<b>s̃β<sub>s</sub>- Open Sets and s̃β<sub>s</sub>- Continuous in Soft Topological Spaces</b>

In this paper, we introduce the concept of a soft [Formula: see text]-open set. We show that the class of soft [Formula: see text]-open sets lies strictly between the classes of [Formula: see text]-open sets (respectively, soft [Formula: see text]-open) and soft semi-open sets. Additionally, we provide several adequate criteria for the equivalence between soft [Formula: see text]-open sets and each of [Formula: see text]-open sets and soft semi-open sets. Moreover, we demonstrate that the family of soft [Formula: see text]-open sets is a supra soft topology. We study the relationships between soft [Formula: see text]-open sets and other types of soft sets such as soft open, soft pre-open, soft [Formula: see text]-open, soft [Formula: see text]-open, soft [Formula: see text]-open, soft [Formula: see text]-open and soft [Formula: see text]-open. Also, we study the behavior of soft [Formula: see text]-open sets under some soft operators. Finally, we clarify the correspondence between the classes of soft [Formula: see text]-open sets in a soft topological space and in its soft topological subspaces.

In this paper, we define the class of soft ω0-open sets. We show that this class forms a soft topology that is strictly between the classes of soft open sets and soft ω-open sets, and we provide some sufficient conditions for the equality of the three classes. In addition, we show that soft closed soft ω-open sets are soft ω0-open sets in soft Lindelof soft topological spaces. Moreover, we study the correspondence between soft ω0-open sets in soft topological spaces and ω0-open sets in topological spaces. Furthermore, we investigate the relationships between the soft α-open sets (respectively, soft regular open sets, soft β-open sets) of a given soft anti-locally countable soft topological space and the soft α-open sets (respectively, soft regular open sets, soft β-open sets) of the soft topological space of soft ω0-open sets generated by it. Finally, we introduce ω0-regularity in topological spaces via ω0-open sets, which is strictly between regularity and ω-regularity, and we also introduce soft ω0-regularity in soft topological spaces via soft ω0-open sets, which is strictly between soft regularity and soft ω-regularity. We investigate relationships regarding ω0-regularity and soft ω0-regularity. Moreover, we study the correspondence between soft ω0-regularity in soft topological spaces and ω0-regularity in topological spaces.

The author devotes this paper to defining a new class of soft open sets, namely soft Rω-open sets, and investigating their main features. With the help of examples, we show that the class of soft Rω-open sets lies strictly between the classes of soft regular open sets and soft open sets. We show that soft Rω-open subsets of a soft locally countable soft topological space coincide with the soft open sets. Moreover, we show that soft Rω-open subsets of a soft anti-locally countable coincide with the soft regular open sets. Moreover, we show that the class of soft Rω-open sets is closed under finite soft intersection, and as a conclusion, we show that this class forms a soft base for some soft topology. In addition, we define the soft δω-closure operator as a new operator in soft topological spaces. Moreover, via the soft δω-closure operator, we introduce soft δω-open sets as a new class of soft open sets which form a soft topology. Moreover, we study the correspondence between soft δω-open in soft topological spaces and δω-open in topological spaces.

In this paper, we introduce two different notions of generalized supra soft sets namely supra A--soft sets and supra soft locally closed sets in supra soft topological spaces, which are weak forms of supra open soft sets and discuss their relationships with each other and other supra open soft sets [{it International Journal of Mathematical Trends and Technology} (IJMTT), (2014) Vol. 9 (1):37--56] like supra semi open soft sets, supra pre open soft sets, supra $alpha$--open sets and supra $beta$--open sets. Furthermore, the soft union and intersection of two supra soft locally closed sets have been obtained. We also introduce two different notions of generalized supra soft continuity namely supra soft A--continuous functions and supra SLC--continuous functions. Finally, we obtain decompositions of supra soft continuity: $f_{pu}$ is a supra soft A--continuous if it is both supra soft semi-continuous and supra SLC--continuous, and also $f_{pu}$ is a supra soft continuous if and only if it is both supra soft pre--continuous and supra SLC--continuous. Several examples are provided to illustrate the behavior of these new classes of supra soft sets and supra soft functions.

As daily problems involve a great deal of data and ambiguity, it has become vital to build new mathematical ways to cope with them, and soft set theory is the greatest tool for doing so. As a result, we study methods of generating soft topologies through several soft set operators. A soft topology is known to be determined by the system of special soft sets, which are called soft open (dually soft closed) sets. The relationship between specific types of soft topologies and their classical topologies (known as parametric topologies) is linked to the idea of symmetry. Under this symmetry, we can study the behaviors and properties of classical topological concepts via soft settings and vice versa. In this paper, we show that soft topological spaces can be characterized by soft closure, soft interior, soft boundary, soft exterior, soft derived set, or co-derived set operators. All of the soft topologies that result from such operators are equivalent, as well as being identical to their classical counterparts under enriched (extended) conditions. Moreover, some of the soft topologies are the systems of all fixed points of specific soft operators. Multiple examples are presented to show the implementation of these operators. Some of the examples show that, by removing any axiom, we will miss the uniqueness of the resulting soft topology.

In this study, we present and study sIIpg*-closed soft sets in a soft topological space.In this article, sIIpg*-closed soft sets are examined using soft set theory and examples. A suitable condition for recording sIIpg*-open soft sets is determined that corresponds to the soft topology.

This article starts with a study of the congruence of soft sets modulo soft ideals. Different types of soft ideals in soft topological spaces are used to introduce new weak classes of soft open sets. Namely, soft open sets modulo soft nowhere dense sets and soft open sets modulo soft sets of the first category. The basic properties and representations of these classes are established. The class of soft open sets modulo the soft nowhere dense sets forms a soft algebra. Elements in this soft algebra are primarily the soft sets whose soft boundaries are soft nowhere dense sets. The class of soft open sets modulo soft sets of the first category, known as soft sets of the Baire property, is a soft σ-algebra. In this work, we mainly focus on the soft σ-algebra of soft sets with the Baire property. We show that soft sets with the Baire property can be represented in terms of various natural classes of soft sets in soft topological spaces. In addition, we see that the soft σ-algebra of soft sets with the Baire property includes the soft Borel σ-algebra. We further show that soft sets with the Baire property in a certain soft topology are equal to soft Borel sets in the cluster soft topology formed by the original one.

In this paper, using the concept of soft topology given in \cite{Mbk} i.e. with our new perspective of soft topology, we give some basic topological concepts such as open soft set, closed soft set, interior and closure of a soft set. We then give the concept of soft continuity of a given function between soft topological spaces, and from here we also define the concept of soft homeomorphism and argue the all obtained results. At the end of the article, we propose a decision-making method using soft topological concepts.

We define soft ωp-openness as a strong form of soft pre-openness. We prove that the class of soft ωp-open sets is closed under soft union and do not form a soft topology, in general. We prove that soft ωp-open sets which are countable are soft open sets, and we prove that soft pre-open sets which are soft ω-open sets are soft ωp-open sets. In addition, we give a decomposition of soft ωp-open sets in terms of soft open sets and soft ω-dense sets. Moreover, we study the correspondence between the soft topology soft ωp-open sets in a soft topological space and its generated topological spaces, and vice versa. In addition to these, we define soft ωp-continuous functions as a new class of soft mappings which lies strictly between the classes of soft continuous functions and soft pre-continuous functions. We introduce several characterizations for soft pre-continuity and soft ωp-continuity. Finally, we study several relationships related to soft ωp-continuity.

This paper defines the base of soft topology, studies the properties of base. We give the definition of the product over the soft topological space, and discuss the relative properties of soft topological product space, and finally generate these results to the soft rough topological space.

Many researchers defined some basic notions on soft topology and studied many properties. In this paper, we define soft regular generalized closed and open sets in soft topological spaces and studied their some properties. We introduce these concepts which are defined over an initial universe with a fixed set of parameters. We investigate behavior relative to union, intersection and soft subspaces of soft regular generalized closed sets. We show that every soft generalized closed set is soft regular generalized closed. Also, we investigate many basic properties of these concepts.

Objectives: Soft topology has developed as a rapidly expanding area of research, offering a wide range of emerging applications and avenues for investigation. This study introduces and examines two novel concepts within this framework: soft 𝑔⋕𝑠 continuous maps and soft 𝑔⋕𝑠 irresolute maps. The paper provides a comprehensive investigation into these newly defined concepts, delving into their foundational properties and exploring their interrelationships. Through this analysis, the work seeks to enhance the understanding of these concepts and lay the groundwork for their application in broader topological and mathematical contexts. By expanding the theoretical base of soft topology, this research contributes to the ongoing development of this vibrant field. Methods: To construct the definition of Soft 𝑔⋕𝑠 continuous maps and soft 𝑔⋕𝑠 irresolute maps, the previously introduced definitions soft b* - continuous functions and soft b* irresolute function have been referred in soft topological spaces. Findings: Some properties of soft 𝑔⋕𝑠 continuous map have been investigated by comparing the newly introduced soft 𝑔⋕𝑠 continuous map with the existing soft continuous functions, soft g s continuous functions, soft 𝛼-continuous functions, soft semi continuous functions and soft gsp continuous functions. Also it has been compared Soft 𝑔⋕𝑠 continuous maps with soft 𝑔⋕𝑠 irresolute maps and an interesting statement has been concluded. Novelty: Some properties of the newly introduced soft 𝑔⋕𝑠 continuous map have been discussed and the converse part of the properties have been verified with suitable examples. Keywords: Soft 𝑔⋕𝑠 closed sets; Soft g s continuous functions; Soft 𝑔⋕𝑠 continuous functions; Soft 𝑔⋕𝑠 irresolute functions

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.

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

Emerging trends in soft set theory and related topics.