The class of $R\omega-$open set was defined by S. Murugesan. He show that the collection of all $R\omega-$open forms abase of some topology on $X$ denoted by $\tau_{\delta-\omega}$. The elements of $\tau_{\delta-\omega}$ are called ${\delta_\omega}-$open sets and the complement of a ${\delta_\omega}-$open set is called a ${\delta_\omega}-$closed set. In this paper we will introduce a new characterization of ${\delta_\omega}-$open and ${\delta_\omega}-$closed sets by using ${\delta \omega}-$cluster point. We show that the set of all ${\delta\omega}-$open sets form a topology denoted by $\tau_{\delta_{\omega}}$ and equal to $\tau_{\delta-\omega}$. We discuss several properties of this topology and we give a characterization for the open sets in $\tau_{\delta-\omega}$. We investigate some of relationship between the separation axioms of $(X,\tau_{\delta_{\omega}})$ and $(X,\tau)$. In the last section we study some of connectedness properties of $(X,\tau_{\delta_{\omega}})$ and some covering properties.