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 various articles, it is said that the class of all soft topologies on a common universe forms a complete lattice, but in this paper, we prove that it is a complete lattice. Some soft topologies are maximal, and some are minimal with respect to specific soft topological properties. We study the properties of soft compact and soft connected topologies that are maximal. Particularly, we prove that a maximal soft compact topology has identical families of soft compact and soft closed sets. We further show that a maximal soft compact topology is soft , while a maximal soft connected topology is soft . Lastly, we establish that each soft connected relative topology to a maximal soft connected topology is maximal.

In a soft environment, we investigated several (classical) structures such as ideals, filters, grills, etc. It is well known that these structures are applied to expand abstract concepts; in addition, some of them offer a vital tool to address some practical issues, especially those related to improving rough approximation operators and accuracy measures. Herein, we contribute to this line of research by presenting a novel type of soft structure, namely “soft primal”. We investigate its basic properties and describe its behaviors under soft mappings with the aid of some counterexamples. Then, we introduce three soft operators ${(·)}^{\diamond }$, $C{l}^{\diamond }$ and ${(·)}^{\square }$ inspired by soft primals and explore their main characterizations. We show that $C{l}^{\diamond }$ satisfies the soft Kuratowski closure operator, which means that $C{l}^{\diamond }$ generates a unique soft topology we call a primal soft topology. Among other obtained results, we elaborate that the set of primal topologies forms a natural class in the lattice of topologies over a universal set and set forth some descriptions for primal soft topology under specific types of soft primals.

This paper begins with an introduction to some soft topological operators that will be used to characterize several soft separation axioms followed by their main properties. Then, we define a new soft separation axiom called "soft $ T_D $-space" and analyze its main properties. We also show that this space precisely lies between soft $ T_0 $ and soft $ T_1 $-spaces. Finally, we characterize soft $ T_i $-spaces, for $ i = 0, 1, D $, in terms of the stated operators.

In this paper, we study the existence, uniqueness, and stability theorems of solutions for a differential equation of mixed Caputo-Riemann fractional derivatives with integral initial conditions in a Banach space. Our analysis is based on an application of the Shauder fixed point theorem with Ulam-Hyers and Ulam-Hyers-Rassias theorems. A couple of examples are presented to illustrate the obtained results.

In this paper, we study soft sets of the first and second Baire categories. The soft sets of the first Baire category are examined to be small soft sets from the point of view of soft topology, while the soft sets of the second Baire category are examined to be large. The family of soft sets of the first Baire category in a soft topological space forms a soft $\sigma$-ideal. This contributes to the development of the theory of soft ideal topology. The main properties of these classes of soft sets are discussed. The concepts of soft points where soft sets are of the first or second Baire category are introduced. These types of soft points are subclasses of non-cluster and cluster soft sets. Then, various results on the first and second Baire category soft points are obtained. Among others, the set of all soft points at which a soft set is of the second Baire category is soft regular closed. Moreover, we show that there is symmetry between a soft set that is of the first Baire category and a soft set in which each of its soft points is of the first Baire category. This is equivalent to saying that the union of any collection of soft open sets of the first Baire category is again a soft set of the first Baire category. The last assertion can be regarded as a generalized version of one of the fundamental theorems in topology known as the Banach Category Theorem. Furthermore, it is shown that any soft set can be represented as a disjoint soft union of two soft sets, one of the first Baire category and the other not of the first Baire category at each of its soft points.