Counterpart of Boundary Topological Operators Within the Framework of Primal Topological Spaces

In this paper, we aim to introduce a new operator via primal topological spaces, called the ⊤-operator, and look into its basic properties. Among the interesting results we prove, for every subset D, we have (1) D⊤⊆FrD, where Fr is the topological boundary operator, and (2) D⊤=∅, whenever either a subset D or its complement lies outside the given primal, which significantly reduces the computational cost of algorithms designed to compute the ⊤-operator. We also demonstrate how the ⊤-operator interacts with some other operators, such as the operator ⋄ and the operator Ψ, in the literature. Although the operator ⊤ does not appear as a boundary operator, we obtain a basis by means of this operator, and so we get a topology, namely, ⊤-topology. By illustrative example, we establish the independence between ⊤-topology and ⋄-topology. Finally, we get a new pseudoclosure operator through the ⊤-operator and determine the relationship between the ⊤-operator and the closure operator when a subset is open or closed.