Closure operators in category of the reflexive relation spaces

This paper investigates the category RRel, whose morphisms are relation-preserving functions and objects are reflexive relation spaces. First of all, it has been shown that this category is a normalized topological. Moreover, the concept of closure in its classical sense was generalized to the topological category and has been adapted to the category RRel. Let (B,R) be the reflexive relational space, and let x∈B. The theorem stating that '{x} is closed if and only if the relation R is antisymmetric at the point x' has been proven. This theorem is associated with the closedness and strong closedness of the subset M⊆B, and has been expressed and proven. Furthermore, our results on closure were compared with those in some well-known topological categories. Moreover, it has been shown that the resulting closure concepts are closure operators in the sense of Dikranjan and Giuli. Then, closure operators such as cl_X, scl_X, q_X and sq_X were investigated, the relationships between them were determined, and it was investigated whether they provided some properties such as weak hereditary, hereditary, productive and idempotent. Subsequently, categories RRel_0c, RRel_1c and RRel_2c, which are subcategories of RRel, were studied, with c being a closure operator. The relationships between these categories and the categories T ̅_0 RRel , T_0'RRel and T_1 RRel were examined. Lastly, our findings were contrasted with those pertaining to closure operators in established topological categories.