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

Abstract

This paper aims to introduce and study two new operators $(.)^{\diamond}_{\omega}$ and $cl^{\diamond}_{\omega}(\cdot)$ 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 $cl^{\diamond}_{\omega}(.)$ satisfied the  Kuratowski closure axioms. Therefore, we obtain a new topology denoted by $\tau^{\diamond}_{\omega},$ which is finer than the original one. Moreover, the topology  $\tau^{\diamond}_{\omega}$ obtained via the operator $cl^{\diamond}_{\omega}(\cdot)$ 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.