Healthcare decision-making (DM) often requires evaluating multiple interdependent criteria under uncertainty, balancing desirable and undesirable outcomes. To address the limitations of traditional methods, we propose the q-rung orthopair fuzzy N-bipolar soft set (q-ROFNBSS) framework, a unified model that integrates multilevel evaluation, bipolar information, and higher-order fuzzy membership. The framework defines key operations—including null and whole sets, complement, subset, equality, union, and intersection—along with their algebraic properties, providing a rigorous foundation for systematic analysis. A step-by-step procedure is introduced to compute aggregated positive and negative scores for alternatives. We apply the model to a telemedicine platform selection case study, evaluating five hypothetical platforms across four positive and four negative criteria. The computed net scores, combining positive and negative impacts, are: z_1 = 0.5954, z_2 = 0.6572, z_3 = 0.7745, z_4 = -0.8369, and z_5 = -1.0852, identifying the mobile-first teleconsultation platform (x_3) as the most suitable choice. Comparative and sensitivity analyses illustrate the framework’s superiority relative to existing models and its flexibility to variations in grading intervals and parameter settings. The proposed framework outperforms existing q-rung orthopair fuzzy set (q-ROFS) and N-bipolar soft set (NBSS) models by providing more balanced, trade-off-aware rankings. Overall, this framework offers a versatile and robust tool for complex DM scenarios in healthcare and other domains.

Cancer remains a leading cause of global mortality, where early and precise diagnosis is critical for effective treatment and patient survival. Conventional diagnostic methods often face challenges due to subjectivity, inter-observer variability, and the uncertainty associated with medical measurements. This study introduces Disc T-Spherical Fuzzy Sets (DT-SFSs), a novel type of spherical fuzzy set incorporating membership, non-membership, hesitancy, and a radius parameter to more effectively model and quantify measurement uncertainty in diagnostic decision-making. The approach is validated through breast cancer diagnosis, comparing DT-SFS performance against a Mamdani-type fuzzy inference system using a synthesized dataset derived from measured imaging features. Comprehensive sensitivity analysis demonstrates the model’s stability and reliability across varying parameters. Results show that DT-SFS achieves superior diagnostic accuracy, precision, and robustness in managing uncertain and conflicting measurement data. These findings highlight DT-SFS as a clinically interpretable and measurement-oriented framework for oncology decision support, offering a promising advancement over conventional fuzzy logic approaches in medical diagnostics.

Cybersecurity risk assessment in financial institutions has become more difficult because cyber threats keep changing, many evaluation factors must be considered, and decisions often depend on the opinions of several experts. To deal with these challenges, an intuitionistic fuzzy N-bipolar soft expert set (IFNBSES) model is introduced. This framework integrates intuitionistic fuzzy concepts with N-bipolar soft expert sets (NBSESs). The IFNBSES model considers both degrees of acceptance and rejection, includes positive and negative evaluations, and allows experts to provide multi-level assessments. The model also provides a clear way to combine expert opinions when there is uncertainty. The IFNBSES model is formally defined, and its structure is explained using practical examples. Basic operations of the model are studied, and their algebraic properties are examined to ensure both theoretical soundness and practical usefulness. The model is then applied to multi-criteria group decision-making (MCGDM) problems to show how it supports clear, reliable, and well-informed selection of cybersecurity strategies. The results are discussed along with their implications for researchers and decision-makers, and the robustness of the proposed approach is evaluated. Finally, a comparison with existing methods shows that IFNBSES handles uncertainty more effectively, making it a strong tool for cybersecurity decision-making (DM).

The q-fractional hesitant fuzzy sets (q-FHFSs) extend the conventional hesitant and intuitionistic fuzzy sets by providing a richer framework for representing uncertainty. This paper proposes an integrated VIKOR–AHP framework based on q-FHFSs to enhance multi-criteria decision-making (MCDM) under uncertain conditions. To address the challenge of comparing fractional and heterogeneous hesitant elements, this study develops an LCM-based generalized distance measure for q-fractional hesitant fuzzy numbers, enabling consistent comparison among hesitant elements with different cardinalities. As criteria weights are inherently subjective, the AHP method is used to systematically derive objective priority weights. The proposed VIKOR–AHP hybrid model improves flexibility in handling fractional hesitancy and conflicting criteria, while preserving mathematical rigor in distance quantification. To demonstrate its applicability, the model is applied to the selection of Green Energy Systems, evaluated under selected subcriteria from Cavallaro and Ciraolo’s list of sustainability criteria. A sensitivity analysis is conducted to examine the influence of parameter q. Finally, a comparative evaluation against q-fractional hesitant fuzzy TOPSIS (q-FHF TOPSIS), hesitant q-rung orthopair fuzzy TOPSIS (Hq-ROF TOPSIS), hesitant q-rung orthopair fuzzy VIKOR (Hq-ROF VIKOR) and hesitant fuzzy VIKOR (HF VIKOR) across varying q values confirms the effectiveness and robustness of the proposed decision-making method.

The rapid advancement of smart healthcare systems demands sophisticated mathematical tools to manage uncertainty and conflicting expert opinions in critical decision-making (DM) processes. Multi-criteria group decision making (MCGDM) plays a pivotal role in synthesizing diverse expert evaluations for complex healthcare challenges. However, existing soft set (SS) extensions often struggle to simultaneously capture multinary evaluations, bipolar reasoning, higher-order fuzzy logic, and multi-expert input. To overcome these limitations, we propose the Fermatean fuzzy N-bipolar soft expert set (FFNBSES), which enhances fuzzy representation while integrating multinary, bipolar, and multi-expert evaluations. We formally define the fundamental operations of FFNBSES and demonstrate its algebraic properties. A DM methodology based on FFNBSES is developed and applied to a healthcare case study, showcasing its superior capability to handle expert consensus and disagreement in multi-criteria evaluations. Comparative analysis within the SS theory framework highlights the enhanced flexibility and robustness of FFNBSES for real-world MCGDM problems. This work provides a powerful and comprehensive approach to support smart healthcare transformation through improved group DM under uncertainty.

The rapid advancement of smart healthcare systems demands sophisticated mathematical tools to manage uncertainty and conflicting expert opinions in critical decision-making (DM) processes. Multi-criteria group decision making (MCGDM) plays a pivotal role in synthesizing diverse expert evaluations for complex healthcare challenges. However, existing soft set (SS) extensions often struggle to simultaneously capture multinary evaluations, bipolar reasoning, higher-order fuzzy logic, and multi-expert input. To overcome these limitations, we propose the Fermatean fuzzy N-bipolar soft expert set (FFNBSES), which enhances fuzzy representation while integrating multinary, bipolar, and multi-expert evaluations. We formally define the fundamental operations of FFNBSES and demonstrate its algebraic properties. A DM methodology based on FFNBSES is developed and applied to a healthcare case study, showcasing its superior capability to handle expert consensus and disagreement in multi-criteria evaluations. Comparative analysis within the SS theory framework highlights the enhanced flexibility and robustness of FFNBSES for real-world MCGDM problems. This work provides a powerful and comprehensive approach to support smart healthcare transformation through improved group DM under uncertainty.

Intuitionistic fuzzy set (IFS), introduced by Atanassov, extends classical fuzzy sets (FSs) by incorporating both membership degree (MD) and non-membership degree (NMD), allowing for a more nuanced representation of uncertainty. N-bipolar soft set (NBSS) extends the traditional N-soft set (NSS) by incorporating bipolarity, while also supporting both binary and multinary evaluations. In this paper, we propose an intuitionistic fuzzy N-bipolar soft set (IFNBSS) model, a hybridization of IFS and NBSS that combines intuitionistic fuzzy elements, multinary evaluation, and bipolarity considerations. We define the fundamental operations of IFNBSS, establish their algebraic properties, and illustrate their applicability through examples. In addition, we embed IFNBSS into a multi-criteria decision-making (MCDM) scenario by providing a structured algorithm for determining the best alternative. A numerical example on the assessment of artificial intelligence (AI) in medical diagnosis illustrates the practical efficacy of the model on handling of uncertain and confounding information. We also demonstrate generality and robustness of IFNBSS through a detailed comparison with existing approaches.

This paper introduces a novel extension of hypersoft sets, termed binary hypersoft sets (Bn-HySSs), representing an advanced generalization of binary soft sets over two universal sets and a parameter set. It presents fundamental operations of BnHySSs such as subset, superset, equality, complement, null and absolute sets, extended union/intersection, union, intersection, difference, AND, and OR operations. We also examine foundational properties and provide a comparative analysis of BnHySSs, HySSs, and BnSSs. Building on this foundation, we introduce the notion of binary hypersoft topology and its corresponding subspace concept. Further, we explore topological notions within this framework, including limit points, neighborhoods, closure, interior, and boundary in the context of BnHySSs.

Urban transportation planning involves evaluating multiple conflicting criteria such as accessibility, cost-effectiveness, and environmental impact, often under uncertainty and incomplete information. These complex decisions require input from various stakeholders, including planners, policymakers, engineers, and community representatives, whose opinions may differ or contradict. Traditional decision-making approaches struggle to effectively handle such bipolar and multivalued expert evaluations. To address these challenges, we propose a novel decisionmaking framework based on Pythagorean fuzzy N-bipolar soft expert sets. This model allows experts to express both positive and negative opinions on a multinary scale, capturing nuanced judgments with higher accuracy. It introduces algebraic operations and a structured aggregation algorithm to systematically integrate and resolve conflicting expert inputs. Applied to a real-world case study, the framework evaluated five urban transport strategies based on key criteria, producing final scores as follows: improving public transit (− 0.70), optimizing traffic signal timing (1.86), enhancing pedestrian infrastructure (3.10), expanding bike lanes (0.59), and implementing congestion pricing (0.77). The results clearly identify enhancing pedestrian infrastructure as the most suitable option, having obtained the highest final score of 3.10. Comparative analysis demonstrates the framework’s superior capability in modeling expert consensus, managing uncertainty, and supporting transparent multi-criteria group decision-making.

Efficient planning in the healthcare sector requires selecting the most suitable facility location based on multiple, often conflicting criteria and expert evaluations. This study proposes Fuzzy N-Bipolar Soft Expert (FN-BSE) set model, which integrates expert input, bipolar assessments, and multinary evaluation within a fuzzy framework. Theoretical structure of the model is developed through fundamental operations and their algebraic properties. Applying FN-BSE set model to a case study of seven potential healthcare locations, expert judgments were aggregated, and alternatives were ranked. The results indicate that location ω3 is the optimal choice. Analysis of positive and negative evaluations shows the model effectively balances conflicting opinions, and comparative evaluation demonstrates its superiority over existing approaches in handling uncertainty and supporting robust Multi-Attribute Group Decision-Making (MAGDM).

Sustainability evaluation in manufacturing industries is increasingly vital for promoting responsible growth and long-term competitiveness amid environmental, social, and economic challenges. Effective decision-making (DM) under uncertainty is crucial for managing multiple, often conflicting sustainability objectives. In this paper, we propose a novel hybrid model, termed Pythagorean fuzzy N-bipolar soft sets (PFNBSSs), which integrates Pythagorean fuzzy sets (PFSs), N-soft sets (NSSs), and bipolar soft sets (BSSs) within a unified multi-criteria decision-making (MCDM) framework. For theoretical purposes, we define basic operations and algebraic properties of PFNBSSs, supported by illustrative examples. To demonstrate practical applicability, the PFNBSS model is applied to assess sustainability practices in manufacturing industries through two numerical examples: one focusing on positive and negative sustainability indicators, and another emphasizing comparative sustainability risk assessment across diverse manufacturing sectors. Detailed interpretations of computational results and their relevance in practical DM are provided. This is followed by a comparative analysis confirming the superior discrimination power and expressive capability of the PFNBSS model over existing alternatives. The paper concludes with a critical evaluation of the model and suggestions for future research.

This study introduces novel q-fractional hesitant fuzzy multicriteria decision-making (MCDM) approaches aimed at optimizing power system performance under uncertain and imprecise conditions. We extend hesitant fuzzy set theory by formulating q-fractional hesitant fuzzy sets (q-FHFSs), integrating with the concept of q-fractional fuzzy sets for more flexible and nuanced decision scenarios. Operational rules for q-fractional hesitant fuzzy numbers (q-FHFNs) are established, and then averaging and geometric aggregation operators dedicated to q-FHFNs are developed. For efficient decision-making, score and accuracy functions are defined to rank alternatives and estimate their relative effectiveness. Moreover, we define a weighted generalized distance measure between multiple q-FHFNs to enable a more accurate assessment of alternatives in intricate decision-making settings. On the basis of these assumptions, two new algorithms for MCDM are formulated; namely, q-fractional hesitant fuzzy weighted average and q-FHF technique for order preference by similarity to ideal solution methods. Their applicability is demonstrated through an illustrative numerical example involving the selection of power systems based on their performance, followed by a sensitivity analysis of the influence of the q and \rho parameters. Then, comparative analysis confirms the superiority of the developed methods to existing MCDM methods. The results highlight the efficiency and reliability of the q-FHFS-based method in dealing with uncertainty in power system decision-making.

This manuscript introduces N-hypersoft topology (N-HST), an innovative extension of the established hypersoft topology (HST). Unlike its predecessor, the N-hypersoft (N-HS) set, N-HST adopts a multi-opinion approach to decision-making, enhancing robustness and adaptability across various domains. In addressing limitations observed in N-soft topology (N-ST), particularly in handling multi-argument approximate functions, N-HST offers a comprehensive framework. The study delves into various operators within the N-HST framework, including closure, interior, exterior, and boundary operators, elucidating their interrelationships and providing a deeper understanding of the structural components. Furthermore, this manuscript explores the practical implications of N-HST in multi-criteria group decision-making (MCGDM), showcasing its applicability in real-world decision scenarios. In addition to its specific applications in MCGDM, N-HST offers generic aspects that contribute to its versatility and efficacy in decision-making processes. By accommodating multiple opinions and handling complex evaluation factors, N-HST emerges as a superior framework for addressing diverse decision-making challenges. Through a numerical example, this manuscript demonstrates the practical utility and potential impact of N-HST in various domains.

This paper introduces fuzzy N-bipolar soft (FN-BS) sets, a novel mathematical framework designed to enhance multi-criteria decision-making (MCDM) processes under uncertainty. The study addresses a significant limitation in existing models by unifying fuzzy logic, the consideration of bipolarity, and the ability to evaluate attributes on a multinary scale. The specific contributions of the FN-BS framework include: (1) a formal definition and set-theoretic foundation, (2) the development of two innovative algorithms for solving decision-making (DM) problems, and (3) a comparative analysis demonstrating its superiority over established models. The proposed framework is applied to a real-world case study on selecting vaccination programs across multiple countries, showcasing consistent DM outcomes and exceptional adaptability to complex and uncertain scenarios. These results position FN-BS sets as a versatile and powerful tool for addressing dynamic DM challenges.

Multiple criteria decision-making (MCDM) is a significant area of decisionmaking theory that certainly warrants attention. It might be difficult to accurately convey the necessary decision facts when navigating decision-making problems since we frequently run into complicated issues and unpredictable situations. To address this, introducing the novel idea of the bipolar pqr-spherical fuzzy set (BpqrSFS), a hybrid structure of the bipolar fuzzy set (BFS) and the pqr-spherical fuzzy set (pqr-SFS), is the main goal of this work. The fundamental (set-theoretic and algebraic) operations on BpqrSFSs are explained as well as their relations to several known models. A distance measure, such as Euclidean distance, among BpqrSFNs, is provided. Afterward, we expand the fundamental aggregation operators to the pqr-spherical fuzzy (BpqrSF) environment by developing bipolar pqr-spherical fuzzy-weighted averaging and bipolar pqr-spherical fuzzy-weighted geometric operators for aggregating BpqrSFNs. According to the aforementioned distance measure and operators, an MCDM approach is established consisting of two algorithms, namely, the TOPSIS method and the method using the proposed operators in the BpqrSF context. Moreover, a numerical example is provided in order to ensure that the presented model is applicable. By using the two algorithms, a comparative analysis of the proposed method with other existing ones is given in order to verify the feasibility of the suggested decision making procedure.

This paper presents N-bipolar soft expert (N-BSE) sets, a novel framework designed to enhance multi-attribute group decision-making (MAGDM) by incorporating expert input, bipolarity, and non-binary evaluations. Existing MAGDM approaches often lack the ability to simultaneously integrate positive and negative assessments, especially in nuanced, multi-valued evaluation spaces. The proposed N-BSE model addresses this limitation by offering a comprehensive, mathematically rigorous structure for decision-making (DM). Fundamental operations of the N-BSE model are defined and analyzed, ensuring its theoretical consistency and applicability. To demonstrate its practical utility, the N-BSE model is applied to a general case study on sustainable energy solutions, illustrating its effectiveness in handling complex DM scenarios. An algorithm is proposed to streamline the DM process, enabling systematic and transparent identification of optimal alternatives. Additionally, a comparative analysis emphasizes the advantages of the N-BSE model over existing MAGDM frameworks, highlighting its capacity to integrate diverse expert opinions, evaluate both positive and negative attributes, and support multi-valued assessments. By bridging the gap between theoretical development and practical application, this paper contributes to advancing DM methodologies.

M-parametrized N-soft set (MPNSS)‎, ‎an extension of N-soft set (N-SS) theory‎, ‎is instrumental in addressing the challenges of assigning non-binary evaluations to both alternatives and attributes‎. ‎Recognizing the inherent duality in human decision-making‎, ‎where choices are influenced by both positive and negative aspects‎, ‎we enhance the MPNSS framework by incorporating bipolarity‎. ‎This addition‎, ‎aimed at capturing the dual nature of decision processes‎, ‎results in the development of bipolar M-parametrized N-soft set (BMPNSS) model‎. ‎In the context of BMPNSS‎, ‎we present some related definitions such as incomplete‎, ‎negatively efficient‎, ‎positively efficient‎, ‎and totally efficient‎. ‎Additionally‎, ‎for the complement of MPNSS‎, ‎we introduce four distinct definitions‎: ‎complement‎, ‎weak complement‎, ‎top weak complement‎, ‎and bottom weak complement‎. ‎Set-theoretic operations‎, ‎including extended and restricted union and intersection‎, ‎are explored accompanied by a discussion of their properties‎, ‎providing a comprehensive understanding of the behavior of these operations within the BMPNSS framework‎. ‎To facilitate understanding‎, ‎we include an illustrative example‎. ‎The decision-making procedure introduces alternative ranking based on extended choice and extended weight choice values‎, ‎demonstrated through a numerical example‎. ‎In our comparative analysis‎, ‎BMPNSS is positioned against existing models‎, ‎emphasizing its distinctive features and advantages in diverse decision-making scenarios‎.

M-parametrized N-soft set (MPNSS)‎, ‎an extension of N-soft set (N-SS) theory‎, ‎is instrumental in addressing the challenges of assigning non-binary evaluations to both alternatives and attributes‎. ‎Recognizing the inherent duality in human decision-making‎, ‎where choices are influenced by both positive and negative aspects‎, ‎we enhance the MPNSS framework by incorporating bipolarity‎. ‎This addition‎, ‎aimed at capturing the dual nature of decision processes‎, ‎results in the development of bipolar M-parametrized N-soft set (BMPNSS) model‎. ‎In the context of BMPNSS‎, ‎we present some related definitions such as incomplete‎, ‎negatively efficient‎, ‎positively efficient‎, ‎and totally efficient‎. ‎Additionally‎, ‎for the complement of MPNSS‎, ‎we introduce four distinct definitions‎: ‎complement‎, ‎weak complement‎, ‎top weak complement‎, ‎and bottom weak complement‎. ‎Set-theoretic operations‎, ‎including extended and restricted union and intersection‎, ‎are explored accompanied by a discussion of their properties‎, ‎providing a comprehensive understanding of the behavior of these operations within the BMPNSS framework‎. ‎To facilitate understanding‎, ‎we include an illustrative example‎. ‎The decision-making procedure introduces alternative ranking based on extended choice and extended weight choice values‎, ‎demonstrated through a numerical example‎. ‎In our comparative analysis‎, ‎BMPNSS is positioned against existing models‎, ‎emphasizing its distinctive features and advantages in diverse decision-making scenarios‎.

This paper investigates N-bipolar hypersoft topology (N-BHST), a novel extension of both the well-established N-hypersoft topology (N-HST) and hypersoft topology (HST). Deviating significantly from its precursor, the N-bipolar hypersoft (N-BHS) set, N-BHST introduces a multi-opinion approach to decision-making, augmenting robustness and adaptability. This innovative framework addresses identified limitations in N-bipolar soft topology (N-BST), especially in managing multi-argument approximate functions. The study analyzes various operators (closure, interior, exterior, and boundary) within the N-BHST framework, elucidating their interrelationships. Additionally, an examination is carried out on the enhancement of multi-criteria group decision-making (MCGDM) using N-BHST, setting it apart from existing models. A numerical example is presented to illustrate its application in real-world decision scenarios.

The purpose of this paper is to define bipolar soft generalized compact sets and bipolar soft generalized compact spaces. The structures of g~~ -centralized bipolar soft generalized closed sets collection in a bipolar soft generalized compact space are given. Moreover, some main properties of bipolar soft generalized compactness are discussed and their relationships are studied. The concept of a bipolar soft generalized compactness is introduced and it investigates under what condition a bipolar soft generalized topological space forms a bipolar soft generalized compact space. The relation between bipolar soft generalized compact space and soft generalized compact space is proposed. Furthermore, some further properties of bipolar soft mappings, such as bipolar soft composite mappings, are presented and some of their characteristics are explained. Additionally, novel classes of bipolar soft mapping such as bipolar soft generalized continuous, bipolar soft generalized open, and bipolar soft generalized closed mappings are defined. Finally, some results and counterexamples are obtained.

The paper develops a novel analysis of mutual interactions between topology and soft topology. It is known that each soft topology produces a system of crisp (parameterized) topologies. The other way round is also possible. Namely, one can generate a soft topology from a system of crisp topologies. Different methods of producing soft topologies are discussed by implementing two formulas. Then, the relationships between the resulting soft topologies are obtained. With the help of an example, it is demonstrated that one formula is more constructible than the other. Now, it is reasonable to ask which (topological) properties of a soft topology can be transferred to the set of crisp topologies, or the opposite. To address this question, we consider the standard separation axioms and show how well these axioms can be preserved when moving from a system of crisp topologies to the soft topology generated by it and contrariwise. Additionally, our findings extend and disprove some results from the literature.

This paper introduces a new fuzzy structure named “fuzzy primal.” Then, it studies the essential properties and discusses their basic operations. By applying the q-neighborhood system in a primal fuzzy topological space and the Łukasiewicz disjunction, we establish a fuzzy operator (·) ⋄ on the family of all fuzzy sets, followed by its core characterizations. Next, we use (·) ⋄ to investigate a further fuzzy operator denoted by Cl⋄. To determine a new fuzzy topology from the existing one, the earlier fuzzy operators are explored. Such a new fuzzy topology is called primal fuzzy topology. Various properties of primal fuzzy topologies are found. Among others, the structure of a fuzzy base that generates a primal fuzzy topology. Furthermore, the concept of compatibility between fuzzy primals and fuzzy topologies is introduced, and some equivalent conditions to that concept are examined. It is shown that if a fuzzy primal is compatible with a fuzzy topology, then the fuzzy base that produces the primal fuzzy topology is itself a fuzzy topology.

In this paper, the concept of $\delta$-cluster point on a set which belongs to the collection of fine open sets generated by the topology $\tau$ on $X$ has been introduced. Using this definition, the idea of $f_\delta$-open sets is initiated and certain properties of these sets have been also studied. On the basis of separation axioms defined over fine topological space, certain types of $f_\delta$-separation axioms on fine space have been also defined along with some illustrative examples.

The aim of this paper is to give an interesting connection between two mathematical approaches to vagueness: binary bipolar soft sets and binary bipolar soft topology. The binary bipolar soft points are defined using binary bipolar soft sets. The binary bipolar soft set will be the binary bipolar soft union of its binary bipolar soft points. Moreover, the notion of binary bipolar soft topological spaces over two universal sets and a parameter set is proposed. Some topological properties of binary bipolar soft sets, such as binary bipolar soft open, binary bipolar soft closed, binary bipolar soft closure, binary bipolar soft interior, and binary bipolar soft boundary, are introduced. Some important properties of these classes of binary bipolar soft sets are investigated. Furthermore, the symmetry relation is compared between binary bipolar soft topology and binary soft topology on a common universe set. Finally, some results and counterexamples are demonstrated to explain this work.

This paper introduces N-hypersoft (N-HS) sets—an enriched and versatile extension of hypersoft (HS) sets—designed to handle evaluations involving both binary and non-binary data while embodying an inherent sense of structural symmetry. The paper presents several algebraic definitions, including incomplete N-HS sets, efficient N-HS sets, normalized N-HS sets, equivalence under normalization, N-HS complements, and HS sets derived from a threshold. These definitions are accompanied by illustrative examples. Additionally, the paper delves into various set-theoretic operations within the framework of N-HS sets, such as relative null/whole N-HS sets, N-HS subsets, and N-HS extended/restricted union and intersection, presented in two different ways. Finally, the paper presents and compares decision-making methodologies regarding N-HS sets.

In this paper, we first explore more properties of soft somewhere dense continuous functions. Then, we discuss the preservation of soft Baire property and soft Baire category. We give some concrete examples to illustrate how our findings extend some conclusions and connections made in the literature.

In this paper, we define a soft somewhat open set using the soft interior operator. We study main properties the class of soft somewhat open sets that is contained in the class soft somewhere dense sets. Then, we introduce the classes of soft somewhat continuous and soft somewhat open functions and soft somewhat homeomorphisms. Moreover, we study properties and characterizations of soft somewhat continuous and soft somewhat open functions. At last, we discuss topological invariants for soft somewhat homeomorphisms. Multiple examples are offered to clarify some invalid results.

For a mathematical model to describe vague (uncertain) problems effectively, it must have the ability to explain the links between the objects and parameters in the problem in the most precise way. There is no suitable model that can handle such scenarios in the literature. This deficiency serves as motivation for this study. In this article, the bipolar hypersoft set (abbreviated, BHSS) is considered since the parameters and their opposite play a symmetrical role. We present a novel theoretical technique for solving decision-making problems using BHSS and investigate parameter reductions for these sets. Algorithms for parameter reduction are provided and explained with examples. The findings demonstrate that our suggested parameter reduction strategies remove unnecessary parameters and still retain the same decision-making options.

Connectedness represents the most significant and fundamental topological property. It highlights the main characteristics of topological spaces and distinguishes one topology from another. There is a constant study of bipolar soft generalized topological spaces (BSGTSs) by presenting BS g ̃ ̃-connected set and BS g ̃ ̃-connected space in BSGTSs as well as it is discussing some properties and results for these topics. Additionally, the notion of bipolar soft disjoint sets is put forward, BS g ̃ ̃-separation set, g-separated BSSs and BS g ̃ ̃-hereditary property. Moreover, there is an extensive study of BS g ̃ ̃-locally connected space and BS g ̃ ̃-component with some related properties and theorems following them, such as the concepts of BS g ̃ ̃-locally connected spaces and BS g ̃ ̃-connected are independent of each other; also determined the conditions under which the BS g ̃ ̃-connected subsets are BS g ̃ ̃-components.

The main objective of this study is to define a new class of bipolar soft (BS) separation axioms known as BS Ti-space (i = 0,1,2,3,4). This type is defined in terms of ordinary points. We prove that BS Ti -space implies BS Ti_1-space for i =1,2 ; however, the opposite is incorrect, as demonstrated by an example. For i=0,1,2,3,4, we investigate that every BS Ti-space is soft Ti-space; and we set up a condition in which the reverse is true. Moreover, we point out that a BS subspace of a BS Ti-space is a BS Ti-space for i=0,1,2,3.

Mappings are significant mathematical tools with many applications in our daily lives. The bipolar hypersoft set is one of the effective tools for dealing with ambiguity and vagueness. The purpose of this article is to define mappings between the classes of bipolar hypersoft sets. The notions of bipolar hypersoft image and bipolar hypersoft inverse image of bipolar hypersoft sets are then defined, and some of their properties are studied. Moreover, we discuss the relations between the bipolar hypersoft image and the bipolar hypersoft inverse image of the bipolar hypersoft sets. This proposed work can be extended to IndetermSoft Set, IndetermHyperSoft Set and TreeSoft Set and their corresponding Fuzzy, Intuitionistic Fuzzy, Neutrosophic forms and other Fuzzy-extension.

In this manuscript, we continue to study the hypersoft topological space (for short, HSTS) by presenting hypersoft (HS) separation axioms, called HS Ti-spaces for i = 0,1,2,3,4. The notions of HS regular and HS normal spaces are explained in detail. We discuss the connections between them and present numerous examplestohelpclarify the interconnections between the different types of these spaces. Wepoint out that HS Ti-axioms imply HS Ti−1 for i = 1,2,3, and with the help of an example we show that HST4 space need no t be HS T3-space. We also clarify that the property that an HS space being HSTi-spaces (i = 0,1,2,3) is HS hereditary. Finally, we provide a diagram to illustrate the relationships between our proposed axioms.

According to its definition, a topological space could be a highly unexpected object. There are spaces (indiscrete space) which have only two open sets: the empty set and the entire space. In a discrete space, on the other hand, each set is open. These two artificial extremes are very rarely seen in actual practice. Most spaces in geometry and analysis fall somewhere between these two types of spaces. Accordingly, the separation axioms allow us to say with confidence whether a topological space contains a sufficient number of open sets to meet our needs. To this end, we use bipolar hypersoft (BHS) sets (one of the efficient tools to deal with ambiguity and vagueness) to define a new kind of separation axioms called BHS Ti-space (i = 0,1,2,3,4). We show that BHS Ti-space (i = 1,2) implies BHS Ti−1-space; however, the converse is false, as shown by an example. For i = 0,1,2,3,4, we prove that BHS Ti-space is hypersoft (HS) Ti-space and we present a condition so that HS Ti-space is BHS Ti-space. Moreover, we study that a BHS subspace of a BHS Ti-space is a BHS Ti-space for i = 0,1,2,3.

The paper starts with the main properties of the class of soft somewhere dense open functions and follows their connections with other types of soft open functions. Then preimages of soft sets with Baire property and images of soft Baire spaces under certain classes of soft functions are discussed. Some examples are presented that support the obtained results. Further properties of somewhere dense open functions related to different types of soft functions are found under some soft topological properties.

Herein, we further contribute and promote topological structures via bipolar hypersoft (BHS) setting by introducing new types of maps called BHS continuous, BHS open, BHS closed, and BHS homeomorphism maps. We investigate their characterizations and establish their main properties. By providing a thorough picture of the proposed maps, we investigate the concept of BHS compact space and obtain several results relating to this concept. We point out that BH compactness preserved under BH continuous map. The relationships among these concepts with their counterparts in hypersoft (HS) structures are discussed.

Hypersoft topology (HST) is the study of a structure based on all hypersoft (HS) sets on a given set of alternatives. In continuation of this concern, in this article, we introduce new maps namely HS continuous, HS open, HS closed, and HS homomorphism. We examine the main characteristics of each of these maps. Furthermore, we study HS compact space and discuss some of its properties. We point out that HS compactness preserved under HS continuous map.

In this article, we start with some properties of several types of soft continuous and soft open functions. We primarily focus on studying soft continuous (soft open) and soft irresolute (soft anti-irresolute) functions. We show that soft continuous and soft irresolute functions are independent and correspondingly soft open and soft anti-irresolute functions. On the other hand, soft bi-continuity implies soft bi-irresoluteness but not the other way round. Moreover, we find conditions under which soft bi-irresoluteness and soft bi-continuity are similar.

The concept of soft set theory can be used as a mathematical tool for dealing with problems that contain uncertainty. Then, a new mixed mathematical model called the bipolar soft set is created by merging soft sets and bipolarity, which gave the concept of a binary model of grading. Bipolar soft set is characterized by two soft sets, one of which provides positive information and the other negative. Bipolar soft generalized topology is a generalization of bipolar soft topology. The importance of limit points in all branches of mathematics cannot be ignored. It forms one of the most significant and fundamental concepts in topology. On this basis, the derived set concept is required in the establishment and continuation of some properties. Accordingly, the limit point in bipolar soft generalized theory is defined. In this paper, we present the notion of bipolar soft generalized limit points. We explained the relation between the bipolar soft generalized derived and the bipolar soft generalized closure set. Added to that, we discussed some structures of a bipolar soft generalized topological space such as: -interior point, -exterior point, -boundary point, -neighborhood point and basis on . Finally, we give comparisons among these concepts of bipolar soft generalized topological spaces () by using bipolar soft point (). Each concept introduced in this paper is explained with clear examples.

Herein, we further contribute and promote topological structures via bipolar hypersoft (BHS) setting by introducing new types of maps called BHS continuous, BHS open, BHS closed, and BHS homeomorphism maps. We investigate their characterizations and establish their main properties. By providing a thorough picture of the proposed maps, we investigate the concept of BHS compact space and obtain several results relating to this concept. We point out that BH compactness preserved under BH continuous map. The relationships among these concepts with their counterparts in hypersoft (HS) structures are discussed.

The most significant and fundamental topological property is connectedness (resp. disconnectedness). This property highlights the most important characteristics of topological spaces and helps to distinguish one topology from another. Taking this into consideration, we investigate bipolar hypersoft connectedness (resp. bipolar hypersoft disconnectedness) for bipolar hypersoft topological spaces. With the help of an example, we show that if there exist a non-null, non-whole bipolar hypersoft sets which is both bipolar hypersoft open and bipolar hypersoft closed over 𝒰, then the bipolar hypersoft space need not be a bipolar hypersoft disconnected. Furthermore, we present the concepts of separated bipolar hypersoft sets and bipolar hypersoft hereditary property.

Smarandache [48] introduced the concept of hypersoft set which is a generalization of soft set. This notion is more adaptable than soft set and more suited to challenges involving decision-making. Consequently, topology defined on the collection of hypersoft sets will be in great importance. In this paper, we introduce hypersoft topological spaces which are defined over an initial universe with a fixed set of parameters. The notions of hypersoft open sets, hypersoft closed sets, hypersoft neighborhood, hypersoft limit point, and hypersoft subspace are introduced and their basic properties are investigated. Finally, we introduce the concepts of hypersoft closure, hypersoft interior, hypersoft exterior, and hypersoft boundary and the relationship between them are discussed.

The basic of bipolar soft set theory stands for a mathematical instrument that brings together the soft set theory and bipolarity. Its definition is based on two soft sets, a set that provides positive information and other that gives negative. This paper mainly aims at defining a new bipolar soft generalized topological space; setting out of the point that the collection of bipolar soft sets forms the basis for the definition of the new concept is defined. Added to that, an investigation has been made of the four concepts of bipolar soft generalized, namely g-interior, g-closure, g-exterior and g-boundary. Furthermore, the main properties of bipolar soft generalized topological space (BSGTS) are established. This paper also attends to the discussion of the relations between these new definitions and the application of the given bipolar soft generalized topological spaces in a decision-making problem where an algorithm for this application has been suggested. Finally, to clarify and substantiate what the current work subsumes, some examples have been provided.

In this article, we introduce bipolar hypersoft topological spaces over the collection of bipolar hypersoft sets. It is proven that a bipolar hypersoft topological space gives a parametrized family of hypersoft topological spaces, but the converse does not hold in general, and this is shown with the help of an example. Furthermore, we give a condition on a given parametrized family of hypersoft topologies, which assure that there is a bipolar hypersoft topology whose induced family of hypersoft topologies is the given family. The notions of bipolar hypersoft neighborhood, bipolar hypersoft subspace, and bipolar hypersoft limit points are introduced. Finally, we define bipolar hypersoft interior, bipolar hypersoft closure, bipolar hypersoft exterior, and bipolar hypersoft boundary, and the relations between them, differing from the relations on hypersoft topology, are investigated.

Hypersoft set theory is an extension of soft set theory and is a new mathematical tool for dealing with fuzzy problems; however, it still suffers from the parametric tools’ inadequacies. In order to boost decision-making accuracy even more, a new mixed mathematical model called the bipolar hypersoft set is created by merging hypersoft sets and bipolarity. It is characterized by two hypersoft sets, one of which provides positive information and the other provides negative information. Moreover, some fundamental properties relative to it such as subset, superset, equal set, complement, difference, relative (absolute) null set and relative (absolute) whole set are defined. Furthermore, some set-theoretic operations such as the extended intersection, the restricted union, intersection, union, AND-operation and OR-operation of two bipolar hypersoft sets with their properties are discussed and supported by examples. Finally, tabular representations for the purposes of storing bipolar hypersoft sets in computer memory are used.

To contribute to soft topology, we originate the notion of soft bioperators \widetilde{\gamma} and \acute{\widetilde{\gamma}}. Then, we apply them to analyze soft (\widetilde{\gamma}, \acute{\widetilde{\gamma}})-open sets and study main properties. We also prove that every soft (\widetilde{\gamma}, \acute{\widetilde{\gamma}})-open set is soft open; however, the converse is true only when the soft space is soft (\widetilde{\gamma}, \acute{\widetilde{\gamma}}))-regular. After that, we define and study two classes of soft closures namely Cl_(\widetilde{\gamma}, \acute{\widetilde{\gamma}}) and \widetilde{\tau}_(\widetilde{\gamma}, \acute{\widetilde{\gamma}})-Cl operators, and two classes of soft interior namely Int_(\widetilde{\gamma}, \acute{\widetilde{\gamma}}) and \widetilde{\tau}_(\widetilde{\gamma}, \acute{\widetilde{\gamma}})-Int operators. Moreover, we introduce the notions of soft (\widetilde{\gamma}, \acute{\widetilde{\gamma}})-g.closed sets and soft (\widetilde{\gamma}, \acute{\widetilde{\gamma}})-T1/2 spaces, and explore their fundamental properties. In general, we explain the relationships between these notions, and give some counterexamples.

Realizing the importance of separation axioms in classifications of topological spaces and studying certain properties of fixed points, we formulate new soft separation axioms, namely tt-soft bTi (i = 0, 1, 2, 3, 4) and tt-soft b-regular spaces. Their definitions depend on three factors: soft b-open sets, total belong and total non-belong relations. In fact, they are genuine generalizations of p-soft Ti-spaces in the cases of i = 0, 1, 2. With the help of examples, we study the relationships between them as well as with soft bTi (i = 0, 1, 2, 3, 4) and soft b-regular spaces. Some interesting properties of them are obtained under the conditions of soft hyperconnected and extended soft topological spaces. Also, we show that they are preserved under finite product soft spaces and soft b*-homeomorphism mappings. Finally, we introduce a concept of b-fixed soft points and investigate its main properties.

One of the considerable subjects in mathematics is the study of topology. Deducing topology from arbitrary binary relations has enticed the attention of many researchers. So, we devote this article to generate some kinds of topologies from ideals and Ej-neighborhoods which are induced from any binary relation. We define new types of approximations and accuracy measures from these topologies and then compare them with their counterparts induced directly from Ej-neighborhoods and ideals. Also, we show that the approximations and accuracy measures given, herein, are better than those introduced in some previous studies under any arbitrary relation.

One of the divergences between soft set and crisp set is the diversity of belong and non-belong relations between soft sets and ordinary points. This widely opens the door to establish several classes of separation axiom on soft setting. In this article, we exploit the relations of total belong and total non-belong to introduce new soft separation axioms with reference to ordinary points, namely tt-soft βTi (i=0,1,2,3,4) and tt-soft β-regular spaces. The advantages behind using these relations are, first, generalization of existing comparable properties on general topology and second eliminating the stability shape of soft open and closed subsets of soft β-regular spaces. With the aid of some examples, we show the relationships between them as well as with soft βTi (i=0,1,2,3,4) and soft β-regular spaces. Also, we explain the role of soft hyperconnected and extended soft topological spaces in obtaining some interesting results. We characterize a β-soft b-regular space and demonstrate that it guarantees the equivalence of tt-soft βTi (i=0,1,2). Further, we investigate these soft separation axioms in terms of product soft spaces and sum of soft topological spaces. Finally, we introduce a concept of β-fixed soft point theorem and study its main properties.

The concept of soft sets was proposed as an effective tool to deal with uncertainty and vagueness. Topologists employed this concept to define and study soft topological spaces. In this paper, we introduce the concepts of soft SD-continuous, soft SD-open, soft SD-closed and soft SD-homeomorphism maps by using soft somewhere dense and soft cs-dense sets. We characterize them and discuss their main properties with the help of examples. In particular, we investigate under what conditions the restriction of soft SD-continuous, soft SD-open and soft SD-closed maps are respectively soft SD-continuous, soft SD-open and soft SD-closed maps. We logically explain the reasons of adding the null and absolute soft sets to the definitions of soft SD-continuous and soft SD-closed maps, respectively, and removing the null soft set from the definition of a soft SD-open map.

In this article, some types of convergence are discussed along with a class of γ-continuous functions. It is known that various classes of generalized continuous functions are closed under the uniform convergence. We show that γ-continuity is closed with respect to a weaker type of convergence. Further properties of such types of convergence related to γ-continuous functions are obtained.

This paper studies the notion of an operator $\gamma$ on supra open subsets of an $STS$ $(X, \mu)$ and then utilizes it to analyze supra $\gamma$-open sets of an $STS$, where $STS$ is an abbreviation of supra topological space. The notions of $\mu_\gamma$-$g$.closed sets and operator on subspace are introduced and investigated. Furthermore, some new $\mu_\gamma$- separation axioms are formulated and relationships among these spaces are shown. Moreover, some characterizations of the new functions via operator $\gamma$ on $\mu$ are presented and investigated. Finally, it gives some properties of $S_{(\gamma,\beta)}$-closed graph and strongly $S_{(\gamma,\beta)}$-closed graph.

The purpose of this article is to introduce new types of limit points and separation axioms on supra topological spaces by using supra β-open sets. We explore some characterizations and explain the relationships between them with the help of examples. Also, we study many features of them and give sufficient conditions for some equivalent relations between them.

Supra topology was defined by neglecting an intersection condition of topology which makes it more flexible to describe some real-life problems and to easily construct some examples whom show some relationships between certain topological concepts. In fact, it is one of the most important developments of topology in the recent years. The purpose of this paper is to introduce new kinds of limit points of a set and separation axioms on supra topological spaces using supra α-open sets. We discuss the main properties of supra α limit points of a set and describe their behaviour on the spaces that possess the difference property. We probe some equivalent conditions for each one of supra α regular, supra α normal and SαTi-spaces (i= 0, 1, 2, 3, 4). For comparison, we prove that every SαTi-space is ST{i-1} for i= 1, 2, 3, 4 and show that SαTi-spaces are weaker than STi-spaces in the cases of i= 0, 1, 2. Moreover, we prove that the finite product of SαTi-spaces is SαTi in the cases of i= 0, 1, 2. Some examples and counterexamples are given. In the end, we draw attention to that the concepts and results obtained in this work will be a guide to investigate their counterparts in other structures such weak and minimal structures.

The authors of [13] formulated a soft topological ordered spaces concept and then they established and studied some ordered mappings [14]. In the present work, we define new ordered mappings via soft topological ordered spaces based on soft β-open sets, namely soft xβ-continuous, soft xβ-open, soft xβ-closed and soft xβ-homeomorphism mappings, for x \in {I, D, B}. We give various characterizations of each one of the introduced soft mappings. One of the most important obtained results is that an extended soft topologies notion guarantees the equivalent between the soft mappings initiated herein and their counterparts of mappings on topological ordered spaces. We provide several interesting examples to examine the relationships among these soft mappings.

This study aims to introduce the concept of sum of soft topological ordered spaces using pairwise disjoint soft topological ordered spaces and investigate its main properties. To link between soft topological spaces and their sum, we define ordered additive, finitely ordered additive and countably ordered additive properties. Then, we demonstrate that the properties of being p-soft Ti-ordered, soft Ti-ordered and strong soft Ti-ordered spaces are ordered additive and prove that the properties of monotonically soft compact and ordered soft compact spaces are finitely ordered additive. In this content, we study some features of soft lambda-continuous, soft lambda-open, soft lambda-closed and soft lambda-homeomorphism, where lambda \in {I, D, B}. Finally, we give some examples and discuss under which conditions a soft topological ordered space represents the sum of some soft ordered topological spaces.

The purpose of this article is to introduce three types of supra compactness and three types of supra Lindelöfness via supra topological spaces based on the supra pre-open sets. With the help of examples, we illustrate the relationships among them and show their relationships with some kinds of supra compactness and supra Lindelöfness given in [3]. We characterize each type of space and investigate the image of them under pre-irresolute mappings. Also, we prove that these spaces are preserved under the finite product spaces, and give a sufficient condition for the equivalence among supra compact, almost supra compact and supra pre-compact spaces. At the end of each section, we provide some examples to demonstrate that the spaces studied and their counterparts, introduced in [9], are independent of each other.

In this paper, we introduce the concept of sum of soft topological spaces using pairwise disjoint soft topological spaces and study its basic properties. Then we define additive and finitely additive properties which are consider a link between soft topological spaces and their sum. In this regard, we show that the properties of being p-soft Ti, soft paracompactness, soft extremally disconnectedness and soft continuity are additive. We provide some examples to elucidate that soft compactness and soft separability are finitely additive, however, soft discrete and soft door spaces are not finitely additive. Also, we prove that soft interior, soft closure, soft limit and soft boundary points are interchangeable between soft topological spaces and their sum. This helps to obtain some results related to some important generalized soft open sets. Finally, we observe under which conditions a soft topological space represents the sum of some soft topological spaces.

By using a notion of soft alpha-open sets, we generalize the concepts of soft compact and soft Lindelof spaces. We define the concepts of soft alpha-compact, soft alpha-Lindelof, almost (approximately, mildly) soft alpha-compact and almost (approximately, mildly) soft alpha-Lindelof spaces. We present two new kinds of the finite intersection property and utilize them to characterize almost soft alpha-compact and approximately soft alpha-compact spaces. To elucidate the relationships among the introduced spaces and to illustrate our main results, we supply several interesting examples. Also, we point out that the initiated spaces are preserved under soft alpha-irresolute mappings and we investigate certain of results which associate an extended soft topology with the introduced soft spaces. In the end, we conclude some findings which associate the introduced spaces with some soft topological notions such as soft alpha-connectedness, soft alpha-T2-spaces, soft alpha-partition and soft subspaces.

This paper defines some types of γ-Ps- separation axioms called γ-Ps-regular and γ-Ps-normal spaces using γ-Ps-open and γ-Ps-closed sets. Some relations, properties and characterizations of these spaces are discussed. Several examples are given to illustrate some of the results.

The authors utilize the notions of increasing, decreasing and balancing supra $b$-open sets to introduce and study several types of supra continuous, supra open, supra closed and supra homeomorphism maps in supra topological ordered spaces. They give the equivalent conditions for each one of these notions and illustrate the relationships among them with the help of examples. Apart from that, they investigate under which conditions these maps preserve some separation axioms between supra topological ordered spaces.

This paper introduces the concept of an operation γ on τf. Using this operation, we define the concept of fγ-open sets, and study some of their related notions. Also, we introduce the concept of fγg.closed sets and then study some of its properties. Moreover, we introduce and investigate some types of fγ-separation axioms and fγβ-continuous functions by utilizing the operation γ on τf. Finally, some basic properties of functions with fβ-closed graphs have been obtained.

The paper considers properties of δ-preopen, δ-semiopen, a-open and e*-open sets in topological spaces. Particularly, we extend various results concerning these types of sets and give some more new results. Furthermore, we introduce two more subclasses of δ-semiopen sets, and study them.

The paper introduces an operation γ on the collection of gα-open subsets of a topological space. Then γ is used to study the concepts of gαγ-open and g\alphaγ-generalized closed sets. Moreover, the separation axioms called gαγ-Ti for i = 0, 1/2, 1, 2, are given and their properties are obtained.

The authors of [13] formulated a soft topological ordered spaces concept and then they established and studied some ordered mappings [14]. In the present work, we define new ordered mappings via soft topological ordered spaces based on soft β-open sets, namely soft xβ-continuous, soft xβ-open, soft xβ-closed and soft xβ-homeomorphism mappings, for x \in {I, D, B}. We give various characterizations of each one of the introduced soft mappings. One of the most important obtained results is that an extended soft topologies notion guarantees the equivalent between the soft mappings initiated herein and their counterparts of mappings on topological ordered spaces. We provide several interesting examples to examine the relationships among these soft mappings.