Acharjee et al. have created a new structure in mathematics called a primal. Therefore, the primary goal of this research was to introduce and explore more primal space features. Additionally, we studied some of the fundamental characteristics of two novel operators that we define using primal spaces. Using these new operators, we were able to create a weaker version of the original topology. Finally, we provide some examples to further illustrate our discussion of some of their characteristics.

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.

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 $\sigma$-algebra. In this work, we mainly focus on the soft $\sigma$-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 $\sigma$-algebra of soft sets with the Baire property includes the soft Borel $\sigma$-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.

Soft continuity can contribute to the development of digital images and computational topological applications other than the field of soft topology. In this work, we study a new class of generalized soft continuous functions defined on the class of soft open sets modulo soft sets of the first category, which is called soft functions with the Baire property. This class includes all soft continuous functions. More precisely, it contains various classes of weak soft continuous functions. The essential properties and operations of the soft functions with the Baire property are established. It is shown that a soft continuous with values in a soft second countable space is identical to a soft function with the Baire property, apart from a topologically negligible soft set. Then we introduce two more subclasses of soft functions with the Baire property and examine their basic properties. Furthermore, we characterize these subclasses in terms of soft continuous functions. At last, we present a diagram that shows the relationships between the classes of soft functions defined in this work and those that exist in the literature.

Our aim in this paper is to define more concepts that are related to primal topological space. We introduce new operators called $ \gamma $-diamond and $ \gamma^\ast $-diamond and explore their main characterizations. We provide results and examples regarding to these operators. Using these new operators, we create a weaker version of the original topology. Additionally, we present some results related to compatibility.

In this paper we continue presenting new types of soft operators for supra soft topological spaces (or SSTSs). Specifically, we investigate more interesting properties and relationships between the supra soft somewhere dense interior (or SS-sd-interior) operator, the SS-sd-closure operator, the SS-sd-cluster operator, and the SS-sd-boundary operator. We prove that the SS-sd-interior operator, SS-sd-boundary operator, and SS-sd-exterior operator form a partition for the absolute soft set. Furthermore, we apply the notion of SS-sd-sets to soft continuity. In addition, we use the SS-sd-interior operator and the SS-sd-closure operator to provide equivalent conditions and many characterizations for SS-sd-continuous, SS-sd-irresolute, SS-sd-open, SS-sd-closed, and SS-sd-homeomorphism maps. Examples include the following: The soft mapping is an SS-sd-homeomorphism if, and only if it is both SS-sd-continuous and an SS-sd-closed if, and only if, the soft mapping in addition to its inverse is SS-sd-continuous. Moreover, a bijective soft mapping is SS-sd-open if, and only if, it is SS-sd-closed. Furthermore, we provide many examples and counterexamples to show our results, which are extensions of previous studies. A diagram summarizing our results is also introduced.