2024-07

Journal of Mathematics and Computer Science (Issue : 1) (Volume : 36)

This article presents a pioneering mathematical model, fuzzy bipolar hypersoft (FBHS) sets, which combines the bipolarity of parameters with the fuzziness of data. Motivated by the need for a comprehensive framework capable of addressing uncertainty and variability in complex phenomena, our approach introduces a novel method for representing both the presence and absence of parameters through FBHS sets. By employing two mappings to estimate positive and negative fuzziness levels, we bridge the gap between bipolarity, fuzziness, and parameterization, allowing for more realistic simulations of multifaceted scenarios. Compared to existing models like bipolar fuzzy hypersoft (BFHS) sets, FBHS sets offer a more intuitive and user-friendly approach to modeling phenomena involving bipolarity, fuzziness, and parameterization. This advantage is underscored by a detailed comparison and a practical example illustrating FBHS sets’ superiority in modeling such phenomena. Additionally, this paper provides an in-depth exploration of fundamental FBHS set operations, highlighting their robustness and applicability in various contexts. Finally, we demonstrate the practical utility of FBHS sets in problem-solving and introduce an algorithm for optimal object selection based on available information sets, further emphasizing the advantages of our proposed framework.

2024-07

Journal of Mathematics and Computer Science (Issue : 1) (Volume : 36)

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.

2024-05

PLoS ONE (Issue : 5) (Volume : 19)

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.

2024-04

Fuzzy Information and Engineering (Issue : 1) (Volume : 16)

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.

2024-04

European Journal of Pure and Applied Mathematics (Issue : 2) (Volume : 17)

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.

2024-03

Journal of Intelligent & Fuzzy Systems

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.

2024-01

Sahand Communications in Mathematical Analysis

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.

2023-12

Symmetry (Issue : 1) (Volume : 16)

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.

2023-09

Symmetry (Issue : 15) (Volume : 9)

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.

2023-07

Journal of Mathematics and Computer Science (Issue : 1) (Volume : 32)

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.

2023-04

TWMS Journal of Applied and Engineering Mathematics (Issue : 2) (Volume : 13)

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.

2023-04

Neutrosophic Sets and Systems (Volume : 55)

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.

2023-02

Journal of Mathematics and Computer Science (Issue : 4) (Volume : 30)

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.

2023-01

Demonstratio Mathematica (Issue : 1) (Volume : 55)

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.

2023-01

Neutrosophic Sets and Systems (Volume : 53)

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.

2023-01

Filomat (Issue : 19) (Volume : 36)

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.

2023-01

International Journal of Neutrosophic Science (Issue : 1) (Volume : 20)

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.

2023-01

Iraqi Journal of Science (Issue : 1) (Volume : 64)

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.

2022-11

International Journal of Neutrosophic Science (Issue : 2) (Volume : 19)

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.

2022-11

International Journal of Neutrosophic Science (Issue : 2) (Volume : 19)

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.

2022-11

Journal of Mathematics and Computer Science (Issue : 1) (Volume : 30)

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.

2022-11

Mathematics and Statistics (Issue : 6) (Volume : 10)

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.

2022-10

Neutrosophic Sets and Systems (Volume : 51)

Connectedness (resp. disconnectedness), which reflects the key characteristic of topological spaces and helps in the differentiation of two topologies, is one of the most significant and fundamental concept in topological spaces. In light of this, we introduce hypersoft connectedness (resp. hypersoft disconnectedness) in hypersoft topological spaces and investigate its properties in details. Furthermore, we present the concepts of disjoint hypersoft sets, separated hypersoft sets, and hypersoft hereditary property. Also, some examples are provided for the better understanding of these ideas.

2022-08

Journal of Intelligent & Fuzzy Systems (Issue : 4) (Volume : 43)

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.

2022-04

Neutrosophic Sets and Systems (Volume : 49)

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.

2022-04

European Journal of Pure and Applied Mathematics (Issue : 2) (Volume : 15)

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.

2022-02

Journal of Mathematics (Volume : 2022)

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.

2021-08

Mathematics (Issue : 9) (Volume : 15)

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.

2021-08

AIMS Mathematics (Issue : 11) (Volume : 6)

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.

2021-07

Fuzzy Information and Engineering

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.

2021-06

Complexity (Volume : 2021)

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.

2021-04

Journal of Interdisciplinary Mathematics

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.

2021-01

Filomat (Issue : 10) (Volume : 34)

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.

2021-01

Journal of Mathematical and Computational Science (Issue : 1) (Volume : 11)

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.

2020-12

Demonstratio Mathematica (Issue : 1) (Volume : 53)

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.

2020-11

Missouri Journal of Mathematical Sciences (Issue : 2) (Volume : 32)

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.

2020-09

Advances in Mathematics: Scientific Journal (Issue : 10) (Volume : 9)

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.

2020-08

Italian Journal of Pure and Applied Mathematics (Volume : 44)

This paper introduces some more γ-Ps-separation axioms called γ-Ps-R0 and γ-Ps-R1 by using γ-PS-open sets and τγ-Ps-closure of a set. Some properties of these spaces are constructed.

2020-07

European Journal of Pure and Applied Mathematics (Issue : 3) (Volume : 13)

Sometimes we need to minimize the conditions of topology for different reasons such as obtaining more convenient structures to describe some real-life problems, or constructing some counterexamples whom show the interrelations between certain topological concepts, or preserving some properties under fewer conditions of those on topology. To contribute this research area, in this paper, we establish some new concepts on supra topological spaces using supra semi-open sets and give some characterizations of them. First, we introduce a concept of supra semi limit points of a set and study main properties, in particular, on the spaces that possess the difference property. Second, we define and investigate new separation axioms, namely supra semi Ti-spaces (i = 0, 1, 2, 3, 4) and give complete descriptions for each one of them. We provide some examples to show the relationships between them as well as with STi-space.

2020-07

Advances in Mathematics: Scientific Journal (Issue : 7) (Volume : 9)

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.

2020-07

Missouri Journal of Mathematical Sciences (Issue : 1) (Volume : 32)

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.

2020-06

Mathematics (Issue : 6) (Volume : 8)

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.

2020-02

Italian Journal of Pure and Applied Mathematics (Volume : 43)

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.

2019-10

International Journal of Applied Mathematics (Issue : 4) (Volume : 32)

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.

2019-09

General Letters in Mathematics (Issue : 1) (Volume : 7)

In this paper, the concepts of Pp-compact spaces by using nets, filter base and Pp-complete accumulation points are introduced and studied.

2019-07

European Journal of Pure and Applied Mathematics (Issue : 3) (Volume : 12)

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.

2019-07

European Journal of Pure and Applied Mathematics (Issue : 3) (Volume : 12)

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.

2019-05

International Journal of Applied Mathematics (Issue : 2) (Volume : 32)

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.

2019-04

New Trends in Mathematical Sciences (Issue : 2) (Volume : 7)

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.

2019-02

European Journal of Pure and Applied Mathematics (Issue : 1) (Volume : 12)

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.

2019-01

General Letters in Mathematics (Issue : 3) (Volume : 4)

Recently, new types of (open) sets have been studied by some topologist. This note shows that these sets are identical to some other sets already exist in the literature.