Soft Topology Research Articles

Between Soft θ-Openness and Soft ω0-Openness

In this paper, we define and investigate soft ωθ-open sets as a novel type of soft set. We characterize them and demonstrate that they form a soft topology that lies strictly between the soft topologies of soft θ-open sets and soft ω0-open sets. Moreover, we show that soft ωθ-open sets and soft ω0-open sets are equivalent for soft regular spaces. Furthermore, we investigate the connections between particular types of soft sets in a given soft anti-locally countable space and the soft topological space of soft ωθ-open sets generated by it. In addition to these, we define soft ωθ,ω-sets and soft ωθ,θ-sets as two classes of sets, and via these sets, we introduce two decompositions of soft θ-open sets and soft ωθ-open sets, respectively. Finally, the relationships between these three new classes of soft sets and their analogs in general topology are examined.

Soft semi-linear uniform spaces and their perceptual application

In this paper, we use the soft set theory and the concept of semi-linear uniform spaces to introduce the notion of soft semi-linear uniform spaces with its generalization, briefly soft-GSLUS. We investigate some properties of soft topology that induced by soft-GSLUS. Also, we use the members of soft-GSLUS to define a soft proximity space and a soft filter then we establish the relationships between them. Finally, we give the perceptual application of soft semi-linear uniform structures by employing the natural transformation of a soft semi-linear uniform space to a soft proximity.

A Novel Framework for Generalizations of Soft Open Sets and Its Applications via Soft Topologies

Soft topological spaces (STSs) have received a lot of attention recently, and numerous soft topological ideas have been created from differing viewpoints. Herein, we put forth a new class of generalizations of soft open sets called “weakly soft semi-open subsets” following an approach inspired by the components of a soft set. This approach opens the door to reformulating the existing soft topological concepts and examining their behaviors. First, we deliberate the main structural properties of this class and detect its relationships with the previous generalizations with the assistance of suitable counterexamples. In addition, we probe some features that are obtained under some specific stipulations and elucidate the properties of the forgoing generalizations that are missing in this class. Next, we initiate the interior and closure operators with respect to the classes of weakly soft semi-open and weakly soft semi-closed subsets and look at some of their fundamental characteristics. Ultimately, we pursue the concept of weakly soft semi-continuity and furnish some of its descriptions. By a counterexample, we elaborate that some characterizations of soft continuous functions are invalid for weakly soft semi-continuous functions.

The soft topology of soft ω ∗ -open sets and soft almost Lindelofness

In this paper, We use the soft closure operator to introduce soft \(\omega ^{\ast }\)-open sets as a new class of soft sets. We prove that this class of soft sets forms a soft topology that lies strictly between the soft topology of soft \(\theta \)-open sets and the soft topology of soft \(\omega \)-open sets. Also, we show that the soft topology of soft \(% \omega ^{\ast }\)-open sets contain the soft co-countable topology and is independent of the topology of soft open sets. Furthermore, several results regarding soft almost Lindelofness are given. In addition to these, we investigate the correspondences between the novel notions in soft topology and their general topological analogs.

Soft Complete Continuity and Soft Strong Continuity in Soft Topological Spaces

In this paper, we introduce soft complete continuity as a strong form of soft continuity and we introduce soft strong continuity as a strong form of soft complete continuity. Several characterizations, compositions, and restriction theorems are obtained. Moreover, several preservation theorems regarding soft compactness, soft Lindelofness, soft connectedness, soft regularity, soft normality, soft almost regularity, soft mild normality, soft almost compactness, soft almost Lindelofness, soft near compactness, soft near Lindelofness, soft paracompactness, soft near paracompactness, soft almost paracompactness, and soft metacompactness are obtained. In addition to these, the study deals with the correlation between our new concepts in soft topology and their corresponding concepts in general topology; as a result, we show that soft complete continuity (resp. soft strong continuity) in soft topology is an extension of complete continuity (resp. strong continuity) in soft topology.

Somewhat omega continuity and somewhat omega openness in soft topological spaces

In this paper, we introduce soft somewhat ω-continuous soft mappings and soft somewhat ω-open soft mappings as two new classes of soft mappings. We characterize these two concepts. Also, we prove that the class of soft somewhat ω-continuous (resp. soft somewhat ω-open) soft mappings contains the class of soft somewhat continuous (resp. soft somewhat open) soft mappings. Moreover, we obtain some sufficient conditions for the composition of two soft somewhat ω-continuous (resp. soft somewhat ω-open) soft mappings to be a soft somewhat ω-continuous (resp. a soft somewhat ω-open) soft mapping. Furthermore, we introduce some sufficient conditions for restricting a soft somewhat ω-continuous (resp. soft somewhat ω-open) soft mapping to being a soft somewhat ω-continuous (resp. soft somewhat ω-open) soft mapping. In addition to these, we introduce extension theorems regarding soft somewhat ω-continuity and soft somewhat ω-openness. Finally, we investigate the correspondences between the novel notions in soft topology and their general topological analogs.

A weak form of soft $ \alpha $-open sets and its applications via soft topologies

<abstract><p>In this work, we present some concepts that are considered unique ideas for topological structures generated by soft settings. We first define the concept of weakly soft $ \alpha $-open subsets and characterize it. It is demonstrated the relationships between this class of soft subsets and some generalizations of soft open sets with the help of some illustrative examples. Some interesting results and relationships are obtained under some stipulations like extended and hyperconnected soft topologies. Then, we introduce the interior and closure operators inspired by the classes of weakly soft $ \alpha $-open and weakly soft $ \alpha $-closed subsets. We establish their master features and derive some formulas that describe the relations among them. Finally, we study soft continuity with respect to this class of soft subsets and investigate its essential properties. In general, we discuss the systematic relations and results that are missing through the frame of our study. The line adopted in this study will create new roads in the branch of soft topology.</p></abstract>

Soft expandable spaces

In this study, we first introduce a new class of spaces, namely soft expandable spaces, which is a generalization of soft paracompact and countably soft compact spaces. Some properties of these spaces that are discussed in this paper the soft expandable space is equivalent to every A?-soft cover has softLF soft open refinement. Also, we discuss under what conditions that countably soft expandable space is soft paracompact. With the help of interesting examples, we elucidate there is no relationship between soft topology and its parametric topologies with respect to possession the property of being a soft expandable space. In this regards, we discuss the role of extended soft topology to inherited this property to classical topology. Second, we define the concept of s-expandable spaces which is stronger than soft expandable spaces. We give some characterizations of this concept, and investigate some equivalent concepts to sexpandable spaces. In the end, we study the behaviours of soft s-expandable spaces under some types of soft mappings.

Novel types of soft compact and connected spaces inspired by soft q-sets

In this work, we make use of soft Q-sets to introduce the concepts of soft Q-compact, soft Q-Lindel?f and soft Q-connected spaces. We explore the essential properties of these concepts and elucidate the relationships between them with the assist of examples and counterexamples. We also give each one of these concepts a complete description and investigate how they behave under specific kinds of soft mappings. Moreover, we demonstrate the unique characterizations of these concepts which are not satisfied for their counterpart notions existing in the published literature; for example, we prove that every soft Qsubset of soft Q-compact and soft Q-Lindel?f spaces is respectively soft Q-compact and soft Q-Lindel?f as well as we discover the conditions under which the concepts of soft connected and soft Q-connected spaces are equivalent. The role of extended and full soft topologies to obtain some relationships between these concepts and their counterparts via parametric topologies is also discussed.

Compactness and connectedness via the class of soft somewhat open sets

<abstract><p>This paper is devoted to study the concepts of compactness, Lindelöfness and connectedness via the class of soft somewhat open sets which represents one of the generalizations of soft open sets. Beside investigation the main properties of these concepts, it is demonstrated, with the help of examples, that some properties of their counterparts via soft open sets are invalid. Also, the relationships between these concepts and their counterparts defined in classical topology (which is studied herein under the name of parametric topology) are discussed in detail. Moreover, we provide the sufficient conditions that guarantee the equivalence between them. In this regard, it is proved that all introduced types of soft compact and Lindelöf spaces are transmitted to all parametric topologies without imposing any conditions, whereas the converse holds true under the conditions of a full soft topology and a finite (countable) set of parameters. These characterizations represent a unique behavior of these spaces compared to the other types defined by celebrated generalizations of soft open sets. Also, there is no relationship associating soft $ sw $-connectedness with its counterparts via parametric topologies. We successfully describe soft $ sw $-disconnectedness using soft open sets instead of soft $ sw $-open sets and consequently prove that the concepts of soft $ sw $-connected and soft hyperconnected spaces are identical. In conclusion, the obtained results show that the framework given in this manuscript enriches and generalizes the previous works, and has a good application prospect.</p></abstract>

A note on strong form of fuzzy soft pre-continuity

We adapt strong θ−pre-continuity into fuzzy soft topology and investigate its properties. Also, the relations with the other types of continuities in fuzzy soft topological spaces are analized. Moreover, we give some new definitions.

Two New Families of Supra-Soft Topological Spaces Defined by Separation Axioms

This paper contributes to the field of supra-soft topology. We introduce and investigate supra pp-soft Tj and supra pt-soft Tj-spaces (j=0,1,2,3,4). These are defined in terms of different ordinary points; they rely on partial belong and partial non-belong relations in the first type, and partial belong and total non-belong relations in the second type. With the assistance of examples, we reveal the relationships among them as well as their relationships with classes of supra-soft topological spaces such as supra tp-soft Tj and supra tt-soft Tj-spaces (j=0,1,2,3,4). This work also investigates both the connections among these spaces and their relationships with the supra topological spaces that they induce. Some connections are shown with the aid of examples. In this regard, we prove that for i=0,1, possessing the Ti property by a parametric supra-topological space implies possessing the pp-soft Ti property by its supra-soft topological space. This relationship is invalid for the other types of soft spaces introduced in previous literature. We derive some results of pp-soft Ti-spaces from the cardinality numbers of the universal set and a set of parameters. We also demonstrate how these spaces behave as compared to their counterparts studied in soft topology and its generalizations (such as infra-soft topologies and weak soft topologies). Moreover, we investigated whether subspaces, finite product spaces, and soft S

Bipolar Spherical Fuzzy Soft Topology with Applications to Multi-Criteria Group Decision-Making in Buildings Risk Assessment

A generalized soft set model that is more accurate, useful, and realistic is the bipolar spherical fuzzy soft set (BSFSs). It is a more developed variant of current fuzzy soft set models that may be applied to characterize erroneous data in practical applications. Bipolar spherical fuzzy soft sets and bipolar spherical fuzzy soft topology are novel ideas that are intended to be introduced in this work. Bipolar spherical fuzzy soft intersection, bipolar spherical fuzzy soft null set, spherical fuzzy soft absolute set, and other operations on bipolar spherical fuzzy soft sets are some of the fundamental ideas defined in this work. The bipolar spherical fuzzy soft open set, the bipolar spherical fuzzy soft close set, the bipolar spherical fuzzy soft closure, and the spherical fuzzy soft interior are also defined. Additionally, the characteristics of this specified set are covered and described using pertinent instances. The innovative notion of BSFSs makes it easier to describe the symmetry of two or more objects. Moreover, a group decision-making algorithm based on the TOPSIS (Technique of Order Preference by Similarity to an Ideal Solution) approach to problem-solving is described. We analyze the symmetry of the optimal decision and ranking of feasible alternatives. A numerical example is used to show how the suggested approach may be used. The extensive benefits of the proposed work over the existing techniques have been listed.

Bipolar Soft Limit Points in Bipolar Soft Generalized Topological Spaces

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: <img src=image/13428354_01.gif>-interior point, <img src=image/13428354_01.gif>-exterior point, <img src=image/13428354_01.gif>-boundary point, <img src=image/13428354_01.gif>-neighborhood point and basis on <img src=image/13428354_02.gif>. Finally, we give comparisons among these concepts of bipolar soft generalized topological spaces (<img src=image/13428354_02.gif>) by using bipolar soft point (<img src=image/13428354_03.gif>). Each concept introduced in this paper is explained with clear examples.

New approach of soft M-open sets in soft topological spaces

The goal of this article is to introduce, investigate and prove several of the properties of M-open and M-closed soft sets in soft topological structure (Sτs). Furthermore, we prove that the collection of soft M-open (SMo) sets is a soft topology by stating and proving the conditions. Finally, we define and study some characteristics of soft M (SM)-continuous function, soft M-irresolute function, soft M-compactness, soft M-connectedness and soft M-separation axioms.

Computing the Cost of Service Projects in Telkaif Town Using Soft Sets and Soft Topology

In this research, we will present a new method of collecting data and performing calculations on it to obtain the required results differently from the usual, at the beginning we will define new concepts of soft sets in soft topological spaces and we will give and explaining many properties of them as soft ii- dense in itself sets, soft ii-connectedness, soft ii-perfection and soft ii-compactness. After that, in the practical part, we will use the concept of soft topology in calculating the cost of service projects in several areas of Telkaif district of Nineveh Governorate in the north of Iraq, whether the cost of projects is in the one area or several areas or all of them using the theory of soft sets in soft topological spaces.

Soft Regular Generalized ω-Closed Sets and Soft ω-T1/2 Spaces

Soft rgω-closed sets are introduced as a new class of soft sets that strictly contain the classes of soft rg-closed sets and soft gω-closed sets. Furthermore, the behavior of soft rgω-closed sets with respect to soft unions, soft intersections, and soft subspaces, as well as induced soft topologies are investigated. Moreover, soft ω-T1/2 spaces which is a weaker form soft T1/2 spaces is defined and investigated. In addition to these, the characterizations of soft rg-T1/2 spaces and soft rgω-T1/2 spaces are discussed. The work also looks at the relationship between our novel notions in soft topological spaces and their analogs in topological spaces.

A different look at the soft topological polygroups

Soft topological polygroups are defined in two different ways. First, it is defined as a usual topology. In the usual topology, there are five equivalent definitions for continuity, but not all of them are necessarily established in soft continuity. Second it is defined as a soft topology including concepts such as soft neighborhood, soft continuity, soft compact, soft connected, soft Hausdorff space and their relationship with soft continuous functions in soft topological polygroups.

Local compactness and paracompactness on bipolar soft topological spaces

A bipolar soft set is given by helping not only a chosen set of “parameters” but also a set of oppositely meaning parameters called “not set of parameters”. It is known that a structure of bipolar soft set is consisted of two mappings such that F : E → P (X) and G :⌉ E → P (X), where F explains positive information and G explains opposite approximation. In this study, we first introduce a new definition of bipolar soft points to overcome the drawbacks of the previous definition of bipolar soft points given in [34]. Then, we explore the structures of bipolar soft locally compact and bipolar soft paracompact spaces. We investigate their main properties and illuminate the relationships between them. Also, we define the concept of a bipolar soft compactification and investigate under what condition a bipolar soft topology forms a bipolar soft compactification for another bipolar soft topology. To elucidate the presented concepts and obtained results, we provide some illustrative examples.

On Certain Type of Semi open Sets in Soft Topological Space

in this item the concept of W-Hausdorff or soft W-T2 construction in soft topological space is announced with relationship to semi-open sets. since via the normal points of soft topology is called soft alpha W-Hausdorff briefly (soft alpha W-T2) construction. Some subspaces of soft alpha W-T2 construction are as well think carefully. Creation of these space is too avail.