EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5845 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Breakthrough Approach to Quadri-Partitioned Neutrosophic Soft Topological Spaces Maha Mohammed Saeed1, Raed Hatamleh2, Alaa M. Abd El-latif3, Abdallah Alhusban4, Husham M. Attaalfadeel3,∗, Takaaki Fujita5, Khaled A. Aldwoah6, Arif Mehmood7 1 Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, 15 Saudi, Arabia 2 Department of Mathematics, Faculty of Science, Jadara University, P.O. Box 733, Irbid 21110, Jordan 3 Department of Mathematics, 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 5 Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan 6 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medinah, Saudi Arabia 7 Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050, KPK, Pakistan Abstract. Neutrosophic set theory, an advanced framework for error reduction, extends fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). It enhances effectiveness by refining the definition of indeterminacy, a concept for situations where values cannot be precisely determined. In this paper, we propose dividing indeterminacy into two components based on membership: relative truth (RT), which leans toward truth, and relative falsehood (RF), which leans toward falsehood. This approach improves accuracy by considering, relative truth and relative falsehood, rather than a single indeterminate value. The modified neutrosophic set, called the quadri-portioned neutrosophic soft set (QPNSS), includes four membership attributes: absolute truth, relative truth, relative falsehood, and absolute falsehood, offering greater clarity in uncertain situations. New operations are introduced on QPNSS, such as the quadri- partitioned neutrosophic soft set, subsets, complement, absolute set, set difference, and null set. AND and OR operations are also defined. Additionally, a quadri-partitioned neutrosophic soft topological space (QPNSTS) is defined, with key results presented. We define and explore new concepts such as pre- open (p-open) sets, interior, and closure. The paper also examines QPNS compactness, reducibility to finite sub-covers, and other important properties like the intersection of QPNS p-closed sets and QPNS p-compact spaces. This work contributes to the theoretical understanding of QPNS spaces, particularly in soft topological spaces. 2020 Mathematics Subject Classifications: 54A05 Key Words and Phrases: Neutrosophic soft set, QPNSS, QPNSTS, QPNS p- open sets, QPNS p-closed sets, QPNS p compact subsets. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5845 Email addresses: (mmmohammed@kau.edu.sa) (M. M. Saeed), (raed@jadara.edu.jo) (R. Hatamleh), (alaa.ali@nbu.edu.sa) (A. M. A. El-latif), (dralhosban@inu.edu.jo) (A. Alhusban), (husham.alhassan@nbu.edu.sa) (H. M. Attaalfadeel), (t171d603@gunma-u.ac.jp) (T. Fujita), (aldwoah@yahoo.com) (K. A. Aldwoah), (mehdaniyal@gmail.com) (A. Mehmood) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 2 of 32 1. Introduction The most significant method for dealing with ambiguity and missing data that arise in many scientific domains is fuzzy set theory (FST) [1]. This theory is only helpful when discussing membership value; it cannot sufficiently address the non-membership value. This challenge was expertly resolved by Atanassov [2], who introduced the concept of intuitionistic fuzzy set theory (IFST). This method captures the value of membership as well as non-membership. The concept of soft set theory (SST) was first proposed by Molodtsov [3] as a novel way to handle nebulous and confusing circumstances. Molodtsov [4] tackled a number of problems, including function smoothness, using soft set theory. Xu et al. [5] initially presented vague soft set theory as an expansion of FST. In their discussion of unclear soft set, Huang et al. [6] identified a few inaccurate findings. The authors provided additional new definitions and provided examples to support the wrong finding. The idea of vague soft topological space was first introduced by Chang Wang and Yaya Li [7] under the heading topological structure of vague soft sets. The authors examined the findings in this area and talked about the fundamental ideas of vague soft topology. Smarandache [8] extended soft set to hyper soft set. The author also presented the hybrids of NHSSs, IFSSs, PHSSs, and crisp FSs. Based on the neutrosophic soft set (NSS). Bera and Mahapatra [9] proposed the idea of a new structure known as neutrosophic soft topology. This concept opened up a whole new area of mathematics. The writers courteously went over each of the foundational concepts before moving on to the fundamental outcomes and providing pertinent instances for greater comprehension. Ozturk et al. [10] installed extremely operations on NSSs and NSTs based on these unique procedures. Based on the operation outlined in [10], Ozturk et al. [11] installed the notions of neutrosophic soft mapping, neutrosophic soft open mapping, and neutrosophic soft homeomorphism. Mehmood et al. [12] created new open sets in neutrosophic soft topological space. AL-Nafee [13] not only created a new family of neutrosophic soft sub sets but also came up with new operations (union and intersection) on neutrosophic sets. Based on the operations described in [14], AL-Nafee et al. [14] constructed NSBTS to finish their previous research. The authors used these fundamental methods to replicate all of the important findings of NSBTS. NSBTS was developed by Dadas and Demiralp [15] using the guidelines provided in [9]. Smarandache [16] generalizes the intuitionistic fuzzy set (IFS), paraconsistent set, and intuitionistic set to the neutrosophic set (NS). Many examples are presented. Distinctions between NS and IFS are underlined. Several aspects of the menger selection principle were explored by Ljubia et al. [17] in their study of the principle in regard to soft sets. The relationships between the recently suggested soft open coverings were examined by the writers. Using soft s-regular open covers, Al-shami et al. [18] provided a detailed description of near SMSs. Additionally; they demonstrated how they fell into the same class as soft regular spaces. Al-shami et al.’s concept of an ISTSs was first presented [19]. Al-shami et al. [20] explored fixed soft points and weak Forms of soft separation axioms. The theory of ISS connected and ISLCS was given by Al-shami et al. [21]. They also discussed the behavior of these spaces as a finite product of soft spaces and under infra soft homeomorphism maps. The authors discussed the essential elements of a soft point and described its main traits. The concepts of open, closed, and homeomorphism mappings in ISTS content were first introduced by Al-shami [22]. Al-shami [23] investigated ISC and ISLSs, demonstrating their main features using a sequence of infra soft closed sets. Examining the transmission of these concepts between classical and IST was the author’s main focus. The concept of soft bi-operators was first introduced by Baravan et al. [24]. An example from Ali et al. was used to build an effective bipolar soft generalization of the q-rung ortho-pair fuzzy set model [25]. A model known as the FBSESs was introduced by Ali et al. [26]. In order to create a unique discretization of these basic mathematical ideas with an essentially continuous character, John [27] proposed a soft structure over a set. The advantage is that it gave rise to new instruments for applying mathematical analysis technology in practical applications to identify uncertainty or incomplete data. S. Kim and Park [28] M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 3 of 32 discussed secure multi-party clustering Protocols. Kandasamy [29] installed DVNSs and used it minimum spanning trees. Mehmood et al. [30] explored many structures and provided new definitions for NSTSs. Mehmood et al. [31] presented the idea of VSTSs in a novel approach and explored various structures with regard to vague soft points using vague soft beta open sets. 1.1. Literature review A novel idea known as vague soft bi-topological space was presented by Mehmood et al. [32], who also looked at its structural characteristics. Generalized vague soft open sets, also known as vague soft β-open sets, are the foundation of this method. These structures are demon- strated with a number of instances. The authors used vague soft β-open sets to study the properties of vague soft bi-topological space with respect to the soft points of the spaces. The Bolzano-Weierstrass property, vague soft compactness and its link to sequences, the characteri- zation of vague soft β-closed and vague soft β-opensets, and the relationship between countable compactness and sequential compactness in VSBTS with respect to soft β − open sets are also discussed. New union, intersection, and complement operations were introduced by Mehmood et al. [33] and serve as the basis for the definition of vague soft bi-topological space. Within vague soft bi-topological space, they created pairwise vague soft closed sets and pairwise vague soft open sets. They also presented generalized vague soft open sets associated with the space’s soft points. They established separation axioms based on these generalized ambiguous soft open sets. Furthermore, other noteworthy findings in vague soft bi-topological space are connected to these separation axioms. The notions of neutrosophic soft b-closed sets and neutrosophic soft b-open sets, as well as their attributes, were first presented by Mehmood et al. [34]. In connection with soft points, they also talked about neutrosophic soft b-neighborhoods and neu- trosophic soft b-separation axioms in NSTS. Numerous findings pertaining to soft points are used to investigate the idea of neutrosophic soft b-separation axioms. Furthermore, the char- acteristics and relationships of neutrosophic soft Ti spaces for (i = 0,1,2,3,4) are investigated. Mehmood et al. [35] discussed NSQS concerning soft points. The novel idea of QPNSS was presented by Ramesh and Mary [36], who also talked about some of its characteristics. For clar- ity and comprehension, the writers picked the best instances and concentrated on a methodical investigation. Shami and Mhemdi [19] explored two families of separation axioms in the context of infra soft topological spaces. Their work aimed at classifying and analyzing different types of separation properties within this specialized topological framework. Shami and Liu [19] intro- duced two classes of infra-soft separation axioms. These axioms are important for defining and studying the separation properties of infra-soft topological spaces, extending classical separation axioms to a more generalized soft-topological setting. Shami and Azzam [37] focused on the concepts of soft semi-open sets and infra soft semi continuity. Their study examined how these notions interact with infra soft topological spaces, offering insights into the continuity proper- ties of such spaces. Ahmad et al. [38] discussed Irreversible k-Threshold conversion number for some graph products and neutrosophic graphs. Hatamleh et al. [39] studied complex tangent trigonometric approach applied to (Ξ, τ)-rung fuzzy set using weighted averaging, geometric operators and its extension. Hatamleh et al. [40] studied different weighted operators such as generalized averaging and generalized geometric based on trigonometric ℘-rung interval-valued approach and in addition to this some examples were given for clear understanding. Shihadeh et al. [41] discussed algebraic structures towards different (α, β) intuitionistic fuzzy ideals and it characterization of an ordered ternary semigroups. Hatamleh et al. [40, 42] studied opera- tors via weighted averaging and geometric approach using trigonometric neutrosophic interval valued set and its extension and characterization of interaction aggregating operators setting interval valued Pythagorean neutrosophic set. Hatamleh et al. [38, 43] discussed applications of complex interval valued picture fuzzy soft relations. . El-Sheikh and Abd El-latif [44] discussed decompositions of some types of supra soft sets and soft continuity and cited some excellent M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 4 of 32 examples for clear understating the concept. Abd El-latif [45] discussed soft supra compactness in supra soft topological spaces. Abd El-latif and Hosny, [46] discussed the eye catching con- cept of soft separation axioms and give examples. Abd El-latif and Hosny discussed some more structures in [45, 47]. 1.2. Research Gap In Neutrosophic set theory, three membership functions truth, indeterminacy, and falsity are commonly used. However, the indeterminacy function presents a significant challenge, as it introduces ambiguity, making it difficult to decisively classify elements. This indeterminate region often leads to errors and confusion in the final results, limiting the applicability and precision of neutrosophic sets in practical scenarios. While some attempts have been made to address this issue, there is still a gap in fully resolving the indeterminacy. A potential solu- tion lies in further subdividing the indeterminacy component into two distinct parts: one that tends toward truth and the other that tends toward falsity. This would effectively eliminate the ambiguity associated with indeterminacy. However, to date, no comprehensive framework has been proposed that incorporates this subdivision. There is a need for the development of a new set structure, which would consist of four components. This could lead to the creation of a quadripartitioned neutrosophic set, and subsequently, the introduction of a novel struc- ture known as the quadripartitioned neutrosophic topological space. This new structure could provide a more precise and error-minimized approach, enabling more accurate operations, prop- erties, and results to be derived from neutrosophic sets. Thus, there exists a clear research gap in formalizing and exploring the properties and applications of quadripartitioned neutrosophic topological spaces. 1.3. Motivation A modified version of a neutrosophic set, called the double-valued neutrosophic Set (DVNS), which consists of two different indeterminate values, was introduced by Kandasamy [31]. Ex- amples are provided to define and demonstrate the related attributes and axioms. In several research domains, such as data mining, pattern recognition, and machine learning, clustering is essential. The cornerstone for building a clustering method is the establishment of a gen- eralized distance measure between DVNSs and the accompanying distance matrix. In order to efficiently cluster data represented by double-valued neutrosophic information, this work suggests the double-valued neutrosophic minimum spanning Tree (DVN-MST) clustering tech- nique. To demonstrate the usefulness and uses of this clustering approach, illustrative examples are given. This work inspired me and served as a stepping stone for my current research. 1.4. Novelty The quadri-portioned neutrosophic soft set (QPNSS) is a novel framework that represents a major breakthrough in neutrosophic set theory. This approach distinguishes between relative truth (RT) and relative falsehood (RF), two essential elements of indeterminacy, improving sen- sitivity and accuracy in uncertain situations. By enabling a more nuanced examination of values that are difficult to categorize as true or untrue, this innovative classification enhances decision- making. Absolute truth, relative truth, relative falsehood, and absolute falsehood are the four membership traits that make up the QPNSS. Along with quadri-partitioned soft sets and their complements, the study defines new operations on QPNSS, such as AND and OR operations. Moreover, eight new definitions, including the semi-open (s-open) set, and the development of quadri-partitioned neutrosophic soft topological spaces (QPNSTS) improve our knowledge of topological features. Crucially, the research investigates compactness in QPNSS, introducing ideas such as reducibility to finite sub-covers and establishing important theorems on inter- section qualities and compact subset separation. These results contribute to the theoretical M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 5 of 32 foundation of QPNS spaces and provide important information about the compactness of soft topological spaces. All things considered, this work opens the door for further study and appli- cations by expanding our knowledge of neutrosophic set theory and offering useful instruments for handling ambiguity. The organization of this paper is structured as follows. We outline basic ideas and findings in Section 2. A new method for operations on quadri-partitioned neu- trosophic soft sets is examined in Section 3. A new viewpoint on heptapartitioned neutrosophic soft topological spaces is presented in Section 4. Results on compactness in quadri-partitioned neutrosophic soft topological spaces are presented in Section 5. Comparative analysis is dis- cussed in Section 6. Applications are given in Section 7. Conclusion and future work is given in Section 8. 2. Preliminaries This section covers foundational ideas that are required for the upcoming research. Firstly, neutrosophic soft set defined by Maji [48] and later this concept has been modified by Deli and Bromi [49] as given below: Definition 1. Let e be a set of parameters and M be an initial universe set. P (M) represents the collection of all neutrosophic soft sets for M. A set defined by a set-valued function Υ̃ expressing a mapping Υ̃ : É → P (M) is then a neutrosophic soft set (Υ̃, É) over M, where Υ̃ is referred to as the approximate function of the NSS (Υ̃, É). Stated differently, the neutrosophic soft set can be expressed as a collection of ordered pairs: (Υ̃, É) = {( e, ⟨s, TΥ̃(e)(s), IΥ̃(e)(s), FΥ̃(e)(s)⟩ ) : s ∈ M, e ∈ É } . Here, TΥ̃(e)(s), IΥ̃(e)(s), and FΥ̃(e)(s) are the truth-membership, indeterminacy-membership, and falsity-membership functions of Υ̃(e), respectively, and they all lie within the interval [0, 1]. The inequality 0 ≤ TΥ̃(e)(s) + IΥ̃(e)(s) + FΥ̃(e)(s) ≤ 3 is evident as the supremum of each T , I, and F is 1 and the infimum is 0. This means that each value is a typical value between 0 and 1. Definition 2. [9] Let (Υ̃, É) be a neutrosophic soft set. Then (Υ̃, É)c is the complement of (Υ̃, É): (Υ̃, É)c = {( e, ⟨s, FΥ̃(e)(s), 1− IΥ̃(e)(s), TΥ̃(e)(s)⟩ ) : s ∈ M, e ∈ É } . It is obvious that: ( (Υ̃, É)c )c = (Υ̃, É). Definition 3. [10] Let (Υ̃, É) and (Λ̃, e) be two neutrosophic soft sets. Then (Υ̃, É) is said to be a neutrosophic soft subset of (Λ̃, e) if: TΥ̃(e)(s) ≤ TΛ̃(e)(s), IΥ̃(e)(s) ≤ IΛ̃(e)(s), FΥ̃(e)(s) ≥ FΛ̃(e)(s), ∀e ∈ É,∀s ∈ M. It is denoted by: (Υ̃, É) ⊆ (Λ̃, e). Definition 4. [10] Let (Υ̃1, e) and (Υ̃2, e) be two neutrosophic soft sets. Then their union is represented by (Υ̃1, e) ⋓ (Υ̃2, e) = (Υ̃3, e) as: (Υ̃3, e) = {( e, ⟨s, TΥ̃3(e) (s), IΥ̃3(e) (s), FΥ̃3(e) (s)⟩ ) : s ∈ M, e ∈ É } . M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 6 of 32 Where: TΥ̃3(e) (s) = max{TΥ̃1(e) (s), TΥ̃2(e) (s)}, IΥ̃3(e) (s) = max{IΥ̃1(e) (s), IΥ̃2(e) (s)}, FΥ̃3(e) (s) = min{FΥ̃1(e) (s), FΥ̃2(e) (s)}. Definition 5. [10] Let (Υ̃1, e) and (Υ̃2, e) be two neutrosophic soft sets. Then their intersection is symbolized by (Υ̃1, e) ⋒ (Υ̃2, e) = (Υ̃3, e) as: (Υ̃3, e) = {( e, ⟨s, TΥ̃3(e) (s), IΥ̃3(e) (s), FΥ̃3(e) (s)⟩ ) : s ∈ M, e ∈ É } . Where: TΥ̃3(e) (s) = min{TΥ̃1(e) (s), TΥ̃2(e) (s)}, IΥ̃3(e) (s) = min{IΥ̃1(e) (s), IΥ̃2(e) (s)}, FΥ̃3(e) (s) = max{FΥ̃1(e) (s), FΥ̃2(e) (s)}. Definition 6. [10] Let (Υ̃1, e) and (Υ̃2, e) be two neutrosophic soft sets. Then the difference operation on them is denoted by (Υ̃1, e) \ (Υ̃2, e) = (Υ̃3, e) and is defined by: (Υ̃3, e) = {( e, ⟨s, TΥ̃3(e) (s), IΥ̃3(e) (s), FΥ̃3(e) (s)⟩ ) : s ∈ M, e ∈ É } . Where: TΥ̃3(e) (s) = min{TΥ̃1(e) (s), TΥ̃2(e) (s)}, IΥ̃3(e) (s) = min{IΥ̃1(e) (s), IΥ̃2(e) (s)}, FΥ̃3(e) (s) = max{FΥ̃1(e) (s), FΥ̃2(e) (s)}. Definition 7. [10] 1. A neutrosophic soft set (Υ̃, É) is said to be a null neutrosophic soft set if: TΥ̃(e)(s) = 0, IΥ̃(e)(s) = 0, FΥ̃(e)(s) = 1, ∀e ∈ É,∀s ∈ M. It is denoted by 0(M,É). 2. A neutrosophic soft set (Υ̃, É) is said to be an absolute neutrosophic soft set if: TΥ̃(e)(s) = 1, IΥ̃(e)(s) = 1, FΥ̃(e)(s) = 0, ∀e ∈ É,∀s ∈ M. It is symbolized as 1(M, É). It is obvious that: 0(M,É) = 1(M,É), 1(M,É) = 0(M,É). 3. A New Approach to 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 accomplished by breaking down the indeterminacy into two distinct categories of possible M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 7 of 32 values: ’relative true’ and ’relative false,’ each representing different aspects of the uncertainty. 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 8. Let e be the set of parameters and M be the key set. Let P (M) represent the power set of M. Then, a quadri-partitioned neutrosophic soft set (F̃ , É) over M is a mapping F̃ : e → P (M), where F̃ is the function of the quadri-partitioned neutrosophic soft set (F̃ , É). Symbolically, (F̃ , É) = {( e, ⟨s,AbTF̃ (e)(s),ReTF̃ (e)(s),ReFF̃ (e)(s),AbFF̃ (e)(s)⟩ ) : s ∈ M, e ∈ É } . Here, AbTF̃ (e)(s), ReTF̃ (e)(s), ReFF̃ (e)(s), and AbFF̃ (e)(s) belong to the interval [0, 1]. Re- spectively, these functions are called the absolute true-membership, relative true-membership, relative false-membership, and absolute false-membership functions of F̃ (e). Since the supre- mum of each function is 1 and the infimum is 0, the following inequality holds: 0 ≤ AbTF̃ (e)(s) + ReTF̃ (e)(s) + ReFF̃ (e)(s) +AbFF̃ (e)(s) ≤ 4. Definition 9. Let (F̃ , É) be a quadri-partitioned neutrosophic soft set over the key set M. Then, the complement of (F̃ , É) is denoted by (F̃ , É)c and is defined as follows: (F̃ , É)c = {( e, ⟨s,AbFF̃ (e)(s),ReFF̃ (e)(s),ReTF̃ (e)(s),AbTF̃ (e)(s)⟩ ) : s ∈ M, e ∈ É } . It follows that ( (F̃ , É)c )c = (F̃ , É). Definition 10. Let (F̃ , É) and (G̃, É) be two quadri-partitioned neutrosophic soft sets over the key set M. Then, (F̃ , É) ⊆ (G̃, É) if AbTF̃ (e)(s) ⪯ AbTG̃(e)(s), ReTF̃ (e)(s) ⪯ ReTG̃(e)(s), ReFF̃ (e)(s) ⪰ ReFG̃(e)(s), AbFF̃ (e)(s) ⪰ AbFG̃(e)(s), for all e ∈ É and s ∈ M. If (F̃ , É) ⊆ (G̃, É) and (F̃ , É) ⊇ (G̃, É), then (F̃ , É) = (G̃, É). Definition 11. Let (F̃ , É) and (G̃, É) be two quadri-partitioned neutrosophic soft sets over the key set M such that (F̃ , É) ̸= (G̃, É). Then their union is denoted by (F̃ , É)⋓ (G̃, É) = (H̃, É) and is defined as: (H̃, e) = {( e, ⟨s,AbTH̃(e)(s),ReTH̃(e)(s),ReFH̃(e)(s),AbFH̃(e)(s)⟩ ) : s ∈ M, e ∈ E } . where AbTH̃(e)(s) = max { AbTF̃ (e)(s),AbTG̃(e)(s) } , ReTH̃(e)(s) = max { ReTF̃ (e)(s),ReTG̃(e)(s) } , ReFH̃(e)(s) = min { ReFF̃ (e)(s),ReFG̃(e)(s) } , AbFH̃(e)(s) = min { AbFF̃ (e)(s),AbFG̃(e)(s) } . M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 8 of 32 Definition 12. Let (F̃ , É) and (G̃, É) be two quadri-partitioned neutrosophic soft sets over the key set M such that (F̃ , É) ̸= (G̃, É). Then their intersection is denoted by (F̃ , É) ⋒ (G̃, É) = (H̃, É) and is defined as: (H̃, É) = {( e, ⟨s,AbTH̃(e)(s),ReTH̃(e)(s),ReFH̃(e)(s),AbFH̃(e)(s)⟩ ) : s ∈ M, e ∈ É } . where AbTH̃(e)(s) = min { AbTF̃ (e)(s),AbTG̃(e)(s) } , ReTH̃(e)(s) = min { ReTF̃ (e)(s),ReTG̃(e)(s) } , ReFH̃(e)(s) = max { ReFF̃ (e)(s),ReFG̃(e)(s) } , AbFH̃(e)(s) = max { AbFF̃ (e)(s),AbFG̃(e)(s) } . Definition 13. Let (F̃ , É) and (G̃, É) be two quadri-partitioned neutrosophic soft sets over the key set M such that (F̃ , É) ̸= (G̃, É). Then, their difference is denoted by (H̃, É) = (F̃ , É) \ (G̃, É) and is defined as: (H̃, É) = (F̃ , É) ⋒ (G̃, É)c, such that (H̃, É) = {( e, ⟨s,AbTH̃(e)(s),ReTH̃(e)(s),ReFH̃(e)(s),AbFH̃(e)(s)⟩ ) : s ∈ M, e ∈ É } . where AbTH̃(e)(s) = min { AbTF̃ (e)(s),AbTG̃(e)(s) } , ReTH̃(e)(s) = min { ReTF̃ (e)(s),ReTG̃(e)(s) } , ReFH̃(e)(s) = max { ReFF̃ (e)(s),ReFG̃(e)(s) } , AbFH̃(e)(s) = max { AbFF̃ (e)(s),AbFG̃(e)(s) } . Definition 14. Let {(F̃i, É) : i ∈ I} be a family of quadri-partitioned neutrosophic soft sets over the key set M. Then, the union and intersection of this family are defined as follows: ⋃ i∈I (F̃i, É) = {( e, ⟨s, sup i∈I AbTF̃i(e) (s), sup i∈I ReTF̃i(e) (s), inf i∈I ReFF̃i(e) (s), inf i∈I AbFF̃i(e) (s)⟩ ) : s ∈ M, e ∈ É } . ⋂ i∈I (F̃i, É) = {( e, ⟨s, inf i∈I AbTF̃i(e) (s), inf i∈I ReTF̃i(e) (s), sup i∈I ReFF̃i(e) (s), sup i∈I AbFF̃i(e) (s)⟩ ) : s ∈ M, e ∈ É } . Definition 15. Let (F̃ , É) and (G̃, É) be two quadri-partitioned neutrosophic soft sets over the key set M. Then, the “AND” operation on them is denoted by (F̃ , É) ∧ (G̃, É) = (H̃, É × e) and is defined as: (H̃, É×e) = {( (e1, e2), ⟨s,AbTH̃(e1, e2)(s),ReTH̃(e1, e2)(s),ReFH̃(e1, e2)(s),AbFH̃(e1, e2)(s)⟩ ) : (e1, e2) ∈ e× e, s ∈ M } . where AbTH̃(e)(s) = min { AbTF̃ (e)(s),AbTG̃(e)(s) } , M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 9 of 32 ReTH̃(e)(s) = min { ReTF̃ (e)(s),ReTG̃(e)(s) } , ReFH̃(e)(s) = max { ReFF̃ (e)(s),ReFG̃(e)(s) } , AbFH̃(e)(s) = max { AbFF̃ (e)(s),AbFG̃(e)(s) } . Definition 16. Let (F̃ , É) and (G̃, É) be two quadri-partitioned neutrosophic soft sets over the key set M. Then, the “OR” operation on them is denoted by (F̃ , É)∨ (G̃, É) = (H̃, É × e) and is defined as: (H̃, É × e) = {( (e1, e2), ⟨s,AbTH̃(e1,e2) (s),ReTH̃(e1,e2) (s), ReFH̃(e1,e2) (s),AbFH̃(e1,e2) (s)⟩ ) : (e1, e2) ∈ e× e, s ∈ M } . where AbTH̃(e)(s) = max { AbTF̃ (e)(s),AbTG̃(e)(s) } , ReTH̃(e)(s) = max { ReTF̃ (e)(s),ReTG̃(e)(s) } , ReFH̃(e)(s) = min { ReFF̃ (e)(s),ReFG̃(e)(s) } , AbFH̃(e)(s) = min { AbFF̃ (e)(s),AbFG̃(e)(s) } . Definition 17. The quadri-partitioned neutrosophic soft set (F̃ , É) over the key set M is said to be a null quadri-partitioned neutrosophic soft set if AbTF̃ (e)(s) = 0, ReTF̃ (e)(s) = 0, ∀e ∈ É,∀s ∈ M, ReFF̃ (e)(s) = 1, AbFF̃ (e)(s) = 1, ∀e ∈ É,∀s ∈ M. It is denoted as 0(M,É). Definition 18. A quadri-partitioned neutrosophic soft set (F̃ , É) over the key set M is called an absolute quadri-partitioned neutrosophic soft set if AbTF̃ (e)(s) = 1, ReTF̃ (e)(s) = 1, ∀e ∈ É,∀s ∈ M, ReFF̃ (e)(s) = 0, AbFF̃ (e)(s) = 0, ∀e ∈ É,∀s ∈ M. Clearly, 0c (M,É) = 1(M, É), 1c (M,É) = 0(M,É). Definition 19. The family of all quadri-partitioned neutrosophic soft sets over M is designated as PNSS(M). A QPNS point for every point s ∈ M is defined as s⋋⟨p1,p2,p3,p4⟩ for 0 ⪯ p1, p2, p3, p4 ⪯ 1, e ∈ É, and is given by: s⋋⟨p1,p2,p3,p4⟩(Y) = { ⟨p1, p2, p3, p4⟩, if e = e′ and Y = s, (0, 0, 0, 1), if e′ ̸= e or Y ̸= s. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 10 of 32 Definition 20. Let (F̃ , É) be a quadri-partitioned neutrosophic soft set over the key set M. A point s⋋⟨p1,p2,p3,p4⟩ ∈ QPNSS(F̃ , É) if p1 ⪯ AbTF̃ (e)(s), p2 ⪯ ReTF̃ (e)(s), p3 ⪰ ReFF̃ (e)(s), p4 ⪰ AbFF̃ (e)(s). Theorem 1. Let (F̃ , É), (G̃, É), and (H̃, É) be quadri-partitioned neutrosophic soft sets over the key set M. Then, the following properties hold: 1. (F̃ , É) ⋓ [ (G̃, É) ⋓ (H̃, É) ] = [ (F̃ , É) ⋓ (G̃, É) ] ⋓ (H̃, É) 2. (F̃ , É) ⋒ [ (G̃, É) ⋒ (H̃, É) ] = [ (F̃ , É) ⋒ (G̃, É) ] ⋒ (H̃, É) 3. (F̃ , É) ⋓ [ (G̃, É) ⋒ (H̃, É) ] = [ (F̃ , É) ⋓ (G̃, É) ] ⋒ [ (F̃ , É) ⋓ (H̃, É) ] 4. (F̃ , É) ⋒ [ (G̃, É) ⋓ (H̃, É) ] = [ (F̃ , É) ⋒ (G̃, É) ] ⋓ [ (F̃ , É) ⋒ (H̃, É) ] 5. (F̃ , É) ⋓ 0(M,É) = (F̃ , É) 6. (F̃ , É) ⋒ 0(M,É) = 0(M,É) 7. (F̃ , É) ⋓ 1(M,É) = 1(M,É) 8. (F̃ , É) ⋒ 1(M,É) = (F̃ , É) Proof. Given (F̃ , É), (G̃, É), and (H̃, É) are quadri-partitioned neutrosophic soft sets, then, (F̃ , É)∨̃[(G̃, É)∨̃(H̃, É)] = [(F̃ , É)∨̃(G̃, É)]∨̃(H̃, É) ⇒ s⋋⟨p1,p2,p3,p4⟩ ∈ (F̃ , É)∨̃[(G̃, É)∨̃(H̃, É)] ⇒ s⋋⟨p1,p2,p3,p4⟩ ∈ (F̃ , É) or s⋋⟨p1,p2,p3,p4⟩ ∈ [(G̃, É)∨̃(H̃, É)] ⇒ s⋋⟨p1,p2,p3,p4⟩ ∈ (F̃ , É) s⋋⟨p1,p2,p3,p4⟩ ∈ ( e, s, ( max [ AbTG̃(e)(s), AbTH̃(e)(s) ] ,max [ ReTG̃(e)(s), ReTH̃(e)(s) ]) , ( min [ ReFG̃(e)(s), ReFH̃(e)(s) ] ,min [ AbFG̃(e)(s), AbFH̃(e)(s) ])) , ∀e ∈ É, s ∈ M ⇒ s⋋⟨p1,p2,p3,p4⟩ ∈ ( e, s, ( max [ AbTF̃ (e)(s), AbTG̃(e)(s) ] ,max [ ReTF̃ (e)(s), ReTG̃(e)(s) ]) , M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 11 of 32 ( min [ ReFF̃ (e)(s), ReFG̃(e)(s) ] ,min [ AbFF̃ (e)(s), AbFG̃(e)(s) ])) , ∀e ∈ É, s ∈ M i.e. s⋋⟨p1,p2,p3,p4⟩ ∈ [(F̃ , É)∨̃(G̃, É)]∨̃(H̃, É) ⇒ (F̃ , É)∨̃[(G̃, É)∨̃(H̃, É)] ⊆ [̃(F̃ , É)∨̃(G̃, É)]∨̃(H̃, É) Similarly, [(F̃ , É)∨̃(G̃, É)]∨̃(H̃, É) ⊆ (F̃ , É)∨̃[(G̃, É)∨̃(H̃, É)] ⇒ (F̃ , É)∨̃[(G̃, É)∨̃(H̃, É)] = [(F̃ , É)∨̃(G̃, É)]∨̃(H̃, É) Thus, in a similar fashion, we can prove the rest of the results. Theorem 2. Let (F̃ , É) and (G̃, É) be quadri-partitioned neutrosophic soft sets over the key set M, then [(F̃ , É) ⋓ (G̃, É)]c = (F̃ , É)c ⋒ (G̃, É)c; [(F̃ , É) ⋒ (G̃, É)]c = (F̃ , É)c ⋓ (G̃, É)c. Proof. Given (F̃ , É) and (G̃, É) be quadri-partitioned neutrosophic soft sets, then ∀e ∈ É,∀s ∈ M, (F̃ , É)⋓(G̃, É) = {( s,max [ AbTF̃ (e)(s),AbTG̃(e)(s) ] ,max [ ReTF̃ (e)(s),ReTG̃(e)(s) ] , min [ ReFF̃ (e)(s),ReFG̃(e)(s) ] ,min [ AbFF̃ (e)(s),AbFG̃(e)(s) ])} . [ (F̃ , É) ⋓ (G̃, É) ]c = {( s,min [ AbFF̃ (e)(s),AbFG̃(e)(s) ] ,min [ ReFF̃ (e)(s),ReFG̃(e)(s) ] , max [ ReTF̃ (e)(s),ReTG̃(e)(s) ] ,max [ AbTF̃ (e)(s),AbTG̃(e)(s) ])} . Now, (F̃ , É)c = {〈 s,AbFF̃ (e)(s),ReFF̃ (e)(s),ReTF̃ (e)(s),AbTF̃ (e)(s) 〉} , (G̃, É)c = {〈 s,AbFG̃(e)(s),ReFG̃(e)(s),ReTG̃(e)(s),AbTG̃(e)(s) 〉} . Thus, (F̃ , É)c ⋒ (G̃, É)c = {( s,min [ AbFF̃ (e)(s),AbFG̃(e)(s) ] ,min [ ReFF̃ (e)(s),ReFG̃(e)(s) ] , max [ ReTF̃ (e)(s),ReTG̃(e)(s) ] ,max [ AbTF̃ (e)(s),AbTG̃(e)(s) ])} . Therefore, [ (F̃ , É) ⋓ (G̃, É) ]c = (F̃ , É)c ⋒ (G̃, É)c. Similarly, ∀e ∈ É,∀s ∈ M, (F̃ , É)⋒(G̃, É) = {( s,min [ AbTF̃ (e)(s),AbTG̃(e)(s) ] ,min [ ReTF̃ (e)(s),ReTG̃(e)(s) ] , M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 12 of 32 max [ ReFF̃ (e)(s),ReFG̃(e)(s) ] ,max [ AbFF̃ (e)(s),AbFG̃(e)(s) ])} . [ (F̃ , É) ⋒ (G̃, É) ]c = (F̃ , É)c ⋓ (G̃, É)c. Theorem 3. Let (F̃ , É) and (G̃, É) be quadri-partitioned neutrosophic soft sets over key set M. Then: 1. [(F̃ , É)∨̃(G̃, É)]c = (F̃ , É)c∧̃(G̃, É)c. 2. [(F̃ , É)∧̃(G̃, É)]c = (F̃ , É)c∨̃(G̃, É)c. Proof. 1. For all (e1, e2) ∈ e× e and for all s ∈ M: (F̃ , É)∨̃(G̃, É) = ( s,max[AbTF̃ (e)(s),AbTG̃(e)(s)],max[ReTF̃ (e)(s),ReTG̃(e)(s)], min[ReFF̃ (e)(s),ReFG̃(e)(s)],min[AbFF̃ (e)(s),AbFG̃(e)(s)] ) . Taking the complement: [(F̃ , É)∨̃(G̃, É)]c = ( s,min[AbFF̃ (e)(s),AbFG̃(e)(s)],min[ReFF̃ (e)(s),ReFG̃(e)(s)], max[ReTF̃ (e)(s),ReTG̃(e)(s)],max[AbTF̃ (e)(s),AbTG̃(e)(s)] ) . By considering the individual complements: (F̃ , É)c = ( s,AbFF̃ (e)(s),ReFF̃ (e)(s),ReTF̃ (e)(s),AbTF̃ (e)(s) ) , (G̃, É)c = ( s,AbFG̃(e)(s),ReFG̃(e)(s),ReTG̃(e)(s),AbTG̃(e)(s) ) . Therefore, (F̃ , É)c∧̃(G̃, É)c = ( s,min[AbFF̃ (e)(s),AbFG̃(e)(s)],min[ReFF̃ (e)(s),ReFG̃(e)(s)], max[ReTF̃ (e)(s),ReTG̃(e)(s)],max[AbTF̃ (e)(s),AbTG̃(e)(s)] ) . Hence, [(F̃ , É)∨̃(G̃, É)]c = (F̃ , É)c∧̃(G̃, É)c. 2. A similar approach can be used to prove: [(F̃ , É)∧̃(G̃, É)]c = (F̃ , É)c∨̃(G̃, É)c. Example 1. Let M = {s1, s2, s3} be the key set and the set of parameters e = {e1, e2}. Let us develop the quadri-partitioned neutrosophic soft sets (F̃ , É) and (G̃, É) over the key set M as follows: (F̃ , É) = [ e1 = ⟨s1, 2 10 , 3 10 , 7 10 , 8 10⟩, ⟨s2, 4 10 , 4 10 , 6 10 , 4 10⟩, ⟨s3, 2 10 , 4 10 , 6 10 , 3 10⟩, e2 = ⟨s1, 3 10 , 2 10 , 6 10 , 6 10⟩, ⟨s2, 1 10 , 5 10 , 6 10 , 5 10⟩, ⟨s3, 4 10 , 3 10 , 6 10 , 5 10⟩ ] Let M = {s1, s2, s3} be the key set and the set of parameters e = {e1, e2}. The quadri-partitioned neutrosophic soft set (G̃, É) over the key set M is defined as: (G̃, É) = [ e1 = ⟨s1, 4 10 , 3 10 , 6 10 , 6 10⟩, ⟨s2, 4 10 , 5 10 , 6 10 , 3 10⟩, ⟨s3, 3 10 , 5 10 , 6 10 , 2 10⟩, e2 = ⟨s1, 3 10 , 4 10 , 6 10 , 5 10⟩, ⟨s2, 2 10 , 6 10 , 6 10 , 4 10⟩, ⟨s3, 4 10 , 6 10 , 6 10 , 3 10⟩ ] Then, their union, intersection, AND, and OR operations are given as follows: M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 13 of 32 (F̃ , É) ⋓ (G̃, É) = [ e1 = ⟨s1, 4 10 , 3 10 , 6 10 , 6 10⟩, ⟨s2, 4 10 , 5 10 , 6 10 , 3 10⟩, ⟨s3, 3 10 , 5 10 , 6 10 , 2 10⟩, e2 = ⟨s1, 3 10 , 4 10 , 6 10 , 5 10⟩, ⟨s2, 2 10 , 6 10 , 6 10 , 4 10⟩, ⟨s3, 4 10 , 6 10 , 6 10 , 3 10⟩ ] (F̃ , É) ⋒ (G̃, É) = [ e1 = ⟨s1, 2 10 , 3 10 , 7 10 , 8 10⟩, ⟨s2, 4 10 , 4 10 , 6 10 , 4 10⟩, ⟨s3, 2 10 , 4 10 , 6 10 , 3 10⟩, e2 = ⟨s1, 3 10 , 2 10 , 6 10 , 6 10⟩, ⟨s2, 1 10 , 5 10 , 6 10 , 5 10⟩, ⟨s3, 4 10 , 3 10 , 6 10 , 5 10⟩ ] The AND and OR operations of the quadri-partitioned neutrosophic soft sets are defined as follows: (F̃ , É) ∧ (G̃, É) =  (e1, e1) = ⟨s1, 2 10 , 3 10 , 7 10 , 8 10⟩, ⟨s2, 4 10 , 4 10 , 6 10 , 4 10⟩, ⟨s3, 2 10 , 4 10 , 6 10 , 3 10⟩, (e1, e2) = ⟨s1, 2 10 , 3 10 , 7 10 , 8 10⟩, ⟨s2, 2 10 , 4 10 , 6 10 , 4 10⟩, ⟨s3, 4 10 , 3 10 , 6 10 , 4 10⟩, (e2, e1) = ⟨s1, 3 10 , 2 10 , 7 10 , 6 10⟩, ⟨s2, 1 10 , 5 10 , 6 10 , 5 10⟩, ⟨s3, 3 10 , 3 10 , 6 10 , 4 10⟩, (e2, e2) = ⟨s1, 3 10 , 2 10 , 6 10 , 6 10⟩, ⟨s2, 1 10 , 5 10 , 6 10 , 5 10⟩, ⟨s3, 4 10 , 3 10 , 6 10 , 5 10⟩  (F̃ , É) ∨ (G̃, É) =  (e1, e1) = ⟨s1, 4 10 , 3 10 , 6 10 , 6 10⟩, ⟨s2, 4 10 , 5 10 , 6 10 , 3 10⟩, ⟨s3, 3 10 , 5 10 , 6 10 , 2 10⟩, (e1, e2) = ⟨s1, 2 10 , 3 10 , 7 10 , 8 10⟩, ⟨s2, 2 10 , 4 10 , 6 10 , 4 10⟩, ⟨s3, 4 10 , 3 10 , 6 10 , 4 10⟩, (e2, e1) = ⟨s1, 4 10 , 3 10 , 6 10 , 6 10⟩, ⟨s2, 4 10 , 5 10 , 6 10 , 3 10⟩, ⟨s3, 4 10 , 5 10 , 6 10 , 2 10⟩, (e2, e2) = ⟨s1, 3 10 , 4 10 , 6 10 , 5 10⟩, ⟨s2, 2 10 , 6 10 , 6 10 , 4 10⟩, ⟨s3, 4 10 , 6 10 , 6 10 , 3 10⟩  4. A New Approach to Operations on Quadri-partitioned Neutrosophic Soft Topological Space The notion of quadri-partitioned neutrosophic soft topological space is presented in this section. The terms QPNS semi-open, QPNS pre-open and QPNS ∗bopen 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 closer, QPNS exterior, QPNS boundary, and QPNS interior. Definition 21. Let the quadri-partitioned neutrosophic soft set (M̃, É) be the family of all quadri-partitioned neutrosophic soft sets, and let τ ⊂ QPNSS(M̃, É). Then, τ is a quadri- partitioned neutrosophic soft topology (QPNST) on M̃ if: (i) 0(⟨M⟩,É), 1(⟨M⟩,É) ∈ τ , (ii) The union of any number of quadri-partitioned neutrosophic soft sets in τ is in τ , (iii) The intersection of a finite number of quadri-partitioned neutrosophic soft sets in τ is in τ . Then, (M̃, τ, É) is said to be a quadri-partitioned neutrosophic soft topological space (QPNSTS) over M̃. Definition 22. A quadri-partitioned neutrosophic soft topological space (M̃, τ, É) over M̃ is denoted as QPNSTS. A quadri-partitioned neutrosophic soft set (F̃ , É) is a QPNS neighbor- hood of a QPNS point s⋋⟨p1,p2,p3,p4⟩ ∈ (F̃ , É), if there exists a QPNS open set (G̃, É) such that s⋋⟨p1,p2,p3,p4⟩ ∈ (G̃, É). Definition 23. Let (M, τ1, É) and (M, τ2, É) be two quadri-partitioned neutrosophic soft topo- logical spaces. Then, (M, τ1, τ2, É) is called a quadri-partitioned neutrosophic soft bitopological space (QPNSBTS). If (M, τ1, τ2, É) is a quadri-partitioned neutrosophic soft topological space, a QPNSS subset (F̃ , É) is open in (M, τ1, τ2, É) if there exist a QPNSS open set (G̃, É) ∈ τ1 and a QPNSS open set (H̃, É) ∈ τ2 such that: (F̃ , É) = (G̃, É) ⋓ (H̃, É). M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 14 of 32 Theorem 4. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft topological space. Then τ1 ⋒ τ2 is a quadri-partitioned neutrosophic soft topological space on M. Proof. The first and third requirements are clear, and we move forward as follows for the second condition. Let {(Υ̃i, É); i ∈ I} ∈ τ1 ⋒ τ2. Then (Υ̃i, É) ∈ τ1 and (Υ̃i, É) ∈ τ2. Since τ1 and τ2 are quadri-partitioned neutrosophic soft topological spaces on M, it follows that:⋃ i∈I (Υ̃i, É) ∈ τ1, ⋃ i∈I (Υ̃i, É) ∈ τ2. Thus, we conclude that: ⋃ i∈I (Υ̃i, É) ∈ τ1 ⋒ τ2. Remark 1. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft topological space. Then τ1 ⋓ τ2 need not be a quadri-partitioned neutrosophic soft topological space on M. This is illustrated in the following Example 2. Example 2. Let M = {s1, s2, s3}, É = {e1, e2}, and define the topologies: τ1 = {0(M, É), 1(M, É), (Υ̃, É), (Ξ̃, É), (Λ̃, É)}, τ2 = {0(M, É), 1(M, É), (Ĩ , É), (J̃ , É)}. where (Υ̃, É), (Ξ̃, É), (Λ̃, É), (Ĩ , É), and (J̃ , É) are quadri-partitioned neutrosophic soft subsets defined as follows: (Υ̃, É) =  e1 = ⟨s1, 2 10 , 3 10 , 8 10 , 8 10 ⟩, ⟨s2, 4 10 , 4 10 , 4 10 , 4 10 ⟩, ⟨s3, 2 10 , 4 10 , 3 10 , 3 10 ⟩; e2 = ⟨s1, 3 10 , 2 10 , 6 10 , 6 10 ⟩, ⟨s2, 1 10 , 5 10 , 5 10 , 5 10 ⟩, ⟨s3, 4 10 , 3 10 , 5 10 , 5 10 ⟩.  (J̃ , É) =[ e1 = ⟨s1, 4 10 , 3 10 , 6 10 , 6 10 ⟩, ⟨s2, 4 10 , 5 10 , 3 10 , 3 10 ⟩, ⟨s3, 3 10 , 5 10 , 2 10 , 2 10 ⟩, e2 = ⟨s1, 3 10 , 4 10 , 5 10 , 5 10 ⟩, ⟨s2, 2 10 , 6 10 , 4 10 , 4 10 ⟩, ⟨s3, 4 10 , 6 10 , 3 10 , 3 10 ⟩ ] . (L̃, É) =[ e1 = ⟨s1, 5 10 , 4 10 , 4 10 , 4 10 ⟩, ⟨s2, 6 10 , 6 10 , 2 10 , 2 10 ⟩, ⟨s3, 4 10 , 6 10 , 1 10 , 1 10 ⟩, e2 = ⟨s1, 4 10 , 6 10 , 3 10 , 3 10 ⟩, ⟨s2, 3 10 , 7 10 , 3 10 , 3 10 ⟩, ⟨s3, 5 10 , 7 10 , 1 10 , 1 10 ⟩ ] . M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 15 of 32 (Ĩ , É) =[ e1 = ⟨s1, 6 10 , 6 10 , 2 10 , 2 10 ⟩, ⟨s2, 6 10 , 6 10 , 2 10 , 2 10 ⟩, ⟨s3, 4 10 , 6 10 , 1 10 , 1 10 ⟩, e2 = ⟨s1, 5 10 , 6 10 , 2 10 , 2 10 ⟩, ⟨s2, 6 10 , 7 10 , 2 10 , 2 10 ⟩, ⟨s3, 5 10 , 5 10 , 1 10 , 1 10 ⟩ ] . (J̃ , É) =[ e1 = ⟨s1, 01 10 , 02 10 , 07 10 , 07 10 ⟩, ⟨s2, 04 10 , 04 10 , 03 10 , 03 10 ⟩, ⟨s3, 02 10 , 04 10 , 02 10 , 02 10 ⟩, e2 = ⟨s1, 03 10 , 02 10 , 05 10 , 05 10 ⟩, ⟨s2, 01 10 , 05 10 , 05 10 , 05 10 ⟩, ⟨s3, 04 10 , 03 10 , 05 10 , 05 10 ⟩ ] . Here, τ1 ⋓ τ2 = {0(M̃,É), 1(M̃,É), (Υ̃, É), (Ξ̃, É), (Λ̃, É), (Ĩ , É), (J̃ , É)} is not a quadri-partitioned neutrosophic soft topological space on M̃ as (Λ̃, É)⋓(Ĩ , É) does not belong to τ1⋓τ2. This justifies that the union of two quadri-partitioned neutrosophic soft topological spaces is not necessarily a quadri-partitioned neutrosophic soft topological space. Definition 24. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft bi-topological space. Then a quadri-partitioned neutrosophic soft set, (Λ̃, É) = [( e, 〈 s,AbTΛ̃(e)(s),ReTΛ̃(e)(s),ReFΛ̃(e)(s),AbFΛ̃(e)(s) 〉 : s ∈ M ) : e ∈ É ] is a pairwise QPNS open set if there exist a QPNS open set (Υ̃, É) in τ1 and a QPNS open set (Ξ̃, É) in τ2 such that for all s ∈ M, (Λ̃, É) = (Υ̃, É)⋓(Ξ̃, É) =  e,  s, AbTH̃(e)(s) = max[AbTF̃ (e)(s),AbTG̃(e)(s)], ReTH̃(e)(s) = max[ReTF̃ (e)(s),ReTG̃(e)(s)], ReFH̃(e)(s) = min[ReFF̃ (e)(s),ReFG̃(e)(s)], AbFH̃(e)(s) = min[AbFF̃ (e)(s),AbFG̃(e)(s)]   : e ∈ É  . This is denoted by QPNSO(M, É). Definition 25. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft bi-topological space. Then a quadri-partitioned neutrosophic soft set, (Λ̃, É) = [( e, 〈〈 s,AbTH̃(e)(s),ReTH̃(e)(s),ReFH̃(e)(s),AbFΛ̃(e)(s) 〉〉 : s ∈ M ) : e ∈ É ] is called a pairwise QPNSCS if (Λ̃, É)c is a pairwise QPNSO. (Λ̃, É) is a QPNS closed set if there exists a QPNS closed set (Υ̃, É) in τ1 and a QPNS closed set (Ξ̃, É) in τ2 such that for all s ∈ M, (Λ̃, É) = (Υ̃, É)⋒ (Ξ̃, É) =  e,  s, AbTH̃(e)(s) = min[AbTF̃ (e)(s),AbTG̃(e)(s)], ReTH̃(e)(s) = min[ReTF̃ (e)(s),ReTG̃(e)(s)], ReFH̃(e)(s) = max[ReFF̃ (e)(s),ReFG̃(e)(s)], AbFH̃(e)(s) = max[AbFF̃ (e)(s),AbFG̃(e)(s)]   : e ∈ É  . This is denoted by QPNSC(M, É). M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 16 of 32 Definition 26. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft bi-topological space over M, and ( ˜̈Υ, É) be a QPNSS. Then: (i) ( ˜̈Υ, É) is QPNS semi-open if ( ˜̈Υ, É) ⊆ NScl(NSint( ˜̈Υ, É)). (ii) ( ˜̈Υ, É) is QPNS pre-open (p-open) if ( ˜̈Υ, É) ⊆ NSint(NScl( ˜̈Υ, É)). (iii) ( ˜̈Υ, É) is QPNS ∗b open if ( ˜̈Υ, É) ⊆ NScl(NSint( ˜̈Υ, É)) ⋓NSint(NScl( ˜̈Υ, É)), and QPNS ∗b close if ( ˜̈Υ, É) ⊇ NScl(NSint( ˜̈Υ, É)) ⋒NSint(NScl( ˜̈Υ, É)). Definition 27. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft bi-topological space over M, and ( ˜̈Υ, É) be a QPNS. The interior of ( ˜̈Υ, É), denoted by ( ˜̈Υ, É)◦, is the union of all QPNS p-open sets of ( ˜̈Υ, É). Clearly, ( ˜̈Υ, É)◦ is the largest QPNS p-open set contained in ( ˜̈Υ, É). Definition 28. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft bi-topological space, and ( ˜̈Υ, É) be a QPNS. The frontier of ( ˜̈Υ, É), denoted by Fr(( ˜̈Υ, É)), is a QPNS point s⋋⟨p1,p2,p3,p4⟩ such that every QPNS p-open set containing s⋋⟨p1,p2,p3,p4⟩ contains at least one point of ( ˜̈Υ, É) and at least one QPNS point of ( ˜̈Υ, É)c. Definition 29. If (M, τ1, τ2, É) is a quadri-partitioned neutrosophic soft bi-topological space and ( ˜̈Υ, É) is a QPNS, then the exterior of ( ˜̈Υ, É), denoted by Ext(( ˜̈Υ, É)), is a QPNS point s⋋⟨p1,p2,p3,p4⟩ such that s⋋⟨p1,p2,p3,p4⟩ is in the interior of ( ˜̈Υ, É)c, i.e., there exists a QPNS p-open set (g̃, É) such that s⋋⟨p1,p2,p3,p4⟩ ∈ (g̃, É) ⊆ ( ˜̈Υ, É)c. Definition 30. If (M̃, τ1, τ2, É) and (⟨Y ⟩,F1,F2, É) are quadri-partitioned neutrosophic soft bi-topological spaces, and (f, ˜̈Υ) : (M̃, τ1, τ2, É) → (⟨Y ⟩,F1,F2, É) is a QPNS mapping, then (f, ˜̈Υ) is said to be a QPNS p-close mapping if the image (f, ˜̈Υ)( ˜̈Υ, É) of each QPNS p-closed set ( ˜̈Υ, É) over M̃ is a QPNS p-closed set in ⟨Y ⟩. Theorem 5. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft bi-topological space over M and ( ˜̈Υ, É) be a QPNS subset. Then, ( ˜̈Υ, É) is a QPNS p-open set if and only if ( ˜̈Υ, É) = ( ˜̈Υ, É)◦. Proof. Let ( ˜̈Υ, É) be a QPNS p-open set. Then, the largest QPNS p-open set contained in ( ˜̈Υ, É) is equal to ( ˜̈Υ, É). Hence, ( ˜̈Υ, É) = ( ˜̈Υ, É)◦. Conversely, it is known that ( ˜̈Υ, É)◦ is a QPNS p-open set, and if ( ˜̈Υ, É) = ( ˜̈Υ, É)◦, then ( ˜̈Υ, É) is a QPNS p-open set. Theorem 6. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft bi-topological space over M, and let (Υ̃, É) and (Ξ̃, É) be QPNS subsets. Then: (i) [(Υ̃, É)◦]◦ = (Υ̃, É)◦, M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 17 of 32 (ii) (0(M̃,É)) ◦ = 0(M̃,É) and (1(M̃,É)) ◦ = 1(M̃,É), (iii) If (Υ̃, É) ⊆ (Ξ̃, É), then (Υ̃, É)◦ ⊆ (Ξ̃, É)◦, (iv) [(Υ̃, É) ⋒ (Ξ̃, É)]◦ = (Υ̃, É)◦ ⋒ (Ξ̃, É)◦, (v) (Υ̃, É)◦ ⋓ (Ξ̃, É)◦ ⊆ [(Υ̃, É) ⋓ (Ξ̃, É)]◦. Proof. (i) Since (Υ̃, É)◦ is a QPNS p-open set, it follows that [(Υ̃, É)◦]◦ = (Υ̃, É)◦. (ii) Since 0(M̃,É) and 1(M̃,É) are always QPNS p-open sets, we have: (0(M̃,É)) ◦ = 0(M̃,É), and (1(M̃,É)) ◦ = 1(M̃,É). (iii) Given that (Υ̃, É)◦ ⊆ (Υ̃, É) ⊆ (Ξ̃, É) and (Ξ̃, É)◦ ⊆ (Ξ̃, É), we conclude that (Υ̃, É)◦ ⊆ (Ξ̃, É)◦. (iv) Since (Υ̃, É) ⋒ (Ξ̃, É) ⊆ (Υ̃, É) and (Υ̃, É) ⋒ (Ξ̃, É) ⊆ (Ξ̃, É), we have: [(Υ̃, É) ⋒ (Ξ̃, É)]◦ ⊆ (Υ̃, É)◦ ⋒ (Ξ̃, É)◦. Conversely, since (Υ̃, É)◦ ⊆ (Υ̃, É) and (Ξ̃, É)◦ ⊆ (Ξ̃, É), we also obtain: (Υ̃, É)◦ ⋒ (Ξ̃, É)◦ ⊆ (Υ̃, É) ⋒ (Ξ̃, É). Since (Υ̃, É)◦ ⋒ (Ξ̃, É)◦ is the largest QPNS p-open set contained in (Υ̃, É) ⋒ (Ξ̃, É), we conclude: [(Υ̃, É) ⋒ (Ξ̃, É)]◦ = (Υ̃, É)◦ ⋒ (Ξ̃, É)◦. (v) Since (Υ̃, É) ⊆ (Υ̃, É) ⋓ (Ξ̃, É) and (Ξ̃, É) ⊆ (Υ̃, É) ⋓ (Ξ̃, É), it follows that: (Υ̃, É)◦ ⊆ [(Υ̃, É) ⋓ (Ξ̃, É)]◦, and (Ξ̃, É)◦ ⊆ [(Υ̃, É) ⋓ (Ξ̃, É)]◦. Thus, (Υ̃, É)◦ ⋓ (Ξ̃, É)◦ ⊆ [(Υ̃, É) ⋓ (Ξ̃, É)]◦. Theorem 7. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft bi-topological space over M, and if (Υ̃, É) is a QPNS subset, then (Υ̃, É) is a QPNS p-closed set if and only if (Υ̃, É) = (Υ̃, É). Proof. Suppose that (Υ̃, É) is a QPNS p-closed set, then we have: (Υ̃, É)d = (Υ̃, É) which implies that (Υ̃, É) ⋓ (Υ̃, É)d = (Υ̃, É) Thus, (Υ̃, É) = (Υ̃, É) Conversely, if (Υ̃, É) = (Υ̃, É), then (Υ̃, É) ⋓ (Υ̃, É)d = (Υ̃, É) which implies that (Υ̃, É)d = (Υ̃, É) Thus, (Υ̃, É) is a QPNS p-closed set. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 18 of 32 Theorem 8. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft bi-topological space over M, and let (Υ̃, É) and (Ξ̃, É) be QPNS subsets. Then: (i) [(Υ̃, É)] = (Υ̃, É), (ii) 0(⟨M⟩,É) = 0(⟨M⟩,É) and 1(⟨M⟩,É) = 1(⟨M⟩,É), (iii) If (Υ̃, É) ⊆ (Ξ̃, É), then (Υ̃, É) ⊆ (Ξ̃, É), (iv) (Υ̃, É) ⋓ (Ξ̃, É) = (Υ̃, É) ⋓ (Ξ̃, É), (v) (Υ̃, É) ⋒ (Ξ̃, É) ⊆ (Υ̃, É) ⋒ (Ξ̃, É). Proof. (i) If (Υ̃, É) = (Ξ̃, É), then (Ξ̃, É) is a QPNS p-closed set. Hence, if (Ξ̃, É) and its closure are equal, then (Υ̃, É) = (Υ̃, É). (ii) Since 0(⟨M⟩,É) and 1(⟨M⟩,É) are always QPNS p-closed sets, it follows that 0(⟨M⟩,É) = 0(⟨M⟩,É) and 1(⟨M⟩,É) = 1(⟨M⟩,É). (iii) Given (Υ̃, É) ⊆ (Υ̃, É) and (Ξ̃, É) ⊆ (Ξ̃, É), we obtain (Υ̃, É) ⊆ (Ξ̃, É) ⊆ (Ξ̃, É). Since (Υ̃, É) is the smallest QPNS p-closed set covering (Υ̃, É), it follows that (Υ̃, É) ⊆ (Ξ̃, É). (iv) Since (Υ̃, É) ⊆ (Υ̃, É) ⋓ (Ξ̃, É) and (Ξ̃, É) ⊆ 0(⟨M⟩,É) ⋓ (Ξ̃, É), we obtain (Υ̃, É) ⊆ (Υ̃, É) ⋓ (Ξ̃, É) and (Ξ̃, É) ⊆ (Υ̃, É) ⋓ (Ξ̃, É). Thus, (Υ̃, É) ⋓ (Ξ̃, É) ⊆ (Υ̃, É) ⋓ (Ξ̃, É). Conversely, since (Υ̃, É) ⊆ (Υ̃, É) and (Ξ̃, É) ⊆ (Ξ̃, É), we conclude that (Υ̃, É) ⋓ (Ξ̃, É) ⊆ (Υ̃, É) ⋓ (Ξ̃, É). Since (Υ̃, É) ⋓ (Ξ̃, É) is the smallest QPNS p-closed set enclosing (Υ̃, É) ⋓ (Ξ̃, É), it follows that (Υ̃, É) ⋓ (Ξ̃, É) ⊆ (Υ̃, É) ⋓ (Ξ̃, É). (v) Using a similar argument, we obtain (Υ̃, É) ⋒ (Ξ̃, É) ⊆ (Υ̃, É) ⋒ (Ξ̃, É). Theorem 9. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft bi-topological space over M, and let (Υ̃, É) be a QPNSS. Then, the following hold: (i) [(Υ̃, É)]c = [(Υ̃, É)c]◦, (ii) [(Υ̃, É)◦]c = [(Υ̃, É)c]. Proof. (i) (Υ̃, É) = ⋒{(H̃, É) ∈ (M, τ1, τ2, É)c : (H̃, É) ⊇ (Υ̃, É)} ⇒ [(Υ̃, É)]c = [ ⋒{(H̃, É) ∈ (M, τ1, τ2, É)c : (H̃, É) ⊇ (Υ̃, É)} ]c = ⋓{(H̃, É)c ∈ (M, τ1, τ2, É) : (H̃, É)c ⊆ (Υ̃, É)c} = [(Υ̃, É)c]◦. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 19 of 32 (ii) (Υ̃, É)◦ = ⋓{(H̃, É) ∈ (M, τ1, τ2, É) : (H̃, É) ⊆ (Υ̃, É)} ⇒ [(Υ̃, É)◦]c = [ ⋒{(H̃, É) ∈ (M, τ1, τ2, É) : (H̃, É) ⊆ (Υ̃, É)} ]c = ⋒{(H̃, É)c ∈ (M, τ1, τ2, É)c : (H̃, É)c ⊇ (Υ̃, É)c} = [(Υ̃, É)c]. Theorem 10. Let (M, τ1, τ2, É) be a quadri-partitioned neutrosophic soft bi-topological space over M. If (Υ̃, É) and (H̃, É) are QPNS subsets, then: (i) Ext((Υ̃, É) ⋓ (H̃, É)) = Ext((Υ̃, É)) ⋓ Ext((H̃, É)). (ii) Ext((Υ̃, É) ⋒ (H̃, É)) ⊇ Ext((Υ̃, É)) ⋓ Ext((H̃, É)). (iii) Fr((Υ̃, É) ⋓ (H̃, É)) ⊆ Fr(Υ̃, É) ⋓ Fr(H̃, É). (iv) Fr((Υ̃, É) ⋒ (H̃, É)) ⊆ Fr(Υ̃, É) ⋓ Fr(H̃, É). Proof. (i) Since Ext((Υ̃, É) ⋓ (H̃, É)) = (((Υ̃, É) ⋓ (H̃, É))C)◦ = ((Υ̃, É)C ⋒ (H̃, É)C)◦ = ((Υ̃, É)C)◦ ⋒ ((H̃, É)C)◦ = Ext((Υ̃, É)) ⋒ Ext((H̃, É)). (ii) We have Ext((Υ̃, É) ⋒ (H̃, É)) = (((Υ̃, É) ⋒ (H̃, É))C)◦ = (((Υ̃, É)C ⋓ (H̃, É)C))◦ ⊇ ((Υ̃, É)C)◦ ⋓ ((H̃, É)C)◦ = Ext((Υ̃, É)) ⋓ Ext((H̃, É)). (iii) For the frontier: Fr((Υ̃, É) ⋓ (H̃, É)) = (Υ̃, É) ⋓ (H̃, É) ⋒ ((Υ̃, É) ⋓ (H̃, É))C = ((Υ̃, É) ⋓ (H̃, É)) ⋒ ((Υ̃, É)C ⋒ (H̃, É)C) ⊆ Fr((Υ̃, É)) ⋓ Fr((H̃, É)). (iv) Similarly, for the intersection: Fr((Υ̃, É) ⋒ (H̃, É)) = (Υ̃, É) ⋒ (H̃, É) ⋒ ((Υ̃, É) ⋒ (H̃, É))C ⊆ ((Υ̃, É) ⋒ (H̃, É)) ⋒ ((Υ̃, É)C ⋓ (H̃, É)C) = Fr((Υ̃, É)) ⋓ Fr((H̃, É)). M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 20 of 32 5. Few results on QPNS P-Compactness In the context of QPNSBTS and QPNS p-compact spaces, the idea of QPNS compactness is examined in this section. The concept of reducibility to finite sub-covers is one of the features of a QPNS cover that we define and study. Important theorems prove that the intersection of QPNS p-closed sets is non-empty and that a QPNS p-compact space preserves the finite intersection property among its QPNS p-closed sets. We also prove the existence of disjoint QPNS p-open sets separating disjoint QPNS p-compact subsets in a QPNS p-Hausdorff space and show that a QPNS p-closed subset of a QPNS p-compact space stays QPNS p-compact. The theoretical underpinning of QPNS spaces is improved by this work, which advances our knowledge of compactness in soft topological spaces. Definition 31. Let (Υ̃, É) = {(Υ̃, É)i} be the class of subsets of M and (Θ̃, É) ⊆ M̃. If (Θ̃, É) ⊆ ⋓̃(Υ̃, É)i, then the class {(Υ̃, É)i} is called a QPNS cover of (Θ̃, É). This cover is known as finite, countable, or QPNS p-open according to whether the members of the above class are finite, countable, or QPNS p-open, respectively. Additionally, let: (Θ̃, É)1 = (Υ̃, É)1 ⋓ (Υ̃, É)2 ⋓ (Υ̃, É)3. If (Θ̃, É)1 = {(Υ̃, É)i : i ∈ Ĩ} and (Θ̃, É)2 = {(Λ̃, É)i : i ∈ Ĩ} are two QPNS covers of a set (Θ̃, É), i.e., (Θ̃, É) ⊆ ⋓̃(Υ̃, É)i and (Θ̃, É) ⊆ ⋓̃(Λ̃, É)i, such that every member of (Θ̃, É)2 is also a member of (Θ̃, É)1, then (Θ̃, É)2 is called the QPNS sub-cover of (Θ̃, É)1. Definition 32. Let (Θ̃, É) = {(Υ̃, É)i : i ∈ Ĩ} be a class of QPNS subsets of M, and suppose (Θ̃, É) ⊆ M̃ such that (Θ̃, É) ⊆ ⋓̃(Υ̃, É)i. Then, as mentioned above, (Θ̃, É) = {(Υ̃, É)i : i ∈ Ĩ} is a QPNS p-cover. If we can select a finite number of QPNS sets from the above class (Θ̃, É), that is, if we can have (Υ̃, É)i1 , (Υ̃, É)i2 , (Υ̃, É)i3 , . . . , (Υ̃, É)ik from (Θ̃, É) such that (Θ̃, É) ⊆ (Υ̃, É)i1 ⋓ (Υ̃, É)i2 ⋓ (Υ̃, É)i3 ⋓ · · · ⋓ (Υ̃, É)ik , or equivalently, (Θ̃, É) ⊆ ⋓̃k n=1(Υ̃, É)in , then we say that the cover (Θ̃, É) = {(Υ̃, É)i : i ∈ Ĩ} is reducible to a finite QPNS sub-cover. Definition 33. Let (M, τ1, τ2, É) be a QPNSBTS over M, and let (Θ̃, É) ⊆ M̃. The set (Θ̃, É) is called QPNS compact if every QPNS p-open cover of (Θ̃, É) is reducible to a finite sub-cover. If we can find a cover of (Θ̃, É) that cannot be reduced to a finite sub-cover, then we say that (Θ̃, É) is not QPNS p-compact. Similarly, if every QPNS p-open cover of M is reducible to a finite sub-cover, then we say that M is QPNS compact. That is, if (Θ̃, É) = {(Υ̃, É)i : i ∈ Ĩ} M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 21 of 32 and (Θ̃, É) ⊆ ⋓̃i∈Ĩ(Υ̃, É)i, then if we can write (Θ̃, É) ⊆ ⋓̃k n=1(Υ̃, É)in , or equivalently, (Θ̃, É) ⊆ (Υ̃, É)i1 ⋓ (Υ̃, É)i2 ⋓ (Υ̃, É)i3 ⋓ · · · ⋓ (Υ̃, É)ik , then (Θ̃, É) is QPNS compact. Definition 34. If M̃ is a QPNS, the collection of QPNS subsets of M̃, say {(Θ̃, É)α : α ∈ I}, is said to have the finite intersection property if we can find a finite soft QPNS sub-collection {(Θ̃, É)α1 , (Θ̃, É)α2 , (Θ̃, É)α3 , . . . , (Θ̃, É)αn} such that (Θ̃, É)α1⋒̃(Θ̃, É)α2⋒̃(Θ̃, É)α3⋒̃ . . . ⋒̃(Θ̃, É)αn ̸= Φ̃ or equivalently, n⋂ i=1 (Θ̃, É)αi ̸= Φ̃. Theorem 11. Suppose (M̃, τ1, τ2, É) is QPNS p-compact if and only if each class of QPNS p-closed sets with the finite intersection property has a non-empty intersection. Proof. If (M̃, τ1, τ2, É) is QPNS p-compact, we need to show that each class of QPNS p-closed sets with the finite intersection property has a non-empty intersection. Let {(Υ̃, É)α : α ∈ I} be a class of QPNS p-closed sets in M̃ satisfying the finite intersection property, i.e., n⋂ i=1 (Υ̃, É)αi ̸= Φ̃. We prove this result by contradiction. Suppose n⋂ i=1 (Υ̃, É)αi = Φ̃. This implies that ( n⋂ i=1 (Υ̃, É)αi )c = Φ̃c. which further leads to ( n⋃ i=1 (Υ̃, É)αi )c = Φ̃c. Thus, we get M̃ = ( n⋃ i=1 (Υ̃, É)αi )c . Since {(Υ̃, É)α : α ∈ I} is a collection of QPNS p-closed sets, their complements {(Υ̃, É)cα} form a QPNS p-open cover of M̃. But M̃ is given to be QPNS p-compact, so the above QPNS p-open cover must be reducible to a finite sub-cover, meaning M̃ = n⋃ k=1 (Υ̃, É)cik . M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 22 of 32 This implies ( n⋂ i=1 (Υ̃, É)ik )c = M̃c. which further implies [( n⋂ k=1 (Υ̃, É)ik )c]c = Φ̃. Thus, n⋂ k=1 (Υ̃, É)ik = Φ̃, which contradicts the finite intersection property. Hence, n⋂ i=1 (Υ̃, É)αi ̸= Φ̃. Conversely, suppose each class of QPNS p-closed sets with the finite intersection property has a non-empty intersection. We now prove that M̃ is QPNS p-compact using a contradiction. Assume that M̃ is not QPNS p-compact. Then, there exists at least one QPNS p-open cover of M̃ that is not reducible to a finite sub-cover. Let this QPNS p-open cover be {(J̃ , É)i : i ∈ I}. Since this is a QPNS p-open cover of M̃, we have M̃ = ⋃ i∈I (J̃ , É)i. However, by assumption, M̃ ≠ m⋃ k=1 (J̃ , É)ik . This implies m⋂ k=1 (J̃ , É)cik ̸= Φ̃. where (J̃ , É)cik are QPNS p-closed. But by hypothesis,⋂ i∈I (J̃ , É)ci ̸= Φ̃. which contradicts our assumption. Therefore, M̃ must be QPNS p-compact. Theorem 12. If (M̃, τ1, τ2, É) is a QPNSTS and a QPNS p-closed subset of a QPNS p-compact space, then it is also QPNS p-compact. Proof. Suppose that (M̃, τ1, τ2, É) is a QPNSTS such that it is QPNS p-compact and (Υ̃, É) is a QPNS p-closed subset of M̃. We aim to show that (Υ̃, É) is also QPNS p-compact. Let {(J̃ , É)i : i ∈ I} be a QPNS p-open cover of (Υ̃, É). That is, (Υ̃, É) ⊆ ⋃ i∈I (J̃ , É)i. Then, (Υ̃, É) will be QPNS p-compact if this QPNS p-open cover is reducible to a finite QPNS sub-cover. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 23 of 32 Since {(J̃ , É)i : i ∈ I} is a QPNS p-open cover of (Υ̃, É), we have: (Υ̃, É) ⊆ ⋃ i∈I (J̃ , É)i. Also, since M̃ can be expressed as the union of (Υ̃, É) and its QPNS p-complement (Υ̃, É)c, we obtain: M̃ = (Υ̃, É) ⋓ (Υ̃, É)c. Thus, taking the union with (Υ̃, É)c on both sides, we get: (Υ̃, É) ⋓ (Υ̃, É)c ⊆ (⋃ i∈I (J̃ , É)i ) ⋓ (Υ̃, É)c. Since (Υ̃, É)c is QPNS p-open (because (Υ̃, É) is given to be QPNS p-closed), the family {(J̃ , É)i : i ∈ I} ⋓ {(Υ̃, É)c} forms a QPNS p-open cover of M̃. Given that M̃ is QPNS p-compact, this cover must be reducible to a finite subcover. That is, there exists a finite subcollection {(J̃ , É)ik : k = 1, 2, . . . , n} such that M̃ = n⋃ k=1 (J̃ , É)ik ⋓ (Υ̃, É)c. Taking the intersection with (Υ̃, É) on both sides, we obtain: (Υ̃, É) = ( n⋃ k=1 (J̃ , É)ik ⋓ (Υ̃, É)c ) ⋒ (Υ̃, É). Since (Υ̃, É) ⋒ (Υ̃, É)c = Φ̃, it follows that: (Υ̃, É) ⊆ n⋃ k=1 (J̃ , É)ik . Thus, the QPNS p-open cover {(J̃ , É)i : i ∈ I} of (Υ̃, É) is reducible to a finite QPNS subcover. Hence, (Υ̃, É) is QPNS p-compact. Theorem 13. Let (Υ̃, É) be a QPNS p-compact subset of a QPNS p-Hausdorff space M̃, and let s⋋⟨p1,p2,p3,p4⟩ ∈ M̃ − (Υ̃, É). Then there exist QPNS p-open sets (J̃ , É) and (L̃, É) such that s⋋⟨p1,p2,p3,p4⟩ ∈ (J̃ , É), (Υ̃, É) ⊆ (L̃, É), and (J̃ , É) ⋒ (L̃, É) ∼= Φ̃. Proof. Since M̃ is a QPNS p-Hausdorff space, for each s⋋⟨p1,p2,p3,p4⟩ ∈ (Υ̃, É), there exist QPNS p-open sets (J̃s⋋⟨p1,p2,p3,p4⟩ , É) and (L̃s⋋⟨p1,p2,p3,p4⟩ , É) such that (J̃s⋋⟨p1,p2,p3,p4⟩ , É) ⋒ (L̃s⋋⟨p1,p2,p3,p4⟩ , É) ∼= Φ̃. Now, {(L̃x, É) : s⋋⟨p1,p2,p3,p4⟩ ∈ (Υ̃, É)} is a QPNS p-open cover of (Υ̃, É). Since (Υ̃, É) is QPNS p-compact, there exist finitely many points s⋋1,⟨p1,p2,p3,p4⟩, s ⋋ 2,⟨p1,p2,p3,p4⟩, . . . , s ⋋ n,⟨p1,p2,p3,p4⟩ M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 24 of 32 of (Υ̃, É) such that (Υ̃, É) ⊆ n⋃ i=1 (L̃s⋋ i,⟨p1,p2,p3,p4⟩ , É). Define (L̃, É) ∼= n⋃ i=1 (L̃s⋋ i,⟨p1,p2,p3,p4⟩ , É), (J̃ , É) ∼= n⋂ i=1 (J̃s⋋ i,⟨p1,p2,p3,p4⟩ , É). Then (L̃, É) and (J̃ , É) are clearly QPNS p-open sets such that s⋋⟨p1,p2,p3,p4⟩ ∈ (J̃ , É) ⊆ (L̃, É). Moreover, (J̃ , É) ⋒ (L̃, É) ∼= Φ̃. If y⋋⟨p1,p2,p3,p4⟩ ∈ (L̃, É), then y⋋⟨p1,p2,p3,p4⟩ ∈ (L̃s⋋ i,⟨p1,p2,p3,p4⟩ , É) for some s⋋i,⟨p1,p2,p3,p4⟩, implying that y⋋⟨p1,p2,p3,p4⟩ /∈ (J̃s⋋ i,⟨p1,p2,p3,p4⟩ , É). Since (J̃s⋋⟨p1,p2,p3,p4⟩ , É) ⋒ (L̃s⋋⟨p1,p2,p3,p4⟩ , É) ∼= Φ̃, it follows that y⋋⟨p1,p2,p3,p4⟩ /∈ n⋂ i=1 (J̃s⋋ i,⟨p1,p2,p3,p4⟩ , É). Thus, y⋋⟨p1,p2,p3,p4⟩ /∈ (J̃ , É). Theorem 14. Let (Υ̃, É) and (ω̃, É) be disjoint QPNS p-compact subsets of a QPNS p- Hausdorff space M̃. Then, there exist disjoint QPNS p-open sets (J̃ , É) and (L̃, É) such that (Υ̃, É) ⊆ ˜ (J , ´ )E and (ω̃, É) ⊆ ˜ (L, ´ )E. Proof. Since s⋋⟨p1,p2,p3,p4⟩ ∈ (Υ̃, É) and (Υ̃, É) ⋒ (J̃ , É) ≈ Φ̃, we have y⋋⟨p1,p2,p3,p4⟩ /∈ (J̃ , É). Now, since (ω̃, É) is a QPNS p-compact subset of the QPNS p-Hausdorff space M̃, and s⋋⟨p1,p2,p3,p4⟩ /∈ (ω̃, É), there exist QPNS p-open sets (J̃s⋋⟨p1,p2,p3,p4⟩ , É) and (L̃s⋋⟨p1,p2,p3,p4⟩ , É) such that s⋋⟨p1,p2,p3,p4⟩ ∈ (J̃s⋋⟨p1,p2,p3,p4⟩ , É), (J̃ , É) ⊆ (̃Ls⋋⟨p1,p2,p3,p4⟩ , É), and (J̃s⋋⟨p1,p2,p3,p4⟩ , É) ⋒ (L̃s⋋⟨p1,p2,p3,p4⟩ , É) ≈ Φ̃. Clearly, {(J̃s⋋⟨p1,p2,p3,p4⟩ , É) : s⋋⟨p1,p2,p3,p4⟩ ∈ (Υ̃, É)} is a QPNS p-open covering of (Υ̃, É). Since (Υ̃, É) is QPNS p-compact, there exist finitely many points s1 ⋋ ⟨p1,p2,p3,p4⟩, s2 ⋋ ⟨p1,p2,p3,p4⟩, . . . , sn ⋋ ⟨p1,p2,p3,p4⟩ in (Υ̃, É) such that (Υ̃, É) ⊆ n⋃ i=1 (J̃si⋋⟨p1,p2,p3,p4⟩ , É). Let (J̃ , É) = n⋃ i=1 (J̃si⋋⟨p1,p2,p3,p4⟩ , É), M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 25 of 32 (L̃, É) = n⋂ i=1 (L̃si⋋⟨p1,p2,p3,p4⟩ , É). Then, we have (Υ̃, É) ⊆ ˜(J , É) and (ω̃, É) ⊆ (̃L, É). Since (J̃ , É) ⊆ (̃Lsi⋋⟨p1,p2,p3,p4⟩ , É) for each i, it follows that (J̃ , É) and (L̃, É) are QPNS p-open sets. Also, (J̃ , É) ⋒ (L̃, É) ≈ Φ̃, because if y⋋⟨p1,p2,p3,p4⟩ ∈ (J̃ , É), then y⋋⟨p1,p2,p3,p4⟩ ∈ (J̃si⋋⟨p1,p2,p3,p4⟩ , É) for some si ⋋ ⟨p1,p2,p3,p4⟩, which implies y⋋⟨p1,p2,p3,p4⟩ /∈ (L̃si⋋⟨p1,p2,p3,p4⟩ , É). Thus, y⋋⟨p1,p2,p3,p4⟩ /∈ n⋂ i=1 (L̃si⋋⟨p1,p2,p3,p4⟩ , É) = (L̃, É). Hence, the result follows. Theorem 15. Let (Ỹ , τ1, τ2, É) be a QPNS sub-space of (M̃, τ3, τ4, É). Then Ỹ is QPNS p- compact with respect to the QPNSBTS τ1 ⋓ τ2 if and only if Ỹ is QPNS p-compact with respect to the QPNSBTS τ3 ⋓ τ4. Proof. To prove that Ỹ is QPNS p-compact with respect to the QPNSTS τ1 ⋓ τ2, let {(L̃, É)i : i ∈ I} be a soft τ1 ⋓ τ2 QPNS p-open cover of Ỹ , then Ỹ ⊆ ⋃ i∈I (L̃, É)i. Since (L̃, É)i ∈ τ1 ⋓ τ2, there exists (J̃ , É)i ∈ τ3 ⋓ τ4 such that (L̃, É)i = (J̃ , É)i ⋒ (τ3 ⋓ τ4), which implies that (L̃, É)i ⊆ (J̃ , É)i. Thus,⋃ i∈I (L̃, É)i ⊆ ⋃ i∈I (J̃ , É)i. Since Ỹ ⊆ ⋃ i∈I(L̃, É)i, it follows that Ỹ ⊆ ⋃ i∈I(J̃ , É)i. As Ỹ is QPNS p-compact with respect to τ3 ⋓ τ4, the cover {(J̃ , É)i : i ∈ I} has a finite subcover {(J̃ , É)ir : r = 1, 2, . . . , n}. Thus, Ỹ ⊆ n⋃ r=1 (J̃ , É)ir , which implies that {(L̃, É)ir : 1 ≤ r ≤ n} is a soft τ1 ⋓ τ2 QPNS p-open cover of Ỹ , proving QPNS p-compactness. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 26 of 32 Conversely, suppose Ỹ is QPNS p-compact with respect to τ1 ⋓ τ2. Let {(J̃ , É)i : i ∈ I} be a soft τ3 ⋓ τ4 QPNS p-open cover of Ỹ . Then, Ỹ ⊆ ⋃ i∈I (J̃ , É)i. Defining (L̃, É)i = Ỹ ⋒ (J̃ , É)i, we get that {(L̃, É)i : i ∈ I} is a τ1 ⋓ τ2 QPNS p-open cover of Ỹ , which must have a finite subcover {(L̃, É)ir : 1 ≤ r ≤ n}. Thus, Ỹ ⊆ n⋃ r=1 (L̃, É)ir . Since (L̃, É)ir ⊆ (J̃ , É)ir , it follows that Ỹ ⊆ n⋃ r=1 (J̃ , É)ir , which proves that Ỹ is QPNS p-compact with respect to τ3 ⋓ τ4. 6. Comparative Analysis The following Table 1 provides a detailed comparative analysis of the proposed methods, contrasting them with the established techniques discussed in [10]. 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: Feature/Aspect Published Work (NSS & NSTS) [10] Proposed Work (QPNSS & QPN- STS) Basic Frame- work Neutrosophic set theory extends fuzzy sets (FS) and intuitionistic fuzzy sets (IFS) by adding indeter- minacy. Enhances neutrosophic set theory by in- troducing a more detailed division of inde- terminacy into two components: relative truth (RT) and relative falsehood (RF). Indeterminacy Component Traditional neutrosophic set in- cludes a single indeterminate value (neutral membership). Divides indeterminacy into relative truth (RT) and relative falsehood (RF) to refine the concept of indeterminacy. Membership Attributes Standard neutrosophic sets have three membership functions: truth, indeterminacy, and falsehood. The proposed QPNSS introduces four membership functions: absolute truth, relative truth, relative falsehood, and ab- solute falsehood. New Opera- tions Operations like intersection, union, complement, and subset are defined, but with three components (truth, indeterminacy, falsehood). New operations defined for QPNSS, in- cluding quadri-partitioned soft set, sub- sets, complement, set difference, null set, AND, and OR operations. Quadri- Partitioned Neutrosophic Soft Set No equivalent concept in standard neutrosophic set theory. The main novelty: QPNSS, which parti- tions the indeterminate value into RT and RF, providing greater clarity in uncertain situations. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 27 of 32 Feature/Aspect Published Work (NSS & NSTS) [10] Proposed Work (QPNSS & QPN- STS) Topological Space Neutrosophic topological spaces (NST) exist, but they deal with three membership functions (truth, indeterminacy, falsehood). A new concept, QPNSTS, is defined, adding topological properties like pre- open sets (p-open), interior, closure, com- pactness, and reducibility to finite sub- covers. Compactness and Reducibil- ity Compactness in neutrosophic topo- logical spaces is studied, but it fo- cuses on three membership func- tions. Explores QPNS compactness, intersec- tion of QPNS p-closed sets, and QPNS p-compact spaces, introducing new compactness-related concepts. Applications and Use Cases Applied in areas like decision- making, multi-criteria decision anal- ysis, and uncertainty modeling. Focus on providing clearer and more pre- cise representations of uncertainty, with potential applications in soft topological spaces and complex decision-making. Theoretical Contribution Introduces a formal framework for handling uncertain, imprecise, or in- determinate information. Advances neutrosophic theory by provid- ing a more granular structure (quadri- partitioned sets) and extending it to topo- logical spaces with additional properties. Key Intro- duced Concept Neutrosophic sets are based on a single indeterminate value for infor- mation representation. The introduction of QPNSS (quadri- partitioned neutrosophic soft set) and QP- NSTS (quadri-partitioned neutrosophic soft topological space) enhances the the- oretical framework for understanding un- certainty. Table 1: Comparison of Published Work (NSS and NSTS) and Proposed Work (QPNSS and QPNSTS) 7. Applications Here are six potential applications of neutrosophic set theory, particularly the concept of the quadri-portioned neutrosophic soft set (QPNSS): (i) Decision-Making in Uncertain Environments: Neutrosophic set theory can be ap- plied in decision-making processes where there is ambiguity or incomplete information. The QPNSS framework allows for more accurate decisions by considering both relative truth and relative falsehood, offering a better representation of uncertainty in fields like engineering, economics, and management. (ii) Fault Diagnosis and Error Detection: In systems where error detection is essential, such as in machine learning or fault diagnosis of complex systems, the QPNSS framework provides a more refined model by dividing indeterminacy into components of relative truth and relative falsehood. This allows for improved detection and error reduction in systems with uncertain or ambiguous data. (iii) Medical Diagnosis and Healthcare: Neutrosophic set theory can be applied to medi- cal diagnosis, where there is often incomplete or ambiguous information about a patient’s condition. The use of QPNSS allows for better handling of indeterminate symptoms, improving diagnostic accuracy by incorporating both positive and negative evidence, en- hancing the healthcare decision-making process. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 28 of 32 (iv) Image and Signal Processing: The QPNSS framework can be applied in image and signal processing where there may be noise or uncertainty in the data. By using the four membership values (absolute truth, relative truth, relative falsehood, and absolute falsehood), the QPNSS model can help enhance image quality, edge detection, or signal filtering, leading to more precise and effective data interpretation. (v) Multi-Criteria Decision Analysis (MCDA): In situations involving multiple conflict- ing criteria, such as in business, policy-making, or resource allocation, neutrosophic set theory can be used to model the uncertainty of criteria weights and evaluations. The QP- NSS allows for a more nuanced understanding of these conflicts, improving the decision- making process by distinguishing between truth, falsehood, and indeterminacy in each criterion. (vi) Knowledge Representation and Reasoning in AI: In artificial intelligence (AI), especially in the domain of knowledge representation and reasoning, neutrosophic set theory can enhance the ability to represent and reason with incomplete, uncertain, or contradictory knowledge. The QPNSS model helps AI systems to better handle cases where information is uncertain or partially known, allowing for more robust reasoning in uncertain environments, such as robotics or expert systems. 8. Conclusion and Future Work Neutrosophic set theory is an important mathematical framework that is recognized as supe- rior to existing theories of error reduction. This theory extends fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). Its effectiveness can be enhanced by improving the definition of indetermi- nacy a mathematical term for situations in which values cannot be precisely determined. In this paper, methods for improving sensitivity and accuracy with respect to indeterminacy are presented. According to the proposed approach, the indeterminate value is divided into two parts based on membership: relative truth (RT), which represents indeterminacy leaning towards truth, and relative falsehood (RF), which represents indeterminacy leaning towards falsehood. Indeterminacy is considered RT when it leans increasingly towards truth without being classified as true. In other words, this component reflects the extent to which the uncertain value tends to be accurate. It represents situations in which there is a greater likelihood of truth indicated by the context or evidence, but the value cannot be definitively determined to be true. Conversely, it is referred to as RF when it leans more towards untruth without being clearly false. This component reflects the extent to which the uncertain value tends to be false. Similarly, it represents situations where the evidence points to a greater probability of falsity, but the value cannot be definitively labeled as such. Uncertainty arises when there is only one indeterminate value, as it remains unclear whether it favors true or false membership. The findings are more accurate when both RT and RF are captured compared to using a single indeterminate value. This differentiation enhances the overall accuracy of the uncertain scenario. The modified neutrosophic set is referred to as a quadri-portioned neutrosophic soft set (QPNSS). This model contains four membership attributes that are extremely relevant in real-world situations: absolute truth, relative truth, relative false, and absolute false, allowing for greater clarity in uncertain situations. Extremely new operations are defined on QPNSS, including quadri-partitioned neutrosophic soft set, quadri-partitioned neutrosophic soft sub-sets, complement of quadri-partitioned neu- trosophic soft set, absolute quadri-partitioned neutrosophic soft set, quadri-partitioned neutro- sophic soft difference of sets, and absolute null quadri-partitioned neutrosophic soft set. In addition to this, AND and OR operations are also defined. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 29 of 32 Quadri-partitioned neutrosophic soft topological space (QPNSTS) is defined, and the basic results related to this structure are addressed. Three new definitions are given, and based on one of these, known as pre-open (p-open) sets, some results are established. Interior, closure, and related results using these concepts are also addressed. Examples are developed for a clear understanding. In the context of QPNSBTS and QPNS p-compact spaces, the idea of QPNS compactness is examined in this section. The concept of reducibility to finite sub-covers is one of the features of a QPNS cover that we define and study. Important theorems prove that the intersection of QPNS p-closed sets is non-empty and that a QPNS p-compact space preserves the finite intersection property among its QPNS p-closed sets. We also prove the existence of disjoint QPNS p-open sets separating disjoint QPNS p-compact subsets in a QPNS p-Hausdorff space and show that a QPNS p-closed subset of a QPNS p-compact space remains QPNS p-compact. The theoretical underpinning of QPNS spaces is improved by this work, which advances our knowledge of compactness in soft topological spaces. In the future, we will examine the real-world applications of QPNSS in domains such as artificial intelligence, risk assessment, and decision-making. We will create algorithms for pat- tern identification and data analysis based on QPNSS. These algorithms may perform better in complex data sets by taking advantage of the increased sensitivity to indeterminacy. We will explore machine learning techniques to examine how QPNSS can be integrated into machine learning models, especially when dealing with ambiguous or insufficient data. 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-2025-1153-02”. Declarations Author Contributions: All authors equally contributed. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this paper. Availability of Data and Materials: All the data and materials are provided in the manuscript. Nomenclature The following abbreviations are used in this manuscript: • CST: Crisp set theory • FST: Fuzzy set theory • IFST: Intuitionistic fuzzy set theory • NS: Neutrosophic set • SVNS: Single valued neutrosophic set • NST: Neutrosophic set theory • SST: Soft set theory M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 30 of 32 • FSST: Fuzzy soft set theory • VSST: Vague soft set theory • NSST: Neutrosophic soft set theory • DVNS: Double-valued neutrosophic set • DVNI: Double-valued neutrosophic information • DRINW: Double refined indeterminacy neutrosophic weighted • TRINS: Triple refined indeterminate neutrosophic set • SNHNS: Single valued heptapartitioned neutrosophic set. References [1] L. A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. [2] K. Atanassov. Intuitionistic fuzzy set. Fuzzy Sets and Systems, 20(1):87–96, 1986. [3] D. Molodtsov. Soft set theory: first results. Computers and Mathematics with Applications, 37(4-5):19–31, 1999. [4] D. Molodtsov. The Theory of Soft Sets. URSS Publishers, Moscow, 2004. In Russian. [5] W. Xu, J. Ma, S. Wang, and G. Hao. Vague soft sets and their properties. Computers and Mathematics with Applications, 59(2):787–794, 2010. [6] X. Huang, H. Lie, and Y. Yin. Notes on vague soft sets and their properties. Computers and Mathematics with Applications, 64(6):2153–2157, 2012. [7] C. Wang and Y. Lie. Topological structure of vague soft sets. Abstract and Applied Analysis, 2014:504021, 2014. [8] F. Smarandache. Extension of soft set to hyper-soft set, and then to plithogenic hyper-soft set. Neutrosophic Sets and Systems, 22:168–170, 2018. [9] T. Bera and N. K. Mahapatra. Introduction to neutrosophic soft topological space. Opsearch, 54(4):841–867, 2017. [10] T. Y. Ozturk, C. G. Aras, and S. Bayramov. A new approach to operations on neutrosophic soft sets and to neutrosophic soft topological spaces. Communications in Mathematics and Applications, 10(3):481–493, 2019. [11] T. Y. Ozturk, E. Karatas, and A. Yolcu. On neutrosophic soft continuous mappings. Turkish Journal of Mathematics, 45(1):81–95, 2021. [12] A. Mehmood, S. Abdullah, M. M. Al-Shomrani, M. I. Khan, and O. Thinnukool. Some results in neutrosophic soft topology concerning soft ∗b open sets. Journal of Function Spaces, 2021:5544319, 2021. [13] A. B. AL-Nafee. New family of neutrosophic soft sets. Neutrosophic Sets and Systems, 38:482–496, 2020. [14] A. B. AL-Nafee, S. Broumi, and F. Smarandache. Neutrosophic soft bitopological spaces. International Journal of Neutrosophic Science, 14(1):47–56, 2021. [15] H. Dadas and S. Demiralp. Introduction to neutrosophic soft bitopological spaces. Inter- national Journal of Neutrosophic Science, 14(2):78–81, 2021. [16] F. Smarandache. Neutrosophic set: a generalization of the intuitionistic fuzzy sets. Inter- national Journal of Pure and Applied Mathematics, 24(3):287–297, 2005. [17] L. D. R. Kocinac, T. M. Al-shami, and V. Çetkin. Selection principles in the context of soft sets: Menger spaces. Soft Computing, 25(20):12665–12677, 2021. [18] T. M. Al-shami and L. D. R. Kocinac. Nearly soft Menger spaces. Journal of Mathematics, 2020:3807418, 2020. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 31 of 32 [19] T. M. Al-shami. Infra soft topological spaces. Mathematical Problems in Engineering, 2021:3361604, 2021. [20] T. M. Al-shami, E. A. Abo-Tabl, and B. A. Asaad. Weak forms of soft separation axioms and fixed soft points. Fuzzy Information and Engineering, 12(4):509–528, 2020. [21] T. M. Al-shami and E.-S. A. A. Tabi. Connectedness and local connectedness on infra soft topological spaces. Mathematics, 9(15):1759, 2021. [22] T. M. Al-shami. Homeomorphism and quotient mappings in infra soft topological spaces. Journal of Mathematics, 2021:3388288, 2021. [23] T. M. Al-shami. Infra soft compact spaces and application to fixed point theorem. Journal of Function Spaces, 2021:3417096, 2021. [24] B. A. Asaad, T. M. Al-shami, and A. Mhemdi. Bi-operators on soft topological spaces. AIMS Mathematics, 6(11):12471–12490, 2021. [25] G. Ali, H. Alolaiyan, D. Pamucar, M. Asif, and N. Lateef. A novel MADM framework under q-rung orthopair fuzzy bipolar soft sets. Mathematics, 9(17):2169, 2021. [26] G. Ali, G. Muhiuddin, A. Adeel, and M. Z. U. Abidin. Ranking effectiveness of COVID- 19 tests using fuzzy bipolar soft expert sets. Mathematical Problems in Engineering, 2021:5874216, 2021. [27] S. J. John. Soft sets. In Soft Sets, volume 400 of Studies in Fuzziness and Soft Computing. Springer, Cham, 2021. [28] S. Kim and J. Park. Secure multi-party clustering protocols. In Conference on Data Protection and Privacy, pages 78–92, 2017. [29] I. Kandasamy. Double-valued neutrosophic sets, their minimum spanning trees, and clus- tering algorithm. Journal of Intelligent Systems, 27(2):163–182, 2018. [30] A. Mehmood, S. A. Ghour, M. Ishfaq, and F. Afzal. A look at soft neutrosophic spaces. Journal of Intelligent and Fuzzy Systems, 40(4):6473–6494, 2021. [31] A. Mehmood, S. A. Ghour, S. Abdullah, C. Park, and R. L. Jung. A new approach to vague soft topological structures concerning soft points. Journal of Intelligent and Fuzzy Systems, 42(3):1483–1499, 2022. [32] A. Mehmood, S. Abdullah, and C. Park. A new approach to vague soft bi-topological spaces. Computer Modeling in Engineering and Sciences, 131(1):411–428, 2022. [33] A. Mehmood, F. Afzal, S. Abdullah, M. I. Khan, and S. Gul. An absolutely new attempt to vague soft bitopological spaces basis at newly defined operations regarding vague soft points. Mathematical Problems in Engineering, 2021:440854, 2021. [34] A. M. Khattak, N. Hanif, F. Nadeem, M. Zamir, C. Park, G. Nordo, and S. Jabeen. Soft b- separation axioms in neutrosophic soft topological structures. Annals of Fuzzy Mathematics and Informatics, 18(1):93–105, 2019. [35] A. Mehmood, F. Afzal, M. I. Khan, S. Abdullah, and S. Gul. Neutrosophic soft quad structures concerning neutrosophic soft points. Mathematical Problems in Engineering, 2022:8483632, 2022. [36] S. Ramesh Kumar and A. Stanis Arul Mary. Quadri-partitioned neutrosophic soft set. International Research Journal on Advanced Science Hub, 3(8):106–112, 2021. [37] T. M. Al-shami and A. A. Azzam. Infra soft semi-open sets and infra soft semi-continuity. Journal of Function Spaces, 2021:5716876, 2021. [38] R. Hatamleh, A. Al-Husban, S. A. M. Zubair, M. Elamin, M. M. Saeed, E. Abdolmaleki, et al. AI-assisted wearable devices for promoting human health and strength using com- plex interval-valued picture fuzzy soft relations. European Journal of Pure and Applied Mathematics, 18(1):5523, 2025. [39] R. Hatamleh, A. S. Heilat, M. Palanikumar, and A. Al-Husban. Different operators via weighted averaging and geometric approach using trigonometric neutrosophic interval- valued set and its extension. Neutrosophic Sets and Systems, 80:194–213, 2025. [40] R. Hatamleh, A. Al-Husban, M. Palanikumar, and K. Sundareswari. Different weighted op- M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5845 32 of 32 erators such as generalized averaging and generalized geometric based on trigonometric ℘- rung interval-valued approach. Communications on Applied Nonlinear Analysis, 32(5):91– 101, 2025. [41] A. Shihadeh, R. Hatamleh, M. Palanikumar, and A. Al-Husban. New algebraic structures towards different (α, β) intuitionistic fuzzy ideals and its characterization of an ordered ternary semigroup. Communications on Applied Nonlinear Analysis, 32(6):568–578, 2025. [42] R. Hatamleh, A. S. Heilat, M. Palanikumar, and A. Al-Husban. Characterization of inter- action aggregating operators setting interval-valued Pythagorean neutrosophic set. Neu- trosophic Sets and Systems, 81:285–305, 2025. [43] R. Hatamleh, A. Al-Husban, K. Sundareswari, G. Balaj, and M. Palanikumar. Complex tangent trigonometric approach applied to (γ, τ)-rung fuzzy set using weighted averaging, geometric operators and its extension. Communications on Applied Nonlinear Analysis, 32(5):133–144, 2025. [44] A. M. Abd El-Latif and R. A. Hosny. Supra soft separation axioms and supra irresoluteness based on supra b-soft sets. Gazi University Journal of Science, 29(4):845–854, 2016. [45] A. M. Abd El-Latif. On soft supra compactness in supra soft topological spaces. Tbilisi Mathematical Journal, 11(1):169–178, 2018. [46] A. M. Abd El-Latif and R. A. Hosny. Supra open soft sets and associated soft separation axioms. International Journal of Advances in Mathematics, 2017(6):68–81, 2017. [47] A. M. Abd El-Latif and R. A. Hosny. Soft supra extra strongly generalized closed. Analele Universitatii Oradea Fascicula Matematica, 24(1):103–112, 2017. [48] P. K. Maji. Neutrosophic soft set. Annals of Fuzzy Mathematics and Informatics, 5(1):157– 168, 2013. [49] I. Deli and S. Broumi. Neutrosophic soft relations and some properties. Annals of Fuzzy Mathematics and Informatics, 9(1):169–182, 2015.