On the Topology $\tau_R^{\diamond}$ of Primal Topological Spaces

Abstract

The main purpose of this paper is to introduce and study two new operators $(\cdot)_R^{\diamond}$ and $cl_R^{\diamond}(\cdot)$ via primal which is a new notion. We also show that the operator $cl_R^{\diamond}(\cdot)$ is a Kuratowski closure operator, while the operator $(\cdot)_R^{\diamond}$ is not. In addition, we prove that the topology on $X$, shown as $\tau_R^{\diamond},$ obtained by means of the operator $cl_R^{\diamond}(\cdot)$ is finer than $\tau_{\delta},$ where $\tau_{\delta}$ is the family of $\delta$-open subsets of a space $(X,\tau).$ Moreover, we not only obtain a base for the topology $\tau_R^{\diamond}$ but also prove many fundamental results concerning this new structure. Furthermore, we give many counterexamples related to our results.