It is commonly known that some topological spaces include structures that may be used to expand abstract notions. Primal structure is one such sort of structure. We provided the primal Hausdorff class of spaces, which included the class of all Hausdorff spaces. Furthermore, we provide the concepts of primal regular spaces and primal normal spaces. We present new theorems and results.

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 introduce new concepts, including $ \mathfrak{P} $-compactness, strongly $ \mathfrak{P} $-compactness, and super $ \mathfrak{P} $-compactness, in view of a primal topological space structure. We provide some results regarding to these concepts. Additionally, some examples are presented to illustrate the relations between these concepts.

The paper develops a novel analysis of mutual interactions between topology and soft topology. It is known that each soft topology produces a system of crisp (parameterized) topologies. The other way round is also possible. Namely, one can generate a soft topology from a system of crisp topologies. Different methods of producing soft topologies are discussed by implementing two formulas. Then, the relationships between the resulting soft topologies are obtained. With the help of an example, it is demonstrated that one formula is more constructible than the other. Now, it is reasonable to ask which (topological) properties of a soft topology can be transferred to the set of crisp topologies, or the opposite. To address this question, we consider the standard separation axioms and show how well these axioms can be preserved when moving from a system of crisp topologies to the soft topology generated by it and contrariwise. Additionally, our findings extend and disprove some results from the literature.

In this paper, we Introduce new operators depending on the definitions of $\gamma$ and $\gamma^\ast$-operatorsthat were defined in \cite{amO}. We provide some examples to illsturate the relation between these new operators.Additionally, we give more results regarding to $\gamma$-diamond operator.

We present new results regarding S2-paracompactness, that we estab-lished in [1], and its relation with other properties such as S-normality, epinormality and L-paracompactness.