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 $c{l}_R^{\diamond }(\cdot )$ via primal which is a new notion. We also show that the operator $c{l}_R^{\diamond }(\cdot )$ 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 $c{l}_R^{\diamond }(\cdot )$ 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.