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.

This paper aimed to introduce novel neutrosophic primal and neutrosophic proximity operators, derived from a new abstract framework called "neutrosophic primal topology". We began by examining the core properties of neutrosophic primal operators. Next, we defined a neutrosophic primal closure operator derived from the neutrosophic primal operator and explored the relationships between them. Based on this neutrosophic primal closure operator, we constructed a neutrosophic topology and identified the conditions under which the image of a neutrosophic primal remained a neutrosophic primal. In the next stage, we defined the neutrosophic point-primal proximity operator and explored a range of fundamental properties characterizing neutrosophic primal proximity topological spaces derived from this operator. We also introduced the concept of neutrosophic proximal closed sets and demonstrated that the collection of complements of neutrosophic primal closed sets constituted a neutrosophic topology. Finally, we defined a neutrosophic operator on a neutrosophic primal proximity topological space that satisfies the neutrosophic Kuratowski closure axioms and used it to construct a neutrosophic topology. All results established in this study were thoroughly supported and clarified through illustrative examples.

Operators serve as fundamental tools in the analysis of topological spaces, prompting extensive research and yielding numerous significant results. In this paper, we introduced a new topological framework called the "diving topological space", which was developed based on the diving structure. Within this framework, several operators were introduced, including one that fulfilled the Kuratowski axioms. We examined the core properties of these operators and explored the interrelations among them. Additionally, two new topologies were formulated and investigated with respect to each other and in comparison to classical topology. The study culminated by introducing concepts of fuzzy diving structures and demonstrating applications of fundamental topological properties, all substantiated with illustrative examples.