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.

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.

In this paper, we provided a new topological structure called co-infra topological space. We provided new results regarding co-infra open and closed sets. Moreover, we investigated a new characterization of Infra-Lindelöf spaces.

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.