More on D α -Closed Sets in Topological Spaces

Abstract

The aim of this paper is to present and study topological properties of $\mathcal{D}\alpha$ -derived, $\mathcal{D}\alpha$ -border, $\mathcal{D}\alpha$ -frontier, and $\mathcal{D}\alpha$ -exterior of a set based on the concept of $\mathcal{D}\alpha$ -open sets. Then, we introduce new separation axioms (i.e., $\mathcal{D}\alpha -R_0$ and $\mathcal{D}\alpha -R_1$ ) by using the notions of $\mathcal{D}\alpha$ -open set and $\mathcal{D}\alpha$ -closure. The space of $\mathcal{D}\alpha -R_0$ (resp., $\mathcal{D}\alpha -R_1$ ) is strictly between the spaces of $\alpha -R_0$ (resp., $\alpha -R_1$ ) and $g-R_0$ (resp., $g-R_1$ ). Further, we present the notions of $\mathcal{D}\alpha$ -kernel and $\mathcal{D}\alpha$ -convergent to a point and discuss the characterizations of interesting properties between $\mathcal{D}\alpha$ -closure and $\mathcal{D}\alpha$ -kernel. Finally, several properties of weakly $\mathcal{D}\alpha -R_0$ space are investigated.