The aim of this study is to introduce and investigate two new classes of separation axioms called supra soft \(R_0\) and supra soft \(R_1\). They are defined in the spaces of supra soft topologies by using the notions of supra soft open sets and supra soft closure operator. We discuss the basic properties and characterizations of them. We also study the relationships between these classes and some other supra soft separation axioms with many results and explanative examples. Moreover, the connections between the properties of these classes and those in some generated soft topologies are presented. Finally, we show that these classes are preserved under subspaces, which means they are supra soft topological properties.

It is common knowledge that fuzzy topology contributes to developing techniques to address real-life applications in various areas like information systems and optimal choices. The building blocks of fuzzy topology are fuzzy open sets, but other extended families of fuzzy open sets, like fuzzy pre-open sets, can contribute to the growth of fuzzy topology. In the present work, we create some classifications of fuzzy topologies which enable us to obtain several desirable features and relationships. At first, we introduce and analyze stronger forms of fuzzy pre-separation and regularity properties in fuzzy topology called fuzzy pre-$T_i,i=0,\frac{1}{2},1,2,3,4,$ fuzzy pre-symmetric, and fuzzy pre-$R_i,i=0,1,2,3$ by utilizing the concepts of fuzzy pre-open sets and quasi-coincident relation. We investigate more novel properties of these classes and uncover their unique characteristics. By presenting a wide array of related theorems and interconnections, we structure a comprehensive framework for understanding these classes and interrelationships with other separation axioms in this setting. Moreover, the relations between these classes and those in some induced topological structures are examined. Additionally, we explore the hereditary and harmonic properties of these classes.

In this article, we apply the concept of fuzzy soft -open sets to define new types of compactness in fuzzy soft topological spaces, namely, -compactness and -compactness. We explore some of their basic properties and reveal the relationships between these types and that which are defined by other authors. Also, we show that -compactness is more general than that which is presented in other papers. To clarify the obtained results and relationships, some illustrative examples are given.

In this article, we prove some new uniqueness and Ulam-Hyers stability results of a nonlinear generalized fractional integro-differential equation in the frame of Caputo derivative involving a new kernel in terms of another function . Our approach is based on Babenko’s technique, Banach’s fixed point theorem, and Banach’s space of absolutely continuous functions. The obtained results are demonstrated by constructing numerical examples.