On the topology $ \tau^{\diamond}_R $ 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 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 provide many counterexamples related to our results.

Keywords

• primal
• primal topological space
• kuratowski closure operator
• the operator $ (\cdot)_r^{\diamond} $
• the operator $ cl^{\diamond}_r $}