EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5900 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Comprehensive Study of Bipolar Vague Soft Expert P-Open Sets in Bipolar Vague Soft Expert Topological Spaces with Applications to Cancer Diagnosis Maha Mohammed Saeed1, Raed Hatamleh2, Ahmad A. Abubaker3, Abdallah Al-Husban4, Jamil J. Hamja5, Giorgio Nordo6, Cris L. Armada7, Takaaki Fujita8, Arif Mehmoodaffil9,∗ 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 Faculty of Computer Studies, Arab Open University, Saudi Arabia 4 Department of Mathematics, Faculty of Science and Technology, Irbid National University, P.O. Box: 2600 Irbid, Jordan 5 Department of Mathematics, College of Arts and Sciences, MSU - Tawi-Tawi College of Technology and Oceanography, 7500 Philippines 6 MIFT Department (Mathematical and Computer Science, Physical Sciences and Earth Sciences) - University of Messina, 98166 Sant’Agata, Messina, Italy 7 Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam and Department of Applied Mathematics, Faculty of Applied Science, Ho Chi Minh City University of Technology (HCMUT), 268 Ly Thuong Kiet, District 10, Ward 14, Ho Chi Minh City, Vietnam 8 Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan 9 Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050, KPK, Pakistan Abstract. We rigorously examine the concept of bipolar vague soft expert sets (BPVSESs) and their defining characteristics. Fundamental operations such as complement, union, and intersection are firmly established as foundational elements of the framework. Additionally, the notion of bipolar vague soft expert topology (BPVSET) is introduced, along with eight innovative definitions. Among these, the definition of the bipolar vague soft expert pre-open set, often abbreviated as the p-open set, is particularly significant for constructing diverse structures. This study also provides a strong and healthy articulation of the concepts of interior and closure, offering a detailed exploration of their interactions. Furthermore, it develops foundational topological concepts in bipolar vague soft expert topology by introducing and analyzing bases, sub-bases, and local bases. The notions of first and second countability in the bipolar vague soft expert topology context are formally defined, while separability is explored via countable dense sets. These results enhance the theoretical framework of bipolar vague soft expert topological space, supporting soft topological modeling under uncertainty and parameterization. A comprehensive investigation into these foundational concepts culminates in a series of compelling results concerning the basis of bipolar vague soft expert topological spaces. Finally, it introduces a decision-making framework based on bi-polar vague soft expert sets to support cancer diagnosis. 2020 Mathematics Subject Classifications: 54A05, 54A10, 06D72, 62P10 Key Words and Phrases: Vague set, bipolar vague set, bipolar vague soft set, bipolar vague soft topology, bipolar vague soft p-open set ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5900 Email addresses: (mmmohammed@kau.edu.sa) (M. M. Saeed), (raed@jadara.edu.jo) (R. Hatamleh),(a.abubaker@arabou.edu.sa) (A. A. Abubaker),(dralhosban@inu.edu.jo) (A. Alhusban),(jamilhamja@msutawi-tawi.edu.ph) (J. J. Hamja),(giorgio.nordo@unime.it) (G. Nordo),(cris.armada@hcmut.edu.vn) (C. L. Armada), (t171d603@gunma-u.ac.jp) (T. Fujita), (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), 5900 2 of 32 1. Introduction Soft set theory, introduced by Molodtsov [1], was developed as a mathematical framework to address uncertainty. Chen et al. [2] critiqued certain irrational and erroneous claims in [3], argu- ing that the characteristic reduction in rough set theory (RST) proposed in [3] was unnecessary for minimizing the parameters required to determine optimal objects. They also highlighted the fundamental differences between characteristic reduction in rough sets and parameterization reduction in soft sets. Maji et al. [3–5] extended the theory by combining soft sets with fuzzy sets to create fuzzy soft sets. Roy and Maji [6] applied this theory to various decision-making problems, thereby enhancing the practical utility and robustness of soft set theory. Alkhazaleh et al. [7] introduced the concept of soft multisets as a generalization of soft sets. They later proposed the fuzzy parameterized interval-valued fuzzy soft set (FPIVFSS) [8] , studying its operations, and further developed the idea of possibility fuzzy soft sets [9] . These sets were demonstrated to be applicable in medical diagnosis by utilizing a similarity measure between two possibility fuzzy soft sets. Building upon these advancements, Alkhazaleh and Salleh [10] integrated fuzzy soft sets with expert sets, resulting in the concept of soft expert sets. This innovation enabled users to access professional opinions even after performing operations on the sets, further broadening the scope of practical applications. O. Dalkilic and I. N. Cangul [11] aimed to analyze decision-making processes involving interactions between elements from two distinct universe sets. To achieve this, they first introduced the concepts of object interaction and inverse object interaction sets for these separate universes. They then extended these con- cepts specifically to binary soft sets, taking into account the presence of two distinct universe sets. By employing a parameter set, their method enabled the determination of interaction values between objects. Furthermore, they proposed two decision-making algorithms based on these concepts within the framework of binary soft sets. The significance and advantages of the proposed algorithms were demonstrated through a practical application. Demirtas et al. [12] proposed several strategies based on soft set theory to address scenarios involving various types of uncertainty in decision-making problems. To develop these algorithms, they introduced several novel concepts to the literature, including object code, personal object code, parameter importance weight, and new distance measures. Additionally, the authors presented application results and provided further illustrative examples. Dalkilic and Demirtac [13] focused on the parameterization tool of soft set theory, introducing factor sets to account for every possible influence on each parameter. This approach aims to yield more accurate results by deter- mining the membership values of parameters in uncertain environments. In addition, several novel hybrid types of soft sets were proposed. One of the key advantages of these new hybrid mathematical tools is their ability to reduce the potential error margin for decision-makers. Furthermore, a decision-making algorithm was developed for the soft set type that offers the most comprehensive data under conditions of uncertainty. Finally, the proposed method was applied to solve an uncertainty problem. 1.1. Literature review According to Bosc and Pivert [14], bi-polarity refers to the human mind’s capacity to think and make decisions based on both positive and negative impacts?. Positive information out- lines what is conceivable, acceptable, permissible, wanted, or thought to be acceptable. Negative statements, on the other hand, express what is impossibly feasible, rejected, or prohibited. Lee [15, 16] introduced the concept of (BFS). Majumder [17] proposed (BVF) sub semigroup, (BVF) bi-idea, bipolar valued fuzzy (1, 2) - ideal and (BVF) ideal. Bipolar fuzzy groups, often referred to as fuzzy d-ideals of groups under (T-S) norm, are applications of bipolar fuzzy sets in groups that Manemaran and Chellappa [18] investigated. The study of m-polar fuzzy sets by Chen et al. [19] demonstrates how many concepts have been defined using(BFSs). Alkhazaleh et al [20].s mapping example used in decision-making showed how the approximate function is defined from M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 3 of 32 a set of fuzzy parameters. Soft expert sets were first proposed by Alkhazaleh and Salleh [21] , allowing users to access all expert sets’ opinions at once. (BVSTS) now have the concepts of vague soft s-open set, vague soft s-interior, vague soft s-closer, and vague soft s-exterior added by Afzal et al. [22]. Saeed et al. [11] established a new idea of operators, such as the interior operator, exterior operator, and closure operator, in (BVSTSs). On the basis of these concepts, a few results in (BVSTS) are addressed. (BVSTS) are used to address some more results based on the appealing idea of a sequence’s limit being the Last. There are four sections to this piece of writing. Dalkilic and Demirtas [23] expanded the existing methodology by incorporat- ing bipolar fuzzy soft set theory, enabling the representation of two distinct types of medical knowledge within a unified framework. They also introduced a novel decision-making algorithm specifically designed for this enhanced model. The effectiveness of the proposed algorithm is demonstrated through practical applications in the medical field, highlighting its potential to enhance diagnostic accuracy and support decision-making in clinical practice. Dalkilic [24] introduced the first type semi-strong (α, β)-cuts, second type semi-strong (α, β)-cuts, strong (α, β)-cuts, inverse (α, β)-cuts, first type semi-weak inverse (α, β)-cuts, second type semi-weak inverse (α, β)-cuts, and weak inverse (α, β)-cuts of bipolar fuzzy soft sets, along with some of their properties. In addition, several distinguishing properties between (α, β)-cuts and in- verse (α, β)-cuts were established. Furthermore, related theorems were formulated and proven. It was also demonstrated that both (α, β)-cuts and inverse (α, β)-cuts of bipolar fuzzy soft sets serve as effective tools in decision-making. Abdullah et al. [25] combined the concepts of bipolar fuzzy sets and soft sets to introduce the notion of a bipolar fuzzy soft set. They investigated its fundamental properties and explored basic operations such as extended union and intersection. Moreover, they demonstrated the applicability of bipolar fuzzy soft sets in solving decision-making problems and proposed a general algorithm for this purpose. Their work laid the groundwork for applying bipolar fuzzy soft sets to real-world scenarios. Mustafa et al. [26] advanced the field by focusing on bipolar fuzzy multicriteria decision-making meth- ods. Their primary objective was to assist students in identifying the most suitable university by evaluating the factors influencing admission decisions. To address the complexities of such decisions, they integrated bipolar fuzzy sets with soft expert sets to create a robust multicriteria decision-making model. The study also involved developing structural hierarchical models of parameters and implementing a new algorithm to enhance the accuracy of decision-making. Their approach proved to be effective, particularly in the context of university selection, and demonstrated strong potential for broader application in the education sector. Hatamleh [27] explored the compactness and continuity of two-variable Uryson operators defined by integral equations in fuzzy functional analysis. The study also examined the convergence of operator sequences using a specific measure and the Carathéodory condition. Rajalakshmi et al. [28] introduced new types of neutrosophic structures in ordered Gamma-semigroups, such as sub- semigroups, ideals, and bi-ideals. Their work extended existing definitions and explored the properties of these structures through level sets. Hatamleh et al. [29] introduced complex cubic intuitionistic fuzzy subbisemirings and studied their properties, including homomorphisms and level sets. The main results were illustrated with examples. Abubaker et al. [30] investigated the use of Lagrange polynomials to find numerical solutions for various neutrosophic boundary value problems. 1.2. Research Gap Given that the bipolar fuzzy soft expert set framework predominantly emphasizes the mem- bership function, it entirely neglects the non-membership function. In essence, it encapsulates only the ”truth” aspect of information while disregarding the ”false” or contradictory component of membership. This inherent limitation underscores the necessity for a more holistic and refined approach-one that can simultaneously capture both dimensions of uncertainty. Broadly speak- ing, the bipolar fuzzy soft expert set is a one-dimensional approach. To study non-membership M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 4 of 32 effectively, an additional dimension is required. As a result, the new set, known as a vague set, emerges as a two-dimensional model capable of handling both membership and non-membership values simultaneously. 1.3. Motivation The following studies have served as pivotal sources of inspiration and intellectual impetus for the present research. Al-Qudah and Hassan [31] extended the concepts of bipolar fuzzy sets and soft expert sets to develop the framework of bipolar fuzzy soft expert sets. They de- fined fundamental theoretical operations?namely complement, union, intersection, AND, and OR on bipolar fuzzy soft expert sets, supported by illustrative examples. The authors also examined several related properties and provided formal proofs. Furthermore, they established basic properties and relevant laws associated with this concept. An algorithm was constructed based on the proposed framework, which was subsequently applied to a decision-making prob- lem to demonstrate its practicality. The results, illustrated through an example, confirmed the effectiveness of the proposed method in solving decision-making problems. M. Akram et al. [32] presented a new multi-criteria group decision-making (MCGDM) model that incorporates criteria evaluation by multiple experts. A novel hybrid framework, termed m-bipolar fuzzy soft expert set, was developed by integrating m-polar fuzzy sets with soft expert sets, thereby enabling the investigation of soft expert sets within an m-polar fuzzy environment. The char- acteristics of this hybrid model are explored through numerical examples. Additionally, its fundamental properties are examined, and operations such as subsethood, complement, inter- section, union, as well as the OR and AND operators, are defined. The proposed model is applied to two well-known real-world problems: site selection for a dam and human traffick- ing analysis across different countries. The algorithm developed for the model demonstrates both efficiency and validity. A comparative analysis with existing mathematical methods is also provided to highlight its advantages. The study of these references leads us to conclude that the absence of a unified topological framework for handling hybrid uncertainty highlights the need to formalize Bipolar Vague Soft Expert Sets (BPVSETS), which integrate bipolarity, vagueness, soft sets, and expert opinions. Fundamental operations such as complement, union, and intersection have not been rigorously defined within this structure, and a corresponding bipolar vague soft expert topology (BPVSET) remains undeveloped. Key topological concepts including p-open sets, interior, closure, basis and local basis lack formal definitions and analysis in this context. Moreover, classical properties like countability and separability have yet to be explored under the BPVSET framework. This research addresses these gaps by constructing a comprehensive topological foundation for BPVSETS, enabling soft topological modeling under uncertainty and parameterization. 1.4. Novelty This study introduces several groundbreaking concepts within the framework of bipolar vague soft expert sets (BPVSETS), marking a significant advancement in the field. Among its notable contributions is the development of a comprehensive bipolar vague soft expert topology, offering a novel perspective for analyzing relationships among uncertain and imprecise data. The introduction of eight innovative definitions, including the pivotal bipolar vague soft expert pre-open set, represents a paradigm shift in the understanding and application of vague soft expert set theory. This foundational definition not only deepens theoretical insights but also paves the way for practical applications across various domains, such as artificial intelligence, decision-making, and fuzzy logic. Furthermore, the study systematically explores the interplay between interior and closure concepts within BPVSETS, adding depth to the existing literature and enabling a more nuanced analysis of their interactions. This dual focus enhances the theoretical framework while simultaneously presenting practical implications for modeling and M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 5 of 32 addressing complex problems under uncertainty. In essence, this research extends the boundaries of traditional set theory, providing novel tools and insights that promise to transform how experts approach and analyze vague information in multifaceted environments. 1.5. Organization of the Paper This paper is organized into seven main sections 2: Introduces Bipolar Vague Soft Sets (BVSS) based on [10], [21], and [22], defining key operations (union, intersection, AND, OR), along with concepts like empty and absolute BVSS, supported by examples. 3: Presents eight new definitions, including the bipolar vague soft expert topology (BVSET). It emphasizes the role of p-open sets in constructing topological structures and explores interior and closure con- cepts in depth. 4: Develops topological foundations in BVSETS through bases, sub-bases, and local bases. Introduces countability and separability with key theorems for comparing and con- structing BVSET topologies. 5: Applies BVSES to cancer diagnosis, handling uncertain, vague, and conflicting expert opinions based on medical parameters. 6: Compares the proposed work with the study in [31], highlighting theoretical advancements and innovations. Summary results are presented in Table 3. 7: Summarizes findings and suggests future directions for extending BVSES in decision-making and soft topological modeling under uncertainty. 2. Preliminaries This section discusses references [10], [21], and [22], introducing bipolar vague soft sets (BVSS), which extend vague soft sets to handle both positive and negative information under uncertainty. It defines BVSS, formalizes key operations and their properties, and introduces concepts such as the empty and absolute BVSS, as well as union, intersection, AND, and OR operators, with examples illustrating their use and consistency. Definition 1. [22] Let X be universal set and E be a set of parameters. Let P ⟨X⟩ denotes power set of X then the vague soft set ⟨⟨f̃ , E⟩⟩ over X is a set given by f̃ : E → P ⟨X⟩ and in other words, ⟨⟨f̃ , E⟩⟩ = [(e,< x, gf̃⟨e⟩⟨x⟩, hf̃⟨e⟩⟨x⟩ >: x∈̃X) : e∈̃X] where gf̃⟨e⟩⟨x⟩∈̃[0, 1] and hf̃⟨e⟩⟨x⟩∈̃[0, 1] with 0≤̃gf̃⟨e⟩⟨x⟩+ hf̃⟨e⟩⟨x⟩≤̃2. This means that each value is a typical value between 0 and 1. Definition 2. [22] Let X be universal set, E be a set of parameters. A bi-polar vague soft set ⟨BV SS⟩ ⟨⟨f̃ , E⟩⟩ = [( e, 〈 x, ( g⊕ f̃⟨e⟩⟨x⟩, h ⊕ f̃⟨e⟩⟨x⟩ g⊖ f̃⟨e⟩⟨x⟩, h ⊖ f̃⟨e⟩⟨x⟩ )〉 : x∈̃X ) : e∈̃E ] . where, g⊕ f̃⟨e⟩⟨x⟩, h ⊕ f̃⟨e⟩ → [0, 1], g⊖ f̃⟨e⟩⟨x⟩, h ⊖ f̃⟨e⟩ → [−1, 0]. Definition 3. [22] Let ⟨⟨f̃ , E⟩⟩ be a bi-polar vague soft over X then complement of a bi-polarr vague rrsoft set ⟨⟨f̃ , E⟩⟩ ,is signified by ⟨⟨f̃ , E⟩⟩c and given as ⟨⟨f̃ , E⟩⟩c = [( e, 〈 x, ( h⊕ f̃⟨e⟩⟨x⟩, g ⊕ f̃⟨e⟩⟨x⟩ h⊖ f̃⟨e⟩⟨x⟩, g ⊖ f̃⟨e⟩⟨x⟩ )〉 : x∈̃X ) : e∈̃E ] . Definition 4. [22] The empty bi-polar vague soft set ⟨⟨f̃null, E⟩⟩ over ?? is defined by; ⟨⟨f̃null, E⟩⟩ = [(e, ⟨x, (0, 0,−1, 0)⟩ : x∈̃X) : e∈̃E] M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 6 of 32 Absolute ⟨BV SS⟩, ⟨⟨Xaboslute, E⟩⟩ over X is defined by; ⟨⟨Xaboslute, E⟩⟩ = [(e, ⟨x, (1, 1, 0,−1)⟩ : x∈̃X) : e∈̃E] Definition 5. [22] Let ⟨⟨f̃1, E⟩⟩ and ⟨⟨f̃2, E⟩⟩ be two bi-polar vague soft sets over X. ⟨⟨f̃1, E⟩⟩ is said to be bi-polar vague soft sub set of ⟨⟨f̃2, E⟩⟩ if g⊕ f̃1⟨e⟩ ⟨x⟩ ≾ g⊕ f̃2⟨e⟩ ⟨x⟩ h⊕ f̃1⟨e⟩ ⟨x⟩ ≿ h⊕ f̃2⟨e⟩ ⟨x⟩ g⊖ f̃1⟨e⟩ ⟨x⟩ ≾ g⊖ f̃2⟨e⟩ ⟨x⟩ h⊖ f̃1⟨e⟩ ⟨x⟩ ≿ h⊖ f̃2⟨e⟩ ⟨x⟩ ∀ ⟨e, x⟩∈̃ (EX⟨M⟩) It is denoted by ⟨⟨f̃1, E⟩⟩⊆̃⟨⟨f̃2, E⟩⟩.⟨⟨f̃1, E⟩⟩ is said to be bi-polar vague soft equal to ⟨⟨f̃2, E⟩⟩ if ⟨⟨f̃1, E⟩⟩ is bi-polar vague soft sub set of ⟨⟨f̃2, E⟩⟩ and ⟨⟨f̃2, E⟩⟩ is bi-polar vague soft sub set of ⟨⟨f̃1, E⟩⟩ and is signified by ⟨⟨f̃1, E⟩⟩ = ⟨⟨f̃2, E⟩⟩ Example 1. [22] Let X = {x1, x2} and E = {e1, e2}, if ⟨⟨f̃1, E⟩⟩ and ⟨⟨f̃2, E⟩⟩ are two bi-polar vague soft sets as ⟨⟨f̃1, E⟩⟩ =  (e1, ⟨x1, (06× 10−1, 05× 10−1,−08× 10−1,−04× 10−1)⟩, ⟨x2, (05× 10−1, 04× 10−1,−06× 10−1,−03× 10−1)⟩), (e2, ⟨x1, (05× 10−1, 07× 10−1,−06× 10−1,−05× 10−1)⟩, ⟨x2, (03× 10−1, 05× 10−1,−04× 10−1,−02× 10−1)⟩),  ⟨⟨f̃1, E⟩⟩ =  (e1, ⟨x1, (07× 10−1, 08× 10−1,−05× 10−1,−06× 10−1)⟩, ⟨x2, (06× 10−1, 06× 10−1,−05× 10−1,−07× 10−1)⟩), (e2, ⟨x1, (06× 10−1, 09× 10−1,−04× 10−1,−07× 10−1)⟩, ⟨x2, (04× 10−1, 07× 10−1,−03× 10−1,−06× 10−1)⟩),  Then, ⟨⟨f̃1, E⟩⟩⊆̃⟨⟨f̃2, E⟩⟩ Definition 6. [22] Let ⟨⟨f̃1, E⟩⟩ = [( e, 〈 x, ( g⊕ f̃1⟨e⟩ ⟨x⟩, h⊕ f̃1⟨e⟩ ⟨x⟩ g⊖ B̃i⟨e⟩ ⟨x⟩, h⊖ B̃i⟨e⟩ ⟨x⟩ )〉 : x∈̃X ) : e∈̃E ] , for i= 1,2 be two bi-polar vague soft sub sets over X. Then, their union is signified by ⟨⟨f̃1, E⟩⟩∪̃⟨⟨f̃2, E⟩⟩ and it is given as; 2∐ i=1 ⟨⟨f̃1, E⟩⟩ = [( e, 〈 x, ( max{g⊕ f̃1⟨e⟩ ⟨x⟩},min{h⊕ f̃1⟨e⟩ ⟨x⟩} max{g⊖ B̃i⟨e⟩ ⟨x⟩},min{h⊖ B̃i⟨e⟩ ⟨x⟩} )〉 : x∈̃X ) : e∈̃E ] . Definition 7. [22] Let ⟨⟨f̃i, E⟩⟩ = [( e, 〈 x, ( g⊕ f̃1⟨e⟩ ⟨x⟩, h⊕ f̃1⟨e⟩ ⟨x⟩ g⊖ f̃i⟨e⟩ ⟨x⟩, h⊖ f̃i⟨e⟩ ⟨x⟩ )〉 : x∈̃X ) : e∈̃E ] , for i= 1,2 be two bi-polar vague soft sub sets over X then, their intersection is signified by ⟨⟨f̃1, E⟩⟩∩̃⟨⟨f̃2, E⟩⟩ and it is given as; 2∏ i=1 ⟨⟨f̃1, E⟩⟩ = [( e, 〈 x, ( min{g⊕ f̃1⟨e⟩ ⟨x⟩},max{h⊕ f̃1⟨e⟩ ⟨x⟩} min{g⊖ B̃i⟨e⟩ ⟨x⟩},max{h⊖ B̃i⟨e⟩ ⟨x⟩} )〉 : x∈̃X ) : e∈̃E ] . M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 7 of 32 Definition 8. [22] Let ⟨⟨f̃i, E⟩⟩ = [( e, 〈 x, ( g⊕ f̃1⟨e⟩ ⟨x⟩, h⊕ f̃1⟨e⟩ ⟨x⟩ g⊖ f̃i⟨e⟩ ⟨x⟩, h⊖ f̃i⟨e⟩ ⟨x⟩ )〉 : x∈̃X ) : e∈̃E ] , for i∈̃I be a family of bi-polar vague soft sub sets over X then ∐ i∈̃I ⟨⟨f̃1, E⟩⟩ = [( e, 〈 x, ( sup{g⊕ f̃1⟨e⟩ ⟨x⟩}, inf{h⊕ f̃1⟨e⟩ ⟨x⟩} sup{g⊖ B̃i⟨e⟩ ⟨x⟩}, inf{h⊖ B̃i⟨e⟩ ⟨x⟩} )〉 : x∈̃X ) : e∈̃E ] . ∏ i∈̃I ⟨⟨f̃1, E⟩⟩ = [( e, 〈 x, ( inf{g⊕ f̃1⟨e⟩ ⟨x⟩}, sup{h⊕ f̃1⟨e⟩ ⟨x⟩} inf{g⊖ B̃i⟨e⟩ ⟨x⟩}, sup{h⊖ B̃i⟨e⟩ ⟨x⟩} )〉 : x∈̃X ) : e∈̃E ] . Proposition 1. [22] Let ⟨⟨f̃null, E⟩⟩ and ⟨⟨Xaboslute, E⟩⟩ be empty bi-polar vague soft sub set and absolutes bi-polar vague soft sub set over X, respectively then, 1. ⟨⟨f̃null, E⟩⟩ ⊆ ⟨⟨Xaboslute, E⟩⟩ 2. ⟨⟨f̃null, E⟩⟩ ∪̃ ⟨⟨Xaboslute, E⟩⟩ = ⟨⟨Xaboslute, E⟩⟩ 3. ⟨⟨f̃null, E⟩⟩ ∩̃ ⟨⟨Xaboslute, E⟩⟩ = ⟨⟨f̃null, E⟩⟩ Proof. Straightforward. Definition 9. [22] Let ⟨⟨f̃1, E⟩⟩ and ⟨⟨f̃2, E⟩⟩ be two bi-polar vague soft sub sets over X then, ⟨⟨f̃1, E⟩⟩ \ ⟨⟨f̃2, E⟩⟩ = ⟨⟨f̃3, E⟩⟩ and is signified by ⟨⟨f̃3, E⟩⟩ = ⟨⟨f̃1, E⟩⟩∩̃⟨⟨f̃2, E⟩⟩c as follows: ⟨⟨f̃3, E⟩⟩ = [( e, 〈 x, ( g⊕ f̃3⟨e⟩ ⟨x⟩, h⊕ f̃3⟨e⟩ ⟨x⟩ g⊖ f̃3⟨e⟩ ⟨x⟩, h⊖ f̃3⟨e⟩ ⟨x⟩ )〉 : x∈̃X ) : e∈̃E ] , Where h⊕ f̃3⟨e⟩ ⟨x⟩ = [min{g⊕ f̃1⟨e⟩ ⟨x⟩, h⊕ f̃2⟨e⟩ ⟨x⟩}, g⊖ f̃3⟨e⟩ ⟨x⟩ = min{g⊖ f̃1⟨e⟩ ⟨x⟩, h⊖ f̃2⟨e⟩ ⟨x⟩}], h⊕ f̃3⟨e⟩ ⟨x⟩ = [max{h⊕ f̃1⟨e⟩ ⟨x⟩, g⊕ f̃2⟨e⟩ ⟨x⟩}, h⊖ f̃3⟨e⟩ ⟨x⟩ = max{h⊖ f̃1⟨e⟩ ⟨x⟩, g⊖ f̃2⟨e⟩ ⟨x⟩}]. Definition 10. [22] Let ⟨⟨f̃1, E⟩⟩ and ⟨⟨f̃2, E⟩⟩ be two bi-polar vague soft sub sets over X then, AND operation is given by ⟨⟨f̃1, E⟩⟩ ∧̃ ⟨⟨f̃2, E⟩⟩ = ⟨⟨f̃3, E × E⟩⟩ and is signified by ⟨⟨B̃3, E × E⟩⟩ = [( (e1, e2), 〈 x, ( g⊕ f̃3⟨e1,e2⟩ ⟨x⟩, h⊕ f̃3⟨e1,e2⟩ ⟨x⟩ g⊖ f̃3⟨e1,e2⟩ ⟨x⟩, h⊖ f̃3⟨e1,e2⟩ ⟨x⟩ )〉 : x∈̃X ) : (e1, e2)∈̃E × E ] , Where g⊕ f̃3⟨e1,e2⟩ ⟨x⟩ = [min{g⊕ f̃1⟨e1⟩ ⟨x⟩, g⊕ f̃2⟨e2⟩ ⟨x⟩}, g⊖ f̃3⟨e1,e2⟩ ⟨x⟩ = min{g⊖ f̃1⟨e⟩ ⟨x⟩, h⊖ f̃2⟨e2⟩ ⟨x⟩}], g⊕ f̃3⟨e1,e2⟩ ⟨x⟩ = [max{h⊕ f̃1⟨e1⟩ ⟨x⟩, h⊕ f̃2⟨e2⟩ ⟨x⟩}, h⊖ f̃3⟨e1,e2⟩ ⟨x⟩ = max{h⊖ f̃1⟨e1⟩ ⟨x⟩, h⊖ f̃2⟨e2⟩ ⟨x⟩}]. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 8 of 32 Definition 11. [22] Let ⟨⟨f̃1, E⟩⟩ and ⟨⟨f̃2, E⟩⟩ be two bi-polar vague soft sub sets over X then, OR operation is given by ⟨⟨f̃1, E⟩⟩ ∨̃ ⟨⟨f̃2, E⟩⟩ = ⟨⟨f̃3, E × E⟩⟩ and is signified by ⟨⟨f̃3, E × E⟩⟩ = [( (e1, e2), 〈 x, ( g⊕ f̃3⟨e1,e2⟩ ⟨x⟩, h⊕ f̃3⟨e1,e2⟩ ⟨x⟩ g⊖ f̃3⟨e1,e2⟩ ⟨x⟩, h⊖ f̃3⟨e1,e2⟩ ⟨x⟩ )〉 : x∈̃X ) : (e1, e2)∈̃E × E ] , Where g⊕ f̃3⟨e1,e2⟩ ⟨x⟩ = [max{g⊕ f̃1⟨e1⟩ ⟨x⟩, g⊕ f̃2⟨e2⟩ ⟨x⟩}, g⊖ f̃3⟨e1,e2⟩ ⟨x⟩ = max{g⊖ f̃1⟨e1⟩ ⟨x⟩, g⊖ f̃2⟨e2⟩ ⟨x⟩}]. h⊕ f̃3⟨e1,e2⟩ ⟨x⟩ = [min{h⊕ f̃1⟨e1⟩ ⟨x⟩, h⊕ f̃2⟨e2⟩ ⟨x⟩}, g⊖ f̃3⟨e1,e2⟩ ⟨x⟩ = min{h⊖ f̃1⟨e1⟩ ⟨x⟩, h⊖ f̃2⟨e2⟩ ⟨x⟩}], Example 2. [22] Let X = {x1, x2} and E = {e1, e2}, if ⟨⟨f̃1, E⟩⟩ and ⟨⟨f̃2, E⟩⟩ are two bi-polar vague soft sets such that ⟨⟨f̃1, E⟩⟩ =  (e1, ⟨x1, (03× 10−1, 05× 10−1,−05× 10−1,−07× 10−1)⟩, ⟨x2, (03× 10−1, 05× 10−1,−05× 10−1,−08× 10−1)⟩), (e2, ⟨x1, (04× 10−1, 04× 10−1,−04× 10−1,−03× 10−1)⟩, ⟨x2, (05× 10−1, 08× 10−1,−09× 10−1,−07× 10−1)⟩),  ⟨⟨f̃2, E⟩⟩ =  (e1, ⟨x1, (04× 10−1, 06× 10−1,−03× 10−1,−09× 10−1)⟩, ⟨x2, (04× 10−1, 06× 10−1,−02× 10−1,−03× 10−1)⟩), (e2, ⟨x1, (03× 10−1, 03× 10−1,−06× 10−1,−08× 10−1)⟩, ⟨x2, (04× 10−1, 05× 10−1,−01× 10−1,−03× 10−1)⟩),  Then, ⟨⟨f̃1, E⟩⟩∪̃⟨⟨f̃2, E⟩⟩ =  (e1, ⟨x1, (04× 10−1, 06× 10−1,−03× 10−1,−09× rr10−1)⟩, ⟨x2, (04× 10−1, 06× 10−1,−02× 10−1,−03× 10−1)⟩), (e2, ⟨x1, (04× 10−1, 04× 10−1,−04× 10−1,−08× 10−1)⟩, ⟨x2, (05× 10−1, 08× 10−1,−01× 10−1,−07× 10−1)⟩),  ⟨⟨f̃1, E⟩⟩∩̃⟨⟨f̃2, E⟩⟩ =  (e1, ⟨x1, (03× 10−1, 05× 10−1,−05× 10−1,−07× rr10−1)⟩, ⟨x2, (03× 10−1, 05× 10−1,−05× 10−1,−03× 10−1)⟩), (e2, ⟨x1, (03× 10−1, 03× 10−1,−06× 10−1,−03× 10−1)⟩, ⟨x2, (05× 10−1, 05× 10−1,−09× 10−1,−03× 10−1)⟩),  ⟨⟨f̃1, E⟩⟩ \ ⟨⟨f̃2, E⟩⟩ =  (e1, ⟨x1, (03× 10−1, 04× 10−1,−07× 10−1,−05× rr10−1)⟩, ⟨x2, (02× 10−1, 04× 10−1,−08× 10−1,−03× 10−1)⟩), (e2, ⟨x1, (04× 10−1, 04× 10−1,−04× 10−1,−03× 10−1)⟩, ⟨x2, (03× 10−1, 05× 10−1,−09× 10−1,−06× 10−1)⟩),  (1) ⟨⟨f̃1, E⟩⟩ ∧̃ ⟨⟨f̃2, E⟩⟩ =  ((e1, e2), ⟨x1, (03× 10−1, 05× 10−1,−05× 10−1,−07× rr10−1)⟩, ⟨x2, (03× 10−1, 05× 10−1,−05× 10−1,−03× 10−1)⟩), ((e1, e2), ⟨x1, (03× 10−1, 03× 10−1,−06× 10−1,−07× 10−1)⟩, ⟨x2, (03× 10−1, 05× 10−1,−05× 10−1,−03× 10−1)⟩), ((e1, e2), ⟨x1, (04× 10−1, 04× 10−1,−04× 10−1,−03× rr10−1)⟩, ⟨x2, (04× 10−1, 06× 10−1,−02× 10−1,−03× 10−1)⟩), ((e1, e2), ⟨x1, (03× 10−1, 03× 10−1,−06× 10−1,−03× rr10−1)⟩, ⟨x2, (04× 10−1, 05× 10−1,−09× 10−1,−03× 10−1)⟩),  M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 9 of 32 ⟨⟨f̃1, E⟩⟩ ∨̃ ⟨⟨f̃2, E⟩⟩ =  ((e1, e2), ⟨x1, (04× 10−1, 06× 10−1,−03× 10−1,−09× 10−1)⟩, ⟨x2, (04× 10−1, 06× 10−1,−02× 10−1,−08× 10−1)⟩), ((e1, e2), ⟨x1, (03× 10−1, 05× 10−1,−05× 10−1,−08× 10−1)⟩, ⟨x2, (04× 10−1, 05× 10−1,−01× 10−1,−08× 10−1)⟩), ((e1, e2), ⟨x1, (04× 10−1, 06× 10−1,−03× 10−1,−09× 10−1)⟩, ⟨x2, (05× 10−1, 08× 10−1,−02× 10−1,−07× 10−1)⟩), ((e1, e2), ⟨x1, (04× 10−1, 04× 10−1,−04× 10−1,−08× rr10−1)⟩, ⟨x2, (05× 10−1, 08× 10−1,−01× 10−1,−07× 10−1)⟩),  Proposition 2. [22] Let ⟨⟨f̃1, E⟩⟩, ⟨⟨f̃2, E⟩⟩ and ⟨⟨f̃3, E⟩⟩ be three bi-polar vague soft sub sets sets over X then, 1. ⟨⟨f̃1, E⟩⟩∪̃[⟨⟨f̃2, E⟩⟩∪̃⟨⟨f̃3, E⟩⟩] = [⟨⟨f̃1, E⟩⟩∪̃⟨⟨f̃2, E⟩⟩]∪̃⟨⟨f̃3, E⟩⟩, ⟨⟨f̃1, E⟩⟩∩̃[⟨⟨f̃2, E⟩⟩∩̃⟨⟨f̃3, E⟩⟩] = [⟨⟨f̃1, E⟩⟩∩̃⟨⟨f̃2, E⟩⟩]∩̃⟨⟨f̃3, E⟩⟩; 2. ⟨⟨f̃1, E⟩⟩∪̃[⟨⟨f̃2, E⟩⟩∩̃⟨⟨f̃3, E⟩⟩] = [⟨⟨f̃1, E⟩⟩∪̃⟨⟨f̃2, E⟩⟩]∪̃[⟨⟨f̃1, E⟩⟩∪̃⟨⟨f̃3, E⟩⟩], ⟨⟨f̃1, E⟩⟩∩̃[⟨⟨f̃2, E⟩⟩∪̃⟨⟨f̃3, E⟩⟩] = [⟨⟨f̃1, E⟩⟩∩̃⟨⟨f̃2, E⟩⟩]∪̃[⟨⟨f̃1, E⟩⟩∩̃⟨⟨f̃3, E⟩⟩] ; 3. ⟨⟨f̃1, E⟩⟩∪̃⟨⟨f̃null, E⟩⟩ = ⟨⟨f̃1, E⟩⟩, ⟨⟨f̃1, E⟩⟩∩̃⟨⟨f̃null, E⟩⟩ = ⟨⟨f̃null, E⟩⟩ ; 4. ⟨⟨f̃1, E⟩⟩∪̃⟨⟨Xabsolute, E⟩⟩ = ⟨⟨Xabsolute, E⟩⟩, ⟨⟨f̃1, E⟩⟩∩̃⟨⟨Xabsolute, E⟩⟩ = ⟨⟨f̃1, E⟩⟩ ; 5. ⟨⟨f̃null, E⟩⟩ \ ⟨⟨Xabsolute, E⟩⟩ = ⟨⟨f̃null, E⟩⟩, ⟨⟨Xabsolute, E⟩⟩ \ ⟨⟨f̃null, E⟩⟩ = ⟨⟨Xabsolute, E⟩⟩ ; Proof. Straightforward. Proposition 3. [22] Let ⟨⟨f̃1, E⟩⟩ and ⟨⟨f̃2, E⟩⟩ be two bi-polar vague soft sub sets sets over X then, 1. [⟨⟨f̃1, E⟩⟩∪̃⟨⟨f̃2, E⟩⟩]c = [⟨⟨f̃1, E⟩⟩]c∩̃[⟨⟨f̃2, E⟩⟩]c, 2. [⟨⟨f̃1, E⟩⟩∩̃⟨⟨f̃2, E⟩⟩]c = [⟨⟨f̃1, E⟩⟩]c∪̃[⟨⟨f̃2, E⟩⟩]c, Proof. (i) For all e∈̃E, x∈̃X 2∐ i=1 ⟨⟨f̃1, E⟩⟩ = [( e, 〈 x, ( max{g⊕ f̃1⟨e⟩ ⟨x⟩, g⊕ f̃2⟨e⟩ ⟨x⟩},min{h⊕ f̃1⟨e⟩ ⟨x⟩, h⊕ f̃2⟨e⟩ ⟨x⟩} max{g⊖ f̃1⟨e⟩ ⟨x⟩, g⊖ f̃2⟨e⟩ ⟨x⟩},min{h⊖ f̃1⟨e⟩ ⟨x⟩, h⊖ f̃2⟨e⟩ ⟨x⟩} )〉 : x∈̃X ) : e∈̃E ] . [ 2∐ i=1 ⟨⟨f̃1, E⟩⟩ ]c = [( e, 〈 x, ( min{h⊕ f̃1⟨e⟩ ⟨x⟩, h⊕ f̃2⟨e⟩ ⟨x⟩},max{g⊕ f̃1⟨e⟩ ⟨x⟩, g⊕ f̃2⟨e⟩ ⟨x⟩} min{h⊖ f̃1⟨e⟩ ⟨x⟩, h⊖ f̃2⟨e⟩ ⟨x⟩},max{g⊖ f̃1⟨e⟩ ⟨x⟩, g⊖ f̃2⟨e⟩ ⟨x⟩} )〉 : x∈̃X ) : e∈̃E ] . Now ⟨⟨f̃1, E⟩⟩c = [( e, 〈 x, ( h⊕ f̃1⟨e⟩ ⟨x⟩, g⊕ f̃1⟨e⟩ ⟨x⟩ h⊖ f̃1⟨e⟩ ⟨x⟩, g⊖ f̃2⟨e⟩ ⟨x⟩ )〉 : x∈̃X ) : e∈̃E ] , M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 10 of 32 ⟨⟨f̃2, E⟩⟩c = [( e, 〈 x, ( h⊕ f̃1⟨e⟩ ⟨x⟩, g⊕ f̃2⟨e⟩ ⟨x⟩ h⊖ f̃1⟨e⟩ ⟨x⟩, g⊖ f̃2⟨e⟩ ⟨x⟩ )〉 : x∈̃X ) : e∈̃E ] . Then, 2∏ i=1 ⟨⟨f̃i, E⟩⟩c = [( e, 〈 x, ( min{h⊕ f̃1⟨e⟩ ⟨x⟩, h⊕ f̃2⟨e⟩ ⟨x⟩},max{g⊕ f̃1⟨e⟩ ⟨x⟩, g⊕ f̃2⟨e⟩ ⟨x⟩} min{h⊖ f̃1⟨e⟩ ⟨x⟩, h⊖ f̃2⟨e⟩ ⟨x⟩},max{g⊖ f̃1⟨e⟩ ⟨x⟩, g⊖ f̃2⟨e⟩ ⟨x⟩} )〉 : x∈̃X ) : e∈̃E ] . = [( e, 〈 x, ( min{h⊕ f̃1⟨e⟩ ⟨x⟩, h⊕ f̃2⟨e⟩ ⟨x⟩},max{g⊕ f̃1⟨e⟩ ⟨x⟩, g⊕ f̃2⟨e⟩ ⟨x⟩} min{h⊖ f̃1⟨e⟩ ⟨x⟩, h⊖ f̃2⟨e⟩ ⟨x⟩},max{g⊖ f̃1⟨e⟩ ⟨x⟩, g⊖ f̃2⟨e⟩ ⟨x⟩} )〉 : x∈̃X ) : e∈̃E ] . Thus, ⟨⟨f̃1, E⟩⟩∪̃⟨⟨f̃2, E⟩⟩]c = [⟨⟨f̃1, E⟩⟩]c∩̃[⟨⟨f̃2, E⟩⟩]c (ii) Obvious. Proposition 4. [22] Let ⟨⟨f̃1, E⟩⟩ and ⟨⟨f̃2, E⟩⟩ be two bi-polar vague soft sub sets sets over X then, 1. [⟨⟨f̃1, E⟩⟩ ∨̃ ⟨⟨f̃2, E⟩⟩]c = [⟨⟨f̃1, E⟩⟩]c ∧̃ [⟨⟨f̃2, E⟩⟩]c, 2. [⟨⟨f̃1, E⟩⟩ ∧̃ ⟨⟨f̃2, E⟩⟩]c = [⟨⟨f̃1, E⟩⟩]c ∨̃ [⟨⟨f̃2, E⟩⟩]c, Proof. (i) For all (e1, e2)∈̃E × E, x∈̃X 2∨ i=1 ⟨⟨f̃i, E⟩⟩ = [ (e1, e2), 〈 x, ( max{g⊕ f̃1⟨e1⟩ ⟨x⟩, g⊕ f̃2⟨e2⟩ ⟨x⟩},min{h⊕ f̃1⟨e1⟩ ⟨x⟩, h⊕ f̃2⟨e2⟩ ⟨x⟩} max{g⊖ f̃1⟨e1⟩ ⟨x⟩, g⊖ f̃2⟨e2⟩ ⟨x⟩},min{h⊖ f̃1⟨e1⟩ ⟨x⟩, h⊖ f̃2⟨e2⟩ ⟨x⟩} )〉] , [ 2∨ i=1 ⟨⟨f̃i, E⟩⟩ ]c = [ (e1, e2), 〈 x, ( min{h⊕ f̃1⟨e1⟩ ⟨x⟩, h⊕ f̃2⟨e2⟩ ⟨x⟩},max{g⊕ f̃1⟨e1⟩ ⟨x⟩, g⊕ f̃2⟨e2⟩ ⟨x⟩} min{h⊖ f̃1⟨e1⟩ ⟨x⟩, h⊖ f̃2⟨e2⟩ ⟨x⟩},max{g⊖ f̃1⟨e1⟩ ⟨x⟩, g⊖ f̃2⟨e2⟩ ⟨x⟩} )〉] . Now ⟨⟨f̃1, E⟩⟩c = [ e1, 〈 x, ( h⊕ f̃1⟨e1⟩ ⟨x⟩, g⊕ f̃2⟨e2⟩ ⟨x⟩ h⊖ f̃1⟨e1⟩ ⟨x⟩, g⊖ f̃2⟨e2⟩ ⟨x⟩ )〉 : e∈̃E ] , ⟨⟨f̃2, E⟩⟩c = [ e2, 〈 x, ( h⊕ f̃1⟨e1⟩ ⟨x⟩, g⊕ f̃2⟨e2⟩ ⟨x⟩ h⊖ f̃1⟨e1⟩ ⟨x⟩, g⊖ f̃2⟨e2⟩ ⟨x⟩ )〉 : e∈̃E ] . Then, 2∧ i=1 ⟨⟨f̃i, E⟩⟩c = [ (e1, e2), 〈 x, ( min{h⊕ f̃1⟨e1⟩ ⟨x⟩, h⊕ f̃2⟨e2⟩ ⟨x⟩},max{g⊕ f̃1⟨e1⟩ ⟨x⟩, g⊕ f̃2⟨e2⟩ ⟨x⟩} min{h⊖ f̃1⟨e1⟩ ⟨x⟩, h⊖ f̃2⟨e2⟩ ⟨x⟩},max{g⊖ f̃1⟨e1⟩ ⟨x⟩, g⊖ f̃2⟨e2⟩ ⟨x⟩} )〉] . = [ (e1, e2), 〈 x, ( min{h⊕ f̃1⟨e1⟩ ⟨x⟩, h⊕ f̃2⟨e2⟩ ⟨x⟩},max{g⊕ f̃1⟨e1⟩ ⟨x⟩, g⊕ f̃2⟨e2⟩ ⟨x⟩} min{h⊖ f̃1⟨e1⟩ ⟨x⟩, h⊖ f̃2⟨e2⟩ ⟨x⟩},max{g⊖ f̃1⟨e1⟩ ⟨x⟩, g⊖ f̃2⟨e2⟩ ⟨x⟩} )〉] .r Thus, ⟨⟨f̃1, E⟩⟩ ∨̃ ⟨⟨f̃2, E⟩⟩]c = [⟨⟨f̃1, E⟩⟩]c ∧̃ [⟨⟨f̃2, E⟩⟩]c (ii) Obvious. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 11 of 32 Definition 12. [21] A pair ⟨⟨f̃ , E⟩⟩ is called a soft expert set (SES) over X , where f̃ is a mapping given by f̃ : E → P (X) where P (X) is power set of X. Definition 13. [21] For two soft expert sets ⟨⟨f̃1, E1⟩⟩ and ⟨⟨f̃2, E2⟩⟩ over X , ⟨⟨f̃ , E⟩⟩ is called a soft expert subset of (G,B) if: 1. E1 ⊆ E2, 2. for all ε∈̃E2, f̃1(ε) ⊆ f̃2(ε) This relationship is denoted by ⟨⟨f̃1, E1⟩⟩ ⊆ ⟨⟨f̃2, E2⟩⟩ In this case ⟨⟨f̃2, E2⟩⟩ is called a (SE) super set of ⟨⟨f̃1, E1⟩⟩ Definition 14. [21] Two soft expert sets ⟨⟨f̃1, E1⟩⟩ and ⟨⟨f̃2, E2⟩⟩ over X are said to be equal if ⟨⟨f̃1, E1⟩⟩ is a (SESs) of ⟨⟨f̃2, E2⟩⟩ and ⟨⟨f̃2, E2⟩⟩ is a (SESs) ⟨⟨f̃1, E1⟩⟩ . Definition 15. [21] Let E be a set of parameters and X a set of experts. The NOT set of Ž = E × X × O denoted by ǏŽ is defined by ǏŽ = {Ǐei, xj , ok} ∀i, j, k, where Ǐei is not ei . Definition 16. [21] The complement of a soft expert set ⟨⟨f̃ , E⟩⟩ is denoted by ⟨⟨f̃ , E⟩⟩c and is defined by ⟨⟨f̃ , E⟩⟩c = ⟨⟨f̃ c, ǏE⟩⟩ where f̃ c : ǏE → P (X) is a mapping given by f̃ c(α) = X− f̃(Ǐα), ∀α∈̃ǏE Definition 17. [21] An absolute soft expert set ⟨⟨f̃ , E⟩⟩1 over X is a (SESS) of ⟨⟨f̃ , E⟩⟩ defined as follows: ⟨⟨f̃ , E⟩⟩1 = {f̃1(α) : α∈̃E ×X × {1} Definition 18. [21] A null soft expert set ⟨⟨f̃ , E⟩⟩0 over X is a soft expert sub set of ⟨⟨f̃ , E⟩⟩ defined as follows: ⟨⟨f̃ , E⟩⟩0 = {f̃0(α) : α∈̃E ×X × {0} Definition 19. [21] The union of two soft expert sub sets ⟨⟨f̃1, E1⟩⟩ and ⟨⟨f̃2, E2⟩⟩ over X denoted by ⟨⟨f̃1, E1⟩⟩∪̃⟨⟨f̃2, E2⟩⟩ is (SES) (H, Č) where Č = E1∪̃E2, ∀ε∈̃Č H(ε) =  f̃1(ε), if ε∈̃E1 − E2, f̃2(ε), if ε∈̃E2 − E1, f̃1(ε)∪̃f̃2(ε), if ε∈̃E1∩̃E2, Definition 20. [21] The intersection of two soft expert sub sets ⟨⟨f̃1, E1⟩⟩ and ⟨⟨f̃2, E2⟩⟩ over X denoted by ⟨⟨f̃1, E1⟩⟩∩̃⟨⟨f̃2, E2⟩⟩ is (SES) (H, Č) where Č = E1∪̃E2, ∀ε∈̃Č H(ε) =  f̃1(ε), if ε∈̃E1 − E2, f̃2(ε), if ε∈̃E2 − E1, f̃1(ε)∩̃f̃2(ε), if ε∈̃E1∩̃E2, Definition 21. [21] If ⟨⟨f̃1, E1⟩⟩ and ⟨⟨f̃2, E2⟩⟩ are two soft expert sub set over X then ⟨⟨f̃1, E1⟩⟩ AND ⟨⟨f̃2, E2⟩⟩ denoted by ⟨⟨f̃1, E1⟩⟩ ∧ ⟨⟨f̃2, E2⟩⟩ , is defined by ⟨⟨f̃1, E1⟩⟩ ∧ ⟨⟨f̃2, E2⟩⟩ = (H,E1 × E2) where H(α, β) = f̃1(α)∩̃f̃2(β) ∀(α, β)∈̃E1 × E2 M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 12 of 32 Definition 22. [21] If ⟨⟨f̃1, E1⟩⟩ and ⟨⟨f̃2, E2⟩⟩ are two soft expert sub set over X then ⟨⟨f̃1, E1⟩⟩ OR ⟨⟨f̃2, E2⟩⟩ denoted by ⟨⟨f̃1, E1⟩⟩ ∨ ⟨⟨f̃2, E2⟩⟩ , is defined by ⟨⟨f̃1, E1⟩⟩ ∨ ⟨⟨f̃2, E2⟩⟩ = (O,E1 × E2) where O(α, β) = f̃1(α)∪̃f̃2(β) ∀(α, β)∈̃E1 × E2 Definition 23. [10] A pair ⟨⟨f̃ , E⟩⟩ is called a fuzzy soft expert sub sets over X where f̃ is a mapping given by f̃ : E → IX where IX denotes the set of all fuzzy soft expert sub set of X. Definition 24. [10] For two fuzzy soft expert sub set that is ⟨⟨f̃1, E1⟩⟩ and ⟨⟨f̃2, E2⟩⟩ over X , ⟨⟨f̃1, E1⟩⟩ called fuzzy soft expert sub set of ⟨⟨f̃2, E2⟩⟩ if: 1. E2 ⊆ E1, 2. ∀ ε∈̃E1, f̃1(ε) is fuzzy soft expert sub set of f̃2(ε). This relationship is denoted by ⟨⟨f̃1, E1⟩⟩ ⊆ ⟨⟨f̃2, E2⟩⟩ In this case ⟨⟨f̃2, E2⟩⟩ is called a (FSE) super-set of ⟨⟨f̃1, E1⟩⟩ Definition 25. [10] Two fuzzy soft expert set that is ⟨⟨f̃1, E1⟩⟩ and ⟨⟨f̃2, E2⟩⟩ over X are said to be equal, if ⟨⟨f̃1, E1⟩⟩ is a fuzzy soft expert sub set of ⟨⟨f̃2, E2⟩⟩ and ⟨⟨f̃2, E2⟩⟩ is a fuzzy soft expert sub set of ⟨⟨f̃1, E1⟩⟩ . Definition 26. [10] An (AFSES) ⟨⟨f̃ , E⟩⟩1 over X is a fuzzy soft expert sub set of ⟨⟨f̃ , E⟩⟩ defined as follows: ⟨⟨f̃ , E⟩⟩1 = {f̃1(α) : α∈̃E ×X × {1} Definition 27. [10] An absolute fuzzy soft expert set ⟨⟨f̃ , E⟩⟩0 over X is a fuzzy soft expert sub set of ⟨⟨f̃ , E⟩⟩ defined as follows: ⟨⟨f̃ , E⟩⟩0 = {f̃0(α) : α∈̃E ×X × {0} Definition 28. [10] The complement of a fuzzy soft expert set ⟨⟨f̃ , E⟩⟩ is denoted by ⟨⟨f̃ , E⟩⟩c and is defined by ⟨⟨f̃ , E⟩⟩c = ⟨⟨f̃ c, ∤ E⟩⟩ where f̃ c : E → P (X) is a mapping given by f̃ c(α) = c(f̃(α)),∀α∈̃E Where c is a fuzzy complement. Definition 29. [10] The union of two fuzzy soft expert sets ⟨⟨f̃1, E1⟩⟩ and ⟨⟨f̃2, E2⟩⟩ over X denoted by ⟨⟨f̃1, E1⟩⟩∪̃⟨⟨f̃2, E2⟩⟩ is the fuzzy soft expert set (H, Č) where Č = E1∪̃E2, ∀ε∈̃Č H(ε) =  f̃1(ε), if ε∈̃E1 − E2, f̃2(ε), if ε∈̃E2 − E1, sf̃1(ε), f̃2(ε), if ε∈̃E1∩̃E2, where s is an s-norm. Definition 30. [10] The intersection of two fuzzy soft expert sets ⟨⟨f̃1, E1⟩⟩ and ⟨⟨f̃2, E2⟩⟩ over X denoted by ⟨⟨f̃1, E1⟩⟩∩̃⟨⟨f̃2, E2⟩⟩ is fuzzy soft expert set (H, Č) where Č = E1∩̃E2, ∀ε∈̃Č H(ε) =  f̃1(ε), if ε∈̃E1 − E2, f̃2(ε), if ε∈̃E2 − E1, tf̃1(ε), f̃2(ε), if ε∈̃E1∩̃E2, where t is a t-norm. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 13 of 32 Definition 31. [10] If ⟨⟨f̃1, E1⟩⟩ and ⟨⟨f̃2, E2⟩⟩ are two fuzzy soft expert sets over X then ⟨⟨f̃1, E1⟩⟩ AND ⟨⟨f̃2, E2⟩⟩ denoted by ⟨⟨f̃1, E1⟩⟩ ∧ ⟨⟨f̃2, E2⟩⟩ , is defined by ⟨⟨f̃1, E1⟩⟩ ∧ ⟨⟨f̃2, E2⟩⟩ = (H,E1 × E2) s.t. H(α, β) = t(f̃1(ε), f̃2(ε)) ∀(α, β)∈̃E1 × E2, where t is a t-norm. Definition 32. [10] If ⟨⟨f̃1, E1⟩⟩ and (G,B) are two fuzzy soft expert sets over X then ⟨⟨f̃1, E1⟩⟩ OR ⟨⟨f̃2, E2⟩⟩ denoted by ⟨⟨f̃1, E1⟩⟩ ∨ ⟨⟨f̃2, E2⟩⟩ , is defined by ⟨⟨f̃1, E1⟩⟩ ∨ ⟨⟨f̃2, E2⟩⟩ = (H,E1 × E2) s.t. H(α, β) = s(f̃1(ε), f̃2(ε)) ∀(α, β)∈̃E1 × E2 , where s is an s-norm. 3. Exhibition of Bipolar Vague Soft Expert Structure This section introduces eight new definitions alongside several original contributions, includ- ing the bipolar vague soft expert topology (BPVST). Among these, the concept of the bipolar vague soft expert pre-open set, abbreviated as ”p-open set,” is highlighted as a pivotal and ver- satile tool for constructing diverse structures. The notions of interior and closure are explored in detail, and outcomes derived from these concepts are thoroughly addressed. Furthermore, additional insights are provided regarding the interaction between interior and closure, adding depth to the analysis. Let X be universal set, E a set of parameter, X a set of expert (agents), and o = {1 = agree, 0 = disagree} a set of opinion. let Ž = E ×X ×O and Ē ⊆ Ž Definition 33. A pair ⟨⟨H, Ē⟩⟩ is called a bipolar vague soft expert set over X, where H is mapping given by H : Ē → P (X) Where P (X) denotes a power set of bipolar vague soft expert set of X and ⟨⟨H, Ē⟩⟩ = ( ⟨u, T+ H(e)(u), f̃ + H(e), T − H(e)(u), f̃ − H(e)(u)⟩ ∀ e∈̃E, u∈̃X ) . Where T+ H(e)(u), f̃ + H(e) : X → [0, 1] and T− H(e)(u), f̃ − H(e) : X → [0, 1] Example 3. Let E = {cheap , expensive} = {(e1, e2)}.X = {p, q, r} be a set of experts such that H(e1, p, 1) =  ⟨u1, 03× 10−1, 07× 10−1, −02× 10−1,−04× 10−1⟩, ⟨u3, 05× 10−1, 03× 10−1, −03× 10−1,−01× 10−1⟩,  .H(e1, q, 1) =  ⟨u2, 08× 10−1, 03× 10−1, −01× 10−1,−05× 10−1⟩, ⟨u3, 09× 10−1, 07× 10−1, −04× 10−1,−02× 10−1⟩,  . H(e1, r, 1) = ( ⟨u1, 04× 10−1, 06× 10−1, −06× 10−1,−04× 10−1⟩ ) . H(e2, p, 1) =  ⟨u1, 04× 10−1, 03× 10−1, −02× 10−1,−01× 10−1⟩, ⟨u2, 07× 10−1, 03× 10−1, −03× 10−1,−05× 10−1⟩,  .H(e2, q, 1) = ( ⟨u3, 03× 10−1, 02× 10−1, −05× 10−1,−04× 10−1⟩, ) . H(e2, r, 1) = ( ⟨u2, 03× 10−1, 09× 10−1, −04× 10−1,−01× 10−1⟩ ) . H(e1, p, 0) = ( ⟨u2, 05× 10−1, 03× 10−1, −05× 10−1,−03× 10−1⟩, ) .H(e1, q, 0) = ( ⟨u1, 06× 10−1, 05× 10−1, −04× 10−1,−06× 10−1⟩, ) . M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 14 of 32 H(e1, r, 0) =  ⟨u2, 07× 10−1, 04× 10−1, −03× 10−1,−05× 10−1⟩ ⟨u3, 09× 10−1, 07× 10−1, −02× 10−1,−05× 10−1⟩  . H(e2, p, 0) = ( ⟨u3, 07× 10−1, 06× 10−1, −02× 10−1,−04× 10−1⟩, ) .H(e2, q, 0) =  ⟨u1, 07× 10−1, 06× 10−1, −03× 10−1,−04× 10−1⟩, ⟨u2, 06× 10−1, 05× 10−1, −03× 10−1,−04× 10−1⟩  . H(e2, r, 0) =  ⟨u1, 06× 10−1, 05× 10−1, −05× 10−1,−02× 10−1⟩ ⟨u3, 07× 10−1, 08× 10−1, −06× 10−1,−01× 10−1⟩  . Definition 34. Let ⟨⟨H, Ē1⟩⟩ and ⟨⟨f̃2, Ē2⟩⟩ be two bipolar vague soft expert sub sets over X. ⟨⟨H, Ē1⟩⟩ is said to be bipolar vague soft expert sub sets of ⟨⟨f̃2, Ē2⟩⟩, if ⟨⟨H, Ē1⟩⟩⊆̃⟨⟨f̃2, Ē2⟩⟩ if and only if T+ H(e)(u) ≾ T+ G(e)(u), f̃ + H(e) ≿ f̃+ G(e) and T+ H(e)(u) ≿ T+ G(e)(u), f̃ + H(e) ≾ f̃+ G(e), ∀eϵ̃Ē1, uϵ̃X.⟨⟨H, Ē1⟩⟩ is said to be bipolar vague soft ex- pert super set of ⟨⟨f̃2, Ē2⟩⟩ if ⟨⟨f̃2, Ē2⟩⟩ is a (BVSE S) of ⟨⟨H, Ē1⟩⟩ denoted by ⟨⟨H, Ē1⟩⟩⊇̃⟨⟨f̃2, Ē2⟩⟩ . Example 4. Let X = {u1, u2, u3} be a master set, E = {u1, u2} a set of parameters where ei(i = 1, 2) denotes the decision ’cheap’ , ’expensive’ respectively and Let X = {u1, u2, u3} be a set of experts. Suppose ⟨⟨H, Ē1⟩⟩ and ⟨⟨f̃2, Ē2⟩⟩ be defined as follows: ⟨⟨H, Ē1⟩⟩ =  [ (e1, p, 1), ⟨u1, 03× 10−1, 06× 10−1,−02× 10−1,−04× 10−1⟩, ⟨u2, 05× 10−1, 03× 10−1, −04× 10−1,−05× 10−1⟩, ] [ (e1, p, 0), ⟨u2, 02× 10−1, 07× 10−1,−05× 10−1,−03× 10−1⟩, ][ (e1, q, 1), ⟨u1, 06× 10−1, 05× 10−1,−06× 10−1,−05× 10−1⟩, ⟨u2, 06× 10−1, 03× 10−1, −05× 10−1,−03× 10−1⟩, ] [ (e1, r, 0), ⟨u1, 02× 10−1, 03× 10−1,−04× 10−1,−05× 10−1⟩, ][ (e2, r, 1), ⟨u2, 03× 10−1, 09× 10−1,−03× 10−1,−04× 10−1⟩, ⟨u3, 07× 10−1, 08× 10−1, −05× 10−1,−06× 10−1⟩, ]  ⟨⟨f̃2, Ē2⟩⟩ =  [ (e1, p, 1), ⟨u1, 03× 10−1, 07× 10−1,−02× 10−1,−06× 10−1⟩, ⟨u2, 05× 10−1, 03× 10−1, −01× 10−1,−07× 10−1⟩, ] [ (e2, p, 0), ⟨u2, 02× 10−1, 07× 10−1,−02× 10−1,−05× 10−1⟩, ][ (e1, q, 1), ⟨u1, 06× 10−1, 05× 10−1,−01× 10−1,−08× 10−1⟩, ⟨u2, 06× 10−1, 03× 10−1, −03× 10−1,−04× 10−1⟩, ]  Therefore ⟨⟨H, Ē1⟩⟩⊇̃⟨⟨f̃2, Ē2⟩⟩ . Definition 35. Let ⟨⟨H, Ē1⟩⟩ and ⟨⟨f̃2, Ē2⟩⟩ be two bipolar vague soft expert sub sets . ⟨⟨H, Ē1⟩⟩ is said to be bipolar vague soft expert equal ⟨⟨f̃2, Ē2⟩⟩ and we write ⟨⟨H, Ē1⟩⟩ = ⟨⟨f̃2, Ē2⟩⟩ if and only if T+ H(e)(u) = T+ G(e)(u), f̃ + H(e) = f̃+ G(e) and T− H(e)(u) = T− G(e)(u), f̃ − H(e) = f̃− G(e), ∀eϵ̃Ē1, uϵ̃X. Definition 36. Let E = {e1, e2, ..., en} be a set of parameters. The NOT set of E is denoted by ¬E = {¬e1,¬e2, ...,¬en} where ¬ei = notei, ∀i = 1, 2, ..., n. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 15 of 32 Example 5. Consider Example ref 3.2 Here ¬E = {notcheap, notexpensive} Definition 37. The complement of a bipolar vague soft expert set ⟨⟨H, Ē1⟩⟩ denoted by ⟨⟨H, Ē1⟩⟩c and is defined as ⟨⟨H, Ē1⟩⟩c = ⟨⟨Hc,¬Ē1⟩⟩ where Hc = ¬Ē1 → P (X) is mapping given by Hc(u) = T+ Hc(u) = f̃+ H(u), f̃ + Hc(u) = T+ Hc(u) and T− Hc(u) = f̃− H(e), f̃ − Hc(u) = T− H(u). Example 6. Consider the 4 Example. Then ⟨⟨H, ¯̌Z⟩⟩c is ⟨⟨H, ¯̌Z⟩⟩c =  [ (¬e1, p, 1), ⟨u2, 03× 10−1, 05× 10−1,−03× 10−1,−05× 10−1⟩, ] ,[ (¬e1, q, 1), ⟨u1, 05× 10−1, 06× 10−1,−04× 10−1,−03× 10−1⟩, ] ,[ (¬e1, r, 1), ⟨u2, 04× 10−1, 07× 10−1,−03× 10−1,−02× 10−1⟩, ⟨u3, 0× 10−1, 09× 10−1, −01× 10−1,−03× 10−1⟩, ] ,[ (¬e2, p, 1), ⟨u3, 06× 10−1, 07× 10−1,−04× 10−1,−02× 10−1⟩, ] ,[ (¬e2, q, 1), ⟨u1, 06× 10−1, 07× 10−1,−05× 10−1,−03× 10−1⟩, ⟨u2, 05× 10−1, 06× 10−1, −03× 10−1,−06× 10−1⟩, ] ,[ (¬e2, r, 1), ⟨u1, 05× 10−1, 06× 10−1,−06× 10−1,−04× 10−1⟩, ⟨u3, 08× 10−1, 07× 10−1, −03× 10−1,−01× 10−1⟩, ] ,[ (¬e1, p, 0), ⟨u1, 07× 10−1, 03× 10−1,−04× 10−1,−03× 10−1⟩, ⟨u3, 03× 10−1, 05× 10−1, −06× 10−1,−05× 10−1⟩, ] ,[ (¬e1, q, 0), ⟨u2, 03× 10−1, 08× 10−1,−03× 10−1,−07× 10−1⟩, ⟨u3, 09× 10−1, 07× 10−1, −07× 10−1,−05× 10−1⟩, ] ,[ (¬e1, r, 0), ⟨u1, 06× 10−1, 04× 10−1,−04× 10−1,−05× 10−1⟩, ] ,[ (¬e2, p, 0), ⟨u1, 03× 10−1, 04× 10−1,−03× 10−1,−04× 10−1⟩, ⟨u2, 03× 10−1, 07× 10−1, −06× 10−1,−01× 10−1⟩, ] ,[ (¬e2, q, 0), ⟨u3, 02× 10−1, 03× 10−1,−07× 10−1,−03× 10−1⟩, ] ,[ (¬e2, r, 0), ⟨u2, 09× 10−1, 03× 10−1,−08× 10−1,−05× 10−1⟩, ]  Definition 38. The set ⟨⟨H, Ē1⟩⟩ is termed to be bi-polar vague soft expert null if T+ H(e)(u) = T+ G(e)(u) = 0, f̃+ H(e)(u) = f̃+ G(e)(u) = 0 and T− H(e)(u) = T− G(e)(u) = 0, f̃− H(e)(u) = f̃− G(e)(u) = 0, ∀ eϵ̃Ē1, uϵ̃X. Example 7. Let X = {u1, u2, u3}, E = {quality} = {e1} and Let X = {p, q} be a set of experts ˇ̄ 1E = (NBNSES) =  [ (e1, p, 1), ⟨u1, 0, 0, 0, 0⟩, ⟨u2, 0, 0, 0, 0⟩, ][ (e1, q, 1), ⟨u1, 0, 0, 0, 0⟩, ⟨u2, 0, 0, 0, 0⟩, ][ (e1, p, 0), ⟨u3, 0, 0, 0, 0⟩, ][ (e1, q, 1), ⟨u3, 0, 0, 0, 0⟩, ]  Definition 39. A bi-polar vague soft expert set ⟨⟨H, Ē1⟩⟩1 soft expert subset of ⟨⟨H, Ē1⟩⟩ is defined as follows: ⟨⟨H, Ē1⟩⟩1 = {H1(u) : uϵ̃E ×X × {1}. Example 8. Consider the 4 Example 3.2. Then bi-polar vague soft expert set ⟨⟨H, Ē1⟩⟩1 over X is ⟨⟨H, Ē1⟩⟩1 =  [ (¬e1, p, 1), ⟨u1, 03× 10−1, 07× 10−1,−02× 10−1,−04× 10−1⟩, ⟨u3, 05× 10−1, 03× 10−1, −03× 10−1,−01× 10−1⟩, ] ,[ (¬e1, q, 1), ⟨u2, 08× 10−1, 03× 10−1,−01× 10−1,−05× 10−1⟩, ⟨u3, 09× 10−1, 07× 10−1, −04× 10−1,−02× 10−1⟩, ] ,[ (¬e1, r, 1), ⟨u1, 04× 10−1, 06× 10−1,−06× 10−1,−04× 10−1⟩ ] ,[ (¬e2, p, 1), ⟨u1, 04× 10−1, 03× 10−1,−02× 10−1,−01× 10−1⟩, ⟨u2, 07× 10−1, 03× 10−1, −03× 10−1,−05× 10−1⟩, ] ,[ (¬e2, q, 1), ⟨u3, 03× 10−1, 02× 10−1,−05× 10−1,−04× 10−1⟩, ] ,[ (¬e2, r, 1), ⟨u2, 03× 10−1, 09× 10−1,−04× 10−1,−01× 10−1⟩ ] ,  M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 16 of 32 Definition 40. A bipolar vague soft expert set ⟨⟨H, Ē1⟩⟩0 over X is a bi-polar vague soft expert set of ⟨⟨H, Ē1⟩⟩ defined as fallows; ⟨⟨H, Ē1⟩⟩0 = {H0(u) : uϵ̃E ×X × {0} Example 9. Consider the 4 Example 3.2. Then the bipolar vague soft expert set ⟨⟨H, Ē1⟩⟩0 over X is ⟨⟨H, Ē1⟩⟩0 =  [ (¬e1, p, 0), ⟨u2, 05× 10−1, 03× 10−1,−05× 10−1,−03× 10−1⟩, ] ,[ (¬e1, q, 0), ⟨u1, 06× 10−1, 05× 10−1,−04× 10−1,−06× 10−1⟩, ] ,[ (¬e1, r, 0), ⟨u2, 07× 10−1, 04× 10−1,−03× 10−1,−05× 10−1⟩⟨u3, 09× 10−1, 07× 10−1, −02× 10−1,−05× 10−1⟩, ] ,[ (¬e2, p, 0), ⟨u3, 07× 10−1, 06× 10−1,−02× 10−1,−04× 10−1⟩, ] ,[ (¬e2, q, 0), ⟨u1, 07× 10−1, 06× 10−1,−03× 10−1,−04× 10−1⟩, ⟨u2, 06× 10−1, 05× 10−1, −03× 10−1,−04× 10−1⟩ ] ,[ (¬e2, r, 0), ⟨u1, 06× 10−1, 05× 10−1,−05× 10−1,−02× 10−1⟩⟨u3, 07× 10−1, 08× 10−1, −06× 10−1,−01× 10−1⟩ ] ,  Definition 41. The union of two bipolar vague soft expert sets. Let ⟨⟨H, Ē1⟩⟩ = { ⟨u, T+ H(e)(u), f̃ + H(e)(u), T − H(e)(u), f̃ − H(e)(u)⟩ : ∀eϵ̃X, uϵ̃X } and ⟨⟨f̃2, Ē2⟩⟩ = { ⟨u, T+ G(e)(u), f̃ + G(e)(u), T − G(e)(u), f̃ − G(e)(u)⟩ : ∀eϵ̃B, uϵ̃X } be two bipolar vague soft expert sets then their union is defined as: ⟨⟨H, Ē1⟩⟩∪̃⟨⟨f̃2, Ē2⟩⟩(u) = ( max(T+ H(e)(u), T + G(e)(u)),min(f̃+ H(e)(u), f̃ + G(e)(u)) min(T− H(e)(u), T − G(e)(u)),max(f̃− H(e)(u), f̃ − G(e)(u)) ) Example 10. Let ⟨⟨H, Ē1⟩⟩ and ⟨⟨f̃2, Ē2⟩⟩ be two bipolar vague soft expert sets ⟨⟨H, Ē1⟩⟩ =  [ (e1, p, 1), ⟨u1, 02× 10−1, 08× 10−1,−04× 10−1,−05× 10−1⟩, ⟨u3, 02× 10−1, 05× 10−1, −02× 10−1,−04× 10−1⟩, ] [ (e1, q, 1), ⟨u1, 05× 10−1, 06× 10−1,−02× 10−1,−03× 10−1⟩, ⟨u2, 08× 10−1, 03× 10−1, −02× 10−1,−01× 10−1⟩, ]  ⟨⟨f̃2, Ē2⟩⟩ = {[ (e1, p, 1), ⟨u1, 01× 10−1, 02× 10−1,−03× 10−1,−04× 10−1⟩, ⟨u2, 04× 10−1, 08× 10−1, −01× 10−1,−05× 10−1⟩, ]} Therefor ⟨⟨H, Ē1⟩⟩∪̃⟨⟨f̃2, Ē2⟩⟩ = ⟨⟨R, C̄⟩⟩ ⟨⟨R, C̄⟩⟩ =   (e1, p, 1), ⟨u1, 02× 10−1, 02× 10−1,−04× 10−1,−04× 10−1⟩, ⟨u2, 04× 10−1, 08× 10−1,−01× 10−1,−05× 10−1⟩, ⟨u3, 02× 10−1, 05× 10−1, −02× 10−1,−04× 10−1⟩,  [ (e1, q, 1), ⟨u1, 05× 10−1, 06× 10−1,−02× 10−1,−03× 10−1⟩, ⟨u2, 04× 10−1, 08× 10−1, −01× 10−1,−05× 10−1⟩, ]  Definition 42. The intersection of two bipolar vague soft expert sets ⟨⟨H, Ē1⟩⟩ = { ⟨u, T+ H(e)(u), f̃ + H(e)(u), T − H(e)(u), f̃ − H(e)(u)⟩ : ∀eϵ̃X, uϵ̃X } M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 17 of 32 and ⟨⟨f̃2, Ē2⟩⟩ = { ⟨u, T+ G(e)(u), f̃ + G(e)(u), T − G(e)(u), f̃ − G(e)(u)⟩ : ∀eϵ̃B, uϵ̃X } be two bipolar vague soft expert sets then their intersection is defined as: ⟨⟨H, Ē1⟩⟩∩̃⟨⟨f̃2, Ē2⟩⟩(u) = ( min(T+ H(e)(u), T + G(e)(u)),max(f̃+ H(e)(u), f̃ + G(e)(u)) max(T− H(e)(u), T − G(e)(u)),min(f̃− H(e)(u), f̃ − G(e)(u)) ) Example 11. Let ⟨⟨H, Ē1⟩⟩ and ⟨⟨f̃2, Ē2⟩⟩ be two bipolar vague soft expert sets ⟨⟨H, Ē1⟩⟩ =  [ (e1, p, 1), ⟨u1, 02× 10−1, 08× 10−1,−04× 10−1,−05× 10−1⟩, ⟨u3, 02× 10−1, 05× 10−1, −02× 10−1,−04× 10−1⟩, ] [ (e1, q, 1), ⟨u1, 05× 10−1, 06× 10−1,−02× 10−1,−03× 10−1⟩, ⟨u2, 08× 10−1, 03× 10−1, −02× 10−1,−01× 10−1⟩, ]  ⟨⟨f̃2, Ē2⟩⟩ = {[ (e1, p, 1), ⟨u1, 01× 10−1, 02× 10−1,−03× 10−1,−04× 10−1⟩, ⟨u2, 04× 10−1, 08× 10−1, −01× 10−1,−05× 10−1⟩, ]} Therefor ⟨⟨H, Ē1⟩⟩∩̃⟨⟨f̃2, Ē2⟩⟩ = ⟨⟨R, C̄⟩⟩ ⟨⟨R, C̄⟩⟩ = {[ (e1, p, 1), ⟨u1, 01× 10−1, 08× 10−1,−03× 10−1,−05× 10−1⟩ ]} Definition 43. Let bi-polar vague soft expert set ⟨⟨X,E⟩⟩ be family of all bipolar vague soft expert sets over X and JBV SES ⊆̃ BV SE set ⟨⟨Xabsolute, E⟩⟩, then JBV SES is said to be a bipolar vague soft expert topology ⟨⟨BV SET ⟩⟩ on X . if 1. ⟨⟨f̃null, E⟩⟩ and ⟨⟨Xabsolute, E⟩⟩ ϵ̃ JBV SES . 2. The union of any number of ⟨⟨BV SE⟩⟩ sets in JBV SES ϵ̃ JBV SES 3. The intersection of finite number of ⟨⟨BV SE⟩⟩ sets in JBV SES ϵ̃ JBV SES Then ⟨⟨Xabsolute, J BV SES , E⟩⟩ is said to be a JBV SESTS over X. Definition 44. Let ⟨⟨X, JBV SES , E⟩⟩ be a bi-polar vague soft expert set topological space over X, ⟨⟨f̃2, Ē2⟩⟩ be a bi-polar vague soft expert set set over X then ⟨⟨f̃2, Ē2⟩⟩ is said to be bi-polar vague soft expert set closed set iff its complemt is a ⟨⟨BV SE⟩⟩ open set. Definition 45. Let ⟨⟨X, JBV SES , E⟩⟩ be a bi-polar vague soft expert set topological space and ⟨⟨f̃ , Ē⟩⟩ be a ⟨⟨BV SE⟩⟩ set over X then ⟨⟨f̃ , Ē⟩⟩ is called bipolar vague soft expert 1. Semi-open if ⟨⟨f̃ , Ē⟩⟩ ⊆̃ BV EScl(BV SEint⟨⟨f̃ , Ē⟩⟩) and BVSE semi-close if ⟨⟨f̃ , Ē⟩⟩ ⊇̃ BV ESinterior(BV SEcl⟨⟨f̃ , Ē⟩⟩) 2. Pre-open if ⟨⟨f̃ , Ē⟩⟩ ⊆̃ BV ESinterior(BV SEcl⟨⟨f̃ , Ē⟩⟩) and BVSE pre close if ⟨⟨f̃ , Ē⟩⟩ ⊇̃ BV EScl(BV SEinterior⟨⟨f̃ , Ē⟩⟩) . M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 18 of 32 3. α−open if ⟨⟨f̃ , Ē⟩⟩ ⊆̃ BV ESinterior(BV SEcl(BV SEinterior)⟨⟨f̃ , Ē⟩⟩) and BVSE αclose if ⟨⟨f̃ , Ē⟩⟩ ⊇̃ BV EScl(BV SEint(BV SEcl⟨⟨f̃ , Ē⟩⟩)) . 4. β − open if ⟨⟨f̃ , Ē⟩⟩ ⊆̃ BV EScl(BV ESinterior(BV SEcl)⟨⟨f̃ , Ē⟩⟩) and BVSE β − open if ⟨⟨f̃ , Ē⟩⟩ ⊇̃ BV ESinterior(BV SEcl(BV ESinterior⟨⟨f̃ , Ē⟩⟩)) . 5. b−open if ⟨⟨f̃ , Ē⟩⟩ ⊆̃ BV EScl(BV ESinterior⟨⟨f̃ , Ē⟩⟩) ∪̃ BV ESinterior(BV SEcl⟨⟨f̃ , Ē⟩⟩) and BVSE b−open if ⟨⟨f̃ , Ē⟩⟩ ⊇̃ BV ESinterior(BV SEcl⟨⟨f̃ , Ē⟩⟩) ∩̃ BV EScl(BV ESinterior⟨⟨f̃ , Ē⟩⟩). 6. ∗b−open if ⟨⟨f̃ , Ē⟩⟩ ⊆̃ BV EScl(BV ESinterior⟨⟨f̃ , Ē⟩⟩) ∩̃ BV ESinterior(BV SEcl⟨⟨f̃ , Ē⟩⟩) and BVSE ∗b−open if ⟨⟨f̃ , Ē⟩⟩ ⊇̃ BV ESinterior(BV SEcl⟨⟨f̃ , Ē⟩⟩) ∪̃ BV EScl(BV ESinterior⟨⟨f̃ , Ē⟩⟩). 7. b∗∗−open if ⟨⟨f̃ , Ē⟩⟩ ⊆̃ BV ESint(BV SEcl(BV SEint⟨⟨f̃ , Ē⟩⟩)) ∪̃ BV EScl(BV SEint(BV SEcl⟨⟨f̃ , Ē⟩⟩)) and BVSE b ∗ ∗ − open if ⟨⟨f̃ , Ē⟩⟩ ⊇̃ BV EScl(BV SEint(BV SEcl⟨⟨f̃ , Ē⟩⟩)) ∩̃ BV ESint(BV SEcl(BV SEint⟨⟨f̃ , Ē⟩⟩)) . 8. ∗∗b−open if ⟨⟨f̃ , Ē⟩⟩ ⊆̃ BV ESint(BV SEcl(BV SEint⟨⟨f̃ , Ē⟩⟩)) ∩̃ BV EScl(BV SEint(BV SEcl⟨⟨f̃ , Ē⟩⟩)) and BVSE ∗ ∗ b− open if ⟨⟨f̃ , Ē⟩⟩ ⊇̃ BV EScl(BV SEint(BV SEcl⟨⟨f̃ , Ē⟩⟩)) ∪̃ BV ESint(BV SEcl(BV SEint⟨⟨f̃ , Ē⟩⟩)) . Proposition 5. Let ⟨⟨X, JBV SES , E⟩⟩ be a bi-polar vague soft expert set topological space over X . Then 1. ⟨⟨f̃null, Ē⟩⟩ , ⟨⟨Xabsolute, E⟩⟩ are ⟨⟨BV SE⟩⟩ p- closed sets over X. 2. The intersection of any number of ⟨⟨BV SE⟩⟩ p- closed sets ⟨⟨ČS⟩⟩ is a ⟨⟨BV S⟩⟩ s-closed sets ⟨⟨ČS⟩⟩ over X. 3. The union of finite number of ⟨⟨BV SE⟩⟩ p- closed sets ⟨⟨ČS⟩⟩ is a ⟨⟨BV SE⟩⟩ p-closed sets ⟨⟨ČS⟩⟩ over X. Proof. Given ⟨⟨X,JBV SES , E⟩⟩ be a ⟨⟨BV SETS⟩⟩ over X. 1. Since ⟨⟨f̃null, Ē⟩⟩ and ⟨⟨Xabsolute, E⟩⟩ are ⟨⟨BV SE⟩⟩ open sets and hence p-open sets because these sets belong to JBV SES . We see that the complement of ⟨⟨f̃null, Ē⟩⟩ is ⟨⟨Xabsolute, E⟩⟩ which is open and hence it is s-open. This implies that ⟨⟨f̃null, Ē⟩⟩ is p- closed and hence it is p-closed. Similarly, the complement of ⟨⟨Xabsolute, E⟩⟩ is ⟨⟨f̃null, Ē⟩⟩ which is open and hence p-open. This implies that ⟨⟨Xabsolute, E⟩⟩ is closed and hence it is s-closed over X 2. Suppose {(f̃ , E)i : iϵ̃I} be collection of s-closed subsets of X then (f̃ , E)i is p-closed for all iϵ̃I this implies that (f̃ , E)ci is s-open for all iϵ̃I . Since the union of any number of p-open sets is p-open, so ⋃ i∈I (f̃ , E)ci is p-open this implies thatc is p-open this implies that ⋂ i∈I(f̃ , E)i is p-closed. 3. Let (f̃ , E)1 , (f̃ , E)2 , (f̃ , E)3, (f̃ , E)4 , ..., (f̃ , E)n, are any finite number of p-closed sets, then (f̃ , E)i is p-closed for all i = 1, 2, 3, ..., n this implies (f̃ , E)ci is s-open for all i = 1, 2, 3, ..., n but the intersection of any finite number of s-open sets is p-open, so ⋂n i∈I (f̃ , E)ci is p-open. This implies that ( ⋃ i∈I(f̃ , E)i) c is p-open this implies that ⋃ i∈I(f̃ , E)i is p-closed. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 19 of 32 Definition 46. The family of all ⟨⟨BV SE⟩⟩ sets over X is denoted by BV SES⟨⟨Xabsolute, E⟩⟩ . 1. If JBV SES = {⟨⟨f̃null, E⟩⟩, ⟨⟨Xabsolute, E⟩⟩} , , then JBV SES is said to be ⟨⟨BV SE⟩⟩ indiscrete topology (X, JBV SES , E) is said to be a ⟨⟨BV SE⟩⟩ indiscrete topological space over X. 2. If JBV SES = BV SS⟨⟨Mabsolute, E⟩⟩ then JBV SES is said to be ⟨⟨BV SE⟩⟩ discrete topol- ogy. (X, JBV SES , E) is said to be a ⟨⟨BV SE⟩⟩ discrete topological space over X. Proposition 6. Let ⟨⟨X, JBV SES , E⟩⟩ and ⟨⟨X,JBV SES 2 , E⟩⟩ be two bi-polar vague soft expert set topological space over X . Then ⟨⟨X, JBV SES 1 , ⋂̃ JBV SES 2 , E⟩⟩ is ⟨⟨BV SETS⟩⟩ over X. Proof. 1. since ⟨⟨f̃null, E⟩⟩ , ⟨⟨Xabsolute, E⟩⟩ ϵ̃JBV SES and ⟨⟨f̃null, E⟩⟩ , ⟨⟨Xabsolute, E⟩⟩ ϵ̃JBV SES 2 then ⟨⟨f̃null, E⟩⟩ , ⟨⟨Xabsolute, E⟩⟩ ϵ̃JBV SES 1 ⋂̃ JBV SES 2 2. Let {⟨⟨f̃ , E⟩⟩ : i ∈ I} be a family of ⟨⟨BV SES⟩⟩ sets in JBV SES 1 ⋂̃ JBV SES 2 . Then (f̃i, E) ϵ̃ JBV SES 1 . (f̃i, E) ϵ̃ JBV SES 2 ∀iϵ̃I , so ∐ i∈I ⟨⟨f̃ , E⟩⟩ ϵ̃ JBV SES 1 and ∐ i∈I ⟨⟨f̃ , E⟩⟩ ϵ̃ JBV SES 2 Thus ∐ i∈I ⟨⟨f̃ , E⟩⟩ ϵ̃ JBV SES 1 ⋂̃ JBV SES 2 . 3. Let {⟨⟨f̃ , E⟩⟩ : i = 1, n} be a family of finite number of ⟨⟨BV SES⟩⟩ sets in JBV SES 1 ⋂̃ JBV SES 2 . Then (f̃i, E) ϵ̃ JBV SES 1 . (f̃i, E) ϵ̃ JBV SES 2 ∀i = 1, n , so ∏n i=1 ⟨⟨f̃ , E⟩⟩ ϵ̃ JBV SES 1 and ∏n i=1 ⟨⟨f̃ , E⟩⟩ ϵ̃ JBV SES 2 Thus ∏n i=1 ⟨⟨f̃ , E⟩⟩ ϵ̃ JBV SES 1 ⋂̃ JBV SES 2 . . Definition 47. Let ⟨⟨X, JBV SES , E⟩⟩ is a bi-polar vague soft expert set topological space over X , ⟨⟨f̃ , E⟩⟩ ϵ̃ BVSES ⟨⟨X,E⟩⟩ be a ⟨⟨BV SES⟩⟩ set. Then, ⟨⟨BV SES⟩⟩ interior of ⟨⟨f̃ , E⟩⟩, , denoted as ⟨⟨f̃ , E⟩⟩o , is defined as ⟨⟨BV SES⟩⟩ union of all ⟨⟨BV SES⟩⟩ p-open subsets of ⟨⟨f̃ , E⟩⟩ . Clearly, ⟨⟨f̃ , E⟩⟩o is biggest ⟨⟨BV SES⟩⟩ p-open set contained by ⟨⟨f̃ , E⟩⟩ Theorem 1. Let ⟨⟨X, JBV SES , E⟩⟩ is a bi-polar vague soft expert set topological space over X , ⟨⟨f̃ , E⟩⟩ ϵ̃ BVSES ⟨⟨X,E⟩⟩ . ⟨⟨f̃ , E⟩⟩ is a ⟨⟨BV SES⟩⟩ p-open set iff ⟨⟨f̃ , E⟩⟩ = ⟨⟨f̃ , E⟩⟩o Proof. Let ⟨⟨f̃ , E⟩⟩ is a ⟨⟨BV SES⟩⟩ p-open set then the biggest ⟨⟨BV SES⟩⟩ p-open set that is contained by ⟨⟨f̃ , E⟩⟩ is equal to ⟨⟨f̃ , E⟩⟩. Hence , ⟨⟨f̃ , E⟩⟩ = ⟨⟨f̃ , E⟩⟩o Conversely, ⟨⟨f̃ , E⟩⟩o is a ⟨⟨BV SES⟩⟩ p-open set, if ⟨⟨f̃ , E⟩⟩ = ⟨⟨f̃ , E⟩⟩o , , then ⟨⟨f̃ , E⟩⟩ is a ⟨⟨BV SES⟩⟩ set. Theorem 2. Let ⟨⟨X,JBV SES , E⟩⟩ is a bi-polar vague soft expert set topological space over X. ⟨⟨f̃1, E⟩⟩ , ⟨⟨f̃2, E⟩⟩ ϵ̃ BVSES ⟨⟨X,E⟩⟩. Then, 1. [⟨⟨f̃1, E⟩⟩o]o = ⟨⟨f̃1, E⟩⟩o, 2. ⟨⟨f̃null, E⟩⟩o = ⟨⟨f̃null, E⟩⟩ and ⟨⟨Xabsolute, E⟩⟩o = ⟨⟨Xabsolute, E⟩⟩, 3. ⟨⟨f̃1, E⟩⟩⊆̃⟨⟨f̃2, E⟩⟩ =⇒ ⟨⟨f̃1, E⟩⟩o⊆̃⟨⟨f̃2, E⟩⟩o, 4. [⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩]o = ⟨⟨f̃1, E⟩⟩o ⋂̃ ⟨⟨f̃2, E⟩⟩o, 5. ⟨⟨f̃1, E⟩⟩o ⋃̃ ⟨⟨f̃2, E⟩⟩o ⊆̃ [⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩]o . Proof. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 20 of 32 1. Let ⟨⟨f̃1, E⟩⟩o = ⟨⟨f̃2, E⟩⟩ . Then ⟨⟨f̃2, E⟩⟩ ϵ̃ τBN iff ⟨⟨f̃2, E⟩⟩ = ⟨⟨f̃2, E⟩⟩o. So, [⟨⟨f̃1, E⟩⟩o]o = ⟨⟨f̃1, E⟩⟩o, 2. Since ⟨⟨f̃null, E⟩⟩ and ⟨⟨Xabsolute, E⟩⟩ are always ⟨⟨BV SES⟩⟩ p-open this implies ⟨⟨f̃null, E⟩⟩o = ⟨⟨f̃null, E⟩⟩ and ⟨⟨Xabsolute, E⟩⟩o = ⟨⟨Xabsolute, E⟩⟩ because BVSES ⟨⟨f̃ , E⟩⟩ of ⟨⟨BV SES⟩⟩ p-open if ⟨⟨f̃1, E⟩⟩o = ⟨⟨f̃2, E⟩⟩ . 3. Let ⟨⟨f̃1, E⟩⟩o ⊆̃ ⟨⟨f̃1, E⟩⟩ ⊆̃ ⟨⟨f̃2, E⟩⟩ , ⟨⟨f̃2, E⟩⟩o ⊆̃ ⟨⟨f̃2, E⟩⟩ . Since ⟨⟨f̃2, E⟩⟩o is the biggest ⟨⟨BV SES⟩⟩ p-open set covered in ⟨⟨f̃2, E⟩⟩ so, ⟨⟨f̃1, E⟩⟩o ⊆̃ ⟨⟨f̃2, E⟩⟩o. 4. Since ⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩ ⊆̃ ⟨⟨f̃1, E⟩⟩ and ⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩ ⊆̃ ⟨⟨f̃2, E⟩⟩, then [⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩]o ⊆̃ ⟨⟨f̃1, E⟩⟩o , [⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩]o ⊆̃ ⟨⟨f̃2, E⟩⟩o , and so [⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩]o ⊆̃ ⟨⟨f̃1, E⟩⟩o⋂̃ ⟨⟨f̃2, E⟩⟩o . On the other hand, since ⟨⟨f̃1, E⟩⟩o ⊆̃ ⟨⟨f̃1, E⟩⟩ and ⟨⟨f̃2, E⟩⟩o ⊆̃ ⟨⟨f̃2, E⟩⟩ , then ⟨⟨f̃1, E⟩⟩o ⋂̃ ⟨⟨f̃2, E⟩⟩o ⊆̃ ⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩ besides [⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩]o ⊆̃ ⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩ and it is the biggest ⟨⟨BV SES⟩⟩ p-open set. Therefore, ⟨⟨f̃1, E⟩⟩o ⋂̃ ⟨⟨f̃2, E⟩⟩o ⊆̃ [⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩]o Thus, [⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩]o = ⟨⟨f̃1, E⟩⟩o ⋂̃ ⟨⟨f̃2, E⟩⟩o 5. Since ⟨⟨f̃1, E⟩⟩ ⊆̃ ⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩ and ⟨⟨f̃2, E⟩⟩ ⊆̃ ⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩ then ⟨⟨f̃1, E⟩⟩o ⊆̃ [⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩]o and ⟨⟨f̃2, E⟩⟩o ⊆̃ [⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩]o. Therefore, ⟨⟨f̃1, E⟩⟩o ⋂̃ ⟨⟨f̃2, E⟩⟩o ⊆̃ [⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩]o. Definition 48. Let ⟨⟨X, JBV SES , E⟩⟩ be a bi-polar vague soft expert set topological space over X . ⟨⟨f̃ , E⟩⟩ ϵ̃ BVSES ⟨⟨X,E⟩⟩ be a ⟨⟨BV SES⟩⟩ set then, ⟨⟨BV SES⟩⟩ p-closure of ⟨⟨f̃ , E⟩⟩ , , denoted ⟨⟨f̃ , E⟩⟩, , is defined as ⟨⟨BV SES⟩⟩ soft intersection of all ⟨⟨BV SES⟩⟩ p-closed supersets of ⟨⟨f̃ , E⟩⟩ . Clearly, ⟨⟨f̃ , E⟩⟩ is the smallest ⟨⟨BV SES⟩⟩ p-closed set covering by ⟨⟨f̃ , E⟩⟩ Definition 49. Let ⟨⟨X, JBV SES , E⟩⟩ be a bi-polar vague soft expert set topological space over X, ⟨⟨f̃ , E⟩⟩ ϵ̃ BVSES ⟨⟨X,E⟩⟩ be a ⟨⟨BV SES⟩⟩ set then the boundary of ⟨⟨f̃ , E⟩⟩ is dented by Fr⟨⟨f̃ , E⟩⟩ is defined as a ⟨⟨BV SES⟩⟩ point τm(ρ1,ρ2) is called boundary of ⟨⟨f̃ , E⟩⟩ if every ⟨⟨BV SES⟩⟩ p- open set containing τm(ρ1,ρ2) contains at least one point of ⟨⟨f̃ , E⟩⟩ and least one ⟨⟨BV SES⟩⟩ point of ⟨⟨f̃ , E⟩⟩c. Definition 50. Let ⟨⟨X, JBV SES , E⟩⟩ be a bi-polar vague soft expert set topological space over X, ⟨⟨f̃ , E⟩⟩ ϵ̃ BVSES ⟨⟨X,E⟩⟩ be a ⟨⟨BV SES⟩⟩ set then ⟨⟨BV SES⟩⟩ exterior of ⟨⟨f̃ , E⟩⟩ s dented by Ext⟨⟨f̃ , E⟩⟩ is defined as a ⟨⟨BV SES⟩⟩ point τm(ρ1,ρ2) is called exterior of ⟨⟨f̃ , E⟩⟩ if τm(ρ1,ρ2) if ⟨⟨BV SES⟩⟩ point τm(ρ1,ρ2) is ⟨⟨BV SES⟩⟩ interior of ⟨⟨f̃ , E⟩⟩c that is there exists ⟨⟨BV SES⟩⟩ p- open set ⟨⟨g̃, E⟩⟩ such that τm(ρ1,ρ2) ϵ̃ ⟨⟨g̃, E⟩⟩ ⊆̃ ⟨⟨f̃ , E⟩⟩c . Theorem 3. Let ⟨⟨X, JBV SES , E⟩⟩ be a bi-polar vague soft expert set topological space over X, ⟨⟨f̃ , E⟩⟩ ϵ̃ BVSES ⟨⟨X,E⟩⟩ . ⟨⟨f̃ , E⟩⟩ is a ⟨⟨BV SES⟩⟩ p- closed set iff ⟨⟨f̃ , E⟩⟩ = ⟨⟨f̃ , E⟩⟩. Proof. If ⟨⟨f̃ , E⟩⟩ is a ⟨⟨BV SES⟩⟩ p- closed set then this means that ⟨⟨f̃ , E⟩⟩ contains all of its limit points that ⟨⟨f̃ , E⟩⟩/ ⊆̃ ⟨⟨f̃ , E⟩⟩ this implies that ⟨⟨f̃ , E⟩⟩ ⋃̃ ⟨⟨f̃ , E⟩⟩/ = ⟨⟨f̃ , E⟩⟩ this implies that ⟨⟨f̃ , E⟩⟩ = ⟨⟨f̃ , E⟩⟩ and conversely, let ⟨⟨f̃ , E⟩⟩ = ⟨⟨f̃ , E⟩⟩ this implies that ⟨⟨f̃ , E⟩⟩ ⋃̃ ⟨⟨f̃ , E⟩⟩/ = ⟨⟨f̃ , E⟩⟩ this implies that ⟨⟨f̃ , E⟩⟩/ ⊆̃ ⟨⟨f̃ , E⟩⟩ this implies that ⟨⟨f̃ , E⟩⟩ is p-closed. Theorem 4. Let ⟨⟨X, JBV SES , E⟩⟩ be a bi-polar vague soft expert set topological space over X, ⟨⟨f̃1, E⟩⟩ , ⟨⟨f̃2, E⟩⟩ ϵ̃ BVSES ⟨⟨X,E⟩⟩. Then, M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 21 of 32 1. [⟨⟨f̃1, E⟩⟩] = ⟨⟨f̃1, E⟩⟩, 2. ⟨⟨f̃null, E⟩⟩ = ⟨⟨f̃null, E⟩⟩ and ⟨⟨Xabsolute, E⟩⟩ = ⟨⟨Xabsolute, E⟩⟩, 3. ⟨⟨f̃1, E⟩⟩ ⊆̃ ⟨⟨f̃2, E⟩⟩ =⇒ ⟨⟨f̃1, E⟩⟩ ⊆̃ ⟨⟨f̃2, E⟩⟩ , 4. [⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩] ⊆̃ ⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩ 5. [⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩] = ⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩ Proof. 1. Let ⟨⟨f̃1, E⟩⟩ = ⟨⟨f̃2, E⟩⟩, then ⟨⟨f̃2, E⟩⟩ is a ⟨⟨BV SETS⟩⟩ p-closed set. Hence ⟨⟨f̃2, E⟩⟩ = ⟨⟨f̃1, E⟩⟩ . So, [⟨⟨f̃1, E⟩⟩] = ⟨⟨f̃1, E⟩⟩. 2. By theorem 3 Let ⟨⟨X, JBV SES , E⟩⟩ be a ⟨⟨BV SETS⟩⟩ overX, ⟨⟨f̃ , E⟩⟩ ϵ̃ BVSES ⟨⟨X,E⟩⟩. ⟨⟨f̃ , E⟩⟩ is a ⟨⟨BV ES⟩⟩ p-closed set iff ⟨⟨f̃1, E⟩⟩ = ⟨⟨f̃1, E⟩⟩. Since ⟨⟨f̃null, E⟩⟩ and ⟨⟨Xabsolute, E⟩⟩ are p-closed sets. So using this results we have ⟨⟨f̃null, E⟩⟩ = ⟨⟨f̃null, E⟩⟩ and ⟨⟨Xabsolute, E⟩⟩ = ⟨⟨Xabsolute, E⟩⟩. 3. It is known that ⟨⟨f̃1, E⟩⟩ ⊆̃ ⟨⟨f̃1, E⟩⟩, ⟨⟨f̃2, E⟩⟩ ⊆̃ ⟨⟨f̃2, E⟩⟩. Since ⟨⟨f̃1, E⟩⟩ ⊆̃ ⟨⟨f̃2, E⟩⟩ ⊆̃ ⟨⟨f̃2, E⟩⟩. Since ⟨⟨f̃1, E⟩⟩ is the smallest ⟨⟨BV SETS⟩⟩ p closed set covering then so, ⟨⟨f̃1, E⟩⟩ ⊆̃ ⟨⟨f̃2, E⟩⟩. 4. Since ⟨⟨f̃1, E⟩⟩ ⊆̃ ⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩, ⟨⟨f̃2, E⟩⟩ ⊆̃ ⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩, then ⟨⟨f̃1, E⟩⟩ ⊆̃ [⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩] , ⟨⟨f̃2, E⟩⟩ ⊆̃ [⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩] and so, ⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩ ⊆̃ [⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩] . Contrary-wise, since ⟨⟨f̃1, E⟩⟩ ⊆̃ ⟨⟨f̃1, E⟩⟩, ⟨⟨f̃2, E⟩⟩ ⊆̃ ⟨⟨f̃2, E⟩⟩ then ⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩ ⊆̃ ⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩. Besides ,[⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩] is the smallest ⟨⟨BV SETS⟩⟩ p-closed set covering ⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩ . Therefore, ,[⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩] ⊆̃ ⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩. Since ⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩ ⊆̃ ⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩ and [⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩] is then small- est ⟨⟨BV SETS⟩⟩ p-closed set covering ⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩ , then [⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩] ⊆̃ ⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩ Theorem 5. Let ⟨⟨M,JBV SES , E⟩⟩ be a bi-polar vague soft expert set topological space over M , ⟨⟨f̃ , E⟩⟩ ϵ̃ BVSETS ⟨⟨M,E⟩⟩. 1. [⟨⟨f̃ , E⟩⟩]c = [⟨⟨f̃ , E⟩⟩c]o, 2. [⟨⟨f̃ , E⟩⟩o]c = [⟨⟨f̃ , E⟩⟩c] . Proof. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 22 of 32 1. ⟨⟨f̃ , E⟩⟩ = ∏ i∈I{⟨⟨f̃i, E⟩⟩ϵ̃(JBV SES)c : ⟨⟨f̃i, E⟩⟩⊇̃⟨⟨f̃ , E⟩⟩}, =⇒ [⟨⟨f̃ , E⟩⟩]c = [ ∏ i∈I{⟨⟨f̃i, E⟩⟩ϵ̃(JBV SES)c : ⟨⟨f̃i, E⟩⟩⊇̃⟨⟨f̃ , E⟩⟩ ∀i ∈ I}]c = ∐ i∈I{⟨⟨f̃i, E⟩⟩cϵ̃(JBV SES) : ⟨⟨f̃i, E⟩⟩c⊆̃⟨⟨f̃ , E⟩⟩c} = [⟨⟨f̃ , E⟩⟩c]o. 2. ⟨⟨f̃ , E⟩⟩o = ∐ i∈I{⟨⟨f̃i, E⟩⟩ϵ̃(JBV SES) : ⟨⟨f̃i, E⟩⟩⊆̃⟨⟨f̃ , E⟩⟩ ∐ i∈I{⟨⟨f̃i, E⟩⟩cϵ̃(JBV SES) : ⟨⟨f̃i, E⟩⟩c⊆̃⟨⟨f̃ , E⟩⟩c}} =⇒ [⟨⟨f̃ , E⟩⟩o]c = [ ∐ i∈I{⟨⟨f̃i, E⟩⟩ϵ̃(JBV SES) : ⟨⟨f̃i, E⟩⟩⊆̃⟨⟨f̃ , E⟩⟩}]c = ∏ i∈I{⟨⟨f̃i, E⟩⟩cϵ̃(JBV SES)c : ⟨⟨f̃i, E⟩⟩c⊇̃⟨⟨f̃ , E⟩⟩c} = [⟨⟨f̃ , E⟩⟩c]. Theorem 6. Let ⟨⟨X, JBV SES , E⟩⟩ be a bi-polar vague soft expert set topological space over X, ⟨⟨f̃1, E⟩⟩ , ⟨⟨f̃2, E⟩⟩ ϵ̃ BVSES ⟨⟨X,E⟩⟩ then 1. Ext(⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩) = Ext⟨⟨f̃1, E⟩⟩ ⋃̃ Ext⟨⟨f̃2, E⟩⟩. 2. Ext(⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩)⊇̃Ext⟨⟨f̃1, E⟩⟩ ⋃̃ Ext⟨⟨f̃2, E⟩⟩. 3. Fr(⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨g, l⟩⟩)⊆̃Fr⟨⟨f̃1, E⟩⟩ ⋃̃ Fr⟨⟨f̃2, E⟩⟩. 4. Fr(⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩)⊆̃Fr⟨⟨f̃1, E⟩⟩ ⋃̃ Fr⟨⟨f̃2, E⟩⟩. Proof. 1. Since , Ext(⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩) = ((⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩)c)o = (⟨⟨f̃1, E⟩⟩c ⋂̃ ⟨⟨f̃2, E⟩⟩c)o = (⟨⟨f̃1, E⟩⟩c)o ⋂̃ (⟨⟨f̃2, E⟩⟩c)9 = Ext⟨⟨f̃1, E⟩⟩ ⋂̃ Ext⟨⟨g, l⟩⟩ 2. Ext(⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩) = ((⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩)c)o = (⟨⟨f̃1, E⟩⟩c ⋃̃ ⟨⟨f̃2, E⟩⟩c)o⊇̃(⟨⟨f̃1, E⟩⟩c)o ⋃̃ (⟨⟨f̃2, E⟩⟩c)o = Ext⟨⟨f̃1, E⟩⟩ ⋃̃ Ext⟨⟨f̃2, E⟩⟩ , that is Ext(⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩)⊇̃Ext⟨⟨f̃1, E⟩⟩ ⋃̃ Ext⟨⟨f̃2, E⟩⟩. 3. Fr(⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩) = ⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩ ⋂̃ (⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩)c = (⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩) ⋂̃ ⟨⟨f̃1, E⟩⟩c ⋂̃ ⟨⟨f̃2, E⟩⟩c ⊆̃ (⟨⟨f̃1, E⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩) ⋂̃ ⟨⟨f̃1, E⟩⟩c ⋂̃ ⟨⟨f̃2, E⟩⟩c = {⟨⟨f̃ , l⟩⟩ ⋃̃ ⟨⟨f̃2, E⟩⟩ ⋂̃ ⟨⟨f̃1, E⟩⟩c} ⋂̃ ⟨⟨g, l⟩⟩c = {(⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃1, E⟩⟩c) ⋃̃ (⟨⟨f̃2, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩c)} ⋃̃ {⟨⟨f̃2, E⟩⟩c ⋂̃ ⟨⟨f̃1, E⟩⟩c ⋂̃ ⟨⟨f̃2, E⟩⟩c} = {Fr⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩c} ⋃̃ {Fr⟨⟨f̃2, E⟩⟩ ⋂̃ ⟨⟨f̃1, E⟩⟩c} ⊆̃ Fr⟨⟨f̃1, E⟩⟩ ⋃̃ Fr⟨⟨f̃2, E⟩⟩. 4. Fr(⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩) = (⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩) ⋂̃ (⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩)c ⊆̃ (⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩)⋂̃ (⟨⟨f̃1, E⟩⟩c ⋃̃ ⟨⟨f̃2, E⟩⟩c) = {(⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩) ⋂̃ ⟨⟨f̃1, E⟩⟩c} ⋃̃ {(⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩) ⋂̃ ⟨⟨f̃2, E⟩⟩c} = {Fr(⟨⟨f̃1, E⟩⟩ ⋂̃ ⟨⟨f̃2, E⟩⟩)} ⋃̃ {⟨⟨f̃1, E⟩⟩ ⋂̃ Fr⟨⟨g, l⟩⟩} ⊆̃ Fr⟨⟨f̃1, E⟩⟩ ⋃̃ Fr⟨⟨f̃2, E⟩⟩. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 23 of 32 4. Characterization of few more results in terms of Basis Concerning P-Open Sets This work develops foundational topological concepts in BVSETS, introducing and analyz- ing bases, sub-bases, and local bases. It defines first and second BVSE countability, explores separability via countable dense sets, and characterizes BVSE limit points using local bases. Key theorems establish criteria for comparing BVSET topologies and constructing subspace topologies, including p-closure operations. These results enhance the theoretical framework of BVSETS, supporting soft topological modeling under uncertainty and parameterization. Definition 51. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological space over X and BBV SES be a sub-family of τBV SES. BBV SES is said to be a BVSE base or p-open base or basis for the BVSET τBV SES if given any non-empty BVS⟨⟨f̃ , E⟩⟩ ϵ̃ τBV SES this implies that there exists B1⊆̃ BBV SES, such that ⟨⟨f̃ , E⟩⟩ = ⋃̃ {B : Bϵ̃B1} . In other words BBV SES is said to be base for BVSET if xe(α) ϵ̃ ⟨⟨f̃ , E⟩⟩ ϵ̃ τBV SES implies that there exists B ϵ̃ B such that xe(α) ϵ̃B ϵ̃ ⟨⟨f̃ , E⟩⟩. Definition 52. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological space over X and SBV SES be a sub-family of τBV SES . SBV SES is said to be a BVSE sub-base or P-open sub-base or basis for the BVSET τBV SES on X if finite intersections of the members of SBV SES form a base for the BVSET τBV SES on X . That is, the union of the members of SBV SES give all the members of τBV SES. The elements of SBV SES are referred to as sub-basic BPVSE P-open sets. If given any non-empty BVSE ⟨⟨f̃ , E⟩⟩ ϵ̃ τBV SES this implies that there exists B1 ϵ̃ BBV SES such that ⟨⟨f̃ , E⟩⟩ = ⋃̃ {B : Bϵ̃B1} In other words BBV SES is said to be base for BVSET if xe(α) ϵ̃ ⟨⟨f̃ , E⟩⟩ ϵ̃ τBV SES implies that there exists B ϵ̃ B such that xe(α) ϵ̃B ϵ̃ ⟨⟨f̃ , E⟩⟩. Definition 53. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological space over X A family Bα of BVSE p-open subsets of X X is said to be BVSE local base at xe(α) ϵ̃ X for the BVSETS on X if 1. Any B ϵ̃ Bxe (α) =⇒ xe(α) ϵ̃ B. 2. Any ⟨⟨f̃ , E⟩⟩ τBV SES with ye(α) ϵ̃ ⟨⟨f̃ , E⟩⟩ =⇒ ∃ B ϵ̃ Bxe (α) such that ye(α) ϵ̃ B ⊆̃ ⟨⟨f̃ , E⟩⟩. Definition 54. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological space over X. The space X is satisfy the first axioms of BVSE soft countability if X has a BVSE countable local base at each xe(α) ϵ̃ X. The BVSE space X, in this case, is called first BVSE countable space. Definition 55. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological space over X. The space X is satisfy the second axioms of BVSE countability if there exists a BVSE countable base for τBV SES on X. The BVSE space X, in this case, is called second BVSE countable space. A second BVSE countable space is also called BVSE completely separable space. Definition 56. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological space over X. A property PNSHP of X is said to be hereditary if the property is possessed by every subspace of X e.g., BVSE first countable, BVSE second countable are hereditary properties whereas BVSE p-closed sets, BVSE p- open sets, are not hereditary properties. Definition 57. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological space over X. This space is said to be if and only if X contains a BVSE countable dense BVSE subset, if and only if there exists BVSE countable subset (k̃, E) of ?? such that (k̃, E) = X M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 24 of 32 Theorem 7. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological space over X and BBV SES be a BVSE basis for τBV SES. Then, τBV SES equals to the collection of all BVSE unions of elements of BBV SES. Proof. This is easily seen from the definition of BVSE basis. Theorem 8. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological space over X. A sub-collection BBV SES of τBV SES is a base for τBV SES If and only if for each BVSE p-open set (f̃ , E) and each BVSE point xe(α) in (f̃ , E) , there exists a basis elements Bxe (α) such that xe(α) ϵ̃ Bxe (α) ⊆̃ (f̃ , E) . Proof. Given that (X, τBV SES , E) be a bi-polar vague soft expert set topological space over X and BBV SES is a collection of BVSE p-open sets. Let BBV SES is base for a BVSET τBV SES , then by definition every BVSE p-open set (f̃ , E) is the union of some members of BBV SES i.e. (f̃ , E) = ⋃ i∈I Bi where Biϵ̃B BV SES for all i ∈ I . Let xe(α) be an arbitrary BVSE point of (f̃ , E), we are to prove that there exists a BVSE basis element Bxe (α) containing xe(α) such that Bxe (α) ⊆̃ (f̃ , E) . Since xe(α) ϵ̃ (f̃ , E) but (f̃ , E) = ⋃ i∈I Bi implies that xe(α) ϵ̃ ⋃ i∈I Bi implies that xe(α) ϵ̃ Bi for some i ∈ I . Let xe(α) ϵ̃ Bi for i = xe(α) , then xe(α) ϵ̃ Bxe (α) and Bxe (α) ⊆̃ ⋃ i∈I Bi as Bi ⊆̃ ⋃ i∈I Bi for all i implies that xe(α) ϵ̃ Bxe (α) ⊆̃ ⋃ i∈I Bi implies that xe(α) ϵ̃ Bxe (α) ⊆̃ (f̃ , E) where Bxe (α) ϵ̃ BBV SES . Conversely, suppose for each BVSE point xe(α) of a BVSE p-open set (f̃ , E), there exists BVSE set xe(α) ϵ̃ B BV SES such that xe(α) ϵ̃ Bxe (α) ⊆̃ (f̃ , E) . We are to prove that BBV SES is a BVSE basis for BVSET τBV SES and for this we will prove that every BVSE P-open set (f̃ , E) can be written as a union of some members of BBV SES . Since xe(α) ϵ̃ Bxe (α) ⊆̃ (f̃ , E) implies that xe(α) ϵ̃ Bxe (α) and Bxe (α) ⊆̃ (f̃ , E) or {xe(α)} ⊆̃ Bxe (α) and Bxe (α) ⊆̃ (f̃ , E) implies that ⋃ xe (α) ∈(f̃ ,E) {xe(α)} ⊆̃ ⋃ i∈I Bi and ⋃ xe (α) ∈(f̃ ,E) Bxe (α) ⊆̃ ⋃ xe (α) ∈(f̃ ,E) (f̃ , E) implies that (f̃ , E) ⊆̃ ⋃ xe (α) ∈(f̃ ,E) Bxe (α) and ⋃ xe (α) ∈(f̃ ,E) Bxe (α,β,γ) ⊆̃ (f̃ , E) therefore (f̃ , E) = ⋃ xe (α) ∈(f̃ ,E) Bxe (α) . Since each Bxe (α) ϵ̃ BBV SES , so (f̃ , E) is the union of some members of BBV SES . But (f̃ , E) is arbitrary BVSE p-open set, so every BVSE p-open set is the union of some members of BBV SES . Therefore, BBV SES is a BVSE basis for the BVSE τBV SES . Theorem 9. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological space over X. A sub-collection BBV SES of τBV SES is a base for τBV SES If and only if , 1. Every BVSE point of X is in some B ϵ̃ BBV SES. 2. For B1, B2 ϵ̃ BBV SES and xe(α) ϵ̃ B1 ⋂ B2 , there is a ?? ϵ̃ BBV SES such that xe(α) ϵ̃ B ⊆̃ B1 ⋂ B2. Proof. Let BBV SES is a base for τBV SES . 1. Let xe(α) be an arbitrary BVSE point of X. Since of X is BVSE p-open set, so there exists Bxe (α) ϵ̃ BBV SES such that xe(α) x e (α) Bxe (α) ⊆̃ X implies that ⋃ xe (α)∈X {xe(α)} ⊆̃ ⋃ xe (α)∈X Bxe (α) ⊆̃ X implies that ⋃ xe (α)∈X Bxe (α) ⊆̃ X this shows that each BVSE point of X is some B ϵ̃ BBV SES M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 25 of 32 2. For B1, B2 ϵ̃ BBV SES and xe(α) ϵ̃ B1 ⋂ B2 since B1 and B2 are BVSE p-open set, so B1 ⋂ B2 is also a BVSE P-open set, and therefore there exists B ϵ̃ BBV SES such that xe(α) ϵ̃ B ⊆̃ B1 ⋂ B2. Conversely let (1) and (2) are true then we prove that A sub-collection BBV SES of τBV SES is a base for τBV SES . For this we prove that sub-collection BBV SES of τBV SES satisfy the three conditions of BVSET. For this we proceed as fallows the BVSE null set 0(π,h) being the union of BVSE null collection of BVSE subsets in BBV SES is in τBV SES . Since X is BVSE P-open, so for any xe(α) ϵ̃ X the condition (1), gives a Bxe (α) ϵ̃ BBV SES such that xe(α) ϵ̃ Bxe (α) ⊆̃ X then ⋃ xe (α)∈X ⊆̃ ⋃ xe (α)∈X Bxe (α) ⊆̃ X implies that X ⊆̃⋃ xe (α)∈X Bxe (α) ⊆̃ X implies that X = ⋃ xe (α)∈X Bxe (α) this shows that X, being the union of members of BBV SES , is in τBV SES . Next we proceed for the second condition as fallows. The union of any number of members of τBV SES , being the union of members of BBV SES is in τBV SES . Next we proceed for the third condition as fallows. Let (f̃ , E)1 , (f̃ , E)2 ϵ̃ τBV SES , then by definition of τBV SES we have (f̃ , E)1 = ⋃ Bα , (f̃ , E)2 = ⋃ Bβ for some α, β ranging over sub-collection of BBV SES . Therefore, (f̃ , E)1∩̃(f̃ , E)2 = ∪̃Bα∩̃Bβ =⇒ (f̃ , E)1∩̃(f̃ , E)2 = ∩̃Bα∪̃Bβ, ...(i) By the (2), for any xe(α) ϵ̃ (f̃ , E)1∩̃(f̃ , E)2, there is a Bxe (α) ϵ̃ BBV SES such that xe(α) ϵ̃ Bxe (α) ⊆̃ Bα∪̃Bβ. implies that,⋃ xe (α)∈Bα∩Bβ {xe(α,β,γ)} ⊆̃ ⋃ xe (α)∈Bα∩Bβ Bxe (α) ⊆̃ Bα∩̃Bβ Bα∩̃Bβ ⊆̃ ⋃ xe (α)∈Bα∩Bβ Bxe (α,β,γ) ⊆̃ Bα∩̃Bβ =⇒ Bα∩̃Bβ = ⋃ xe (α)∈Bα∩Bβ Bxe (α) . Putting this value in (i) , we have (f̃ , E)1∩̃(f̃ , E)2 = ∪̃Bα∩̃Bβ = ∪̃( ⋃ xe (α)∈Bα∩Bβ Bxe (α) ) This shows that (f̃ , E)1∩̃(f̃ , E)2 is the union of members of BBV SES , so in in τBV SES . In the same way, we prove that the intersection of any finite number of members of τBV SES is in τBV SES . Since all the conditions of the is BVSET are satisfied, so τBV SES is BVSET on X. Consequently, BBV SES is a BVSE p-base for τBV SES . Theorem 10. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological space over ??. A BVSE point xe(α) in BVSETS is a BVSE limit point of (F̃ , E) ⊆̃ X if and only if every member of any BVSE p-local base Bxe (α) at xe(α) contains a point of (F̃ , E) different from xe(α) . Proof. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological space over ?? and (F̃ , E) ⊆̃ X . Let xe(α) ϵ̃ X be a BVSE limit point of (F̃ , E) . Let Bxe (α) be a local base at xe(α) for BVSET on X. To prove that (B−xe(α))∩̃(F̃ , E) ̸= 0(X,E) ∀ B ϵ̃ Bxe (α) . By hypothesis, xe(α) BVSE limit point of (F̃ , E) and so ((F̃ , E)−xe(α))∩̃(F̃ , E) ̸= 0(X,E) ∀ BVSE p-open sets (f̃ , E) that is (f̃ , E) ϵ̃ τBV SES . By definition of BVSE p-local base, (f̃ , E) ϵ̃ Bxe (α) implies that (f̃ , E) ϵ̃ τBV SES with xe(α) ϵ̃ (f̃ , E) then the foregoing statement takes the form ((F̃ , E) − xe(α))∩̃(F̃ , E) ̸= 0(X,E) ∀ (f̃ , E) ϵ̃ Bxe (α) that is (B−xe(α))∩̃(F̃ , E) ̸= 0(X,E) ∀ B ϵ̃ Bxe (α) . Conversely, suppose Bxe (α) be BVSE p-local base at xe(α) ϵ̃ X for some BVSET τBV SES on X. Also suppose that (B− xe(α))∩̃(F̃ , E) ̸= 0(X,E) ∀ B ϵ̃ Bxe (α) . Where ((F̃ , E) ϵ̃ X. Let (f̃ , E) ϵ̃ τBV SES be an arbitrary such that xe(α) ϵ̃ (f̃ , E) M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 26 of 32 then by definition of BVSE P-local base there exists B ϵ̃ Bxe (α) such that xe(α) ϵ̃ B ϵ̃ ((F̃ , E) . consequently, ((f̃ , E)− xe(α))∩̃(F̃ , E) ⊇̃ (B − xe(α))∩̃(F̃ , E) ̸= 0(X,E) . This implies that ((f̃ , E)− xe(α))∩̃(F̃ , E) ̸= 0(X,E) , meaning that there by xe(α) is a BVSE limit point of (F̃ , E) Theorem 11. Let τBV SES 1 and τBV SES 2 be two bi-polar vague soft expert set topological spaces over X generated by BVSE p-bases BBV SES 1 and BBV SES 2 , respectively. Then τBV SES 1 ⊆̃ τBV SES 2 iff for each xe(α) ϵ̃ BVSES (X,E) and for each (B̃1, E) ⊆̃ BBV SES 1 containing xe(α) there exists (B̃2, E) ⊆̃ BBV SES 2 such that xe(α) ϵ̃ (B̃2, E) ⊆̃ (B̃1, E). Proof. Let τBV SES 1 ⊆̃ τBV SES 2 and xe(α) ⊆̃ BVSES (X,E) , (B̃1, E) ϵ̃ BBV SES 1 such that xe(α) ϵ̃ (B̃1, E). Since BBV SES 1 is a BVSE p-basis for BVSET τBV SES 1 over X, then (B̃1, E) ⊆̃ τBV SES 1 =⇒ xe(α) ϵ̃ (B̃1, E) ϵ̃ BBV SES 2 ⊆̃ τBV SES 1 i.e, xe(α) ϵ̃ (B̃1, E) ϵ̃ τBV SES 2 . Since BBV SES 2 is a BVSE P-basis for τBV SES 2 , so for (B̃2, E) ϵ̃ BBV SES 2 we have xe(α) ϵ̃ (B̃2, E) ⊆̃ (B̃1, E). Conversely, assume that the hypothesis holds. Let (F̃ , E) ϵ̃ τBV SES 1 . Since BBV SES 1 is a BVSE p-basis for BVSET τBV SES 1 , then for xe(α) ϵ̃ (F̃ , E) there exist (B̃1, E) ϵ̃ BBV SES 1 such that xe(α) ϵ̃ (B̃1, E) ⊆̃ (F̃ , E). No by hypothesis, there exist (B̃2, E) ϵ̃ BBV SES 2 such that (B̃2, E) ⊆̃ (B̃1, E) =⇒ (B̃2, E) ⊆̃ (B̃1, E) ⊆̃ (F̃ , E) =⇒ (B̃2, E) ⊆̃ (F̃ , E) =⇒ (F̃ , E) ϵ̃ τBV SES 2 .This show that τBV SES 1 ⊆̃ τBV SES 2 . Theorem 12. Let (X, τBV SES , E) be a bi-polar vague soft expert set topological spaces over of (F̃ , E) , (K̃, E) ϵ̃ BVSES (X,E) . 1. If BBV SES is a BVSE P-base for τBV SES , then BBV SES (F̃ ,E) = {(B̃, E)∩̃(F̃ , E) : (B̃, E)ϵ̃BBV SES} is a BVSE p-base for the BVSSET τBV SES (F̃ ,E) , 2. If (F̃ , E) is a BVSE in τBV SES (F̃ ,E) and (F̃ , E) is a BVSE p-closed set in τBV SES (F̃ ,E) , then (F̃ , E) is a BVSE P-closed in τBV SES (F̃ ,E) 3. Let (F̃ , E) ⊆̃ (F̃ , E). If (f̃ , E) is BVSE p-closure (X, τBV SES , E), then (f̃ , E) ∩̃ (F̃ , E) is a BVSE p-closure in (X(F̃ ,E), τ BV SES (F̃ ,E) , E). Proof. 1. Since BBV SES is a BVSE p-base for τBV SES so for arbitrary (Ũ , E) ϵ̃ τBV SES , we have (Ũ , E) = ⋃ (B̃,E)ϵBBV SES (B̃, E) .In case, (Ũ , E)∩̃(F̃ , E) = ( ⋃ (B̃,E)ϵBBV SES (B̃, E) ) ∩̃(F̃ , E) = ( ⋃ (B̃,E)ϵBBV SES ((B̃, E)∩̃(F̃ , E)) for (Ũ , E)∩̃(F̃ , E) ϵ̃ τBV SES (F̃ ,E) . Since arbitrary member τBV SES (F̃ ,E) can be expressed as the union of members of BBV SES (F̃ ,E) 2. We first show that if (f̃ , E) is a BVSE p-closed set in τBV SES (F̃ ,E) then there exist a closed set (Ṽ , E) ⊆̃ (K̃, E) i.e, (Ṽ , E) /∈ τBV SES such that (f̃ , E) = (F̃ , E) ∩̃ (F̃ , E). Let (f̃ , E) be a p-closed in τBV SES (F̃ ,E) . Then (g̃, E)c is a BVSE p-open set in τBV SES (F̃ ,E) i.e, (f̃ , E)c , (f̃ , E)c = (Ũ , E) ∩̃ (F̃ , E) for (Ũ , E) ϵ̃τBV SES =⇒ ((f̃ , E)c)c = (F̃ , E) ∩̃ ((Ũ , E)∩̃(F̃ , E))c = (Ũ , E)c ∩̃ (F̃ , E) . Here (Ũ , E)c /∈ τBV SES M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 27 of 32 i.e, (Ũ , E)c is a p-closed in τBV SES . So here acts as (Ṽ , E) ⊆̃ (K̃, E). Conversely, sup- pose that (f̃ , E) = (Ṽ , E) ∩̃ (F̃ , E) where (F̃ , E) ⊆̃ (K̃, E) and (Ṽ , E) is P-closed in τBV SES (F̃ ,E) . Clearly, (Ṽ , E)c ϵ̃ τBV SES so that (Ṽ , E)c ∩̃ (F̃ , E) ϵ̃ τBV SES (F̃ ,E) Now, (Ṽ , E)c ∩̃ (F̃ , E) = ((K̃, E)\(Ṽ , E)) ∩̃ (F̃ , E) = ((K̃, E)∩̃(Ṽ , E))\((Ṽ , E)∩̃(F̃ , E)) = (F̃ , E)\(f̃ , E). This implies (F̃ , E) \ (f̃ , E) is a BVSE in (F̃ , E) i,e., (f̃ , E) is a BVSE p-closed set in τBV SES (F̃ ,E) . (f̃ , E) = ∩̃ {(f̃i, E) : (f̃i, E)isBV SEp − closedand(f̃i, E)⊇̃(f̃ , E)} is the BVSETS P-closure of (f̃ , E) and so (f̃ , E) is a BVSETS p-closed set. Now, (f̃ , E) ∩̃ (F̃ , E) = ∩̃{(f̃i, E) : (f̃i, E), (f̃i, E)⊇̃(f̃ , E)} ∩̃ (F̃ , E) = ∩̃ ((f̃i, E)∩̃F̃ , E)). Since each (f̃i, E) is p-closed then each (f̃i, E)∩̃F̃ , E) is p-closed in τBV SES (F̃ ,E) . Now (f̃ , E) ⊆̃ (f̃i, E) and (f̃ , E) ⊆̃ (F̃ , E). So ((f̃ , E)∩̃F̃ , E)) ⊆̃ ((f̃i, E)∩̃F̃ , E)) =⇒ (f̃ , E) ⊆̃ (f̃i, E)∩̃F̃ , E) Therefore, (f̃ , E) ∩̃ (F̃ , E) = ∩̃ {((f̃i, E)∩̃F̃ , E)) : (f̃i, E)∩̃F̃ , E)isp− closedand((f̃i, E)∩̃F̃ , E))⊇̃((f̃i, E)}. Thus (f̃ , E) ∩̃ (F̃ , E) is a BVSE p- closure of (f̃ , E) in τBV SES (F̃ ,E) 5. Decision-Making Problem Using Bipolar Vague Soft Expert Sets in Cancer Diagnosis In the medical field, diagnosing cancer often involves analyzing uncertain, vague, and con- flicting expert opinions based on various medical parameters (e.g., symptoms, test results, imaging). Bipolar Vague Soft Expert Sets (BVSES) provide a powerful tool to represent such complex information involving multiple experts and both positive (supportive) and negative (opposing) evaluations. Let X = {u1, u2, u3} is a set of patients. E = {e1, e2} is a set of parameters. 1. e1 : Tumor Marker Level 2. e2 : Imaging Results (e.g., CT Scan) 3. Experts: {p, q, r} 4. Decision values: 1 (Yes, indication of cancer), 0 (No, less likely to have cancer). 5. Experts: p,q,r Decision values: 1 (Yes, indication of cancer), 0 (No, less likely to have cancer). The bipolar vague soft expert set ??H,??? assigns a tuple to each patient under each parameter and expert. 6. (T̃+, f̃+, T̃−, f̃−) = (truth, indeterminacy, counter-truth, counter-indeterminacy). Sample Evaluations: Let?s extract only the evaluations for decision-making purposes. We’ll simplify a subset: For decision 1(Yes, cancer suspected): Expert Parameter Patient (T̃+, f̃+, T̃−, f̃−) p e1 u1 (0.3, 0.7,−0.2,−0.4) p e2 u1 (0.4, 0.3,−0.2,−0.1) q e1 u2 (0.8, 0.3,−0.1,−0.5) q e2 u3 (0.3, 0.2,−0.5,−0.4) r e1 u1 (0.4, 0.6,−0.6,−0.4) M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 28 of 32 Table 1: Decision-Making Step: Score Function We define a score function for each tuple (T̃+, f̃+, T̃−, f̃−) as: Score We define a score function for each tuple (T̃+− f̃+− T̃−− f̃−). A higher score suggests stronger positive evidence with lower uncertainty and opposition. Score Calculation: Patient u1 1. From p, e1 : S1 = 0.3− 0.2− 0.7− 0.4 = −1.0 2. From p, e2 : S2 = 0.4− 0.2− 0.3− 0.1 = −0.2 3. From r, e1 : S3 = 0.4− 0.6− 0.6− 0.4 = −1.2 Total score for u1 : S(u1) = −1.0 + (−0.2) + (−1.2) = −2.4 Patient u2 1. From q, e1 : S = 0.8− 0.1− 0.3− 0.5 = −0.1 Patient u3 1. From q, e2 : S = 0.8− 0.1− 0.3− 0.5 = −0.1 S = 0.3− 0.5− 0.2− 0.4 = −0.8 Total score for u3 : S(u3) = −0.8 Final decision scores of patients is given in Table 1. Patient Total score S(ui) Diagnosis sugges- tion u2 e1 Most likely to have can- cer u3 e2 Moderate likelihood u1 e1 Least likely to have can- cer Conclusion : : Patient u2 has the least negative score, meaning the evaluations show less opposition and lower uncertainty about the presence of cancer, hence u2 is more likely to have cancer based on the bipolar vague soft expert decision model. Interpretation :This examp demonstrates how Bipolar Vague Soft Expert Sets can model real-world uncertainty in cancer diagnosis by incorporating multiple expert opinions with varying confidence levels, and allowing decision-makers to derive conclusions using aggregated scoring mechanisms. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 29 of 32 6. Comparative Analysis This section presents a comprehensive and critical comparison between the published study referenced in [31] and the proposed research, highlighting the advancements, innovations, and theoretical depth introduced in the current work. This given in Table 3 Aspect Published Work [31] Proposed Work Core Concept Bipolar Fuzzy Soft Expert Sets (BF- SES) Bipolar Vague Soft Expert Sets (BPVSES) Theoretical Fo- cus Defined fundamental operations: complement, union, intersection, AND, OR Defined operations: complement, union, intersection, and introduced new topolog- ical structures Innovations Algorithm development and appli- cation to decision-making TIntroduction of BPVSET, 8 new defini- tions, and the novel concept of p-open sets Topological Ex- tension Not addressed. Extensive development of bipolar vague soft expert topology (BPVSET) Interior and Closure Not explored Defined and analyzed interior, closure, and their interactions Advanced Con- cepts Basic properties and laws of opera- tions. Bases, sub-bases, local bases; first and sec- ond countability; separability via dense sets Main Theme MCDM using bipolar fuzzy soft sets. Topological modeling in vague bipolar soft expert sets Application Area University selection. Cancer diagnosis Theoretical Depth Moderate High. Innovation Algorithm + hierarchy model TNew definitions + topological frame- work. Real-World Im- pact Education guidance Medical diagnostics under uncertainty 7. Conclusion and Future Work In this study, we have rigorously developed the theoretical foundation of bipolar vague soft expert sets (BPVSESs) by introducing and formalizing their essential operations, such as complement, union, and intersection. Building upon these core elements, we advanced the framework further by proposing the concept of bipolar vague soft expert topology (BPVSET), supported by eight novel and meaningful definitions. Notably, the introduction of the p-open set emerged as a powerful and versatile construct for shaping complex topological structures under uncertainty. We presented a thorough articulation of interior and closure operations, examining their properties and mutual interactions, which underpin the topological behavior of BPVSETS. Furthermore, we extended classical topological notions?such as bases, sub-bases, local bases, countability, and separability?into the context of bipolar vague soft expert topol- ogy. These contributions significantly enrich the theoretical framework, making it robust for applications involving parameterized and uncertain environments. The study culminates in a practical decision-making framework, with an application in cancer diagnosis, illustrating the capability of BPVSESs to model and resolve real-world problems involving vague, bipolar, and expert-driven information. The work not only strengthens the mathematical foundations of soft expert set theory but also opens new directions for research in soft topology, uncertainty M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5900 30 of 32 modeling, and intelligent decision-making systems. Future work will focus on developing new structures within bipolar vague soft expert bi-topological spaces, with an emphasis on the role of soft points and the integration of bipolar vague soft expert semi-open sets. Results concerning the basis of these structures will be outlined. Furthermore, we will plan to apply the Encrypted K-Mean Clustering method, a novel approach for K-mean clustering on encrypted data, which ensures the confidentiality of sensitive information. We will also apply the Elbow method, a strategy to determine the optimal value of ?? (the number of clusters) in clustering analysis. This method enhances consistency in cluster design and helps identify natural groupings within datasets, enabling a deeper understanding of data diversity. In our work, we will try to imple- ment an encrypted version of the Elbow method on our dataset. 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