Special Properties of Operators in Neutrosophic Crisp Topological Spaces

Neutrosophic crisp sets are an important and recent topic that has entered both pure and applied mathematics, especially general topology. A neutrosophic crisp topological space is defined as a generalization of classical topology in a broad sense, without specifying the type of neutrosophic crisp family or the algebraic operations of union and intersection, in addition to the kind of complement and the kinds of inclusion. In this research on the neutrosophic crisp topological space, we constructed such a space in which the intersection is fixed as Type I, the union as Type II, and the complement as Type II for all kinds of neutrosophic crisp families, and we considered the two kinds of inclusion. Within this framework, we defined two kinds of closure for a neutrosophic crisp set and two kinds of interior for a neutrosophic crisp set. We studied all relations, results, and theorems from general topology on them, as well as the properties related to them and the relationship between the closure and the interior of a neutrosophic crisp set. We proved all relations, results, and theorems that hold and gave examples of those that do not; however, many properties failed to hold. We also defined continuity in the constructed neutrosophic crisp topological space and proved all corresponding relations, results, and theorems in topology that hold, while providing examples of those that do not hold.