q−X+κN (X−p)

q−p , for p ⪯ X ⪯ q,
X−r+κN (s−X)

s−r , for q ⪯ X ⪯ r,

τN , for r ⪯ X ⪯ s,

0, otherwise.

Definition 9. The (i, j, k, l)-cut of a neutrosophic set N is denoted by N(i, j, k, l), where (i, j, k, l) ∈
[0, 1]. Then,

N(i, j, k, l) =
{
⟨TN (X), ReTN (X), ReFN (X), FN (X)⟩ : X ∈ U, TN (X) ⪰ i, ReTN (X) ⪯ j, ReFN (X) ⪯ k, FN (X) ⪯ l

}
.

4. Characterization of Few Results in Quadri-partitioned Neutrosophic
Space

In this section, some properties of single valued quadri-partitioned neutrosophic numbers
and operations of single valued quadri-partitioned neutrosophic numbers are presented. In
addition to this few results are also addressed, which are necessary for the up-coming sections.

Definition 10. (i) A neutrosophic number ñ is called a closed single valued quadri-partitioned
neutrosophic number if ñ is a quadri-partitioned neutrosophic number and the truth mem-
bership function and relative truth membership functions are upper semi-continuous, while
the false and relative false membership functions are lower semi-continuous.

(ii) A neutrosophic number ñ is called a bounded single valued quadri-partitioned neutrosophic
number if ñ is a quadri-partitioned neutrosophic number and the truth membership, relative
truth membership, relative false membership, and false membership functions have compact
support.


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(iii) Let m̃ and ñ be two single valued quadri-partitioned neutrosophic numbers. Then, m̃ and
ñ are said to be equal, denoted as m̃ = ñ, if and only if m̃(i,j,k,l) and ñ(i,j,k,l). Here, m̃(i,j,k,l)

and ñ(i,j,k,l) denote the (i, j, k, l)-cut of m̃ and ñ, respectively.

m̃(i,j,k,l) = ñ(i,j,k,l).

Proposition 1. If ñ is a closed single-valued quadri-partitioned neutrosophic number, then the
(i, j, k, l)-cut of ñ, denoted as

ñ(i,j,k,l) =
〈
[ñL

i , ñ
U
i ], [ñ

L
j , ñ

U
j ], [ñ

L
k , ñ

U
k ], [ñ

L
l , ñ

U
l ]
〉

is a closed interval Quadri-Neutrosophic Single Number (QNSN) or Interval Quadri-Neutrosophic
Number (IQNN), where [ñL

k , ñ
U
k ] and [ñL

l , ñ
U
l ] are all closed intervals. Here,

ñL
i , ñ

U
i , ñ

L
j , ñ

U
j , ñ

L
k , ñ

U
k , ñ

L
l , ñ

U
l

denote inf TÑ (X), supTÑ (X), inf IÑ (X), sup IÑ (X), inf FÑ (X), supFÑ (X), infHÑ (X), and
supHÑ (X), respectively.

Proof. If TÑ (X) is upper semi-continuous, and IÑ (X) and FÑ (X) are lower semi-continuous,
then the (i, j, k, l)-level set of ñ, i.e.,

ñ(i,j,k,l) =
{
TÑ (X) ⪰ i, IÑ (X) ⪯ j, FÑ (X) ⪯ k, HÑ (X) ⪯ l, X ∈ R

}
is a closed set. Then, from Definition 7, ñ is an interval Quadri-Neutrosophic Single Number

(QNSN), and intervals are closed intervals.

Proposition 2. Let m̃ and ñ be two single-valued quadri-partitioned neutrosophic numbers,
then:

(i) (m̃⊙ ñ)(i,j,k,l) = m̃(i,j,k,l) ⊙ ñ(i,j,k,l)

(ii) (λm̃)(i,j,k,l) = λm̃(i,j,k,l), where λ ̸= 0 is any real number.

Proof.

1 Since

m̃⊙ ñ =

{
Z,max

X
Z = (X)⊛ y {min(Tm̃(X), Tñ(y))} ,

minZ = (X)⊛ y {max(Im̃(X), Iñ(y))} ,

minZ = (X)⊛ y {max(Fm̃(X), Fñ(y))} ,minZ = (X)⊛ y {max(Hm̃(X), Hñ(y))}} .

And

(m̃⊙ ñ)(i, j, k, l) = {Z : Tm̃⊙ñ(z) ⪰ i, Im̃⊙ñ(z) ⪰ j, Fm̃⊙ñ(z) ⪰ k, Hm̃⊙ñ(z) ⪰ l} .

Let Z ∈ (m̃ ⊙ ñ)(i, j, k, l), then there exists at least one X ∈ m̃ and y ∈ ñ such that
Z = m̃⊙ ñ. Then,

Tm̃⊙ñ(z) = max {min(Tm̃(X), Tñ(y))} ⪰ i.

Tm̃(X) ⪰ i and Tñ(y) ⪰ i

Im̃⊙ñ(z) = min {max(Im̃(X), Iñ(y))} ⪯ j


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Im̃(X) ⪯ j and Iñ(y) ⪯ j

Fm̃⊙ñ(z) = min {max(Fm̃(X), Fñ(y))} ⪯ k

Fm̃(X) ⪯ k and Fñ(y) ⪯ k

Hm̃⊙ñ(z) = min {max(Hm̃(X), Hñ(y))} ⪯ l

Hm̃(X) ⪯ l and Hñ(y) ⪯ l

This implies X ∈ m̃(i, j, k, l) and y ∈ ñ(i, j, k, l). Therefore,

z = X⊙ y ∈ m̃(i, j, k, l)⊙ ñ(i, j, k, l)

Again, let z∗ ∈ m̃(i, j, k, l)⊙ ñ(i, j, k, l). Then there exists at least one X∗ ∈ m̃(i, j, k, l) and
y∗ ∈ ñ(i, j, k, l) such that

z∗ = X∗ ⊙ y∗

Then, we have

Tm̃(X∗) ⪰ i and Tñ(y
∗) ⪰ i

Im̃(X∗) ⪯ j and Iñ(y
∗) ⪯ j

Fm̃(X∗) ⪯ k and Fñ(y
∗) ⪯ k

Hm̃(X∗) ⪯ l and Hñ(y
∗) ⪯ l

This implies that

min(Tm̃(X∗), Tñ(y
∗)) ⪰ i, max(Im̃(X∗), Iñ(y

∗)) ⪯ j,

max(Fm̃(X∗), Fñ(y
∗)) ⪯ k, max(Hm̃(X∗), Hñ(y

∗)) ⪯ l

Then, we have

max
X

z∗ = X∗ ⊙ y∗ {min(Tm̃(X∗), Tñ(y
∗))} ⪰ i

⇒ Tm̃⊙ñ(z
∗) = Tm̃⊙ñ(X

∗ ⊙ y∗) ⪰ i

Similarly,


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 10 of 54

Im̃⊙ñ(z
∗) = Im̃⊙ñ(X

∗ ⊙ y∗) ⪯ j

Fm̃⊙ñ(z
∗) = Fm̃⊙ñ(X

∗ ⊙ y∗) ⪯ k

Hm̃⊙ñ(z
∗) = Hm̃⊙ñ(X

∗ ⊙ y∗) ⪯ l

Therefore,

z∗ ∈ (m̃⊙ ñ)(i,j,k,l).

2 Since

m̃ =

{
Z | max

ζ
z = λ(ζ)Tm̃(ζ),min

ζ
z = λ(ζ)Im̃(ζ),min

ζ
z = λ(ζ)Fm̃(ζ),min

ζ
z = λ(ζ)Hm̃(ζ)

}
Let

Zλm̃(i,j,k,l) = {Z | Tm̃(ζ) ⪰ i, Im̃(ζ) ⪯ j, Fm̃(ζ) ⪯ k, Hm̃(ζ) ⪯ l}

Let Z ∈ (λm), then there exists ζ ∈ m̃ such that

z = λ(ζ)

which implies

Tm̃(ζ) ⪰ i, Im̃(ζ) ⪯ j, Fm̃(ζ) ⪯ k, Hm̃(ζ) ⪯ l

Thus,

ζ ∈ m̃(i,j,k,l) ⇒ z = λ(ζ) ∈ (λm)(i,j,k,l)

Again, let Z∗ ∈ (λm), then there exists ζ∗ ∈ m̃ such that

Z∗ = λ(ζ∗)

which implies

Tm̃(ζ∗) ⪰ i, Im̃(ζ∗) ⪯ j, Fm̃(ζ∗) ⪯ k, Hm̃(ζ∗) ⪯ l

Then we have

Z∗ = λ(ζ)∗ Tm̃(ζ∗) ⪰ i ⇒ T (λm)m̃(λ(ζ)∗) = T (λm)m̃(z∗) ⪰ i

Z∗ = λ(ζ)∗ Im̃(ζ∗) ⪯ j ⇒ I(λm)m̃(λ(ζ)∗) = I(λm)m̃(z∗) ⪯ j

Z∗ = λ(ζ)∗ Fm̃(ζ∗) ⪯ k ⇒ F (λm)m̃(λ(ζ)∗) = F (λm)m̃(z∗) ⪯ k


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 11 of 54

Z∗ = λ(ζ)∗ Hm̃(ζ∗) ⪯ l ⇒ H(λm)m̃(λ(ζ)∗) = H(λm)m̃(z∗) ⪯ l

Therefore,
Z∗ ∈ (λm)m̃(i,j,k,l)

Proposition 3. Let m̃ and ñ be two closed quadri-partitioned neutrosophic numbers. Then,
m̃+ ñ, m̃− ñ, m̃× ñ, and λm̃ are also closed quadri-partitioned neutrosophic numbers, where
0 ̸= λ ∈ R and λ is any real number.

Proof. Since Tm̃⊙ñ and T (λm̃) are upper semi-continuous, and Im̃⊙ñ and I(λm̃) are lower
semi-continuous, it follows that (m̃⊙ ñ)(i,j,k,l) and (λm̃)(i,j,k,l), along with Fm̃⊙ñ, F (λm̃), Hm̃⊙ñ,
and H(λm̃), are closed sets for all (i, j, k, l).

Proposition 4.

Let m̃, ñ be closed quadri-partitioned neutrosophic numbers, then

(m̃+ñ)(i,j,k,l) =
〈[
mL

i + nL
i ,m

U
i + nU

i

]
,
[
mL

j + nL
j ,m

U
j + nU

j

]
,
[
mL

k + nL
k ,m

U
k + nU

k

]
,
[
mL

l + nL
l ,m

U
l + nU

l

]〉

(m̃−ñ)(i,j,k,l) =
〈[
mL

i − nL
i ,m

U
i − nU

i

]
,
[
mL

j − nL
j ,m

U
j − nU

j

]
,
[
mL

k − nL
k ,m

U
k − nU

k

]
,
[
mL

l − nL
l ,m

U
l − nU

l

]〉

(λm̃)(i,j,k,l) =

{〈[
λmL

i , λm
U
i

]
,
[
λmL

j , λm
U
j

]
,
[
λmL

k , λm
U
k

]
,
[
λmL

l , λm
U
l

]〉
for λ > 0〈[

λmU
i , λm

L
i

]
,
[
λmU

j , λm
L
j

]
,
[
λmU

k , λm
L
k

]
,
[
λmU

l , λm
L
l

]〉
for λ < 0

Since (m̃+ ñ)(i,j,k,l) = (m̃)(i,j,k,l) + (ñ)(i,j,k,l)

=
〈[
mL

i ,m
U
i

]
,
[
mL

j ,m
U
j

]
,
[
mL

k ,m
U
k

]
,
[
mL

l ,m
U
l

]〉
+
〈[
nL
i , n

U
i

]
,
[
nL
j , n

U
j

]
,
[
nL
k , n

U
k

]
,
[
nL
l , n

U
l

]〉

=
〈[
mL

i + nL
i ,m

U
i + nU

i

]
,
[
mL

j + nL
j ,m

U
j + nU

j

]
,
[
mL

k + nL
k ,m

U
k + nU

k

]
,
[
mL

l + nL
l ,m

U
l + nU

l

]〉
Let me know if you need further modifications!

Proof. Trivial.

Theorem 1.
Let N be a quadri-partitioned neutrosophic set. Then,

(N)(i,j,k,l) = {(X) : TN(X) ⪰ i, IN(X) ⪯ j, FN(X) ⪯ k, HN(X) ⪯ l}

1. N(i2, j2, k2, l2) ⊆ N(i1, j1, k1, l1)

where i1 ≺ i2, j1 ≻ j2, k1 ≻ k2, l1 ≻ l2.

2.
∞⋂
n=1

N(in, jn, kn, ln) = (N)(i, j, k, l)

where i1 ≺ i2, j1 ≻ j2, k1 ≻ k2, l1 ≻ l2.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 12 of 54

Proof. 1. Let (X) ∈ N(i2, j2, k2, l2), then we have:

TN(X) ⪰ i2, IN(X) ⪯ j2, FN(X) ⪯ k2, HN(X) ⪯ l2.

Now,

TN(X) ⪰ i2 ≻ i1, IN(X) ⪯ j2 ≺ j1, FN(X) ⪯ k2 ≺ k1, HN(X) ⪯ l2 ≺ l1

this implies that

TN(X) ⪰ i1, IN(X) ⪯ j1, FN(X) ⪯ k1, HN(X) ⪯ l1

Therefore, X ∈ N(i1, j1, k1, l1) ⇒ N(i2, j2, k2, l2) ⊆ N(i1, j1, k1, l1).

2. Let X ∈
∞⋂
n=1

N(in, jn, kn, ln) ⇒ X ∈ N(in, jn, kn, ln).

Since
lim
n→∞

in = i, lim
n→∞

jn = j, lim
n→∞

kn = k, lim
n→∞

ln = l

then

TN(X) ⪰ lim
n→∞

in = i, IN(X) ⪯ lim
n→∞

jn = j, FN(X) ⪯ lim
n→∞

kn = k, HN(X) ⪯ lim
n→∞

ln = l

Thus,
TN(X) ⪰ i, IN(X) ⪯ j, FN(X) ⪯ k, HN(X) ⪯ l

Therefore, X ∈ (N)(i, j, k, l) ⇒ N(i1, j1, k1, l1) ⊆ (N)(i, j, k, l).
Again, let X ∈ (N)(i, j, k, l), then

TN(X) ⪰ i, IN(X) ⪯ j, FN(X) ⪯ k, HN(X) ⪯ l

Since in ↑ i, jn ↓ j, kn ↓ k, ln ↓ l, we obtain:

TN(X) ⪰ i ⪰ in, IN(X) ⪯ j ⪯ jn, FN(X) ⪯ k ⪯ kn, HN(X) ⪯ l ⪯ ln

for all n. This implies that

X ∈
∞⋂
n=1

N(in, jn, kn, ln) ⇒ (N)(i, j, k, l) ⊆ N(i1, j1, k1, l1)

Thus, we conclude:
∞⋂
n=1

N(in, jn, kn, ln) = (N)(i, j, k, l).

Proposition 5. Let
N(i,j,k,l) = ⟨[li, Ui], [lj, Uj], [lk, Uk], [ll, Ul]⟩,

where 0 ⪯ i ⪯ 1, 0 ⪯ j ⪯ 1, 0 ⪯ k ⪯ 1, 0 ⪯ l ⪯ 1 be a domain of interval quadri-neutrosophic
sets (QNS), and each interval [li, Ui], [lj, Uj], [lk, Uk], [ll, Ul] is closed.

Suppose N(i,j,k,l) is decreasing with respect to (i, j, k, l) and ñ is a closed QNSN. Then {N(i,j,k,l)}
can induce ñ, and

ñ(i, j, k, l) = N(i, j, k, l)

where ñ(i, j, k, l) and N(i, j, k, l) denote the (i, j, k, l)-cut of ñ and N , respectively.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 13 of 54

Proof. Let ñ be a single-valued quadri-partitioned neutrosophic number, and supposeN(i,j,k,l)

is decreasing with respect to (i, j, k, l). From Theorem 1, we have

∞⋂
n=1

N(in, jn, kn, ln) = N(i,j,k,l)

for in ↑ i, jn ↓ j, kn ↓ k, ln ↓ l.
Now, from Definitions 2 and 3, it follows that {N(i,j,k,l)} can induce ñ. From Definition 4,

we have
{X : Tñ(X) ⪰ i, Iñ(X) ⪯ j, Fñ(X) ⪯ k, Hñ(X) ⪯ l}

= N(i,j,k,l) = ⟨[li, Ui], [lj, Uj], [lk, Uk], [ll, Ul]⟩

is a closed single-valued quadri-partitioned neutrosophic number.

Proposition 6. If ñ is a closed single-valued quadri-partitioned neutrosophic number, then

ñ and ñ

U ↓ ñU
i for in ↑ i, ñL

jn ↑ ñL
j , ñU

jn ↓ ñU
j for jn ↓ j,

ñL
kn ↑ ñL

k , ñU
kn ↓ ñU

k for kn ↓ k,

ñL
ln ↑ ñL

l , ñU
ln ↓ ñU

l for ln ↓ l.

Proof. Since (N)(i,j,k,l) = ñ(i, j, k, l) = ⟨[ñL
i , ñ

U
i ], [ñ

L
j , ñ

U
j ], [ñ

L
k , ñ

U
k ], [ñ

L
l , ñ

U
l ]⟩, then (N)(i,j,k,l) is

decreasing with respect to (i, j, k, l). We have:

lim ñL
an ⪯ ñL

i , lim ñU
an ⪰ ñU

i , lim ñL
jn ⪯ ñL

j , lim ñU
jn ⪰ ñU

j ,

lim ñL
kn ⪯ ñL

k , lim ñU
kn ⪰ ñU

k , lim ñL
ln ⪯ ñL

l , lim ñU
ln ⪰ ñU

l .

This implies that:

⟨[lim ñL
an, lim ñU

an], [lim ñL
jn, lim ñU

jn], [lim ñL
kn, lim ñU

kn], [lim ñL
ln, lim ñU

ln]⟩ ⊆ ⟨[ñL
i , ñ

U
i ], [ñ

L
j , ñ

U
j ], [ñ

L
k , ñ

U
k ], [ñ

L
l , ñ

U
l ]⟩.

Thus, we conclude:

lim ñL
an ⪰ ñL

i , lim ñU
an ⪯ ñU

i , lim ñL
jn ⪰ ñL

j , lim ñU
jn ⪯ ñU

j ,

lim ñL
kn ⪰ ñL

k , lim ñU
kn ⪯ ñU

k , lim ñL
ln ⪰ ñL

l , lim ñU
ln ⪯ ñU

l .


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 14 of 54

5. Characterization of Riemann Integration in terms of Quadri-partitioned
Neutrosophic Structure

This section is devoted to Riemann integral theory based on quadri-partitioned neutrosophic
sets, and all the fundamentals are defined according to this new theory. Some new effects are
given, and this whole scenario has been established by introducing interesting examples.

Definition 11. Let Q be the set of all Quadri-Partitioned Neutrosophic Numbers (QNSN), Qcl

be the set of all closed (QNSN), and Qb signify the set of all bounded (QNSN).

(i) f̃Q(x) is a Q-NSVF if f̃Q : X → Q.

(ii) f̃Q(x) is a closed Q-NSVF if f̃Q : X → Qcl.

(iii) f̃Q(x) is a bounded Q-NSVF if f̃Q : X → Qb.

Definition 12. Let f̃Q(X) be a closed bounded Q-NSVF on a closed interval [a1, b1]. Let

f̃L
Qi(X), f̃U

Qi(X), f̃L
Qj(X), f̃U

Qj(X), f̃L
Qk(X), f̃U

Qk(X), f̃L
Ql(X), and f̃U

Ql(X)

are all quadri-neutrosophic Riemann integrable (QNRI) on [a1, b1]. For all (i, j, k, l), let

I(i,j,k,l) =



[∫ b1
a1

f̃L
Qi(X)dX,

∫ b1
a1

f̃U
Qi(X)dX

]
,[∫ b1

a1
f̃L
Qj(X)dX,

∫ b1
a1

f̃U
Qj(X)dX

]
,[∫ b1

a1
f̃L
Qk(X)dX,

∫ b1
a1

f̃U
Qk(X)dX

]
,[∫ b1

a1
f̃L
Ql(X)dX,

∫ b1
a1

f̃U
Ql(X)dX

]


Here,

[
f̃L
Qi(X), f̃

U
Qi(X)

]
,
[
f̃L
Qj(X), f̃

U
Qj(X)

]
,
[
f̃L
Qk(X), f̃

U
Qk(X)

]
, and

[
f̃L
Ql(X), f̃

U
Ql(X)

]
denote

the (i, j, k, l)-cut of f̃Q(X), respectively.

Proposition 7. [56] If g(x) is defined on [a1, b1] and is a bounded function over [a1, b1], then
g(x) is also Lebesgue integrable over [a1, b1].

Proposition 8. [56] If g(x) is a bounded function defined on [a1, b1], then g(x) is Riemann
integrable on [a1, b1] if and only if g(x) is continuous on the closed bounded interval [a1, b1].

Proposition 9. [56] If g(x) is Riemann integrable on [a1, b1] and µ ∈ R, then µg(X) is also
Riemann integrable on [a1, b1] and∫ b1

a1

µg(X) dx = µ

∫ b1

a1

g(X) dx.

Theorem 2. Let f̃Q(x) be a closed bounded quadri-neutrosophic valued function on the closed
quadri-neutrosophic bounded interval [a1, b1]. If f̃Q(x) ∈ QRI on the closed quadri-neutrosophic
bounded interval [a1, b1], then the quadri-neutrosophic Riemann integral∫ b1

a1

f̃Q(x)dx

is a closed quadri-neutrosophic single number (QNSN). The (i, j, k, l)-cut of∫ b1

a1

f̃Q(x)dx


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 15 of 54

is given by:

(∫ b1

a1

f̃Q(x)dx

)
(i,j,k,l)

=



[∫ b1
a1

f̃LQi(X)dX,
∫ b1
a1

f̃UQi(X)dX
]
,[∫ b1

a1
f̃LQj(X)dX,

∫ b1
a1

f̃UQj(X)dX
]
,[∫ b1

a1
f̃LQk(X)dX,

∫ b1
a1

f̃UQk(X)dX
]
,[∫ b1

a1
f̃LQl(X)dX,

∫ b1
a1

f̃UQl(X)dX
]


where [̃fLQi(X), f̃

U
Qi(X)], [̃f

L
Qj(X), f̃

U
Qj(X)], [̃fLQk(X), f̃

U
Qk(X)], and [̃fLQl(X), f̃

U
Ql(X)] denote the (i, j, k, l)-

cut of f̃Q(X), respectively.

Proof. Let

Q(i, j, k, l) =


∫
f̃
L(X)
Qi d(X),

∫
f̃
U(X)
Qi d(X)∫

f̃
L(X)
Qj d(X),

∫
f̃
U(X)
Qj d(X)∫

f̃
L(X)
Qk d(X),

∫
f̃
U(X)
Qk d(X)∫

f̃
L(X)
Ql d(X),

∫
f̃
U(X)
Ql d(X)


For i2 ≻ i1, j2 ≺ j1, k2 ≺ k1, l2 ≺ l1, we have:

f̃
L(X)
Qi1

⪯ f̃
L(X)
Qi2

,

f̃
U(X)
Qi1

⪰ f̃
U(X)
Qi2

,

f̃
L(X)
Qj1

⪰ f̃
L(X)
Qj2

,

f̃
U(X)
Qj1

⪯ f̃
U(X)
Qj2

,

f̃
L(X)
Qk1

⪰ f̃
L(X)
Qk2

,

f̃
U(X)
Qk1

⪯ f̃
U(X)
Qk2

,

f̃
L(X)
Ql1

⪰ f̃
L(X)
Ql2

,

f̃
U(X)
Ql1

⪯ f̃
U(X)
Ql2

.

Then,
Q(i2, j2, k2, l2) ⊆ Q(i1, j1, k1, l1)

for 0 ⪯ in1 ⪯ 1, we get:
f̃Q0L(X) ⪯ f̃Qin1

(X) ⪯ f̃Q1L(X).

Thus,
|̃fQin1

(X)| ⪯ max{|̃fQ0L(X)|, |̃fQ1L(X)|} = h(X).

Since f̃Q0L(X) and f̃Q1L(X) are qudri-neutrosophic Riemann integrable on [a1, b1], |̃fQ0L(X)|,
|̃fQ1L(X)|, and h(X) are also qudri-neutrosophic Riemann integrable on [a1, b1].

From Proposition 9, it follows that h(X) is qudri-neutrosophic Lebesgue integrable on [a1, b1].
Now we should apply the quadri-neutrosophic Lebesgue dominated convergence theorem. For
in1 ↑ i1, we have:

lim
n→∞

∫ b1

a1

f̃LQin1
(X)d(X) =

∫ b1

a1

f̃UQi1(X)d(X).

Since f̃Q(X) is a closed QNVF, we have

lim
n→∞

f̃LQin1
(X) = f̃LQin1

(X),


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 16 of 54

by Definition 9. Again, for 0 ⪯ in1 ⪯ 1, we have:

f̃UQ1(X) ⪯ f̃LQin1
(X) ⪯ f̃UQ0(X).

Then,
|̃fUQin1

(X)| ⪯ max{|̃fUQ1(X)|, |̃fUQ0(X)|} = g(X).

By a similar fashion, we can argue:

lim
n→∞

∫ b1

a1

f̃UQin1
(X)d(X) =

∫ b1

a1

f̃UQi1(X)d(X).

Since f̃Q(X) is a closed QNVF, then:

lim
n→∞

f̃LQin1
(X) = f̃LQin1

(X).

By a similar process, we have:

lim
n→∞

∫ b1

a1

f̃LQjn1
(X)d(X) =

∫ b1

a1

f̃LQj1(X)d(X),

lim
n→∞

∫ b1

a1

f̃UQjn1
(X)d(X) =

∫ b1

a1

f̃UQj1(X)d(X),

lim
n→∞

∫ b1

a1

f̃LQkn1
(X)d(X) =

∫ b1

a1

f̃LQk1(X)d(X),

lim
n→∞

∫ b1

a1

f̃UQkn1
(X)d(X) =

∫ b1

a1

f̃UQk1(X)d(X).

Theorem 3. If f̃Q(X) is a closed bounded QNVF on the closed quadri-neutrosophic bounded
interval [a1, b1], and the functions f̃LQi

(X), f̃UQi
(X), f̃LQj

(X), f̃UQj
(X), f̃LQk

(X), f̃UQk
(X), f̃LQl

(X), f̃UQl
(X)

are all continuous on the interval [a1, b1], then we have:

f̃Q(X) ∈ QRI on the closed bounded interval [a1, b1]

Thus, the integral is given by:

(∫ b1

a1

f̃Q(x)dx

)
(i,j,k,l)

=


∫ b1
a1

f̃LQi
(X)dX,

∫ b1
a1

f̃UQi
(X)dX∫ b1

a1
f̃LQj

(X)dX,
∫ b1
a1

f̃UQj
(X)dX∫ b1

a1
f̃LQk

(X)dX,
∫ b1
a1

f̃UQk
(X)dX∫ b1

a1
f̃LQl

(X)dX,
∫ b1
a1

f̃UQl
(X)dX


Proof. From Proposition 8, the functions

f̃L
Qi(X), f̃U

Qi(X), f̃L
Qj(X), f̃U

Qj(X), f̃L
Qk(X), f̃U

Qk(X), f̃L
Ql(X), f̃U

Ql(X)

are Riemann integrable on [a1, b1]. Then, f̃Q(X) is Riemann integrable on [a1, b1], and we have

(∫ b1

a1

f̃Q(x) dx

)
i,j,k,l

=



[∫ b1
a1

f̃L
Qi(X) dX,

∫ b1
a1

f̃U
Qi(X) dX

][∫ b1
a1

f̃L
Qj(X) dX,

∫ b1
a1

f̃U
Qj(X) dX

][∫ b1
a1

f̃L
Qk(X) dX,

∫ b1
a1

f̃U
Qk(X) dX

][∫ b1
a1

f̃L
Ql(X) dX,

∫ b1
a1

f̃U
Ql(X) dX

]




A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 17 of 54

Theorem 4. Let f̃Q(X) and g̃Q(X) be closed bounded quadri-neutrosophic valued functions on
[a1, b1]. If f̃Q(X) · g̃Q(X) ∈ QRI, then f̃Q(X) + g̃Q(X) ∈ QRI and f̃Q(X)− g̃Q(X) ∈ QRI.

Moreover, we have∫ b1

a1

(f̃Q(X) + g̃Q(X)) dx =

∫ b1

a1

f̃Q(X) dx+

∫ b1

a1

g̃Q(X) dx

∫ b1

a1

(f̃Q(X)− g̃Q(X)) dx =

∫ b1

a1

f̃Q(X) dx−
∫ b1

a1

g̃Q(X) dx

Proof.
Let h̃Q(X) = f̃Q(X) + g̃Q(X), then h̃Q(X) is a closed quadri-neutrosophic valued function.(∫ b1

a1

f̃Q(X) dx+

∫ b1

a1

g̃Q(X) dx

)
(i,j,k,l)〈[∫ b1

a1

f̃ i
Q,L(X) dX,

∫ b1

a1

g̃iQ,L(X) dX,

∫ b1

a1

f̃ i
Q,U (X) dX,

∫ b1

a1

g̃iQ,U (X) dX
]〉

[∫ b1

a1

f̃ j
Q,L(X) dX,

∫ b1

a1

g̃jQ,L(X) dX,

∫ b1

a1

f̃ j
Q,U (X) dX,

∫ b1

a1

g̃jQ,U (X) dX
]

[ ∫ b1

a1

f̃ k
Q,L(X) dX,

∫ b1

a1

g̃kQ,L(X) dX,

∫ b1

a1

f̃ k
Q,U (X) dX,

∫ b1

a1

g̃kQ,U (X) dX
]

〈[ ∫ b1

a1

f̃ l
Q,L(X) dX,

∫ b1

a1

g̃lQ,L(X) dX,

∫ b1

a1

f̃ l
Q,U (X) dX,

∫ b1

a1

g̃lQ,U (X) dX
]〉

(∫ b1

a1

f̃Q(X) dx+

∫ b1

a1

g̃Q(X) dx

)
(i,j,k,l)[ [∫ b1

a1
f̃ i
Q,L(X) dX+

∫ b1
a1

g̃iQ,L(X) dX,
∫ b1
a1

f̃ i
Q,U (X) dX+

∫ b1
a1

g̃iQ,U (X) dX
] ]

[∫ b1

a1

f̃ j
Q,L(X) dX+

∫ b1

a1

g̃jQ,L(X) dX,

∫ b1

a1

f̃ j
Q,U (X) dX+

∫ b1

a1

g̃jQ,U (X) dX

]

[∫ b1

a1

f̃ k
Q,L(X) dX+

∫ b1

a1

g̃kQ,L(X) dX,

∫ b1

a1

f̃ k
Q,U (X) dX+

∫ b1

a1

g̃kQ,U (X) dX

]

[∫ b1

a1

f̃ l
Q,L(X) dX+

∫ b1

a1

g̃lQ,L(X) dX,

∫ b1

a1

f̃ l
Q,U (X) dX+

∫ b1

a1

g̃lQ,U (X) dX

]
⟨
∫ b1

a1

(f̃Q(X) + g̃Q(X))
i
L dx,

∫ b1

a1

(f̃Q(X) + g̃Q(X))
i
U dx,

∫ b1

a1

(f̃Q(X) + g̃Q(X))
j
L dx,

∫ b1

a1

(f̃Q(X) + g̃Q(X))
j
U dx,

∫ b1

a1

(f̃Q(X) + g̃Q(X))
k
L dx,

∫ b1

a1

(f̃Q(X) + g̃Q(X))
k
U dx,


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 18 of 54

∫ b1

a1

(f̃Q(X) + g̃Q(X))
l
L dx,

∫ b1

a1

(f̃Q(X) + g̃Q(X))
l
U dx⟩

(∫ b1

a1

f̃Q(X) dx+

∫ b1

a1

g̃Q(X) dX

)
(i,j,k,l)

Theorem 5. Let f̃Q(X) be a closed bounded quadri-neutrosophic valued function on the closed
interval [a1, b1].

If f̃Q(X) ∈ QRI, then λf̃Q(X) ∈ QRI. Moreover,∫ b1

a1

λf̃Q(X) dx = λ

∫ b1

a1

f̃Q(X) dx

where λ ̸= 0 is any real number.

Proof.
For λ ≻ 0, let g̃Q(X) = λf̃Q(X). Also, g̃Q(X) is a closed quadri-neutrosophic valued function

(QNVF). (
λ

∫ b1

a1

f̃Q(X) dx

)
(i,j,k,l)〈[

λ

∫ b1

a1

f̃LQi(X) dX, λ

∫ b1

a1

f̃UQi(X) dX
]
,

[
λ

∫ b1

a1

f̃LQj(X) dX, λ

∫ b1

a1

f̃UQj(X) dX
]
,

[
λ

∫ b1

a1

f̃LQk(X) dX, λ

∫ b1

a1

f̃UQk(X) dX
]
,

[
λ

∫ b1

a1

f̃LQl(X) dX, λ

∫ b1

a1

f̃UQl(X) dX
]〉

〈[∫ b1

a1

λf̃LQi(X) dX,

∫ b1

a1

λf̃UQi(X) dX
]
,

[ ∫ b1

a1

λf̃LQj(X) dX,

∫ b1

a1

λf̃UQj(X) dX
]
,

[ ∫ b1

a1

λf̃LQk(X) dX,

∫ b1

a1

λf̃UQk(X) dX
]
,

[ ∫ b1

a1

λf̃LQl(X) dX,

∫ b1

a1

λf̃UQl(X) dX
]〉

(∫ b1

a1

λf̃Q(X) dx

)
(i,j,k,l)

Since, ∫ b1

a1

λf̃Q(X) dx = λ

∫ b1

a1

f̃Q(X) dx

the same holds for λ ≺ 0.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 19 of 54

Example 1. If
f̃(Q(X)) = ñ(X)2 on [0, 1]

where
ñ =

〈
(0, 1, 2, 3); 8× 10−1, 6× 10−1, 4× 10−1, 4× 10−1

〉
is a quadri-neutrosophic valued number. Now, we integrate the quadri-neutrosophic valued func-
tion on [0, 1] and try to find ∫ 1

0
f̃Q(X) dX.

Now, taking the (i, j, k, l)-cut of the integral, we obtain:(∫ 1

0
f̃(Q(X)) dX

)
(i,j,k,l)

=

(∫ 1

0
ñ(X)2 dX

)
(i,j,k,l)

.

This results in:〈[∫ 1

0

5i

4
X2 dX,

∫ 1

0

8− 5i

4
X2 dX

]
,

[∫ 1

0

5

2
(1− j)X2 dX,

∫ 1

0

1

2
(5j− 1)X2 dX

]
,

[∫ 1

0

5

3
(1− k)X2 dX,

∫ 1

0

1

3
(5k− 1)X2 dX

]
,

[∫ 1

0

5

3
(1− l)X2 dX,

∫ 1

0

1

3
(5l− 1)X2 dX

]〉
.

Evaluating the integrals:〈[
5i

12
,
8− 5i

12

]
,

[
5

6
(1− j),

1

6
(5j− 1)

]
,

[
5

9
(1− k),

1

9
(5k− 1)

]
,

[
5

9
(1− l),

1

9
(5l− 1)

]〉
.

Thus, we obtain: ∫ 1

0
f̃LQi(X) dX =

5i

12
,

∫ 1

0
f̃UQi(X) dX =

8− 5i

12
,∫ 1

0
f̃LQj(X) dX =

5

6
(1− j),

∫ 1

0
f̃UQj(X) dX =

1

6
(5j− 1).∫ 1

0
f̃LQk(X) dX =

5

9
(1− k),

∫ 1

0
f̃UQk(X) dX =

1

9
(5k− 1).

∫ 1

0
f̃LQl(X) dX =

5

9
(1− l),

∫ 1

0
f̃UQl(X) dX =

1

9
(5l− 1).

For parameter ranges:

i ∈ [0, 8× 10−1], j ∈ [6× 10−1, 1], k ∈ [4× 10−1, 1], l ∈ [4× 10−1, 1].

From the table, it is observed that as the value of α increases, the value of∫ 1

0
f̃LQi(X) dX

also increases, whereas the value of ∫ 1

0
f̃UQi(X) dX

decreases. At i = 0.8, we obtain: ∫ 1

0
f̃LQi(X) dX.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 20 of 54

i
∫ 1
0 f̃LQi(X)dX

∫ 1
0 f̃UQi(X)dX j

∫ 1
0 f̃LQj(X)dX

∫ 1
0 f̃UQj(X)dX k

∫ 1
0 f̃LQk(X)dX

∫ 1
0 f̃UQk(X)dX l

0× 10−1 0× 10−1 0.66667× 10−5 6× 10−1 0.33333× 10−5 0.33333× 10−5 4× 10−1 0.33333× 10−5 0.33333× 10−5 4× 10−1

1.6667× 10−5 5× 10−1 8× 10−1 0.16667× 10−5 5× 10−1 6× 10−1 0.22222× 10−5 0.44444× 10−5 6× 10−1 0.22222× 10−5

0.44444× 10−5 6× 10−1 0.25× 10−2 0.41667× 10−5 9× 10−1 0.08333× 10−5 0.58333× 10−5 8× 10−1 0.11111× 10−5 0.66667× 10−5

8× 10−1 0.11111× 10−5 0.66667× 10−5 8× 10−1 0.33333× 10−5 0.33333× 10−5 1× 100 0× 10−1 0.66667× 10−5 1× 10−1

Table 1: Solution for different values of i, j, k, l for Example 1

∫ 1

0
f̃uQi

(X) dX

gives the same solution. Again, when the value of j increases, the value of∫ 1

0
f̃LQj

(X) dX

decreased, and the value of ∫ 1

0
f̃uQj

(X) dX

increased. At j = 0.6, ∫ 1

0
f̃LQj

(X) dX

and ∫ 1

0
f̃uQj

(X) dX

give the same solution.
When k increases, the value of ∫ 1

0
f̃LQ∥

(X) dX

decreases, and the value of ∫ 1

0
f̃UQ∥

(X) dX

increases. Similarly, when l increases, the value of∫ 1

0
f̃LQl

(X) dX

decreases, and the value of ∫ 1

0
f̃UQl

(X) dX

increases. This implies that the approximate solution in Table 1 provides a quadri-neutrosophic-
valued number. In the following example, we are going to show that how one can use the existing
numerical integration methods to solve the quadri-neutrosophic integral. So, in the following
example, we consider the same quadri-neutrosophic function, but the parameter was taken in
the form of trapezoidal quadri-neutrosophic number.

Example 2. Let us consider the Quadri-Neutrosophic Valued Function (QNVF) given by:

f̃(Q(X)) = ñ(X)2 on [0, 1]

where

ñ = ⟨(0, 1, 2, 2); 0.8, 0.6, 0.4, 0.4⟩

is a single-valued triangular quadri-neutrosophic number.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 21 of 54

Now, we integrate the quadri-neutrosophic valued function over the interval [0, 1], aiming to
find: ∫ 1

0
f̃(Q(X)) dX

Next, we take the (i, j, ∥, l)-cut of the integral:∫ 1

0
f̃(Q(X)) dX

Then, we have: (∫ 1

0
f̃(Q(X)) dX

)
(i,j,∥,l)

=



[∫ 1
0 f̃Qi

L(X) dX,
∫ 1
0 f̃Qi

U(X) dX
]
,[∫ 1

0 f̃Qj
L(X) dX,

∫ 1
0 f̃Qj

U(X) dX
]
,[∫ 1

0 f̃Q∥L(X) dX,
∫ 1
0 f̃Q∥U(X) dX

]
,[∫ 1

0 f̃Ql
L(X) dX,

∫ 1
0 f̃Ql

U(X) dX
]


Since each of the integral ∫ 1

0
f̃Qi

L(X) dX,

∫ 1

0
f̃Qi

U(X) dX,∫ 1

0
f̃Qj

L(X) dX,

∫ 1

0
f̃Qj

U(X) dX,∫ 1

0
f̃Qk

L(X) dX,

∫ 1

0
f̃Qk

U(X) dX,∫ 1

0
f̃Ql

L(X) dX,

∫ 1

0
f̃Ql

U(X) dX

are Riemann integrable on [0, 1]. Then we can use the trapezoidal rule to approximate the
integral: ∫ 1

0
x2 dX

with the help of the trapezoidal rule.
Let

p =

{
0,

1

4
,
1

2
,
3

4
, 1

}
be the set of all elements at the endpoints of the sub-intervals, and

∆X =
1− 0

4
=

1

4
.

Then, ∫ 1

0
x2 dX ≈ 1

4
× 1

4

[
f(0) + 2f

(
1

4

)
+ 2f

(
1

2

)
+ 2f

(
3

4

)
+ f(1)

]


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 22 of 54

=
1

8

(
0 +

1

8
+

1

2
+

9

8
+ 1

)
=

11

32
.

Therefore, (∫ 1

0
f̃(Q(X)) dx

)
i,j,k,l

=

(∫ 1

0
ñ(X)2 dx

)
i,j,k,l

∫ 1
0

5i
4 (X)

2 dx,
∫ 1
0

(
2− 5i

4

)
(X)2 dx∫ 1

0
5
2(1− j)(X)2 dx,

∫ 1
0

1
2(5j− 1)(X)2 dx∫ 1

0
5
3(1− k)(X)2 dx,

∫ 1
0

1
3(5k− 1)(X)2 dx∫ 1

0
5
3(1− l)(X)2 dx,

∫ 1
0

1
3(5l− 1)(X)2 dx




5i
4

∫ 1
0 (X)

2 dx, 12−5i
4

∫ 1
0 (X)

2 dx
5
2(1− j)

∫ 1
0 (X)

2 dx, 1
2(5j− 1)

∫ 1
0 (X)

2 dx
5
3(1− k)

∫ 1
0 (X)

2 dx, 1
3(5k− 1)

∫ 1
0 (X)

2 dx
5
3(1− l)

∫ 1
0 (X)

2 dx, 1
3(5l− 1)

∫ 1
0 (X)

2 dx


∫
f̃QL

i (X)dX
∫
f̃QU

i (X)dX
∫
f̃QL

j (X)dX
∫
f̃QU

j (X)dX
∫
f̃QL

k (X)dX
∫
f̃QU

k (X)dX
∫
f̃QL

l (X)dX
∫
f̃QU

l (X)dX

0 0× 10−1 0× 10−1 1.03125× 10−5 6× 10−1 0.34375× 10−5 0.6875× 10−4 4× 10−1 0.34375× 10−5

0.4 0.34375× 10−5 0.6875× 10−4 4× 10−1 0.34375× 10−5 0.6875× 10−4 4× 10−1 0.171875× 10−6 0.859375× 10−6

0.8 0.17185× 10−5 0.859375× 10−6 6× 10−1 0.229167× 10−6 0.802083× 10−6 6× 10−1 0.229167× 10−6 0.802083× 10−6

0.6 0.257813× 10−6 0.773438× 10−6 9× 10−1 0.0859375× 10−7 0.945313× 10−6 8× 10−1 0.114583× 10−6 0.916667× 10−6

0.8 0.114583× 10−6 0.916667× 10−6 8× 10−1 0.34375× 10−5 0.6875× 10−4 1× 10−1 0× 10−1 1.03125× 10−5

1.0 0× 10−1 1.03125× 10−5 1× 10−1 0× 10−1 1.03125× 10−5 1× 10−1 0× 10−1 1.03125× 10−5

Table 2: Value of Example 2 by the Quadri-Trapezoidal Rule for Different Values of i, j, k, and l
55i
128 ,

132−55i
128

55
64(1− j), 11

64(5j− 1)
55
96(1− k), 11

96(5k− 1)
55
96(1− l), 11

96(5l− 1)


Where: ∫

f̃QL
i (X) dx

∣∣∣1
0
=

55i

128
,

∫
f̃QU

i (X) dx
∣∣∣1
0
=

132− 55i

128∫
f̃QL

j (X) dx
∣∣∣1
0
=

55

64
(1− j),

∫
f̃QU

j (X) dx
∣∣∣1
0
=

11

64
(5j− 1)

∫
f̃QL

k (X) dx
∣∣∣1
0
=

55

96
(1− k),

∫
f̃QU

k (X) dx
∣∣∣1
0
=

11

96
(5k− 1)

∫
f̃QL

l (X) dx
∣∣∣1
0
=

55

96
(1− l),

∫
f̃QU

l (X) dx
∣∣∣1
0
=

11

96
(5l− 1).

For:

i ∈ [0, 0.8], j ∈ [0.6, 1], k ∈ [0.4, 1], l ∈ [0.4, 1]


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 23 of 54

6. Comparative Analysis

The following Table 3 provides a detailed comparative analysis of the proposed methods,
contrasting them with the established techniques discussed in [43]. This comparison highlights
the strengths and weaknesses of each approach, offering insights into how the proposed methods
perform relative to the established techniques across various key factors.

Aspect Published Work (Reference [43]) Proposed Method/Work

1. Foundation Introduces Neutrosophic Riemann Inte-
gration, focusing on neutrosophic numbers
and functions, extending classical integra-
tion theory.

Based on Quadri-Partitioned Neutro-
sophic Set Theory (QPNST), extending
Neutrosophic Set Theory (NST) and In-
tuitionistic Fuzzy Set Theory (IFST). In-
troduces a fourth component for refined
set representation.

2. Theoretical
Basis

Uses neutrosophic numbers with three
membership values (truth, indeterminacy,
falsity) and (α, β, γ)-level sets.

Extends NST by introducing a fourth
component, forming Quadri-Partitioned
Neutrosophic Set Theory (QPNST), with
Riemann Integral Theory (RIT) adapted
to a four-tuple (i, j, k, l) level cut.

3. Conceptual
Innovation

Introduces Neutrosophic Riemann Inte-
gration to manage uncertainty in classical
Riemann integration.

Proposes Quadri-Partitioned Neu-
trosophic Riemann Integral Theory
(QPNRIT), adding a fourth uncertainty
dimension for a more comprehensive
uncertainty model.

4. Integration
Framework

Uses neutrosophic numbers with (α, β, γ)-
level sets to define Neutrosophic Riemann
Integration.

Incorporates Riemann Integral Theory
(RIT) into Quadri-Partitioned Neutro-
sophic Set Theory (QPNST), allowing for
four uncertainty possibilities (true, false,
indeterminacy, and a new fourth possibil-
ity).

5. Numerical
Approach

Uses numerical methods like the trape-
zoidal rule to compute Neutrosophic Rie-
mann integrals, validating results with ex-
amples.

Applies numerical analysis to Quadri-
Partitioned Neutrosophic Riemann Inte-
grals (QPNRIT), presenting systematic
numerical results in tables to assess the
impact of the fourth possibility.

6. Level Set
Representation

Uses (α, β, γ)-level sets to represent truth,
indeterminacy, and falsity in neutrosophic
numbers.

Introduces a four-tuple level cut (i, j, k, l),
enhancing the representation of uncer-
tainty within QPNST and providing a
more detailed integral calculation frame-
work.

7. Output Rep-
resentation

Displays results in tables and figures to
validate Neutrosophic Riemann Integra-
tion, using the trapezoidal rule for approx-
imation.

Presents Quadri-Partitioned Neutro-
sophic Riemann Integral Theory (QP-
NRIT) results in tables, showcasing
numerical insights and the fourth possi-
bility’s impact on integration.

8. Scope and
Extensiveness

Introduces and numerically validates
Neutrosophic Riemann Integration with
graphical representations.

Extends NST and Riemann integration
into a more sophisticated framework (QP-
NST), exploring a deeper theoretical foun-
dation with advanced numerical tech-
niques.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 24 of 54

Aspect Published Work (Reference [43]) Proposed Method/Work

9. Application
Focus

The primary application in the published
work is the numerical verification of neu-
trosophic Riemann integration, focusing
on how the trapezoidal rule can be used
to approximate the neutrosophic integral.

The proposed method’s application fo-
cuses on improving the accuracy and
handling of uncertainty in Riemann in-
tegration, with potential applications in
decision-making, engineering, economics,
and artificial intelligence. It also lays the
groundwork for more sophisticated models
and techniques in multi dimensional and
dynamic systems.

10. Foundation The published work introduces Neutro-
sophic Riemann Integration for the first
time, focusing on neutrosophic numbers
and functions. It builds on the concept of
fuzziness and uncertainty in classical inte-
gration theory.

The proposed method is grounded in
Quadri-Partitioned Neutrosophic Set
Theory (QPNST), an extension of Neu-
trosophic Set Theory (NST), which itself
is an extension of Intuitionistic Fuzzy Set
Theory (IFST). It introduces a fourth
possibility (along with true, false, and
indeterminacy) for set representation,
allowing for more refined descriptions of
sets and their behavior.

11. Theoretical
Basis

Integration framed within neutrosophic
numbers with three membership values
(truth, indeterminacy, and falsity), using
(α, β, γ)-level sets to represent fuzzy mem-
bership.

Extends NST with a fourth possibility,
defining Quadri-Partitioned Neutrosophic
Set Theory (QPNST). Develops Riemann
Integral Theory (RIT) within QPNST us-
ing a four-tuple (i, j, k, l) level cut.

12. Conceptual
Innovation

Introduces neutrosophic Riemann integra-
tion, applying neutrosophic numbers to
handle uncertainty in classical Riemann
integration.

Proposes Quadri-Partitioned Neutro-
sophic Riemann Integral Theory (QPN-
RIT), extending neutrosophic integration
by incorporating a fourth dimension of
uncertainty for a more comprehensive
uncertainty model.

13. Integration
Framework

Defines Neutrosophic Riemann Integra-
tion using (α, β, γ)-level sets and fuzzy
membership functions to represent uncer-
tainty.

Incorporates extended Riemann Integral
Theory (RIT) within QPNST, enhancing
uncertainty modeling with four possibili-
ties (true, false, indeterminacy, and a new
fourth possibility).

14. Numerical
Approach

Uses numerical methods like the trape-
zoidal rule to calculate neutrosophic Rie-
mann integrals, validated with numerical
examples, tables, and figures.

Conducts numerical exploration of
Quadri-Partitioned Neutrosophic Rie-
mann Integrals (QPNRIT). Systematic
numerical results in tables illustrate
the impact of the fourth possibility on
integral computation.

15. Level Set
Representation

Uses (α, β, γ)-level sets, where each pa-
rameter represents different membership
degrees (truth, indeterminacy, falsity) in
the neutrosophic number.

Introduces a more refined four-tuple level
cut (i, j, k, l), providing a more granu-
lar representation of multiple possibilities
within QPNST, enhancing neutrosophic
Riemann integral calculation.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 25 of 54

Aspect Published Work (Reference [43]) Proposed Method/Work

16. Output
Representation

Presents numerical results in tables and
figures to validate neutrosophic Riemann
integration, with the trapezoidal rule used
for approximation.

Presents Quadri-Partitioned Neutro-
sophic Riemann Integral Theory (QPN-
RIT) results in tables, offering advanced
integral representations to analyze the
impact of the fourth possibility on
integration.

17. Scope and
Extensiveness

Focuses on basic introduction and numeri-
cal validation of neutrosophic Riemann in-
tegration, including examples and graphi-
cal representations.

Extends neutrosophic set theory within
Riemann integration to a more complex
framework (QPNST), deepening theoreti-
cal foundations and numerical techniques
for broader applications.

18. Application
Focus

Primarily applies to numerical verification
of neutrosophic Riemann integration, em-
phasizing the trapezoidal rule for approx-
imation.

Enhances accuracy and uncertainty han-
dling in Riemann integration with poten-
tial applications in decision-making, engi-
neering, economics, and AI, forming the
basis for sophisticated multidimensional
and dynamic system models.

Table 3: Comparison between Published Work and Proposed
Method

7. Operations on Quadri-Partitioned Neutrosophic Soft Sets

Neutrosophic set theory (NST), a generality of vague set theory (VST), is regarded as the
most appealing theory since it considers the three possible membership values: true, false, and
indeterminacy. The principles are all quite obvious, but the third one is particularly fascinating
since it addresses uncertainty, which arises in all aspects of daily life. One can make the situation
more certain and free of error if the indeterminacy membership is refined. This can be done
by splitting the indeterminacy into five pieces that is possible values. These are relative true,
relative false, contradiction, unknown (undefined) and ignorance. This section is devoted to the
most basic operations of union, intersection, difference, and absolute null, absolute HPNNNs.
Theorems and examples are given for better understanding the situation.

Definition 13. Let Ω be the set of parameters and X be the key set. Let P (X) represent the
power set of X. Then, a Quadri-Partitioned Neutrosophic Set Structure (QPNSS) (F̃ ,Ω) over
X is a mapping

F̃ : Ω → P (X)

where F̃ is the function of QPNSS (F̃ ,Ω). Symbolically,

(F̃ ,Ω) =
[(

θ,
〈
x,AbTF̃ (θ)(x), ReTF̃ (θ)(x), ReFF̃ (θ)(x), AbFF̃ (θ)(x) : x ∈ X

〉)
: θ ∈ Ω

]
.

Here, AbTF̃ (θ)(x), ReTF̃ (θ)(x), ReFF̃ (θ)(x), and AbFF̃ (θ)(x) belong to the interval [0, 1]. These
functions are referred to as:

• AbTF̃ (θ)(x): Absolute true-membership function,

• ReTF̃ (θ)(x): Relative true-membership function,

• ReFF̃ (θ)(x): Relative false-membership function,

• AbFF̃ (θ)(x): Absolute false-membership function of F̃ (θ).


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 26 of 54

Since the supremum of each function is 1 and the infimum of each function is 0, the following
inequality holds automatically:

0 ≤ AbTF̃ (θ)(x) +ReTF̃ (θ)(x) +ReFF̃ (θ)(x) +AbFF̃ (θ)(x) ≤ 4.

Definition 14. Let (F̃ ,Ω) be a Quadri-Partitioned Neutrosophic Soft Set (QPNSS) over the
key set X. Then, the complement of (F̃ ,Ω) is denoted by (F̃ ,Ω)c and is defined as follows:

(F̃ ,Ω)c =
[(

θ,
〈
x,AbFF̃ (θ)(x), ReFF̃ (θ)(x), ReTF̃ (θ)(x), AbTF̃ (θ)(x) : x ∈ X

〉)
: θ ∈ Ω

]
Furthermore, the double complement satisfies:(

(F̃ ,Ω)c
)c

= (F̃ ,Ω).

Definition 15. Let (F̃ ,Ω) and (G̃,Ω) be two Quadri-Partitioned Neutrosophic Soft Sets (QP-
NSSs) over the key set X. Then, (F̃ ,Ω) ⊆ (G̃,Ω) if

AbTF̃ (θ)(x) ⪯ AbTG̃(θ)(x), ReTF̃ (θ)(x) ⪯ ReTG̃(θ)(x),

ReFF̃ (θ)(x) ⪰ ReFG̃(θ)(x), AbFF̃ (θ)(x) ⪰ AbFG̃(θ)(x),

for all θ ∈ Ω and for all x ∈ X.
If (F̃ ,Ω) ⊆ (G̃,Ω) and (F̃ ,Ω) ⊇ (G̃,Ω), then

(F̃ ,Ω) = (G̃,Ω).

Definition 16. Let (F̃ ,Ω) and (G̃,Ω) be two QPNSSs over key set X such that (F̃ ,Ω) ̸= (G̃,Ω).
Then their union is denoted by (F̃ ,Ω)∪̃(G̃,Ω) = (H̃,Ω) and is defined as:

(H̃,Ω) =
[(

θ, ⟨x,AbTH̃(θ)(x), ReTH̃(θ)(x), ReFH̃(θ)(x), AbFH̃(θ)(x) : x ∈ X⟩
)
: θ ∈ Ω

]
where,

AbTH̃(θ)(x) = max
[
AbTF̃ (θ)(x), AbTG̃(θ)(x)

]
,

ReTH̃(θ)(x) = max
[
ReTF̃ (θ)(x), ReTG̃(θ)(x)

]
,

ReFH̃(θ)(x) = min
[
ReFF̃ (θ)(x), ReFG̃(θ)(x)

]
,

AbFH̃(θ)(x) = min
[
AbFF̃ (θ)(x), AbFG̃(θ)(x)

]
.

Definition 17. Let (F̃ ,Ω), (G̃,Ω) be two QPNSSs over key set X such that (F̃ ,Ω) ̸= (G̃,Ω),
then their intersection is denoted by

(F̃ ,Ω) ∩ (G̃,Ω) = (H̃,Ω)

and is defined as

(H̃,Ω) =
[(

θ, ⟨x,AbTH̃(θ)(x), ReTH̃(θ)(x), ReFH̃(θ)(x), AbFH̃(θ)(x) : x ∈ X⟩
)
: θ ∈ Ω

]
where,

AbTH̃(θ)(x) = min
[
AbTF̃ (θ)(x), AbTG̃(θ)(x)

]
,

ReTH̃(θ)(x) = min
[
ReTF̃ (θ)(x), ReTG̃(θ)(x)

]
,

ReFH̃(θ)(x) = max
[
ReFF̃ (θ)(x), ReFG̃(θ)(x)

]
,

AbFH̃(θ)(x) = max
[
AbFF̃ (θ)(x), AbFG̃(θ)(x)

]
.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 27 of 54

Definition 18. Let (F̃ ,Ω), (G̃,Ω) be two QPNSSs over key set X such that (F̃ ,Ω) ̸= (G̃,Ω),
then their difference is given by

(H̃,Ω) = (F̃ ,Ω) \ (G̃,Ω)

and is defined as
(H̃,Ω) = (F̃ ,Ω) ∩ (G̃,Ω)c

such that

(H̃,Ω) =
[(

θ, ⟨x,AbTH̃(θ)(x), ReTH̃(θ)(x), ReFH̃(θ)(x), AbFH̃(θ)(x) : x ∈ X⟩
)
: θ ∈ Ω

]
where,

AbTH̃(θ)(x) = min
[
AbTF̃ (θ)(x), AbTG̃(θ)(x)

]
,

ReTH̃(θ)(x) = min
[
ReTF̃ (θ)(x), ReTG̃(θ)(x)

]
,

ReFH̃(θ)(x) = max
[
ReFF̃ (θ)(x), ReFG̃(θ)(x)

]
,

AbFH̃(θ)(x) = max
[
AbFF̃ (θ)(x), AbFG̃(θ)(x)

]
.

Definition 19. Let {(F̃i,Ω) : i ∈ I} be a family of QPNSSs over the key set X. Then,⋃
i∈I

(F̃i,Ω) ∩
⋂
i∈I

(F̃i,Ω)

is given by[
θ,

(
x, sup

i∈I
AbTF̃i(θ)(x)

, sup
i∈I

ReTF̃i(θ)(x)
, inf
i∈I

ReFF̃i(θ)(x)
, inf
i∈I

AbFF̃i(θ)(x)

)
: θ ∈ Ω, x ∈ X

]
.

Definition 20. A quadripartitioned neutrosophic soft set (F̃ ,Ω) over key set X is said to be a
null QPNSS if

AbTF̃ (θ)(x) = 0, ReTF̃ (θ)(x) = 0, ∀θ ∈ Ω,∀x ∈ X,

ReFF̃ (θ)(x) = 1, AbFF̃ (θ)(x) = 1, ∀θ ∈ Ω,∀x ∈ X.

It is signified as 0(X,Ω).

Definition 21. A quadripartitioned neutrosophic soft set (F̃ ,Ω) over key set X is an absolute
QPNSS if

AbTF̃ (θ)(x) = 1, ReTF̃ (θ)(x) = 1, ∀θ ∈ Ω,∀x ∈ X,

ReFF̃ (θ)(x) = 0, AbFF̃ (θ)(x) = 0, ∀θ ∈ Ω,∀x ∈ X.

Clearly,
0(X,Ω)c = 1(X,Ω), 1(X,Ω)c = 0(X,Ω).

Definition 22. The family of all quadripartitioned neutrosophic soft sets over X is designated
as QPNSS(X). Then, xθ⟨r1, r2, r3, r4⟩ is called a QPNS point for every point x ∈ X, θ ∈ Ω,
and is defined as follows:

xθ⟨r1, r2, r3, r4⟩θ′/(Y) =

{
⟨r1, r2, r3, r4⟩, if θ′ = θ and Y = x,

(0, 0, 0, 1), if θ′ ̸= θ or Y ̸= x.

Definition 23. Let (F̃ ,Ω) be a quadripartitioned neutrosophic soft set over key set X. Then,
xθ⟨r1, r2, r3, r4⟩ ∈ QPNSS(F̃ ,Ω) if

r1 ⪯ AbTF̃ (θ)(x), r2 ⪯ ReTF̃ (θ)(x), r3 ⪰ ReFF̃ (θ)(x), r4 ⪰ AbFF̃ (θ)(x).


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 28 of 54

Theorem 6. Let (F̃ ,Ω), (G̃,Ω), and (H̃,Ω) be quadripartitioned neutrosophic soft sets over
key set X. Then, the following properties hold:

(i) (F̃ ,Ω) ∪∼ [(G̃,Ω) ∪∼ (H̃,Ω)] = [(F̃ ,Ω) ∪∼ (G̃,Ω)] ∪∼ (H̃,Ω).

(ii) (F̃ ,Ω) ∩∼ [(G̃,Ω) ∩∼ (H̃,Ω)] = [(F̃ ,Ω) ∩∼ (G̃,Ω)] ∩∼ (H̃,Ω).

(iii) (F̃ ,Ω) ∪∼ [(G̃,Ω) ∩∼ (H̃,Ω)] = [(F̃ ,Ω) ∪∼ (G̃,Ω)] ∩∼ [(F̃ ,Ω) ∪∼ (H̃,Ω)].

(iv) (F̃ ,Ω) ∩∼ [(G̃,Ω) ∪∼ (H̃,Ω)] = [(F̃ ,Ω) ∩∼ (G̃,Ω)] ∪∼ [(F̃ ,Ω) ∩∼ (H̃,Ω)].

(v) (F̃ ,Ω) ∪∼ 0(X,Ω) = (F̃ ,Ω).

(vi) (F̃ ,Ω) ∩∼ 0(X,Ω) = 0(X,Ω).

(vii) (F̃ ,Ω) ∪∼ 1(X,Ω) = 1(X,Ω).

(viii) (F̃ ,Ω) ∩∼ 1(X,Ω) = (F̃ ,Ω).

Proof. Obvious.

Theorem 7. Let (F̃ ,Ω) and (G̃,Ω) be QPNSSs over key set X. Then, the following De Morgan’s
laws hold:

(i) [(F̃ ,Ω) ∪∼ (G̃,Ω)]c = (F̃ ,Ω)c ∩∼ (G̃,Ω)c.

(ii) [(F̃ ,Ω) ∩∼ (G̃,Ω)]c = (F̃ ,Ω)c ∪∼ (G̃,Ω)c.

Proof. Obvious.

Theorem 8. Let (F̃ ,Ω) and (G̃,Ω) be QPNSSs over key set X. Then, the following De Morgan’s
laws hold:

(i) [(F̃ ,Ω) ∨∼ (G̃,Ω)]c = (F̃ ,Ω)c ∧∼ (G̃,Ω)c.

(ii) [(F̃ ,Ω) ∩∼ (G̃,Ω)]c = (F̃ ,Ω)c ∪∼ (G̃,Ω)c.

Proof. 1. ∀(θ1, θ2) ∈ Ω× Ω,∀x ∈ X,

(F̃ ,Ω) ∨ (G̃,Ω) =

{(
x,max

[
AbTF̃ (θ)(x),AbTG(θ)(x)

]
,

max
[
ReTF̃ (θ)(x),ReTG(θ)(x)

]
,

min
[
ReFF̃ (θ)(x),ReFG(θ)(x)

]
,

min
[
AbFF̃ (θ)(x),AbFG(θ)(x)

])}
.

[(F̃ ,Ω) ∨ (G̃,Ω)]c =

{(
x,min

[
AbFF̃ (θ)(x),AbFG(θ)(x)

]
,

min
[
ReFF̃ (θ)(x),ReFG(θ)(x)

]
,

max
[
ReTF̃ (θ)(x),ReTG(θ)(x)

]
,


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 29 of 54

max
[
AbTF̃ (θ)(x),AbTG(θ)(x)

])}
.

Now,

(F̃ ,Ω)c =
{
⟨x,AbFF̃ (θ)(x),ReFF̃ (θ)(x),ReTF̃ (θ)(x),AbTF̃ (θ)(x)⟩

}
,

(G̃,Ω)c =
{
⟨x,AbFG̃(θ)(x),ReFG̃(θ)(x),ReTG̃(θ)(x),AbTG̃(θ)(x)⟩

}
.

Thus,

(F̃ ,Ω)c ∧ (G̃,Ω)c =

{(
x,min

[
AbFF̃ (θ)(x),AbFG(θ)(x)

]
,

min
[
ReFF̃ (θ)(x),ReFG(θ)(x)

]
,

max
[
ReTF̃ (θ)(x),ReTG(θ)(x)

]
,

max
[
AbTF̃ (θ)(x),AbTG(θ)(x)

])}
.

Therefore,

[(F̃ ,Ω) ∨ (G̃,Ω)]c = (F̃ ,Ω)c ∧ (G̃,Ω)c.

2. ∀(θ1, θ2) ∈ Ω× Ω,∀x ∈ X

(F̃ ,Ω) ∧ (G̃,Ω) =

{(
x,min

[
TF̃ (θ)(x),TG(θ)(x)

]
,

min
[
ReTF̃ (θ)(x),ReTG(θ)(x)

]
,

max
[
ReFF̃ (θ)(x),ReFG(θ)(x)

]
,

max
[
AbFF̃ (θ)(x),AbFG(θ)(x)

])}
.

[(F̃ ,Ω) ∧ (G̃,Ω)]c =

{(
x,max

[
AbFF̃ (θ)(x),AbFG(θ)(x)

]
,

max
[
ReFF̃ (θ)(x),ReFG(θ)(x)

]
,

min
[
ReTF̃ (θ)(x),ReTG(θ)(x)

]
,

min
[
AbTF̃ (θ)(x),AbTG(θ)(x)

])}
.

Now,

(F̃ ,Ω)c =
{
⟨x,AbFF̃ (θ)(x),ReFF̃ (θ)(x),ReTF̃ (θ)(x),AbTF̃ (θ)(x)⟩

}
,

(G̃,Ω)c =
{
⟨x,AbFG̃(θ)(x),ReFG̃(θ)(x),ReTG̃(θ)(x),AbTG̃(θ)(x)⟩

}
.

Thus,


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 30 of 54

(F̃ ,Ω)c ∨ (G̃,Ω)c =

{(
x,max

[
AbFF̃ (θ)(x),AbFG(θ)(x)

]
,

max
[
ReFF̃ (θ)(x),ReFG(θ)(x)

]
,

min
[
ReTF̃ (θ)(x),ReTG(θ)(x)

]
,

min
[
AbTF̃ (θ)(x),AbTG(θ)(x)

])}
.

Therefore,

[(F̃ ,Ω) ∧ (G̃,Ω)]c = (F̃ ,Ω)c ∨ (G̃,Ω)c.

8. A New Approach to Operations on Quadripartitioned Neutrosophic Soft
Topological Space

The notion of QPNSTS is presented in this section. The terms QPNS semi-open, QPNS pre-
open, and QPNS ∗b-open sets are defined. One of these intriguing QPNS generalized open sets,
referred to as the QPNS pre-open set, is selected, and certain fundamentals are then produced
based on this description. These consist of the QPNS closure, QPNS exterior, QPNS boundary,
and QPNS interior.

Definition 24. Let QNSS(X̃,Ω) be the family of all QPNSSs and τ ⊂ QPNSS(X̃,Ω), then
τ is a quadri-partitioned neutrosophic soft topology (QPNST) on X̃ if

(i) 0(⟨X⟩,Ω), 1(⟨X⟩,Ω) ∈ τ ,

(ii) The union of any number of QPNSSs in τ belongs to τ ,

(iii) The intersection of a finite number of QPNSSs in τ belongs to τ .

Then, (X̃, τ,Ω) is said to be a QPNSTS over X̃.

Definition 25. (X̃, τ,Ω) is a QPNSTS over X. A QPNSS (F̃ ,Ω) is a QPNS neighborhood of
a QPNS point xλ⟨r1,r2,r3,r4⟩ ∈ (F̃ ,Ω), if there is a QPNS open set (G̃,Ω) such that xλ⟨r1,r2,r3,r4⟩ ∈
(G̃,Ω).

Definition 26. Let (X, τ1,Ω) and (X, τ2,Ω) be two QPNSBTSs. Then, (X, τ1, τ2,Ω) is a QP-
NSBTS.

If (X, τ1, τ2,Ω) is a QPNSBTS, a QPNSS subset (F̃ ,Ω) is open in (X, τ1, τ2,Ω) if there
exists a QPNSS open set (G̃,Ω) belonging to τ1 and a QPNSS open set (H̃,Ω) belonging to τ2
such that

(F̃ ,Ω) = (G̃,Ω) ∪ (H̃,Ω).

Example 3. Let X = {x1, x2, x3} and Ω = {θ1, θ2}, τ1 = {0(X,Ω), 1(X,Ω), (F̃ ,Ω), (G̃,Ω)} and

τ2 = {0(X,Ω), 1(X,Ω), (H̃,Ω), (Ĩ ,Ω)},
where (F̃ ,Ω), (G̃,Ω), (H̃,Ω), and (Ĩ ,Ω) being QPNSSs are as follows:

(F̃ ,Ω) =

[
θ1 = ⟨x1, 2

10 ,
3
10 ,

7
10 ,

8
10⟩, ⟨x2,

4
10 ,

4
10 ,

6
10 ,

4
10⟩, ⟨x3,

2
10 ,

4
10 ,

6
10 ,

2
10⟩,

θ2 = ⟨x1, 3
10 ,

2
10 ,

6
10 ,

6
10⟩, ⟨x2,

1
10 ,

5
10 ,

6
10 ,

5
10⟩, ⟨x3,

4
10 ,

3
10 ,

6
10 ,

5
10⟩

]


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 31 of 54

(G̃,Ω) =

[
θ1 = ⟨x1, 4

10 ,
3
10 ,

6
10 ,

6
10⟩, ⟨x2,

4
10 ,

5
10 ,

6
10 ,

3
10⟩, ⟨x3,

3
10 ,

5
10 ,

6
10 ,

2
10⟩,

θ2 = ⟨x1, 3
10 ,

4
10 ,

6
10 ,

5
10⟩, ⟨x2,

2
10 ,

6
10 ,

6
10 ,

4
10⟩, ⟨x3,

4
10 ,

6
10 ,

6
10 ,

3
10⟩

]

(H̃,Ω) =

[
θ1 = ⟨x1, 6

10 ,
6
10 ,

6
10 ,

2
10⟩, ⟨x2,

6
10 ,

6
10 ,

6
10 ,

2
10⟩, ⟨x3,

4
10 ,

6
10 ,

6
10 ,

1
10⟩,

θ2 = ⟨x1, 5
10 ,

6
10 ,

6
10 ,

2
10⟩, ⟨x2,

6
10 ,

7
10 ,

6
10 ,

2
10⟩, ⟨x3,

5
10 ,

5
10 ,

6
10 ,

1
10⟩

]
(Ĩ ,Ω) =

[
θ1 = ⟨x1, 1

10 ,
2
10 ,

6
10 ,

7
10⟩, ⟨x2,

4
10 ,

4
10 ,

6
10 ,

3
10⟩, ⟨x3,

2
10 ,

4
10 ,

6
10 ,

2
10⟩,

θ2 = ⟨x1, 3
10 ,

2
10 ,

6
10 ,

5
10⟩, ⟨x2,

1
10 ,

5
10 ,

6
10 ,

5
10⟩, ⟨x3,

4
10 ,

3
10 ,

6
10 ,

5
10⟩

]
Theorem 9. Let (X, τ1, τ2,Ω) be a QPNSBTS. Then τ1 ∩ τ2 is a QPNSBTS on X.

Proof. The first and third requirements are clear, and we move forward as follows for the
second condition.

Let {(F̃i,Ω); i ∈ I} ∈ τ1 ∩ τ2, then (F̃i,Ω) ∈ τ1 and (F̃i,Ω) ∈ τ2.
As τ1 and τ2 are QPNSBTSs on X, then

⋃
i(F̃i,Ω) ∈ τ1 and

⋃
i(F̃i,Ω) ∈ τ2.

So,
⋃

i(F̃i,Ω) ∈ τ1 ∩ τ2.

Definition 27. Let (X, τ1, τ2,Ω) be a QPNSBTS over X, and let ( ˜̈Υ,Ω) be a QPNSS. Then,

(i) ( ˜̈Υ,Ω) is QPNS semi-open if

( ˜̈Υ,Ω) ⊆ NScl(NSint( ˜̈Υ,Ω))

(ii) ( ˜̈Υ,Ω) is QPNS pre-open (p-open) if

( ˜̈Υ,Ω) ⊆ NSint(NScl( ˜̈Υ,Ω))

(iii) ( ˜̈Υ,Ω) is QPNS ∗b-open if

( ˜̈Υ,Ω) ⊆ NScl(NSint( ˜̈Υ,Ω)) ∪NSint(NScl( ˜̈Υ,Ω))

(iv) ( ˜̈Υ,Ω) is QPNS ∗b-close if

( ˜̈Υ,Ω) ⊇ NScl(NSint( ˜̈Υ,Ω)) ∩NSint(NScl( ˜̈Υ,Ω))

Definition 28. Let (X, τ,Ω) be a quadripartitioned neutrosophic soft topological space over X
and (F̃ ,Ω) ∈ QPNSS(X,Ω). Then, the collection

τ(F̃ ,Ω) = {(F̃ ,Ω) ∩ (G̃,Ω) : (G̃,Ω) ∈ τ for i ∈ I}

is called a quadripartitioned neutrosophic soft subspace topology on (F̃ ,Ω), and (τ(F,Ω), τ(F̃ ,Ω),Ω)
is called a quadripartitioned neutrosophic soft topological subspace of (X, τ,Ω).

In order for the above definition to be consistent, we must prove that (τ(F,Ω), τ(F̃ ,Ω),Ω)
is actually a quadripartitioned neutrosophic soft topology for (F̃ ,Ω).

Theorem 10. Let (X, τ,Ω) be a quadripartitioned neutrosophic soft topological space over X
and (F̃ ,Ω) ∈ QPNSS(X,Ω). Then, the collection

τ(F̃ ,Ω) = {(F̃ ,Ω) ∩ (G̃,Ω) : (G̃,Ω) ∈ τ}

is a quadripartitioned neutrosophic soft topology on (F̃ ,Ω), and (X(F,Ω), τ(F̃ ,Ω),Ω) is a
quadripartitioned neutrosophic soft topological space of (X, τ,Ω).


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 32 of 54

Proof. (1) Since 0(F̃ ,Ω), 1(F̃ ,Ω) ∈ (X, τ,Ω), so by definition:

0(F̃ ,Ω) ∩ (F̃ ,Ω) = 0(F̃ ,Ω),

1(F̃ ,Ω) ∩ (F̃ ,Ω) = (F̃ ,Ω).

Thus, 0(F̃ ,Ω), (F̃ ,Ω) ∈ τ(F̃ ,Ω).
(2) Let {(H̃i,Ω) : i ∈ I} be a quadripartitioned neutrosophic soft sub-collection of τ(F̃ ,Ω).

Then (H̃i,Ω) ∈ τ(F̃ ,Ω) for all i ∈ I, so by definition, this implies that:

(H̃i,Ω) = (F̃ ,Ω) ∩ (G̃i,Ω) for some (G̃i,Ω) ∈ τ.

Taking the union, we obtain:⋃
i∈I

(H̃i,Ω) =
⋃
i∈I

(
(F̃ ,Ω) ∩ (G̃i,Ω)

)
= (F̃ ,Ω) ∩

(⋃
i∈I

(G̃i,Ω)

)
.

Since (G̃i,Ω) ∈ τ for all i ∈ I and τ is a quadripartitioned neutrosophic soft topological space,
we conclude that: ⋃

i∈I
(G̃i,Ω) ∈ τ.

Thus,

(F̃ ,Ω) ∩

(⋃
i∈I

(G̃i,Ω)

)
∈ τ(F̃ ,Ω),

implying that
⋃

i∈I(H̃i,Ω) ∈ τ(F̃ ,Ω).

(3) Now, let (H̃1,Ω), (H̃2,Ω), . . . , (H̃n,Ω) ∈ τ(F̃ ,Ω). Then, for all i = 1, 2, . . . , n, we have:

(H̃i,Ω) = (F̃ ,Ω) ∩ (G̃i,Ω) for some (G̃i,Ω) ∈ τ.

Taking the intersection, we obtain:

n⋂
i=1

(H̃i,Ω) =
n⋂

i=1

(
(F̃ ,Ω) ∩ (G̃i,Ω)

)
= (F̃ ,Ω) ∩

(
n⋂

i=1

(G̃i,Ω)

)
.

Since (G̃i,Ω) ∈ τ for all i = 1, 2, . . . , n and τ is a quadripartitioned neutrosophic soft topology,
we conclude that:

n⋂
i=1

(G̃i,Ω) ∈ τ.

Therefore,

(F̃ ,Ω) ∩

(
n⋂

i=1

(G̃i,Ω)

)
∈ τ(F̃ ,Ω),

implying that
⋂n

i=1(H̃i,Ω) ∈ τ(F̃ ,Ω).
Thus,

τ(F̃ ,Ω) = {(F̃ ,Ω) ∩ (G̃,Ω) : (G̃,Ω) ∈ τ}

is a quadripartitioned neutrosophic soft topology on (F̃ ,Ω).


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 33 of 54

Definition 29. Let (X, τ1, τ2,Ω) be a QPNSBTS over X and ( ˜̈Υ,Ω) be a QPNS. Then the

interior of ( ˜̈Υ,Ω), designated by ( ˜̈Υ,Ω)◦, is the union of all QPNS s-open sets of ( ˜̈Υ,Ω). Clearly,

( ˜̈Υ,Ω)◦ is the largest QPNS s-OS that is contained in ( ˜̈Υ,Ω).

Definition 30. Let (X, τ1, τ2,Ω) be a QPNSBTS, and let ( ˜̈Υ,Ω) be a QPNS. The frontier of

( ˜̈Υ,Ω), denoted as Fr(( ˜̈Υ,Ω)), is a QPNS point x1
λ
⟨r1,r2,r3,r4⟩. A point x1λ⟨r1,r2,r3,r4⟩ is in the

frontier of ( ˜̈Υ,Ω) if every QPNS s-open set containing x1λ⟨r1,r2,r3,r4⟩ contains at least one point

of ( ˜̈Υ,Ω) and at least one QPNS point of ( ˜̈Υ,Ω)c.

Definition 31. If (X, τ1, τ2,Ω) is a QPNSBTS and ( ˜̈Υ,Ω) is a QPNS, then the exterior of

( ˜̈Υ,Ω), denoted by Ext(( ˜̈Υ,Ω)), is a QPNS point x1
λ
⟨r1,r2,r3,r4⟩.

A QPNS point x1
λ
⟨r1,r2,r3,r4⟩ is in the exterior of ( ˜̈Υ,Ω) if and only if it is in the interior of

( ˜̈Υ,Ω)c, meaning there exists a QPNS s-open set (g̃,Ω) such that

x1
λ
⟨r1,r2,r3,r4⟩ ∈ (g̃,Ω) ⊆ ( ˜̈Υ,Ω)c.

Definition 32. If (X̃, τ1, τ2,Ω) and (⟨Ỹ ⟩,F1,F2,Ω) are QPNSBTSs, and ({, φ) : (X̃, τ1, τ2,Ω) →
(⟨Ỹ ⟩,F1,F2,Ω) is a QPNS mapping, then if the image ({, φ)( ˜̈Υ,Ω) of each QPNS s-closed set

( ˜̈Υ,Ω) over X̃ is a QPNS s-closed set in ⟨Ỹ ⟩, the mapping ({, φ) is said to be a QPNS s-closed
mapping.

Theorem 11. Let (X, τ1, τ2,Ω) be a QPNSBTS over X and ( ˜̈Υ,Ω) be a QPNS subset. Then,

( ˜̈Υ,Ω) is a QPNS s-open set if and only if ( ˜̈Υ,Ω) = ( ˜̈Υ,Ω)◦.

Proof. Let ( ˜̈Υ,Ω) be a HQPNS s-open set. Then, the largest QPNS s-open set that is

contained within ( ˜̈Υ,Ω) is equal to ( ˜̈Υ,Ω). Hence, ( ˜̈Υ,Ω) = ( ˜̈Υ,Ω)◦.

Conversely, it is known that ( ˜̈Υ,Ω)◦ is a QPNS s-open set, and if ( ˜̈Υ,Ω) = ( ˜̈Υ,Ω)◦, then

( ˜̈Υ,Ω) is a QPNS p-open set.

Theorem 12. Let (X, τ1, τ2,Ω) be a QPNSBTS over X, and let (F̃ ,Ω), (G̃,Ω) be QPNS subsets.
Then,

1.
[
(F̃ ,Ω)◦

]◦
= (F̃ ,Ω)◦,

2.
(
0(⟨X⟩,Ω)

)◦
= 0(⟨X⟩,Ω) and

(
1(⟨X⟩,Ω)

)◦
= 1(⟨X⟩,Ω),

3. (F̃ ,Ω) ⊆ (G̃,Ω) ⇒ (F̃ ,Ω)◦ ⊆ (G̃,Ω)◦,

4.
[
(F̃ ,Ω) ∩ (G̃,Ω)

]◦
= (F̃ ,Ω)◦ ∩ (G̃,Ω)◦,

5. (F̃ ,Ω)◦ ∪ (G̃,Ω)◦ ⊆
[
(F̃ ,Ω) ∪ (G̃,Ω)

]◦
.

Proof. Let (X, τ1, τ2,Ω) be a QPNSBTS over X, and let (F̃ ,Ω), (G̃,Ω) be QPNS subsets.
Then,

1. If (F̃ ,Ω)◦ = (G̃,Ω), then (G̃,Ω) ∈ τ if and only if (G̃,Ω) = (F̃ ,Ω)◦. Thus,
[
(F̃ ,Ω)◦

]◦
=

(F̃ ,Ω)◦.
2. Since 0(⟨X⟩,Ω) and 1(⟨X⟩,Ω) are always QPNS s-open sets, we have(

0(⟨X⟩,Ω)

)◦
= 0(⟨X⟩,Ω) and

(
1(⟨X⟩,Ω)

)◦
= 1(⟨X⟩,Ω).

3. It is known that (F̃ ,Ω)◦ ⊆ (F̃ ,Ω) ⊆ (G̃,Ω) and (G̃,Ω)◦ ⊆ (G̃,Ω). Since (G̃,Ω)◦ is the
largest QPNS s-open set contained in (G̃,Ω), it follows that

(F̃ ,Ω)◦ ⊆ (G̃,Ω)◦.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 34 of 54

4. Since (F̃ ,Ω) ∩ (G̃,Ω) ⊆ (F̃ ,Ω) and (F̃ ,Ω) ∩ (G̃,Ω) ⊆ (G̃,Ω), we have[
(F̃ ,Ω) ∩ (G̃,Ω)

]◦
⊆ (F̃ ,Ω)◦ and

[
(F̃ ,Ω) ∩ (G̃,Ω)

]◦
⊆ (G̃,Ω)◦.

Therefore, [
(F̃ ,Ω) ∩ (G̃,Ω)

]◦
⊆ (F̃ ,Ω)◦ ∩ (G̃,Ω)◦.

Conversely, since (F̃ ,Ω)◦ ⊆ (F̃ ,Ω) and (G̃,Ω)◦ ⊆ (G̃,Ω), we obtain

(F̃ ,Ω)◦ ∩ (G̃,Ω)◦ ⊆ (F̃ ,Ω) ∩ (G̃,Ω).

Since
[
(F̃ ,Ω) ∩ (G̃,Ω)

]◦
is the largest QPNS s-open set contained in (F̃ ,Ω)∩ (G̃,Ω), it follows

that
(F̃ ,Ω)◦ ∩ (G̃,Ω)◦ ⊆

[
(F̃ ,Ω) ∩ (G̃,Ω)

]◦
.

Thus, [
(F̃ ,Ω) ∩ (G̃,Ω)

]◦
= (F̃ ,Ω)◦ ∩ (G̃,Ω)◦.

5. Since (F̃ ,Ω) ⊆ (F̃ ,Ω) ∪ (G̃,Ω) and (G̃,Ω) ⊆ (F̃ ,Ω) ∪ (G̃,Ω), we have

(F̃ ,Ω)◦ ⊆
[
(F̃ ,Ω) ∪ (G̃,Ω)

]◦
and (G̃,Ω)◦ ⊆

[
(F̃ ,Ω) ∪ (G̃,Ω)

]◦
.

Thus,

(F̃ ,Ω)◦ ∪ (G̃,Ω)◦ ⊆
[
(F̃ ,Ω) ∪ (G̃,Ω)

]◦
.

Theorem 13. Let (X, τ1, τ2,Ω) be a QPNSBTS over X, and let (F̃ ,Ω) be a QPNS subset.
Then, (F̃ ,Ω) is a QPNS p-closer set if and only if

(F̃ ,Ω) = (F̃ ,Ω).

Proof.
Let (F̃ ,Ω) be a QPNS p-closer set. Then,

(F̃ ,Ω)d = (F̃ ,Ω)

which implies
(F̃ ,Ω)∪̃(F̃ ,Ω)d ∼= (F̃ ,Ω)

⇒ (F̃ ,Ω) ∼= (F̃ ,Ω)

Conversely, let

(F̃ ,Ω) ∼= (F̃ ,Ω)

this implies
(F̃ ,Ω)∪̃(F̃ ,Ω)d ∼= (F̃ ,Ω)

⇒ (F̃ ,Ω)d ∼= (F̃ ,Ω)

which implies that (F̃ ,Ω) is a QPNS p-closer set.

Theorem 14. Let (X, τ1, τ2,Ω) be a QPNSBTS over X, and let (F̃ ,Ω) and (G̃,Ω) be QPNS
subsets. Then:

(i) (F̃ ,Ω) = (F̃ ,Ω),


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 35 of 54

(ii) 0(⟨X̃,Ω⟩) = 0(⟨X̃,Ω⟩) and 1(⟨X̃,Ω⟩) = 1(⟨X̃,Ω⟩),

(iii) (F̃ ,Ω) ⊆ ⟨(G̃,Ω)⟩ ⇒ (F̃ ,Ω) ⊆ (G̃,Ω),

(iv) (F̃ ,Ω) ∪ (G̃,Ω) = (F̃ ,Ω) ∪ (G̃,Ω),

(v) (F̃ ,Ω) ∩ (G̃,Ω) ⊆ (F̃ ,Ω) ∩ (G̃,Ω).

Proof. Let (X, τ1, τ2,Ω) be a QPNSBTS over X, and let (F̃ ,Ω) and (G̃,Ω) be QPNS subsets.
Then:

(i) If (F̃ ,Ω) = (G̃,Ω), then (G̃,Ω) is a QPNS s-CS. Hence, if (G̃,Ω) and (G̃,Ω) are equal,
then:

(F̃ ,Ω) = (F̃ ,Ω)

(ii) Since 0(⟨X̃,Ω⟩) and 1(⟨X̃,Ω⟩) are always QPNS s-CS, by the above result (1), we get:

0(⟨X̃,Ω⟩) = 0(⟨X̃,Ω⟩) and 1(⟨X̃,Ω⟩) = 1(⟨X̃,Ω⟩).

(iii) Since (F̃ ,Ω) ⊆ (F̃ ,Ω) and (G̃,Ω) ⊆ (G̃,Ω), it follows that:

(F̃ ,Ω) ⊆ (G̃,Ω) ⊆ (G̃,Ω).

Since (F̃ ,Ω) is the smallest QPNS p-CS covering (F̃ ,Ω), we obtain:

(F̃ ,Ω) ⊆ (G̃,Ω).

(iv) Since (F̃ ,Ω) ⊆ (F̃ ,Ω) ∪ (G̃,Ω) and (G̃,Ω) ⊆ (F̃ ,Ω) ∪ (G̃,Ω), we have:

(F̃ ,Ω) ⊆ (F̃ ,Ω) ∪ (G̃,Ω)

and
(G̃,Ω) ⊆ (F̃ ,Ω) ∪ (G̃,Ω).

Thus,

(F̃ ,Ω) ∪ (G̃,Ω) ⊆ (F̃ ,Ω) ∪ (G̃,Ω).

Conversely, since (F̃ ,Ω) ⊆ (F̃ ,Ω) and (G̃,Ω) ⊆ (G̃,Ω), we have:

(F̃ ,Ω) ∪ (G̃,Ω) ⊆ (F̃ ,Ω) ∪ (G̃,Ω).

Since (F̃ ,Ω) ∪ (G̃,Ω) is the smallest QPNS p-closed set enclosing (F̃ ,Ω) ∪ (G̃,Ω), we
obtain:

(F̃ ,Ω) ∪ (G̃,Ω) ⊆ (F̃ ,Ω) ∪ (G̃,Ω).

Hence,

(F̃ ,Ω) ∪ (G̃,Ω) = (F̃ ,Ω) ∪ (G̃,Ω).

(v) Since ⟨0(⟨X̃,Ω⟩)⟩ ∩ (G̃,Ω) ⊆ (F̃ ,Ω) ∩ (G̃,Ω) and (F̃ ,Ω) ∩ (G̃,Ω) is the smallest QPNS p-

closed set enclosing (F̃ ,Ω) ∩ (G̃,Ω), we conclude:

(F̃ ,Ω) ∩ (G̃,Ω) ⊆ (F̃ ,Ω) ∩ (G̃,Ω).


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 36 of 54

9. Characterization of few more results in terms of basis concerning P-open
sets

Definition 33. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X, and let BNSS be a sub-family of τQPNSS.

BNSS is said to be a quadripartitioned neutrosophic soft base (or p-open base or basis) for
the quadripartitioned neutrosophic soft topology τQPNSS if for any non-empty quadripartitioned
neutrosophic soft set (G̃,Ω) ∈ τQPNSS, there exists B1 ⊆ BNSS such that:

(G̃,Ω) =
⋃

{B : B ∈ B1}.

In other words, BNSS is said to be a base for the quadripartitioned neutrosophic soft topology if
for every x(r1,r2,r3,r4) ∈ (G,Ω), there exists B ∈ B such that:

x(r1,r2,r3,r4) ∈ B ⊆ (G̃,Ω).

Moreover, if (G̃,Ω) ∈ τQPNSS, then there must exist some base element B ∈ BNSS satisfy-
ing:

x(r1,r2,r3,r4) ∈ B ⊆ (G̃,Ω).

Definition 34. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X, and let SNSS be a sub-family of τQPNSS.

SNSS is said to be a quadripartitioned neutrosophic soft sub-base (or p-open sub-base or sub-
basis) for the quadripartitioned neutrosophic soft topology τQPNSS on X if finite intersections of
the members of SNSS form a base for the quadripartitioned neutrosophic soft topology τQPNSS

on X.
That is, the union of the members of SNSS generates all the members of τQPNSS. The

elements of SNSS are referred to as sub-basic quadripartitioned neutrosophic soft p-open sets.
If, for any non-empty quadripartitioned neutrosophic soft set (G̃,Ω) ∈ τQPNSS, there exists

B1 ⊆ BNSS such that:
(G̃,Ω) =

⋃
{B : B ∈ B1},

then BNSS is said to be a base for the quadripartitioned neutrosophic soft topology. In other
words, BNSS is a base for the quadripartitioned neutrosophic soft topology if, for every point
x(r1,r2,r3,r4) ∈ (G̃,Ω), there exists B ∈ BNSS such that:

x(r1,r2,r3,r4) ∈ B ⊆ (G̃,Ω).

Definition 35. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X.

A family Bα of quadripartitioned neutrosophic soft p-open subsets of X is said to be a
quadripartitioned neutrosophic soft local base at xθ(r1,r2,r3,r4) in the neutrosophic soft topology on
X if:

(i) For any B ∈ Bxθ
(r1,r2,r3,r4)

, we have xθ(r1,r2,r3,r4) ∈ B.

(ii) For any (G̃,Ω) ∈ τQPNSS with yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ B ⊆ (G̃,Ω), there exists B ∈ Bxθ
(r1,r2,r3,r4)

such that:
yθ

′

(r′1,r
′
2,r

′
3,r

′
4)

∈ B ⊆ (G̃,Ω).

Definition 36. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X. The space X satisfies the first axiom of quadripartitioned neutrosophic soft countability
if X has a quadripartitioned neutrosophic soft countable local base at each xθ(r1,r2,r3,r4) ∈ X. A
quadripartitioned neutrosophic soft space X satisfying this condition is called a first quadripar-
titioned neutrosophic soft countable space.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 37 of 54

Definition 37. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X. The space X satisfies the second axiom of quadripartitioned neutrosophic soft count-
ability if there exists a quadripartitioned neutrosophic soft countable base for τQPNSS on X.

A quadripartitioned neutrosophic soft space X satisfying this condition is called a second
quadripartitioned neutrosophic soft countable space.

A second quadripartitioned neutrosophic soft countable space is also called a quadriparti-
tioned neutrosophic soft completely separable space.

Definition 38. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X. A property PQPNSS of X is said to be hereditary if the property is possessed by every
subspace of X.

For example, quadripartitioned neutrosophic soft first countability and quadripartitioned neu-
trosophic soft second countability are hereditary properties. However, quadripartitioned neutro-
sophic soft p-closed sets and quadripartitioned neutrosophic soft p-open sets are not hereditary
properties.

Definition 39. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X. This space is said to be quadripartitioned neutrosophic soft separable if and only if X
contains a quadripartitioned neutrosophic soft countable dense subset.

That is, there exists a quadripartitioned neutrosophic soft countable subset (k̃,Ω) of X such
that

(k̃,Ω) = X.

Theorem 15. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X, and let BQPNSS be a quadripartitioned neutrosophic soft basis for τQPNSS.

Then, the topology τQPNSS is given by the collection of all quadripartitioned neutrosophic
soft unions of elements of BQPNSS, that is,

τQPNSS =

{⋃
α∈I

Bα | Bα ∈ BQPNSS , I is an index set

}
.

Proof. This is easily seen from the definition of quadripartitioned neutrosophic soft basis.

Theorem 16. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X. A sub-collection BQPNSS of τQPNSS is a basis for τQPNSS if and only if, for each
quadripartitioned neutrosophic soft p-open set (G̃,Ω) ∈ τQPNSS and for each quadripartitioned
neutrosophic soft point xθ(r1,r2,r3,r4) ∈ (G̃,Ω), there exists a basis element Bxθ

(r1,r2,r3,r4)
∈ BQPNSS

such that
xθ(r1,r2,r3,r4) ∈ Bxθ

(r1,r2,r3,r4)
⊆ (G̃,Ω).

Proof. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space,
and let BQPNSS be a collection of quadripartitioned neutrosophic soft p-open sets. Sup-
pose that BQPNSS forms a basis for the quadripartitioned neutrosophic soft topology τQPNSS .
Then, for every quadripartitioned neutrosophic soft p-open set (G̃,Ω) ∈ τQPNSS and for ev-
ery quadripartitioned neutrosophic soft point xθ(r1,r2,r3,r4) ∈ (G̃,Ω), there exists a basis element

Bxθ
(r1,r2,r3,r4)

∈ BQPNSS such that

xθ(r1,r2,r3,r4) ∈ Bxθ
(r1,r2,r3,r4)

⊆ (G̃,Ω).

Given that (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space and
BQPNSS is a collection of quadripartitioned neutrosophic soft p-open sets. Let BQPNSS be
a base for a quadripartitioned neutrosophic soft topology τQPNSS , then by definition every


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 38 of 54

quadripartitioned neutrosophic soft p-open set (G̃,Ω) is the union of some members of BQPNSS ,
i.e.,

(G̃,Ω) =
⋃
i∈I

Bi,

whereBi ∈ BQPNSS for all i ∈ I. Let xθ(r1,r2,r3,r4) be an arbitrary quadripartitioned neutrosophic

soft point of (G̃,Ω). We are to prove that there exists a quadripartitioned neutrosophic soft
basis element Bxθ

(r1,r2,r3,r4)
containing xθ(r1,r2,r3,r4) such that

Bxθ
(r1,r2,r3,r4)

⊆ (G̃,Ω).

Since xθ(r1,r2,r3,r4) ∈ (G̃,Ω) but (G̃,Ω) =
⋃

i∈I Bi, it follows that

xθ(r1,r2,r3,r4) ∈
⋃
i∈I

Bi,

which implies that xθ(r1,r2,r3,r4) ∈ Bi for some i ∈ I.

Let xθ(r1,r2,r3,r4) ∈ Bi for i = xθ(r1,r2,r3,r4), then

xθ(r1,r2,r3,r4) ∈ Bxθ
(r1,r2,r3,r4)

⊆
⋃

Bi, i ∈ I.

Since Bi ⊆
⋃
Bi for all i, it implies that

xθ(r1,r2,r3,r4) ∈ Bxθ
(r1,r2,r3,r4)

⊆
⋃

Bi, i ∈ I.

This further implies that
xθ(r1,r2,r3,r4) ∈ Bxθ

(r1,r2,r3,r4)
⊆ (G̃,Ω),

where xθ(r1,r2,r3,r4) ∈ BQPNSS .

Conversely, suppose for each quadripartitioned neutrosophic soft point xθ(r1,r2,r3,r4) of a

quadripartitioned neutrosophic soft p-open set (G̃,Ω), there exists a quadripartitioned neu-
trosophic soft set xθ(r1,r2,r3,r4) ∈ BQPNSS such that

xθ(r1,r2,r3,r4) ∈ Bxθ
(r1,r2,r3,r4)

⊆ (G̃,Ω).

We are to prove that BQPNSS is a quadripartitioned neutrosophic soft basis for the quadriparti-
tioned neutrosophic soft topology τQPNSS . For this, we will prove that every quadripartitioned
neutrosophic soft p-open set (G̃,Ω) can be written as a union of some members of BQPNSS .

Since xθ(r1,r2,r3,r4) ∈ Bxe
(r1,r2,r3,r4)

⊆ (G̃,Ω), it follows that

xθ(r1,r2,r3,r4) ∈ Bxθ
(r1,r2,r3,r4)

,

and
Bxθ

(r1,r2,r3,r4)
⊆ (G̃,Ω).

Thus, we have⋃
{xθ(r1,r2,r3,r4)} ⊆

⋃
xθ
(r1,r2,r3,r4)

∈(G̃,Ω)

Bxθ
(r1,r2,r3,r4)

⊆
⋃

xθ
(r1,r2,r3,r4)

∈(G̃,Ω)

(G̃,Ω).

This implies that

(G̃,Ω) ⊆
⋃

xθ
(r1,r2,r3,r4)

∈(G̃,Ω)

Bxθ
(r1,r2,r3,r4)

,


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 39 of 54

and ⋃
xθ
(r1,r2,r3,r4)

∈(G̃,Ω)

Bxθ
(r1,r2,r3,r4)

⊆ (G̃,Ω).

Therefore,

(G̃,Ω) =
⋃

xθ
(r1,r2,r3,r4)

∈(G̃,Ω)

Bxθ
(r1,r2,r3,r4)

.

Since each Bxθ
(r1,r2,r3,r4)

∈ BQPNSS , it follows that (G̃,Ω) is the union of some members

of BQPNSS . Since (G̃,Ω) is an arbitrary quadripartitioned neutrosophic soft p-open set, we
conclude that every quadripartitioned neutrosophic soft p-open set is the union of some members
of BQPNSS .

Thus, BQPNSS is a quadripartitioned neutrosophic soft basis for the quadripartitioned neu-
trosophic soft topology τQPNSS .

Theorem 17. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X. A sub-collection BQPNSS of τQPNSS is a base for τQPNSS if and only if:

(i) Every quadripartitioned neutrosophic soft point of X is in some B ∈ BQPNSS.

(ii) For B1, B2 ∈ BQPNSS and xθ(r1,r2,r3,r4) ∈ B1 ∩ B2, there is a B ∈ BQPNSS such that

xθ(r1,r2,r3,r4) ∈ B ⊆ B1 ∩B2.

Proof. Let BQPNSS be a base for τQPNSS .
1. Let xθ(r1,r2,r3,r4) be an arbitrary quadripartitioned neutrosophic soft point of X.
2. Since X is a quadripartitioned neutrosophic soft p-open set, there exists

Bxθ
(r1,r2,r3,r4)

∈ BQPNSS

such that
xθ(r1,r2,r3,r4) ∈ Bxθ

(r1,r2,r3,r4)
⊆ X.

3. This implies that: ⋃
{xθ(r1,r2,r3,r4) | x

θ
(r1,r2,r3,r4)

∈ X} ⊆ X.

4. Furthermore, since Bxθ
(r1,r2,r3,r4)

is a base element, we get:⋃
x(α,β,γ)∈X

Bxθ
(r1,r2,r3,r4)

⊆ X.

5. Therefore,

X =
⋃

xθ
(r1,r2,r3,r4)

∈X

Bxθ
(r1,r2,r3,r4)

.

This shows that each quadripartitioned neutrosophic soft point of X belongs to some B ∈
BQPNSS . 2. Let B1, B2 ∈ BQPNSS and xθ(r1,r2,r3,r4) ∈ B1 ∩B2.

Since B1 and B2 are quadripartitioned neutrosophic p-open sets, it follows that B1 ∩ B2 is
also a quadripartitioned neutrosophic p-open set. Therefore, there exists B ∈ BQPNSS such
that

xθ(r1,r2,r3,r4) ∈ B ⊆ B1 ∩B2.

Conversely, suppose that (1) and (2) hold. We now prove that a sub-collection BQPNSS of
τQPNSS forms a base for τQPNSS .


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 40 of 54

To establish this, we show that the sub-collection BQPNSS of τQPNSS satisfies the three
conditions of quadripartitioned neutrosophic soft topology. We proceed as follows:

- The quadripartitioned neutrosophic soft null set 0(X,Ω), being the union of a quadri-
partitioned neutrosophic soft null collection of quadripartitioned neutrosophic soft subsets in
BQPNSS , belongs to τQPNSS . - Since X is quadripartitioned neutrosophic soft p-open, for any

xθ(r1,r2,r3,r4) ∈ X

the condition (1) provides a
Bxθ

(r1,r2,r3,r4)
∈ BQPNSS

such that
xθ(r1,r2,r3,r4) ∈ Bxθ

(r1,r2,r3,r4)
⊆ X.

- This implies: ⋃
xθ
(r1,r2,r3,r4)

∈X

Bxθ
(r1,r2,r3,r4)

⊆ X.

- Since X ⊆
⋃

xθ
(r1,r2,r3,r4)

∈X Bxθ
(r1,r2,r3,r4)

⊆ X, it follows that:

X =
⋃

xθ
(r1,r2,r3,r4)

∈X

Bxθ
(r1,r2,r3,r4)

.

This implies that

X =
⋃

xθ
(r1,r2,r3,r4)

∈X

Bxe
(r1,r2,r3,r4)

.

This shows that X, being the union of members of BQPNSS , is in τQPNSS .
Next, we proceed with the second condition as follows: The union of any number of members

of τQPNSS , being the union of members of BQPNSS , is in τQPNSS .
Now, we proceed with the third condition as follows: Let

(G ,Ω)1, (G ,Ω)2, (G ,Ω)3, (G ,Ω)4 ∈ τQPNSS .

By the definition of τQPNSS , we have

(G ,Ω)1 =
⋃

Br1 , (G ,Ω)2 =
⋃

Br2 , (G ,Ω)3 =
⋃

Br3 , (G ,Ω)4 =
⋃

Br4

for some α, γ ranging over a sub-collection of BQPNSS .
Therefore,

(G ,Ω)1 ∩ (G ,Ω)2 ∩ (G ,Ω)3 ∩ (G ,Ω)4 =
⋃

(Br1 ∩Br2 ∩Br3 ∩Br4).

(i)
By condition (2), for any xθ(r1,r2,r3,r4) ∈ B1 ∩B2, there exists

Bxe
(r1,r2,r3,r4)

∈ BQPNSS

such that
Bxθ

(r1,r2,r3,r4)
⊆ Br1 ∩Br2 ∩Br3 ∩Br4 .

This implies that ⋃
{xθ(r1,r2,r3,r4)} ⊆

⋃
xθ
(r1,r2,r3,r4)

∈Bα∩Bγ

Bxθ
(r1,r2,r3,r4)

.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 41 of 54

Since ⋃
xθ
(r1,r2,r3,r4)

∈(Br1∩Br2∩Br3∩Br4 )

Bxθ
(r1,r2,r3,r4)

⊆ Br1 ∩Br2 ∩Br3 ∩Br4 ,

it follows that

Br1 ∩Br2 ∩Br3 ∩Br4 ⊆
⋃

xθ
(r1,r2,r3,r4)

∈Bα∩Bγ

Bxθ
(r1,r2,r3,r4)

⊆ Br1 ∩Br2 ∩Br3 ∩Br4 .

Thus, we conclude that

Br1 ∩Br2 ∩Br3 ∩Br4 =
⋃

xθ
(r1,r2,r3,r4)

∈Bα∩Bγ

Bxθ
(r1,r2,r3,r4)

.

Substituting this value in equation (i), we obtain:

(G ,Ω)1 ∩ (G ,Ω)2 ∩ (G ,Ω)3 ∩ (G ,Ω)4 =
⋃

(Br1 ∩Br2 ∩Br3 ∩Br4).

Since
Br1 ∩Br2 ∩Br3 ∩Br4 =

⋃
xθ
(r1,r2,r3,r4)

∈Br1∩Br2∩Br3∩Br4

Bxθ
(r1,r2,r3,r4)

we conclude that

(G ,Ω)1 ∩ (G ,Ω)2 ∩ (G ,Ω)3 ∩ (G ,Ω)4 =
⋃

xθ
(r1,r2,r3,r4)

∈Br1∩Br2∩Br3∩Br4

Bxθ
(r1,r2,r3,r4)

.

This shows that
(G ,Ω)1 ∩ (G ,Ω)2 ∩ (G ,Ω)3 ∩ (G ,Ω)4

is the union of members of BQPNSS , which implies it is in τQPNSS .
Similarly, we can prove that the intersection of any finite number of members of BQPNSS is

in τQPNSS . Since all the conditions of quadripartitioned neutrosophic soft topology are satisfied,
we conclude that τQPNSS is a quadripartitioned neutrosophic soft topology on X.

Consequently, BQPNSS is a quadripartitioned neutrosophic soft base for τQPNSS .

Theorem 18. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X. A quadripartitioned neutrosophic soft point xθ(r1,r2,r3,r4) in a quadripartitioned neutro-

sophic soft topological space is a quadripartitioned neutrosophic soft limit point of (F̃ ,Ω) ⊆ X
if and only if every member of any quadripartitioned neutrosophic soft local base Bxθ

(r1,r2,r3,r4)

at xθ(r1,r2,r3,r4) contains a point of (F̃ ,Ω) different from xθ(r1,r2,r3,r4).

Proof.
Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space over X, and

let (F̃ ,Ω) ⊆ X. Let xθ(r1,r2,r3,r4) ∈ X be a quadripartitioned neutrosophic soft limit point of

(F̃ ,Ω). Let Bxθ
(r1,r2,r3,r4)

be a local base at xθ(r1,r2,r3,r4). We need to prove that:

(B − xθ(r1,r2,r3,r4)) ∩ (F̃ ,Ω) ̸= ∅ ∀B ∈ Bxθ
(r1,r2,r3,r4)

Since xθ(r1,r2,r3,r4) is a quadripartitioned neutrosophic soft limit point of (F̃ ,Ω), we have:

((G̃,Ω)− xθ(r1,r2,r3,r4)) ∩ (F̃ ,Ω) ̸= ∅ ∀(G̃,Ω) ∈ τQPNSS .

By the definition of a quadripartitioned neutrosophic soft local base, we know that:


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 42 of 54

(G̃,Ω) ∈ Bxθ
(r1,r2,r3,r4)

=⇒ (G̃,Ω) ∈ τQPNSS .

Thus, since xθ(r1,r2,r3,r4) ∈ (G̃,Ω), it follows that:

((G̃,Ω)− xθ(r1,r2,r3,r4)) ∩ (F̃ ,Ω) ̸= ∅.

Since Bxθ
(r1,r2,r3,r4)

consists of members of τQPNSS , we conclude:

(B − xθ(r1,r2,r3,r4)) ∩ (F̃ ,Ω) ̸= ∅ ∀B ∈ Bxθ
(r1,r2,r3,r4)

.

Conversely, suppose that for some quadripartitioned neutrosophic soft topology τQPNSS on
X, we have:

(B − xθ(r1,r2,r3,r4)) ∩ (F̃ ,Ω) ̸= ∅ ∀B ∈ Bxθ
(r1,r2,r3,r4)

.

Let (G̃,Ω) ∈ τQPNSS be an arbitrary set such that xθ(r1,r2,r3,r4) ∈ (G̃,Ω). By the definition
of a quadripartitioned neutrosophic soft local base, there exists B ∈ Bxθ

(r1,r2,r3,r4)
such that:

B ⊆ (G̃,Ω).

Consequently, we have:

((G̃,Ω)− xθ(r1,r2,r3,r4)) ∩ (F̃ ,Ω) ̸= ∅.

Thus, xθ(r1,r2,r3,r4) is a quadripartitioned neutrosophic soft limit point of (F̃ ,Ω), completing
the proof.

Theorem 19. Let τQPNSS
1 and τQPNSS

2 be two quadripartitioned neutrosophic soft topologies

over X generated by the quadripartitioned neutrosophic soft bases BQPNSS
1 and BQPNSS

2 , re-

spectively. Then τQPNSS
1 ⊆ τQPNSS

2 if and only if for each xθ(r1,r2,r3,r4) ∈ QPNSS(X,Ω) and

for each (B̃1,Ω) ∈ BQPNSS
1 containing xθ(r1,r2,r3,r4), there exists (B̃2,Ω) ∈ BQPNSS

2 such that

xθ(r1,r2,r3,r4) ∈ (B̃2,Ω) ⊆ (B̃1,Ω).

Proof. Suppose that τQPNSS
1 ⊆ τQPNSS

2 and xθ(r1,r2,r3,r4) ∈ QPNSS(X,Ω), (B̃1,Ω) ∈
BQPNSS

1 such that xθ(r1,r2,r3,r4) ∈ (B̃1,Ω). Since BQPNSS
1 is a quadripartitioned neutrosophic

soft basis for quadripartitioned neutrosophic soft topology τQPNSS
1 over X, then BQPNSS

1 ⊆
τQPNSS
1 .

Thus, xθ(r1,r2,r3,r4) ∈ (B̃1,Ω) ∈ BQPNSS
2 ⊆ τQPNSS

1 , i.e., xθ(r1,r2,r3,r4) ∈ (B̃1,Ω) ∈ τQPNSS
2 .

Since BQPNSS
2 is a quadripartitioned neutrosophic soft basis for τQPNSS

2 , there exists (B̃2,Ω) ∈
BQPNSS

2 such that xθ(r1,r2,r3,r4) ∈ (B̃2,Ω) ⊆ (B̃1,Ω).

Conversely, assume that the hypothesis holds. Let (F̃ ,Ω) ∈ τQPNSS
1 . Since BQPNSS

1 is
a quadripartitioned neutrosophic soft basis for quadripartitioned neutrosophic soft topology
τQPNSS
1 , then for xθ(r1,r2,r3,r4) ∈ (F̃ ,Ω), there exists (B̃1,Ω) ∈ BQPNSS

1 such that xθ(r1,r2,r3,r4) ∈
(B̃1,Ω) ⊆ (F̃ ,Ω).

By hypothesis, there exists (B̃2,Ω) ∈ BQPNSS
2 such that (B̃2,Ω) ⊆ (B̃1,Ω) ⊆ (F̃ ,Ω). This

implies (B̃2,Ω) ⊆ (F̃ ,Ω), which means (F̃ ,Ω) ∈ τQPNSS
2 . Thus, we conclude that τQPNSS

1 ⊆
τQPNSS
2 .

Theorem 20. Let (X, τQPNSS , E) be a quadripartitioned neutrosophic soft topological space
over (F̃ ,Ω), (K̃,Ω) ∈ QPNSS(X,Ω).


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 43 of 54

(i) If BQPNSS is a quadripartitioned neutrosophic soft base for τQPNSS, then

B(F̃ , E)QPNSS = {(B̃,Ω) ∩ (F̃ ,Ω) : (B̃,Ω) ∈ BQPNSS}

is a quadripartitioned neutrosophic soft base for the quadripartitioned neutrosophic soft
sub-topology τQPNSS

(F̃ ,E)
.

(ii) If (G̃,Ω) is a quadripartitioned neutrosophic soft p-closed set in τQPNSS

(F̃ ,E)
and (F̃ ,Ω) is a

quadripartitioned neutrosophic soft p-closed set in τQPNSS

(F̃ ,E)
, then (F̃ ,Ω) is a quadriparti-

tioned neutrosophic soft p-closed set in τQPNSS

(F̃ ,E)
.

(iii) Let (F̃ ,Ω) ⊆ (K̃,Ω). If (G̃,Ω) ∈ τQPNSS, then (G̃,Ω) ∩ (F̃ ,Ω) is the quadripartitioned
neutrosophic soft closure in (X(F̃ ,E), τ

QPNSS

(F̃ ,E)
,Ω).

Proof. 1. Since BQPNSS is a quadripartitioned neutrosophic soft base for τQPNSS , for
arbitrary (Ũ ,Ω) ∈ τQPNSS , we have:

(Ũ ,Ω) =
⋃

(B̃,Ω)∈BQPNSS

(B̃,Ω).

Thus,

(Ũ ,Ω) ∩ (F̃ ,Ω) =
⋃

(B̃,Ω)∈BQPNSS

(B̃,Ω) ∩ (F̃ ,Ω).

Since (Ũ ,Ω) ∩ (F̃ ,Ω) ∈ τQPNSS

(F̃ ,E)
, it follows that⋃

(B̃,Ω)∈BQPNSS

((B̃,Ω) ∩ (F̃ ,Ω)) ∈ τQPNSS

(F̃ ,E)
.

Since an arbitrary member of τQPNSS

(F̃ ,E)
can be expressed as the union of members of BQPNSS

(F̃ ,E)
,

it follows that BQPNSS

(F̃ ,E)
is a quadripartitioned neutrosophic soft base for τQPNSS

(F̃ ,E)
.

2. We first show that if (G̃,Ω) is a quadripartitioned neutrosophic soft p-closed set in
τQPNSS

(F̃ ,E)
then there exists a p-closed set (Ṽ ,Ω) ⊆ (K̃,Ω) i.e., (Ṽ ,Ω) /∈ τQPNSS such that

(G̃,Ω) = (Ṽ ,Ω) ∩ (F̃ ,Ω).

Let (G̃,Ω) be p-closed in τQPNSS

(F̃ ,E)
. Then (G̃i,Ω)

c is a quadripartitioned neutrosophic soft

p-open set in τQPNSS

(F̃ ,E)
, i.e., (G̃i,Ω)

c can be written as

(G̃′,Ω)c = (Ũ ,Ω) ∩ (F̃ ,Ω),

where (Ũ ,Ω) ∈ τQPNSS .
Thus,

((G̃′,Ω)c)c = (F̃ ,Ω) ∩ ((Ũ ,Ω) ∩ (F̃ ,Ω))c = (Ũ ′,Ω)c ∩ (F̃ ,Ω).

Here, (Ũ ,Ω)c is p-closed in τQPNSS . So, it acts as (Ṽ ,Ω) ⊆ (K̃,Ω). Conversely, suppose
that

(G̃,Ω) = (Ṽ ,Ω) ∩ (F̃ ,Ω),

where (F̃ ,Ω) ⊆ (K̃,Ω) and (Ṽ ,Ω) is p-closed in τQPNSS

(K̃,E)
. Clearly, (Ṽ ,Ω)c ∈ τQPNSS , so that

(Ṽ ,Ω)c ∩ (F̃ ,Ω) ∈ τQPNSS(K̃, E).


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 44 of 54

Now,
(Ṽ ,Ω)c ∩ (F̃ ,Ω) = ((K̃,Ω) \ (Ṽ ,Ω)) ∩ (F̃ ,Ω)

= ((K̃,Ω) ∩ (F̃ ,Ω)) \ ((Ṽ ,Ω) ∩ (F̃ ,Ω)) = (F̃ ,Ω) \ (G̃,Ω).

This implies that (F̃ ,Ω) \ (G̃,Ω) is a quadripartitioned neutrosophic soft set in (F̃ ,Ω), i.e.,
(G̃,Ω) is a neutrosophic soft p-closed set in τQPNSS

(K̃,E)
.

⋂
{(G̃i,Ω) | (G̃i,Ω) is closed and (G̃i,Ω) ⊇ (G̃,Ω)}

is the quadripartitioned neutrosophic soft closure of (G̃,Ω) and so (G̃,Ω) is a quadripartitioned
neutrosophic soft p-closed set. Now,

(G̃,Ω) ∩ (F̃ ,Ω) =
⋂

{(G̃i,Ω) | (G̃i,Ω) is closed and (G̃i,Ω) ⊇ (G̃,Ω)} ∩ (F̃ ,Ω)

=
⋂

((G̃i,Ω) ∩ (F̃ ,Ω)).

Since each (G̃i,Ω) is p-closed, then each (G̃i,Ω) ∩ (F̃ ,Ω) is p-closed in τQPNSS

(F̃ ,E)
. Now,

(G,Ω) ⊆ (G̃i,Ω) and (G,Ω) ⊆ (F̃ ,Ω).

So,

(G̃,Ω) ∩ (F̃ ,Ω) ⊆ (G̃i,Ω) ∩ (F̃ ,Ω) ⇒ (G̃,Ω) ⊆ (G̃i,Ω) ∩ (F̃ ,Ω).

Therefore,

(G̃,Ω)∩(F̃ ,Ω) =
⋂

{(G̃i,Ω)∩(F̃ ,Ω) | (G̃i,Ω)∩(F̃ ,Ω) is p-closed and (G̃i,Ω)∩(F̃ ,Ω) ⊇ (G̃i,Ω)}.

Thus, (G̃,Ω) ∩ (F̃ ,Ω) is a quadripartitioned neutrosophic soft closure of (G̃,Ω) in τQPNSS

(F̃ ,E)
.

Theorem 21. Let (X(F̃ , E), τQPNSS ,Ω) be a quadripartitioned neutrosophic soft subspace of

a quadripartitioned neutrosophic soft topological space (X, τQPNSS , E) over X. If (F̃ ,Ω) is
a quadripartitioned neutrosophic soft p-open set in (X, τQPNSS , E) if and only if (F̃1,Ω) is a
quadripartitioned neutrosophic soft p-open set in (X, τQPNSS ,Ω).

Proof. Suppose that (F̃ ,Ω) is a quadripartitioned neutrosophic soft p-open set in (X, τQPNSS , E)
such that a quadripartitioned neutrosophic soft subset (F̃1,Ω) of (F̃ ,Ω) is a p-open set in
(X(F̃ ,E), τ

QPNSS

(F̃ ,E)
,Ω).

Then (F̃1,Ω) ∈ τQPNSS

(F̃ ,E)
and so (F̃1,Ω) = (Ũ ,Ω) ∩ (F̃ ,Ω) for some (Ũ ,Ω) ∈ τQPNSS .

But (F̃1,Ω) is a quadripartitioned neutrosophic soft p-open set in (X, τQPNSS , E) as (Ũ ,Ω)
and (F̃ ,Ω) are both quadripartitioned neutrosophic soft p-open sets in (X, τQPNSS , E).

Conversely, assume that (F̃1,Ω) is a quadripartitioned neutrosophic soft p-open set in
(X, τQPNSS , E) when (F̃ ,Ω) is a quadripartitioned neutrosophic soft p-open set in (X, τQPNSS , E)
and (F̃1,Ω) ⊆ (F̃ ,Ω).

Then (F̃1,Ω) ∈ τQPNSS . But (F̃1,Ω)∩(F̃ ,Ω) = (F̃1,Ω), and so (F̃1,Ω) is a quadripartitioned
neutrosophic soft set in (X, τQPNSS , E).

Therefore, the first part is proved.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 45 of 54

Theorem 22. Let (X(K̃,E), τ
QPNSS ,Ω) be a quadripartitioned neutrosophic soft subspace of a

quadripartitioned neutrosophic soft topological space (X, τQPNSS , E) over X.
If (K̃,Ω) is a quadripartitioned neutrosophic soft p-closed set in (X, τQPNSS , E), then a

quadripartitioned neutrosophic soft set (K̃1,Ω) ⊆ (K̃,Ω) is a quadripartitioned neutrosophic
soft p-closed set in (X(K̃,E), τ

QPNSS

(K̃,E)
,Ω)

if and only if (K̃1,Ω) is a quadripartitioned neutrosophic soft p-closed set in (X, τQPNSS , E).

Proof. Suppose that (K̃,Ω) is a quadripartitioned neutrosophic soft p-closed set in (X, τQPNSS , E)
such that a quadripartitioned neutrosophic soft subset (K̃1,Ω) or (K̃,Ω) is a quadripartitioned
neutrosophic soft p-closed set in (X(K̃,E), τ

QPNSS

(K̃,E)
,Ω).

Since (K̃1,Ω) is p-closed in (X(K̃,E),τ
QPNSS

(K̃,E)
,Ω), it follows that

(K̃1,Ω) = (Ṽ ,Ω) ∩ (K̃,Ω)

for some (Ṽ ,Ω), which is a quadripartitioned neutrosophic soft p-closed set in (X, τQPNSS , E).
But (K̃1,Ω) is also a quadripartitioned neutrosophic soft p-closed set in (X, τQPNSS , E) be-

cause both (Ṽ ,Ω) and (K̃,Ω) are quadripartitioned neutrosophic soft p-closed sets in (X, τQPNSS , E).
Conversely, assume that (F̃1,Ω) is a quadripartitioned neutrosophic soft p-open set in

(X, τQPNSS , E), where (K̃,Ω) is a quadripartitioned neutrosophic soft p-closed set in (X, τQPNSS , E)
and (K̃1,Ω) ⊆ (K̃,Ω).

Then
(K̃1,Ω) ∩ (K̃,Ω) = (K̃1,Ω)

which implies that (K̃1,Ω) is a quadripartitioned neutrosophic soft p-closed set in (X(K̃,E), τ
QPNSS

(K̃,E)
,Ω).

Hence, the first part is proved.

Theorem 23. Every quadripartitioned neutrosophic soft second countable space is always quadri-
partitioned Neutrosophic soft first countable space.

Proof. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space over
X which satisfies the second axiom of quadripartitioned neutrosophic soft countability. That
is, (X, τQPNSS ,Ω) is quadripartitioned neutrosophic soft second countable. To prove that
(X, τQPNSS ,Ω) is quadripartitioned neutrosophic soft first countable, we proceed as follows.
By hypothesis, there exists a quadripartitioned neutrosophic soft countable base BQPNSS for
the quadripartitioned neutrosophic soft topology τQPNSS on X. The countability of BQPNSS

implies that BQPNSS can be expressed as

BQPNSS = {Bn : n ∈ N}.

Let xθ(r1,r2,r3,r4) ∈ X be arbitrary. Define the set

Lxθ
(r1,r2,r3,r4)

= {Bn ∈ BQPNSS : xθ(r1,r2,r3,r4) ∈ Bn}.

(i) Since Lxθ
(r1,r2,r3,r4)

is a quadripartitioned neutrosophic soft subset of the countable set

BQPNSS , it follows that Lxθ
(r1,r2,r3,r4)

is also countable.

(ii) Since members of BQPNSS are τQPNSS-open sets, the members of Lxθ
(r1,r2,r3,r4)

are also

contained in τQPNSS , i.e.,
Lxθ

(r1,r2,r3,r4)
⊆ τQPNSS .

(iii) Any (G̃,Ω) ∈ Lxθ
(r1,r2,r3,r4)

implies that xθ(r1,r2,r3,r4) ∈ (G̃,Ω).


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 46 of 54

(iv) Let (G̃,Ω) ∈ τQPNSS be arbitrary such that xθ(r1,r2,r3,r4) ∈ (G̃,Ω). Then, by the definition
of a quadripartitioned neutrosophic soft base, there exists some Br ∈ BQPNSS such that

xθ(r1,r2,r3,r4) ∈ Br ⊆ (G̃,Ω).

Since Br ∈ BQPNSS and contains xθ(r1,r2,r3,r4), it follows that Br ∈ Lxθ
(r1,r2,r3,r4)

.

Thus, Lxθ
(r1,r2,r3,r4)

forms a quadripartitioned neutrosophic soft countable local base at

xθ(r1,r2,r3,r4) ∈ X. Hence, by definition, (X, τQPNSS ,Ω) is quadripartitioned neutrosophic soft
first countable.

Theorem 24. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X such that it is quadripartitioned neutrosophic soft first countable, and let (Y,J QPNSS ,Ω)
be a quadripartitioned neutrosophic soft subspace of (X, τQPNSS ,Ω). Then (Y,J QPNSS ,Ω) is
quadripartitioned neutrosophic soft first countable.

Proof. Let yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ Y be arbitrary, then yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ Y as Y ⊆ X. Since X is quadri-

partitioned neutrosophic soft first countable, it guarantees that there exists a quadripartitioned
neutrosophic soft countable local base at xθ(r1,r2,r3,r4) ∈ X and hence, in particular, there exists

a quadripartitioned neutrosophic soft countable local base BQPNSS at yθ
′

(r′1,r
′
2,r

′
3,r

′
4)
.

Members of BQPNSS can be enumerated as B1, B2, B3, B4, B5, . . . , that is,

BQPNSS = {Bn : n ∈ N}.

Evidently, yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ X. Since yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ Bn for all n ∈ N, we write

B1 = {Y ∩Bn : n ∈ ν/N} (1).

Since yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ Y and yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ Bn for all n ∈ N, it follows that

yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ Bn for all n ∈ N (2).

Since Bn ∈ BQPNSS for all n ∈ N, we have Bn ∈ τQPNSS , which implies that

Y ∩Bn ∈ J QPNSS .

Theorem 25. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X such that it is quadripartitioned neutrosophic soft second countable, and let (Y,J QPNSS ,Ω)
be a quadripartitioned neutrosophic soft subspace of (X, τQPNSS ,Ω). Then (Y,J QPNSS ,Ω) is
quadripartitioned neutrosophic soft second countable.

Proof. Let (Y,J QPNSS ,Ω) be a quadripartitioned neutrosophic soft subspace of (X, τQPNSS ,Ω),
a quadripartitioned neutrosophic soft topological space over X which is quadripartitioned neu-
trosophic soft second countable. This implies that there exists a quadripartitioned neutrosophic
soft countable base BQPNSS for τQPNSS . If we prove that (Y,J QPNSS ,Ω) is quadripartitioned
neutrosophic soft countable, the result will automatically follow.

Since BQPNSS is quadripartitioned neutrosophic soft countable, it follows that BQPNSS ∼
N, which implies that BQPNSS can be expressed as

BQPNSS = {Bn : n ∈ N}.

Define
B1 = {Y ∩Bn : n ∈ N} (i).


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 47 of 54

Evidently, B1 ∼ N under the quadripartitioned neutrosophic soft map Y ∩Bn → n. Therefore,
B1 is quadripartitioned neutrosophic soft countable.

(ii) B1 is a quadripartitioned neutrosophic soft family of all J QPNSS-quadripartitioned
neutrosophic soft p-open sets. Since Bn ∈ BQPNSS implies Bn ∈ τQPNSS , it follows that
B ⊆ τQPNSS , implying that Y ∩Bn ∈ J QPNSS .

(iii) For any yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ (G̃,Ω) ∈ J QPNSS , there existsBr∩Y ∈ B1 such that yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈
Br ∩ Y ⊆ (G̃,Ω).

To prove this, let (G̃,Ω) ∈ J QPNSS such that yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ (G̃,Ω). Then there exists

(H̃,Ω) ∈ τQPNSS such that (G̃,Ω) = (H̃,Ω) ∩ Y . Since yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ (H̃,Ω) ∩ Y , we have

yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ (G̃,Ω).

By the definition of the quadripartitioned neutrosophic soft base, any yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ Y be-

longing to J QPNSS implies that there exists Br ∈ BQPNSS such that yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ Br ⊆ (H̃,Ω).

From the above, it follows that B1 is a quadripartitioned neutrosophic soft countable base for
the quadripartitioned neutrosophic soft topology J QPNSS on Y . Consequently, (Y,J QPNSS ,Ω)
is quadripartitioned neutrosophic soft second countable.

Theorem 26. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space
over X such that it is a quadripartitioned neutrosophic soft second countable space. Then it
also has the characteristics of another quadripartitioned neutrosophic soft space known as a
quadripartitioned neutrosophic soft separable space. Interestingly, the converse is not always
true.

Proof. Let (X, τQPNSS ,Ω) be a quadripartitioned neutrosophic soft topological space over
X such that it is a quadripartitioned neutrosophic soft second countable space. Since X is
a quadripartitioned neutrosophic soft second countable space, there exists a quadripartitioned
neutrosophic soft countable base BQPNSS for the quadripartitioned neutrosophic soft topology
τQPNSS . Members of BQPNSS may be enumerated as B1, B2, B3, B4, . . .

Choose any quadripartitioned neutrosophic soft point xθ(r1,r2,r3,r4) from each Bi and take Y
as a collection of all these quadripartitioned neutrosophic soft points:

Y = {xθ(r1,r2,r3,r4) | ∀i ∈ N}. (1)

That is to say,
xθ(r1,r2,r3,r4) ∈ Bi ∈ BQPNSS , ∀i ∈ N. (2)

Evidently, N ∼ Y under the quadripartitioned neutrosophic soft map i → xθ(r1,r2,r3,r4), therefore
Y is enumerable. Clearly, Y ⊆ X.

We claim that Y = X. Suppose not. Then X −Y ̸= ∅:

X −Y ̸= ∅. (3)

Let yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ X −Y be arbitrary. Since Y is quadripartitioned neutrosophic soft p-closed,

X −Y is quadripartitioned neutrosophic soft p-open. That is,

yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ X −Y ∈ τQPNSS . (4)

By the definition of the quadripartitioned neutrosophic soft base, there exists Bn0 ∈ BQPNSS

such that
yθ

′

(r′1,r
′
2,r

′
3,r

′
4)n0

⊆ X −Y. (5)

But xθ(r1,r2,r3,r4)n0

∈ Y according to (1) and (2), contradicting (4). Thus, our assumption

X −Y ̸= ∅ was incorrect. Consequently,

X −Y = ∅ ⇒ X = Y. (6)


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 48 of 54

Thus, we have proved that there exists Y ⊆ X such that X = Y and Y is quadripartitioned neu-
trosophic soft enumerable. By definition, this proves that X is quadripartitioned neutrosophic
soft separable. Let us now discuss the converse. Suppose X be an infinite quadripartitioned
neutrosophic soft set. Let τQPNSS be a family consisting of 0(X,Ω) and all those quadriparti-
tioned neutrosophic soft subsets Y of X such that Yc is quadripartitioned neutrosophic soft
finite. Then τQPNSS is a quadripartitioned neutrosophic soft topology.

We claim that τQPNSS is quadripartitioned neutrosophic soft separable. Since X is an infi-
nite quadripartitioned neutrosophic soft set, there existsY ⊆ X such thatY is quadripartitioned
neutrosophic soft enumerable.

To prove thatX = Y, we note thatY ⊆ X, and hence all the quadripartitioned neutrosophic
soft closure points of Y will reside in X, which implies that Y ⊆ X.

If (G̃,Ω) ∈ τQPNSS , then (G̃,Ω)c is quadripartitioned neutrosophic soft p-closed. Thus,
(G̃,Ω) ∈ τQPNSS implies that (G̃,Ω)c is quadripartitioned neutrosophic soft finite and quadri-
partitioned neutrosophic soft p-closed. This means that all the quadripartitioned neutrosophic
soft p-closed subsets of X are finite quadripartitioned neutrosophic subsets of X and X itself.

Thus, the only infinite quadripartitioned neutrosophic soft p-closed subset of X is X, which
contains Y. Therefore, X = Y. Since Y is the smallest quadripartitioned neutrosophic soft
p-closed set containing Y and X is the only such set, we conclude that:

X = Y.

To prove that (X, τQPNSS ,Ω) is not quadripartitioned neutrosophic soft second countable,
suppose the contrary. Then X is quadripartitioned neutrosophic soft second countable, so
there exists a quadripartitioned neutrosophic soft countable base BQPNSS for the quadripar-
titioned neutrosophic soft topology on X. The members of BQPNSS may be enumerated as
B1, B2, B3, B4, . . ..

Let xθ(r1,r2,r3,r4) ∈ X be arbitrary but fixed. Define:

BQPNSS
0 = {Br ∈ BQPNSS | xθ(r1,r2,r3,r4) ∈ Br} (1)

Y =
⋂

{Br | Br ∈ BQPNSS
0 } (2)

From (2), it is clear that xθ(r1,r2,r3,r4) is common in all the members of BQPNSS
0 and hence:

xθ(r1,r2,r3,r4) ∈ Y.

We claim that:

Y = {xθ(r1,r2,r3,r4)} (3).

Let yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ X be arbitrary such that:

xθ(r1,r2,r3,r4) ̸= yθ
′

(r′1,r
′
2,r

′
3,r

′
4)
.

Since {xθ(r1,r2,r3,r4)} is a quadripartitioned neutrosophic soft finite set, its complement is in

τQPNSS , i.e.,

{xθ(r1,r2,r3,r4)}
c ∈ τQPNSS .

Clearly, yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ {xθ(r1,r2,r3,r4)}
c. By the definition of the quadripartitioned neutro-

sophic soft base, there exists B
yθ

′
(r′1,r

′
2,r

′
3,r

′
4)

∈ BQPNSS such that:

yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ B
yθ

′
(r′1,r

′
2,r

′
3,r

′
4)

⊆ {xθ(r1,r2,r3,r4)}
c.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 49 of 54

This implies that:

yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

/∈ BQPNSS
0 .

From (2), this means:

yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

/∈ Y.

Thus, we have shown that:

yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

∈ X, yθ
′

(r′1,r
′
2,r

′
3,r

′
4)

/∈ Y.

This proves (3), and consequently:

Yc is quadripartitioned neutrosophic soft countable. (4)

Since X is quadripartitioned neutrosophic soft uncountable and Y is a quadripartitioned
neutrosophic soft singleton set, we get:

Yc = X −
⋂

{Br | Br ∈ BQPNSS
0 } =

⋃
{Bc

r | Br ∈ BQPNSS
0 } (5).

Since Br ∈ BQPNSS
0 , it follows that Br ∈ BQPNSS , so:

BQPNSS
0 ⊆ BQPNSS ⊆ τQPNSS .

Thus, Br ∈ τQPNSS implies that Bc
r is a quadripartitioned neutrosophic soft finite set. Being

a quadripartitioned neutrosophic soft countable union of finite quadripartitioned neutrosophic
soft sets,

⋃
{Bc

r | Br ∈ BQPNSS
0 } is quadripartitioned neutrosophic soft countable, which means:

Yc is quadripartitioned neutrosophic soft countable, in accordance with (5).

This contradicts (4). Hence, our supposition was wrong, and we conclude that:

(X, τQPNSS ,Ω) is not quadripartitioned neutrosophic soft second countable.

10. Advantages of Quadri-Partitioned Neutrosophic Set Theory

(i) More Complex and Comprehensive Framework: By adding a fourth partition,
QPNST provides a more detailed structure for handling sets, capturing more possibilities
and allowing for a more comprehensive exploration of complex systems. This expanded
framework is particularly useful in situations where traditional fuzzy logic or intuitionistic
sets may fall short.

(ii) Improved Mathematical Tools: The introduction of Quadri-Partitioned Neutrosophic
Riemann Integral Theory (QPNRIT) offers new insights into the Riemann integral, an
essential concept in analysis. By extending the theory into the QPNST context, this
work opens new avenues for the study of integration and its properties under uncertainty,
helping to reveal behaviors and characteristics not previously observable.

(iii) Level Cut (Four-Tuple Representation): The definition of the level cut as a four-
tuple (i, j, k, l) captures the multiple possibilities inherent in QPNST. This enables more
precise modeling and analysis, reflecting the different dimensions of truth, indeterminacy,
and falsity in set membership more clearly. This richer structure supports more accurate
decision-making and problem-solving in uncertain contexts.


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 50 of 54

(iv) Numerical Study and Practical Insights: The numerical study conducted within the
QPNST framework provides a concrete understanding of the behavior of the Riemann
integral in this extended context. Organizing the findings in tabular form makes it easier
for researchers and practitioners to grasp the implications and applications of the new
theory.

11. Limitations of Quadri-Partitioned Neutrosophic Set Theory

(i) Increased Complexity: While the added complexity of QPNST allows for more detailed
and accurate representations of uncertainty, it also makes the theory more challenging to
apply. Researchers may find it difficult to work with the four-part structure, especially in
cases where the additional partitions add little value in practical applications.

(ii) Computational Challenges: With the introduction of additional partitions and the
need to calculate more complex level cuts, the computational effort required to implement
QPNST and QPNRIT could be significantly higher. This might limit its use in large-scale
or real-time applications where computational efficiency is critical.

(iii) Interpretation and Practical Implementation: The four-dimensional nature of the
model may pose interpretative challenges. While it provides a richer model for uncer-
tainty, translating the abstract concepts into real-world decision-making processes can be
difficult. Practitioners may struggle to interpret the results or apply the theory in prac-
tical situations, especially in industries not traditionally familiar with higher-order logic
systems.

(iv) Limited Existing Tools and Resources: The extension of Riemann integral theory
into the QPNST context is a relatively new development, and as such, there may be
limited resources, tools, and research available to fully support its application. This could
slow down its adoption and the development of practical applications based on QPNRIT.

(v) Need for Further Theoretical Validation: Although the framework holds promise,
the full range of theoretical properties of QPNST and QPNRIT needs to be further ex-
plored. Many questions regarding the consistency, stability, and broader applicability of
these theories remain unanswered, requiring further research and validation.

12. Conclusion and Future Work

Finally, by adding a third option for set representation, Neutrosophic Set Theory (NST)
expands on Intuitionistic Fuzzy Set Theory (IFST) and improves the theory’s ability to han-
dle uncertainty. By introducing Quadri-Partitioned Neutrosophic Set Theory (QPNST), which
adds a fourth alternative and offers a more complex and all-encompassing framework for set
representation, this work significantly develops NST. Within this framework, we define the
Riemann Integral Theory (RIT), opening new avenues for exploring the properties and charac-
teristics of the Riemann integral in an extended context. A central concept in this work is the
level cut, defined as a four-tuple (i, j, k, l), which captures the multiple possibilities inherent in
QPNST. Furthermore, we conduct a numerical study of the Quadri-Partitioned Neutrosophic
Riemann Integral Theory (QPNRIT) and provide the findings in an organized tabular man-
ner. This numerical study enhances our comprehension of the integral’s behavior within the
QPNST framework and offers insightful information about its characteristics. This study ex-
plores quadripartitioned neutrosophic soft topological spaces, extending neutrosophic set theory
(NST), which incorporates three membership values: true, false, and indeterminacy. The study
introduces new concepts such as QPNS semi-open, QPNS pre-open, and QPNS ∗b open sets, and
builds on these to define QPNS closure, exterior, boundary, and interior. A key development is


A. Shihadeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5839 51 of 54

the definition of a quadripartitioned neutrosophic soft base, which plays a central role in these
topological structures. The paper also explores the concept of a quadripartitioned neutrosophic
soft sub-base and discusses local bases, as well as the first- and second-countability axioms. The
study further examines hereditary properties of these spaces, distinguishing between inherited
and non-inherited properties. Key results include that a quadripartitioned neutrosophic soft
subspace of a first-countable space is also first-countable, and a second countable subspace of
a second-countable space remains second-countable. It also highlights the relationship between
second countability and separability in these spaces, asserting that a second-countable quadri-
partitioned neutrosophic soft space is separable, though the converse is not always true. This
work lays the foundation for further research in neutrosophic soft topologies. With the possibil-
ity of incorporating it into other sophisticated theoretical frameworks, future studies will seek to
broaden the use of Quadri-Partitioned Neutrosophic Set Theory (QPNST) in a variety of math-
ematical fields. Investigating a generalized Riemann Integral Theory (RIT) employing QPNST
in more intricate contexts, like multi-dimensional spaces and dynamic systems, is one exciting
avenue. In order to increase computational accuracy and efficiency, more effort will be done to
enhance numerical methods for computing Quadri-Partitioned Neutrosophic Riemann Integrals
(QPNRIT). The creation of decision-making models that incorporate QPNST will be another
crucial topic. These models could provide more reliable frameworks for dealing with uncertainty
in real-world applications including artificial intelligence, engineering, and economics. Further-
more, it is crucial to conduct additional research on the theoretical characteristics of QPNST,
such as its algebraic and topological features. These studies could provide deeper insights into
the behavior of neutrosophic sets in various contexts, enriching the overall understanding of this
extended theory.

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at Northern
Border University, Arar, KSA for funding this research work through the project number “NBU-
FFR-2025-2727-03”..
Authors contributions: All authors contributed equally.
Conflicts of Interests: The authors have no conflicts of interest.

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