On a New Operator Based on a Primal and its Associated Topology

Abstract

This paper aims to introduce and study two new operators $(.{)}_{\omega }^{\diamond }$ and $c{l}_{\omega }^{\diamond }(\cdot )$ by utilizing the notion of primal defined by S. Acharjee et al. Also, we investigate some fundamental properties of them. In addition, we showed that the operator $c{l}_{\omega }^{\diamond }(.)$ satisfied the  Kuratowski closure axioms. Therefore, we obtain a new topology denoted by ${\tau }_{\omega }^{\diamond },$ which is finer than the original one. Moreover, the topology  ${\tau }_{\omega }^{\diamond }$ obtained via the operator $c{l}_{\omega }^{\diamond }(\cdot )$ is finer than ${\tau }_{\omega }$, where ${\tau }_{\omega }$ is the family of all $\omega$-open subsets of a primal topological space $(X,\tau ,\mathcal{P}).$ Furthermore, we not only examine the fundamental properties of this class of sets but also provide some counterexamples.