This study presented a comprehensive multi-attribute decision-making framework founded on bipolar complex intuitionistic fuzzy numbers (BCIFNs) to enhance decision reliability under conditions of uncertainty and dual (positive-negative) evaluation. First, new operational laws were formulated for BCIFNs, providing a rigorous mathematical foundation for developing two aggregation mechanisms: the bipolar complex intuitionistic fuzzy weighted average (BCIFWA) and the bipolar complex intuitionistic fuzzy weighted geometric (BCIFWG) operators. These operators were systematically derived to ensure logical consistency, robustness, and suitability for handling uncertain, imprecise, and bipolar information. Building on these operators, a structured multi-attribute decision- making methodology was proposed to evaluate alternatives across multiple conflicting attributes. The approach was applied to a real-world case study on waste-to-energy (WtE) and circular economy solutions, focusing on identifying the most sustainable strategy for integrated waste management. Six WtE alternatives were assessed across environmental, economic, and social dimensions. The analysis revealed that incineration with energy recovery emerged as the most effective solution, offering a balance of high energy recovery, substantial waste volume reduction, and technological reliability. A comparative analysis with existing fuzzy decision-making methods highlighted the superior ranking stability and adaptability of the proposed BCIFWA and BCIFWG operators, particularly in complex evaluation scenarios with bipolar uncertainty. The results demonstrated the framework’s flexibility, scalability, and enhanced decision precision, contributing to both the theoretical development of fuzzy multi-attribute decision-making and the practical advancement of sustainable waste management and circular economy strategies.

Graph construction is a powerful tool for solving complex problems across vari- ous fields of computer science and computational intelligence. When dealing with uncertainty, fuzzy graph structures are preferred over crisp graph structures, as they effectively represent the inherent uncertainty within a network. To better cap- ture vague and imprecise concepts, the (n,m)-rung orthopair fuzzy set serves as a highly effective framework, as its models provide greater flexibility than other fuzzy models when handling human judgment data. In this paper, we introduce a new framework, referred to as (n,m)-rung orthopair fuzzy graphs ((n,m)-ROFGs), by integrating the concept of graphs with (n,m)-rung orthopair fuzzy sets. The key advantage of this framework is that it meets the demands of certain applica- tions that cannot be effectively modeled using previous approaches to handling uncertainty, as they either require equal degrees of uncertainty (such as IFSs, PFSs, and q-ROFSs) or impose restrictions on the powers of uncertainty degrees through fixed parameter values n and m( as in (2,1)-FSs and (3,2)-FSs). We start by in- troducing the degree and whole degree of a vertex in (n,m)-ROFGs, and illustrate their significance using road networks. Understanding the degree and whole degree of a vertex also provides insight into the characteristics of product operations on (n,m)-ROFGs. Building on this, we define the strong product and ∗-product as product operations on (n,m)-ROFGs. These operations become particularly useful when dealing with a large number of (n,m)-ROFGs. We also introduce the concept of vertex degree and whole degree in the strong product and ∗-product, and estab- lish general theorems concerning the degree and whole degree of (n,m)-ROFGs under the proposed product operations, and provide several numerical examples to illustrate the introduced concepts. Furthermore, we explore an application of the degree and whole degree of the ∗-product of two (n,m)-ROFGs in multi-criteria decision-making concerning the selection of the most likely team to compete in games and using optimal performance measures. Finally, we make comparisons to highlight the shortcomings of previous techniques in modeling certain practical cases that are effectively addressed by our approach, as well as we discuss the mi- nor limitations of the proposed method associated with excessively large values of n and m that can generally be mitigated by selecting moderate settings that better reflect typical decision-making contexts.

The newly introduced n$^{th}$ power root fuzzy set is a useful tool for expressing ambiguity and vagueness. It has an improved ability to manage uncertain situations compared to intuitionistic fuzzy set and Pythagorean fuzzy set theories, making n$^{th}$ power root fuzzy sets applicable in various everyday decision-making contexts. The notions of n$^{th}$ power root fuzzy sets and complex fuzzy sets are integrated in this study to offer complex n$^{th}$ power root fuzzy sets (CnPR-FSs), explaining its fundamental ideas and useful applications. The proposed CnPR-FS integrates the advantages of n$^{th}$ power root fuzzy set and captures both quantitative and qualitative analyses of decision-makers. It is shown that CnPR-FSs are a crucial tool that can describe uncertain data better than complex intuitionistic fuzzy sets and complex Pythagorean fuzzy sets. A key characteristic of CnPR-FSs is a constraint that guarantees the summation of the nth power of the real (and imaginary) part of the complex-valued membership degree and the 1/n power of the real (and imaginary) part of the complex-valued non-membership degree to be equal to or less than one. This allows for a broader representation of uncertain information. The study also explores the creation of customized comparison techniques, accuracy functions, and scoring functions for two complex n$^{th}$ power root fuzzy numbers. Furthermore, it investigates novel aggregation operators by providing in-depth descriptions of their characteristics, such as complex n$^{th}$ power root fuzzy weighted averaging (CnPR-FWA) as well as complex n$^{th}$ power root fuzzy weighted geometric (CnPR-FWG) operators based on CnPR-FSs. Through an in-depth analysis, this paper aims to determine the selection of the most suitable caterer and optimal venue for corporate events. The study's outcomes highlight the suggested method's effectiveness and practical application as compared to other approaches, providing insight into its practical applicability and efficacy.

The circular Pythagorean fuzzy set is an expansion of the circular intuitionistic fuzzy set (CIFS), in which each component is represented by a circle. Nevertheless, even though CIFS improves the intuitionistic fuzzy set representation, it is still restricted to the inflexible intuitionistic fuzzy interpretation triangle (IFIT) space, where the square sum of membership and nonmembership in a circular Pythagorean fuzzy environment and the sum of membership and nonmembership in a circular intuitionistic fuzzy environment cannot exceed one. To overcome this restriction, we provide a fresh extension of the CIFS called the circular n,m-rung orthopair fuzzy set (Cn,m-ROFS), which allows the IFIT region to be expanded or contracted while maintaining the features of CIFS. Consequently, decision makers can assess items over a wider and more flexible range when using a Cn,m-ROFS, allowing for the making of more delicate decisions. In addition, we define several basic algebraic and arithmetic operations on Cn,m-ROFS, such as intersection, union, multiplication, addition, and scalar multiplication, and we discuss their key characteristics together with some of the known relations over Cn,m-ROFS. In addition, we present and study the new circular n,m-rung orthopair fuzzy weighted average/geometric aggregation operators and their properties. Further, a strategy for resolving multicriteria decision-making problems in a Cn,m-ROF environment is provided. The suggested strategy is tested on two situations: the best teacher selection problem and the best school selection problem. To confirm and illustrate the efficacy of the suggested methodology, a comparative analysis with the intuitionistic fuzzy weighted average, intuitionistic fuzzy weighted geometric, q-rung orthopair fuzzy weighted averaging, q-rung orthopair fuzzy geometric averaging, circular PFWAmax , and circular PFWAmin operators approaches is also carried out. Ultimately, in the final section, there are discussions and ideas for future research.

The recently introduced n,m-rung orthopair fuzzy set emerges as a valuable means to articulate ambiguity and vagueness. With its enhanced capacity to handle uncertain scenarios in contrast to q-rung orthopair fuzzy set theories, n,m-rung orthopair fuzzy sets find diverse applications in everyday decision-making processes. This study presents a fresh approach to handling ambiguity and uncertainty in multi-attribute decision-making by introducing a novel tool termed as complex n,m-rung orthopair fuzzy set (Cn,m-ROFS). The proposed Cn,m-ROFS integrates the advantages of n,m-rung orthopair fuzzy set and captures both quantitative and qualitative analyses of decision-makers. This model represents a significant advancement over complex q-rung orthopair fuzzy sets and offers a fruitful avenue for addressing such complexities in decision-making processes. Moreover, the fundamental laws of Cn,m-ROFSs are outlined. Additionally, the paper develops specialized score functions, accuracy functions, and comparison methods tailored for two complex n,m-rung orthopair fuzzy numbers. Furthermore, it explores novel aggregation operators such as complex n,m-rung orthopair fuzzy weighted averaging (Cn,m-ROFWA) and complex n,m-rung orthopair fuzzy weighted geometric (Cn,m-ROFWG) oper- ators based on Cn,m-ROFSs, providing comprehensive descriptions of their properties. Utilizing these operators, a new method for multi-attribute decision making problems within the complex n,m-rung orthopair fuzzy set theory framework is presented and validated through practical examples. The paper’s findings underscore the effectiveness and practical applicability of the proposed method compared to existing methodologies, shedding light on its practical utility and efficacy.

A useful tool for expressing fuzziness and uncertainty is the recently developed n,m-rung orthopair fuzzy set (n,m-ROFS). Due to their superior ability to manage uncertain situations compared to theories of q-rung orthopair fuzzy sets, the n,m-rung orthopair fuzzy sets have variety of applications in decision-making in daily life. To deal with ambiguity and unreliability in multi-attribute group decision-making, this study introduces a novel tool called kn m-rung picture fuzzy set (kn m-RPFS). The suggested knm-RPFS incorporates all of the benefits of n,m-ROFS and represents both the quantitative and qualitative analyses of the decision-makers. The presented model is a fruitful advancement of the q-rung picture fuzzy set (q-RPFS). Furthermore,numerous of its key notions, including as complement, intersection, and union are explained and illustrated with instances. In many decision-making situations, the main benefit of knm-rung picture fuzzy sets is the ability to express more uncertainty than q-rung picture fuzzy sets. Then, along with their numerous features, we discover the basic set of operations for the kn m-rung picture fuzzy sets. Importantly, we present a novel operator, knm-rung picture fuzzy weighted power average (kn m-RPFWPA) over knm-rung picture fuzzy sets, and use it to multi-attribute decision-making issues for evaluating alternatives with kn m-rung picture fuzzy information. Additionally, we use this operator to pinpoint the countries with the highest expat living standards and demonstrate how to choose the best option by comparing aggregate findings and applying score values. Finally, we compare the outcomes of the q-RPFEWA, SFWG, PFDWA, SFDWA, and SFWA operators to those of the kn m-RPFWPA operator.

Since intuitionistic and Pythagorean fuzzy sets have not always worked properly in real-life situations when dealing with high levels of uncertainty, the theory of q-rung orthopair fuzzy sets has appeared as a key component of many mathematical hybrid decision-making models for dealing with these difficult situations. Following that, the n,m-rung orthopair fuzzy set theory created as a fresh mathematical tool for handling uncertainty in a variety of applications. The purpose of the n,m-rung orthopair fuzzy sets was to promote mathematical tractability of uncertain data from broadly applicable real-world decision-making situations. In this regard, n,m-rung orthopair fuzzy sets outperform intuitionistic, Pythagorean, and q-rung orthopair fuzzy sets in terms of flexibility and dependability. The situations can be handled more successfully employing bipolar fuzzy sets and its extensions, enabling decisions to be more accurate. The aim of this study is to greatly improve decision-makers' ability to gather knowledge across a wider variety of contexts. In this paper, a new tool called the interval-valued bipolar n,m-rung orthopair fuzzy set (IVBn,m-ROFS) is put forth as a solution to the imprecision and haziness that exist in practical applications. This suggested set completely involve both quantitative and qualitative evaluations and make use of the n,m-rung orthopair fuzzy set to its fullest extent. The model designed is an effective extension of interval-valued bipolar q-rung orthopair fuzzy set (IVBq-ROFS). Additionally, the proposed IVBn,m-ROFS technique essential properties, such as intersection, union, complement, and subset are studied, and numerical examples are given to support their validity. Furthermore, we address several popular similarity measures that are present on the interval-valued bipolar n,m-rung orthopair fuzzy set. The indicated similarity measures have the significant benefit of taking into account the relationships between interval-valued bipolar n,m-rung orthopair fuzzy numbers. Moreover, in order to select the best supplier selection in supply chain management and analyze its acquired consequences, we create a multi-criteria decision-making algorithm that is based on the recommended similarity measures. Finally, we evaluate the advantages and limitations of the novel methodologies in comparison to some of the existing similarity measures based on fuzzy extensions.

The n,m-rung orthopair fuzzy set theory has appeared as a fresh mathematical tool for dealing with uncertainty in several fields of the real world. The n,m-rung orthopair fuzzy sets were introduced to make uncertain data from broadly applicable real-world decision-making scenarios tractable analytically. In this regard, n,m-rung orthopair fuzzy sets outperform intuitionistic, Pythagorean, Fermatean, and q-rung orthopair fuzzy sets in terms of flexibility and dependability. Bipolar fuzzy sets are useful tools for handling fuzziness and bipolarity. The bipolar n,m-rung orthopair fuzzy set (Bn,m-ROFS) model is presented in this study as a generic extension of two powerful existing models, namely the bipolar intuitionistic fuzzy set and the bipolar Pythagorean fuzzy set models. This suggested set completely reflects both quantitative and qualitative evaluations and fully exploits the n,m-rung orthopair fuzzy set. Moreover, subset, equal, complement, inter- section, and union are some key characteristics of the proposed Bn,m-ROFS model that are examined along with numerical examples. Additionally, certain fundamental operations are established here, including the bipolar n,m-rung orthopair fuzzy weighted average operator and the bipolar n,m-rung orthopair fuzzy weighted geometric operator. Furthermore, a Bn,m- ROFS application is investigated to handle various multiattribute decision-making situations, such as the choice of the best manager. The suggested methodology is backed up by an algorithm. Finally, the comparison of the indicated hybrid model with a known model, such as the bipolar Pythagorean fuzzy set, is offered.

A significant addition to fuzzy set theory for expressing uncertain data is an n,m-th power root fuzzy set. Compared to the nth power root, Fermatean, Pythagorean, and intuitionistic fuzzy sets, n,m-th power root fuzzy sets can cover more uncertain situations due to their greater range of displayed membership grades. When discussing the symmetry between two or more objects, the innovative concept of an n,m-th power root fuzzy set over dual universes is more flexible than the current notion of an intuitionistic fuzzy set, a Pythagorean fuzzy set, and a nth power root fuzzy set. In this study, we demonstrate a number of additional operations on n,m-th power root fuzzy sets along with a number of their special aspects. Additionally, to deal with choice information, we create a novel weighted aggregated operator called the n,m-th power root fuzzy weighted power average (FWPAn m) across n,m-th power root fuzzy sets and demonstrate some of its fundamental features. To rank n,m-th power root fuzzy sets, we also define the score and accuracy functions. Moreover, we use this operator to identify the countries with the best standards of living and show how we can select the best option by contrasting aggregate results using score values. Finally, we contrast the results of the FWPAnm operator with the square-root fuzzy weighted power average (SR-FWPA), the nth power root fuzzy weighted power average (nPR-FWPA), the Fermatean fuzzy weighted power average (FFWPA), and the n,m-rung orthopair fuzzy weighted power average (n,m-ROFWPA) operators.

In many practical situations, the theory of q-rung orthopair fuzzy sets has proven to be a more effective tool for handling high levels of uncertainty than intuitionistic, Pythagorean, and Fermatean fuzzy sets; as a result, it is a fundamental part of many mathematicalhybriddecision-makingmodelsforhandlingsuchchallengingdecisivesituations.Thefuzzyparameterizedsoft sets and their extensions are more effective at handling situations where some of the parameters are strongly preferred over others, which allows for more precise decision-making. Similar to this, soft expert sets, as an effective approach, offer the capability of taking into account many experts’ opinions in one place for multiattribute group decision-making situations. The goal of this study is to substantially improve decision-makers’ capacity for collecting their judgment over a wider range of situations. To accomplish this, we present the idea of the n,m-rung orthopair fuzzy soft expert set (n,m-ROFSES) model by combining n,m-rung orthopair fuzzy sets and soft expert sets in order to handle more generalized information. The created model is a successful extension of fuzzy parameterized q-rung orthopair fuzzy soft expert sets (FPqROFSESs). In addition, several of its fundamental concepts, such as subset, complement, union, and intersection are discussed and demonstrated with examples. Due to its benefits over other warm production methods and the world’s rapidly expanding population, heating technology is having an ever-growing global influence. As a result of the increased demand for heating systems, numerous new manufacturers have emerged. Choosing the best technology solution is a difficult decision for suppliers or sale dealers. In order to tackle these issues, we offer a multi-criteria group decision-making algorithm whose authenticity and efficacy are carefully verified. We investigate the use of this approach in reality, that is, the selection of a suitable brand of heating system. At last, a comparison of the initiated model with some existing approaches is presented to verify its advantages over them.

An nth power root fuzzy set is a useful extension of a fuzzy set for expressing uncertain data. Because of their wider range of showing membership grades, nth power root fuzzy sets can cover more ambiguous situations than intuitionistic fuzzy sets. In this article, we present several novel operations on nth power root fuzzy sets, as well as their various features. Besides, we develop a new weighted aggregated operator, namely, nth power root fuzzy weighted power average (nPR-FWPA) over nth power root fuzzy sets to deal with choice information and show some of their basic properties. In addition, we defne a scoring function for nth power root fuzzy sets ranking. Furthermore, we use this operator to determine the optimal location for constructing a home and demonstrate how we may choose the best alternative by comparing aggregate outputs using score values. Finally, we compare the nPR-FWPA operator outcomes to those of other well-known operators.

One of the most useful expansions of fuzzy sets for coping with information uncertainties is the q-rung orthopair fuzzy sets. In such circumstances, in this article, we define a novel extension of fuzzy sets called nth power root fuzzy set (briefly, nPR-fuzzy set) and elucidate their relationship with intuitionistic fuzzy sets, SR-fuzzy sets, CR-fuzzy sets and q-rung orthopair fuzzy sets. Then, we provide the necessary set of operations for nPR-fuzzy sets as well as study their various features. Furthermore, we familiarize the concept of nPR-fuzzy topology and investigate the basic aspects of this topology. In addition, we define separated nPR-fuzzy sets and then present the concept of disconnected nPR-fuzzy sets. Moreover, we study and characterize nPR-fuzzy continuous maps in great depth. Finally, we establish T0 and T1 in nPR-fuzzy topologies and discover the links between them

An n,m-rung orthopair fuzzy set is the one of the most useful expansions of fuzzy sets for coping with information uncertainties. In such circumstances, in this article, we define an n,m-rung orthopair fuzzy topology and investigate the basic aspects of this topology. We introduce their relationship with Fermatean fuzzy topology, Pythagorean fuzzy topology and intuitionistic fuzzy topology, and provide some examples. In addition, we introduce separated n,m-rung orthopair fuzzy sets andthenwepresenttheconceptofdiconnectedn,m-rungorthopairfuzzysets.Moreover,westudyandcharacterizen,m-rung orthopair fuzzy continuous maps in great depth. Furthermore, we establish T0and T1in n,m-rung orthopair fuzzy topology anddiscoverthelinksbetweenthem.Finally,wecreateanewrelationextensiononn,m-rungorthopairfuzzysetsandprovide a method for classifying children with learning disabilities.

In this paper, we provide a new mapping class termed hesitant fuzzy weakly continuous, which includes hesitant fuzzy continuous mappings, and obtain several characterizations of hesitant fuzzy weakly continuous mappings. Furthermore, we propose the concepts of hesitant fuzzy preopen and hesitant fuzzy semiopen mappings in hesitant fuzzy topological spaces, as well as various features of these mappings.

In this article, we define an γδ∗ -open sets via operation γ- defined on δ and investigate some of their properties. Moreover, we study the notion of γδ∗-generalized closed set in topological spaces and present some of their respective features. Ultimately, we introduce and characterize ed γδ∗-disconnect and component γδ∗-sets.

One of the most helpful fuzzy set extensions for dealing with information uncertainties is an n th power root fuzzy set. In light of this, in this paper, we define an n,m th power root fuzzy set, which is a new type of fuzzy set extension and introduce their relationship with n-rung orthopair fuzzy set, n th power root fuzzy set and n,m-rung orthopair fuzzy set. There is a symmetry between the values of this membership and non-membership functions. To broaden the scope of the decision-making problems, any power function scales are used here. When disputing the symmetry between two or more objects, the innovative concept of an n,mth power root fuzzy set over dual universes is more flexible than the current notion of an intuitionistic fuzzy set, as well as Pythagorean fuzzy set. Then, for the n,m th power root fuzzy sets, we present the essential set of operations as well as their varied characteristics. In addition, we develop the idea of n,m th power root fuzzy topology and study its fundamental properties. Furthermore, we describe the concept of disconnected n,m th power root fuzzy sets after introducing separated n,m th power root fuzzy sets. Moreover, we thoroughly investigate and characterize n,m th power root fuzzy continuous maps. Also we build T 0 and T 1 in n,m th power root fuzzy topology and find their connections. Finally, we construct a new concept of relation in n,m th power root fuzzy set, and based on sufficient experience, we provide the candidate’s decision-making technique via the proposed relation to determine the suitability of companies to applicants.

A q-rung orthopair fuzzy set is one of the effective generalizations of fuzzy set for dealing with uncertainties in information. Under this environment, in this study, we define a new type of extensions of fuzzy sets called n,m-rung orthopair fuzzy sets and investigate their relationship with Fermatean fuzzy sets, Pythagorean fuzzy sets and intuitionistic fuzzy sets. The n,m-rung orthopair fuzzy sets can supply with more doubtful circumstances than Fermatean fuzzy sets, Pythagorean fuzzy sets and intuitionistic fuzzy sets because of their larger range of depicting the membership grades. There is a symmetry between the values of this membership function and non-membership function. Here, any power function scales are utilized to widen the scope of the decision-making problems. In addition, the novel notion of an n,m-rung orthopair fuzzy set through double universes is more flexible when debating the symmetry between two or more objects that are better than the diffusing concept of an n-rung orthopair fuzzy set, as well as m-rung orthopair fuzzy set. The main advantage of n,m-rung orthopair fuzzy sets is that it can describe more uncertainties than Fermatean fuzzy sets, which can be applie in many decision-making problems. Then, we discover the essential set of operations for the n,m-rung orthopair fuzzy sets along with their several properties. Finally, we introduce a new operator, namely, n,m-rung orthopair fuzzy weighted power average (n,m-ROFWPA) over n,m-rung orthopair fuzzy sets and apply this operator to the MADM problems for evaluation of alternatives with n,m-rung orthopair fuzzy information.

In this paper, a new kind of set called a Micro b -open set in micro topological space is introduced. Basic properties of Micro b -open sets are analyzed and also this set used to introduce and study the new types of functions called Micro b -irresolute, Micro b -continuous and Micro W b -continuous functions.

This paper is devoted to introduce a new kind of topological vector spaces called strongly a-topological vector space and study their relationship with other types of topological vector spaces. The a-bounded and totally a-bounded sets are defined by using a-open in strongly a -topological vector space and some of their characterizations are established.

Pythagorean fuzzy set is one of the successful extensions of the fuzzy set for handling uncertainties in information. Underthis environment, in this paper, we introduce a new type of generalized fuzzy sets is called CR-fuzzy sets and compare CR-fuzzysets with Pythagorean fuzzy sets and Fermatean fuzzy sets. The set operations, score function and accuracy function of CR-fuzzysets will study along with their several properties

The aim of this paper is to introduce the concept of (3, 4)-fuzzy sets. We compare (3, 4)-fuzzy sets with intuitionistic fuzzy sets, Pythagorean fuzzy sets, and Fermatean fuzzy sets. We focus on the complement of (3, 4)-fuzzy sets. We construct some of the fundamental set of operations of the (3, 4)-fuzzy sets. Due to their larger range of describing membership grades, (3, 4)-fuzzy sets can deal with more uncertain situations than other types of fuzzy sets. For ranking (3, 4)-fuzzy sets, we define a score function and an accuracy function. In addition, we introduce the concept of (3, 4)-fuzzy topological space. Ultimately, we define (3, 4)-fuzzy continuity of a map defined between (3, 4)-fuzzy topological spaces and we characterize this concept.

An intuitionistic fuzzy set is one of the efficient generalizations of a fuzzy set for dealing with vagueness/uncertainties ininformation. Under this environment, in this manuscript, we familiarize a new type of extensions of fuzzy sets called square-root fuzzy sets (briefly, SR-Fuzzy sets) and contrast SR-Fuzzy sets with intuitionistic fuzzy sets and Pythagorean fuzzy sets. Wediscover the essential set of operations for the SR-Fuzzy sets along with their several properties. In addition, we define a scorefunction for the ranking of SR-Fuzzy sets. To study multiattribute decision-making problems, we introduce four new weightedaggregated operators, namely, SR-Fuzzy weighted average (SR-FWA) operator, SR-Fuzzy weighted geometric (SR-FWG)operator, SR-Fuzzy weighted power average (SR-FWPA) operator, and SR-Fuzzy weighted power geometric (SR-FWPG)operator over SR-Fuzzy sets. We apply these operators to select the top-rank university and show how we can choose the bestoption by comparing the aggregate outputs through score values.

In this paper, we study different properties of δ ∗-semiopen set. We define the concept of δ ∗-semi generalized closed sets andpresent some characteristics. In addition, as applications to δ ∗-semi generalized closed set, we introduce δ ∗-semi T12space and obtainsome of their basic properties. Moreover, we defined the notions of δ ∗-semi symmetric space, δ ∗-semi difference sets and δ ∗-semikernel of sets, and investigate some of their fundamental properties. At the latest, some new types of spaces are introduced and therelationships of these spaces are studied

The purpose of this paper is to define some new classes of soft generalizedclosed sets, namely soft β-∼γg.closed, soft b-∼γg.closed, soft p-∼γg.closed, soft s-∼γg.closed,soft α-∼γg.closed and soft ∼γg.closed by using soft ∼γ open set. The relationships betweenthese soft sets and the related concepts are investigated. Basic properties of soft β-∼γg.closed, soft b-∼γg.closed, soft p-∼γg.closed, soft s-∼γg.closed, soft α-∼γg.closed and soft∼γg.closed sets are analyzed. Finally, the new soft separation axiom, namely soft ∼γ-T 12space is introduced and its basic properties are investigated

The purpose of this paper is to introduce the notion of Fer-matean fuzzy topological space by motivating from the notion of intuition-istic fuzzy topological space, and define Fermatean fuzzy continuity of afunction defined between Fermatean fuzzy topological spaces. For this pur-pose, we define the notions of image and the pre-image of a Fermatean fuzzysubset with respect to a function and we investigate some basic propertiesof these notions. We also construct the coarsest Fermatean fuzzy topologyon a non-empty set X which makes a given function f from X into Y a Fer-matean fuzzy continuous where Y is a Fermatean fuzzy topological space.Finally, we introduce the concept of Fermatean fuzzy points and studysome types of separation axioms in Fermatean fuzzy topological space.

/e purpose of this paper is to define the concept of (3, 2)-fuzzy sets and discuss their relationship with other kinds of fuzzy sets. We describe some of the basic set operations on (3, 2)-fuzzy sets. (3, 2)-Fuzzy sets can deal with more uncertain situations than Pythagorean and intuitionistic fuzzy sets because of their larger range of describing the membership grades. Furthermore, we familiarize the notion of (3, 2)-fuzzy topological space and discuss the master properties of (3, 2)-fuzzy continuous maps. /en, we introduce the concept of (3, 2)-fuzzy points and study some types of separation axioms in (3, 2)-fuzzy topological space. Moreover, we establish the idea of relation in (3, 2)-fuzzy set and present some properties. Ultimately, on the basis of academic performance, the decision-making approach of student placement is presented via the proposed (3, 2)-fuzzy relation to ascertain the suitability of colleges to applicants.

The structure of an α(β,β) -topological ring is richer in comparison with the structure of an α(β,β)-topological group. The theory of α(β,β)-topological rings has many common features with the theoryof α(β,β) -topological groups. Formally, the theory of α(β,β) -topological abelian groups is included in the theory of α(β,β) -topological rings. The purpose of this paper is to introduce and study the concepts of α(β,β) -topological rings and α(β,γ) -topological R-modules. we show how they may be introduced by specifying the neighborhoods of zero, and present some basic constructions. We provide fundamental concepts and basic results on α(β,β) -topological rings and α(β,γ)-topological Rmodules.

The purpose of this paper is to define and study some new types of hesitant fuzzy open sets namely, hesitant fuzzy α-open, hesitant fuzzy preopen, hesitant fuzzy semiopen, hesitant fuzzy b-open and hesitant fuzzy β-open in hesitant fuzzy topological space. Some properties and the relationships between these hesitant fuzzy sets are investigated. Furthermore, some relationships between them in hesitant fuzzy subspace are introduced.

The purpose of this paper is to define and study some new classes of sets by using nano operation namely, ζ-nano regular open, ζ-nano open, ζ-nano α-open, ζ-nano pre-open, ζ-nano semi-open, ζ-nano b-open and ζ-nano β-open in nano topology. Some properties and the relationships between these sets and the related concepts are investigated. Also, we found the deciding factors for the most common disease fever.

The purpose of this paper is to define and study a new class of set called τ∗-b-closed set, and investigating the characteristics of τ∗-b-closed set. Furthermore, the new type of continuous function is introduce and find some of its basic properties.

The aim of this paper is to introduce and study different properties of Micro generalized closed set in micro topological space. As applications to Micro generalized closed set, we introduce Micro $T_{1\over 2}$ space and obtain some of their basic properties. We analyze the behavior of Micro generalized closed set and Micro $T_{1\over 2}$ space under Micro-continuous and Micro-closed functions. Also, we introduce the notions of Micro difference sets and Micro kernel of sets and investigate some of their fundamental properties.

In this paper, some new types of spaces are defined and studied in micro topological spaces namely, Micro T0, Micro T1, Micro T2, Micro R0and Micro R1spaces. Properties and the relationships of these spaces are introduced. Finally, the relationships between these spaces and the related concepts are investigated.

In this paper, Micro β-open in micro topological space is introduced and basic properties of Micro β-open sets are analyzed and also this set used to introduce and study the new types of functions called Micro β-irresolute, Micro β-continuous and Micro Wb-continuous functions

In this paper, we begin the study of β-α-continuous multiplication map. Here we begin by stating some properties of β-α-continuous multiplication and define the β-α-continuity of the inversion map as well. We then introduce the concept of β-α-topological groups and we go over some important theorems and examples of such spaces.

The purpose of the present paper is to introduce and study some a given operation. Also, we study some topological properties of such subspaces operations defined on subspaces and subspace-operation by using α-open sets for and investigate some relationships among the families which are obtained by such subspace operation. Finally, we study some types of operation-closures on subsets of subspace-operation by means of α-open sets.

The purpose of this paper is to introduce and study new notions of continuity, compactness and stability in ditopological texture spaces based on the notions of β-g-open and β-g-closed sets and some of their characterizations are obtained. Finally, the relationships between these concepts and the other related concepts are investigated.

In this paper, we studied the subgroup in β-α-topological group and several basic theorems are introduced. From normal subgroups, the study naturally follows to quotient groups, which are decomposition spaces topologically. The quotient group space of β-α-topological groups defined with the quotient topology. Also, the definition and some basic results of isomorphism are presented.

The purpose of this paper is to introduce and study new separation axioms by using the notions of α-open and α-bioperations. Also, we analyze the relations with some well known separation axioms.

The purpose of the present paper is to define and study $\alpha_{(\gamma, \gamma^{'})}$-semiopen in topological spaces via bioperations. Using this set to introduce the concepts of $(\alpha_{(\gamma, \gamma^{'})}, \alpha_{(\beta, \beta^{'})})$-semicontinuous, $(\alpha_{(\gamma, \gamma^{'})}, \alpha_{(\beta, \beta^{'})})$-semiopen and $(\alpha_{(\gamma, \gamma^{'})}, \alpha_{(\beta, \beta^{'})})$-irresolute functions and investigate some of their properties.

The purpose of this paper is to introduce the notion of $\alpha_{(\gamma, \gamma^{'})}$-separated sets and study their properties in topological spaces, then we introduce the notions of $\alpha_{(\gamma, \gamma^{'})}$-connected and $\alpha_{(\gamma, \gamma^{'})}$-disconnected sets. We discuss the characterizations and properties of $\alpha_{(\gamma, \gamma^{'})}$-connected sets and then properties under $(\alpha_{(\gamma, \gamma^{'})}$, $\alpha_{(\beta, \beta^{'})})$-continuous functions. The $\alpha_{(\gamma, \gamma^{'})}$-components in a space $X$ is also introduced.

The purpose of the present paper is to introduce and study two strong forms of $\alpha_{(\gamma, \gamma^{'})}$-semiopen sets called $\alpha_{(\gamma, \gamma^{'})}$-semiregular sets and $\alpha_{(\gamma, \gamma^{'})}$-$\theta$-semiopen sets. And also introduce a new class of functions called $(\alpha_{(\gamma, \gamma^{'})}, \alpha_{(\beta, \beta^{'})})$-$\theta$-semi continuous functions and obtain several properties of such functions.

The purpose of this paper is to introduce the concept of $\alpha_{(\gamma, \gamma^{'})}$-open sets in topological spaces and study some of their properties. It also studies the class of $\alpha_{(\gamma, \gamma^{'})}$-generalized closed set in topological spaces and some of their properties.

The purpose of this paper is to define and study the notions of new classes of functions, namely $(\alpha_{(\gamma, \gamma^{'})}$, $\alpha_{(\beta, \beta^{'})})$-continuous, $(\alpha_{(\gamma, \gamma^{'})}$, $\alpha_{(\beta, \beta^{'})})$-closed and $(\alpha_{(\gamma, \gamma^{'})}$, $\alpha_{(\beta, \beta^{'})})$-homeomorphism functions and investigate their fundamental properties.

In this paper, we introduce two strong forms of $\alpha_{[\gamma, \gamma^{'}]}$-semiopen sets called $\alpha_{[\gamma, \gamma^{'}]}$-semiregular sets and $\alpha_{[\gamma, \gamma^{'}]}$-$\theta$-semiopen sets. we also introduce a new class of functions called $(\alpha_{[\gamma, \gamma^{'}]}, \alpha_{[\beta, \beta^{'}]})$-$\theta$-semicontinuous functions. Moreover, we obtain some characterizations and several properties of such functions.

We introduce and study new notions of continuity, compactness and stability in ditopological texture spaces based on the notions of pre-g-open and pre-g-closed sets and some of their characterizations are obtained.

The aim of the present paper is to define and study a new class of groups, namely Wm-groups with a single binary operation based on axioms of semi commutativity, right identity and left inverse. Moreover, we introduce the notions of right cosets, quotient Wm-groups, homomorphisms, kernel and normal Wm-subgroups in terms of Wm-groups, and investigate some of their properties.

The aim of this paper is to introduce and study two new classes of spaces, called weakly $\alpha_{\gamma}$-regular and weakly $\alpha_{\gamma}$-normal spaces. Some basic properties of these separation axioms are studied by utilizing $\alpha_{\gamma}$-open and $\alpha_{\gamma}$-closed sets.

In this paper, the author introduce and study new notions of continuity, compactness and stability in ditopological texture spaces based on the notions of semi-g-open and semi-g-closed sets and some of their characterizations are obtained.

The main objective of this paper is to present the study of α-topological vector spaces. α-topological vector spaces are defined by using α-open sets and α-irresolute mappings. Notions of convex, balanced and bounded set are introduced and studied for α-topological vector spaces. Along with other results, it is proved that every α-open subspace of an α-topological vector space is an α-topological vector space. A homomorphism between α-topological vector spaces is α-irresolute if it is α-irresolute at the identity element. In α-topological vector spaces, the scalar multiple of α-compact set is α-compact and αCl(C) as well as αInt(C) is convex if C is convex. And also, in α-topological vector spaces, αCl(E) is balanced (resp. bounded) if E is balanced (resp. bounded), but αInt(E) is balanced if E is balanced and 0 ∈ αInt(E).

The aim of the present paper is to introduce the concept of α-γ-compactness by means of γ-operation defined on the family of α-open sets of a topological space. We define the α-γ-convergence, α- γ-accumulation points of a filterbase and give some of their properties. Some characterizations and properties of α-γ-compact spaces are obtained

The main purpose of the present paper is to define and study the notions of new classes of functions, namely α-β-continuous, α-(γ, β)-continuous and α(γ,β)-open functions. Moreover, we introduce the concept of α-(γ, β)-homeomorphism and give some properties of these functions in topological spaces. Also functions with α-β-closed graphs are defined and investigated.

The purpose of this paper is to introduce and study the concept of $\alpha_{(\beta, \beta)}$-topological abelian group. Also we study the subgroup in $\alpha_{(\beta, \beta)}$-topological group and several basic theorems are introduced. The factor group space of $\alpha_{(\beta, \beta)}$-topological groups defined and some basic results are presented.

The purpose of this paper is to investigate several types of separation axioms in topological spaces and study some of the essential properties of such spaces. Moreover, we investigate their relationship to some other known separation axioms and some counterexamples.

In this paper, we consider the class of $\alpha_{[\gamma, \gamma^{'}]}$-generalized closed set in topological spaces and investigate some of their properties. We also present and study new separation axioms by using the notions of $\alpha$-open and $\alpha$-bioperations. Also, we analyze the relations with some well known separation axioms.

The aim of this paper is to introduce and study the concept of $\alpha_{[\gamma, \gamma^{'}]}$-semiopen sets. Using this set, we introduce and study the concept of $(\alpha_{[\gamma, \gamma^{'}]}, \alpha_{[\beta, \beta^{'}]})$-semicontinuous and $(\alpha_{[\gamma, \gamma^{'}]}, \alpha_{[\beta, \beta^{'}]})$-irresolute functions.

In this paper, we introduce the notions of αγ-interior, αγ-neighbourhood, αγ-derived, αγ- boundary, αγ-kernel and α-γ-g.closed set defined by γ-operation on αO(X) and investigate some of their properties.

In this paper, we introduce and study the notion of semi-γ-I-open sets in ideal topological spaces and investigate some of their properties. Further, we study continuous functions on the above set and derive some of their properties.

The main purpose of the present paper is to introduce a new class of functions called almost β-γ-continuous functions which is contained in the class of almost β-continuous functions and contains the class of β-γ-continuous functions.

In this paper, we introduce the concept of $\alpha_{[\gamma, \gamma^{'}]}$-open sets in topological spaces and study some of their properties. Furthermore, we offer a new class of functions called $(\alpha_{[\gamma, \gamma^{'}]}$, $\alpha_{[\beta, \beta^{'}]})$-continuous functions and investigate their fundamental properties.

This article examines the beliefs of teachers who teach at scientific departments about teaching mathematics in schools. In exploring their perceptions about the contributing factors that help kurd students to achieve high-scores in math’s subject. Therefore, forty seven teachers (who teach at scientific departs) questionnaire made by the researcher. The data collected was analyzed using descriptive analysis. As a result, it was revealed that the majority of teachers agreed that real life applications, processing skills, using concrete instructional manipulative and conceptual knowledge are very important in teaching mathematics. However, most of the teachers who have participated in this survey was aware of the fact that kurd students have high achievements compared to other students around the world. On the other hand, the results have also shown that kurd students usually concentrate on private tuition to improve their math’s skills, as well as, the parent’s involvement in stimulating them for better results has an active role in their high achievements.

The purpose of this paper is to introduce the new concepts namely, α-g-closed, pre-g-closed, semi-g-closed, b- g-closed, β-g-closed, α-g-open, pre-g-open, semi-g-open, b-g-open and β-g-open sets in ditopological texture spaces. The relationships between these classes of sets are also obtained. Also some properties and several characterizations are investigated.

The main goal of this paper is to introduce and study new notions of continuity, compactness and stability in ditopological texture spaces based on the notions of b-g-open and b-g-closed sets and some of their characterizations are obtained.

In this paper, the author introduce and study new notions of continuity, compactness and stability in ditopological texture spaces based on the notions of α-g-open and α-g-closed sets and some of their characterizations are obtained.

The main goal of this paper is to study some new classes of sets by using the open sets and functions in topological spaces. For this aim, the notions of τ*-open, τ-τ*-α-open, τ-τ*-preopen, τ-τ*-semiopen, τ-τ*-b-open, τ-τ*-β-open, τ*-τ-α-open, τ*-τ-preopen, τ*-τ-semiopen, τ*-τ-b-open, τ*-τ-β-open, τ*-α-open, τ*-preopen, τ*-semiopen, τ*-b-open and τ*-β-open sets are introduced. Properties and the relationships of τ*-open sets are investigated. Finally, the relationships between these sets and the related concepts are investigated.

The purpose of this present paper is to study some new classes of sets by using the open sets and functions in topological spaces. For this aim, the notions of $\delta^{*}$-open, $\delta$-$\delta^{*}$-$\alpha$-open, $\delta$-$\delta^{*}$-preopen, $\delta$-$\delta^{*}$-semiopen, $\delta$-$\delta^{*}$-b-open, $\delta$-$\delta^{*}$-$\beta$-open, $\delta^{*}$-$\delta$-$\alpha$-open, $\delta^{*}$-$\delta$-preopen, $\delta^{*}$-$\delta$-semiopen, $\delta^{*}$-$\delta$-b-open, $\delta^{*}$-$\delta$-$\beta$-open, $\delta^{*}$-$\alpha$-open, $\delta^{*}$-preopen, $\delta^{*}$-semiopen, $\delta^{*}$-b-open and $\delta^{*}$-$\beta$-open sets are introduced. Properties and the relationships of $\delta^{*}$-open sets are investigated. Finally, the relationships between these sets and the related concepts are investigated.

In this paper we introduce and study the notion of $(i, j)$-$\beta$-$I$-$i$-open sets in ideal bitopological spaces and investigate some of their properties. Also we present and study $(i, j)$-$\beta$-$I$-$i$-continuous and $(i, j)$-$\beta$-$I$-$i$-almost continuous functions. Furthermore, we obtain basic properties and preservation theorems of $(i, j)$-$\beta$-$I$-$i$-continuous and $(i, j)$-$\beta$-$I$-$i$-almost continuous.

We characterize minimal $P_{\gamma}$-open sets in topological spaces. We show that any nonempty subset of a minimal $P_{\gamma}$-open set is pre $P_{\gamma}$-open. As an application of a theory of minimal $P_{\gamma}$-open sets, we obtain a sufficient condition for a $P_{\gamma}$-locally finite space to be a pre $P_{\gamma}$-Hausdorff space.

The concept of semi-Bc-continuous functions in topological spaces is introduced and studied. Some of their characteristic properties are considered. Also we investigate the relationships between these classes of functions and other classes of functions.

In this paper, we introduce a new class of sets called γ-generalized b-closed sets in topological spaces (briefty γgb-closed set). Also we study some of its basic properties and investigate the relations between the associated topology.

In this paper, we introduce the concept of $R_{\gamma}$-open sets as a strong of $\gamma$-open sets in a topological space $(X,\tau)$. Using this set, we introduce $R_{\gamma}$-$T_0$, $R_{\gamma}$-$T_{\frac{1}{2}}$, $R_{\gamma}$-$T_1$, $R_{\gamma}$-$T_2$, $R_{\gamma}$-$T_3$, $R_{\gamma}$-$T_4$, $R_{\gamma}$-$D_0$, $R_{\gamma}$-$D_1$ and $R_{\gamma}$-$D_2$ spaces and study some of its properties. Finally we introduce $R_{(\gamma,{\gamma^{'}})}$-continuous mappings and give some properties of such mappings.

In this paper, a new kind of set called an $\alpha^{\gamma}$-open set is introduced and investigated using the $\gamma$-operator due to Ogata. Such sets are used for studying new types of mappings, viz. $\alpha^{\gamma}$-continuous, $\alpha^{(\gamma,\beta)}$-irresolute, etc. Finally, new separation axioms: $\alpha^{\gamma}$-$T_i$ ($i=0, \frac{1}{2}, 1, 2$), $\alpha^{\gamma}$-$D_i$ ($i=0, 1, 2$), and a new notion of the graph of a function called an $\alpha^{\gamma}$-closed graph.

In this paper, a new kind of set called a γ-semi-open set is introduced and investigated using the γ-operator due to Ogata. Such sets are used for studying new types of mappings, viz. γ-semi-continuous, (γ,β)-semi-irresolute and weakly γ-semi-continuous.

In this paper, we introduce the concept of αγ-open sets as a generalization of γ-open sets in a topological space (X, τ). Using this set, we introduce αγT0, αγ-T½, αγT1, αγT2, αγD0, αγD1 and αγD2 spaces and study some of its properties. Finally we introduce α(γ,γ')-continuous mappings and give some properties of such mappings.

In this paper, we introduce the notion of pre-$\gamma$-g.closed sets and some weak separation axioms. Also we show that some basic properties of pre-$\gamma$-$T_{\frac 1{2}}$, pre-$\gamma$-$T_i$, pre-$\gamma$-$D_i$ for $i=0, 1, 2$ spaces and we ofer a new class of functions called pre-$\gamma$-irresolute, pre-$\gamma$-continuous functions and a new notion of the graph of a function called a pre-$\gamma$-closed graph and investigate some of their fundamental properties.

In this paper, we introduce the notion of $\beta$-$\gamma$-g.closed sets and some weak separation axioms. Also we show that some basic properties of $\beta$-$\gamma$-$T_{\frac 1{2}}$, $\beta$-$\gamma$-$T_i$, $\beta$-$\gamma$-$D_i$ for $i=0, 1, 2$ spaces and we ofer a new class of functions called $\beta$-$\gamma$-irresolute, $\beta$-$\gamma$-continuous functions and a new notion of the graph of a function called a $\beta$-$\gamma$-closed graph and investigate some of their fundamental properties.

In this paper, we introduce a new class of sets called γ-PS-open sets. This class of sets lies strictly between the classes of δ-open and γ-p-open sets. We also study its fundamental properties and compare it with some other types of sets and we investigate further topological properties of sets and we introduce and investigate new class of continuous named γ-PS-continuous.

In this paper, we introduce and explore fundamental properties of maximal Pγ-open sets in topological spaces such as decomposition theorem for maximal Pγ-open set. Basic properties of intersection of maximal Pγ-open sets are established.

In this paper, we introduce and study the notions of $S_{\gamma_{1}}$-open sets, $S_{\gamma_{1}}$-continuous and $12$-almost $S_{\gamma_{1}}$-continuous functions in bitopological space. We also investigated the fundamental properties of such functions.

In this paper, we introduce a new class of b-open sets called Bc-open, this class of sets lies strictly between the classes of θ-semi open and b-open sets. We also study its fundamental properties and compare it with some other types of sets and we investigate further topological properties of sets and we introduce and investigate new class of space named Bc-compact.

In this paper, we introduce the notion of $\alpha$-$\gamma$-g.closed sets and some weak separation axioms. Also we show that some basic properties of $\alpha$-$\gamma$-$T_{\frac 1{2}}$, $\alpha$-$\gamma$-$T_i$, $\alpha$-$\gamma$-$D_i$ for $i=0, 1, 2$ spaces and we ofer a new class of functions called $\alpha$-$\gamma$-irresolute, $\alpha$-$\gamma$-continuous functions and a new notion of the graph of a function called a $\alpha$-$\gamma$-closed graph and investigate some of their fundamental properties.

In this paper, we introduce the notion of b-$\gamma$-g.closed sets and some weak separation axioms. Also we show that some basic properties of b-$\gamma$-$T_{\frac 1{2}}$, b-$\gamma$-$T_i$, b-$\gamma$-$D_i$ for $i=0, 1, 2$ spaces and we ofer a new class of functions called b-$\gamma$-irresolute, b-$\gamma$-continuous functions and a new notion of the graph of a function called a b-$\gamma$-closed graph and investigate some of their fundamental properties.

In this paper, we introduce and explore fundamental properties of maximal b-γ-open sets in topological spaces such as decomposition theorem for maximal b-γ-open set. Basic properties of intersection of maximal b-γ-open sets are established.

We characterize minimal b-γ-open sets in topological spaces. We show that any nonempty subset of a minimal b-γ-open set is pre b-γ-open. As an application of a theory of minimal b-γ-open sets, we obtain a sufficient condition for a b-γ-locally finite space to be a pre b-γ-Hausdorff space.

In this paper we introduce the concept of Pγ-open sets as a generalization of γ-open sets. By using this set we introduce Pγ-T1/2 spaces and study some of its properties. Finally, we introduce Pγ,β-continuous mappings and give some properties of such mappings.

In this paper, we introduce two new classes of topological spaces called γ-R0 and γ-R1 spaces in terms of the concept of γ-open sets and investigate some of their fundamental properties.

The concept of preopen sets and precontinuous functions in topological spaces are introduced by A. S. Mashhour et al. [2]. Ogata [3] introduced the notion of γ-open sets which are weaker than open sets. In this paper, we introduce the notion of γ-p- open sets, and γ-p-irresolute in topological spaces. By utilizing these notions we introduce some weak separation axioms. Also we show that some basic properties of γ-p-Ti, γ-p-Di for i = 0, 1, 2 spaces and we offer a new class of functions called γ-p- continuous functions and a new notion of the graph of a function called a γ-p-closed graph and investigate some of their fundamental properties.