Theta Omega Topological Operators and Some Product Theorems

Abstract

We introduce and investigate the concepts of ${\theta }_{\omega }$ -limit points and ${\theta }_{\omega }$ -interior points, and we use them to introduce two new topological operators. For a subset $B$ of a topological space $\left(Y,\sigma \right)$ , denote the set of all limit points of $B$ (resp. $\theta$ -limit points of $B$ , ${\theta }_{\omega }$ -limit points of $B$ , interior points of $B$ , $\theta$ -interior points of $B$ , and ${\theta }_{\omega }$ -interior points of $B$ ) by $D\left(B\right)$ (resp. $D_{\theta }\left(B\right)$ , $D_{{\theta }_{\omega }}\left(B\right)$ , $\text{Int}\left(B\right)$ , ${\text{Int}}_{\theta }\left(B\right)$ , and ${\text{Int}}_{{\theta }_{\omega }}\left(B\right)$ ). Several results regarding the two new topological operators are given. In particular, we show that $D_{{\theta }_{\omega }}\left(B\right)$ lies strictly between $D\left(B\right)$ and $D_{\theta }\left(B\right)$ and ${\text{Int}}_{{\theta }_{\omega }}\left(B\right)$ lies strictly between ${\text{Int}}_{\theta }\left(B\right)$ and $\text{Int}\left(B\right)$ . We show that $D\left(B\right)=D_{{\theta }_{\omega }}\left(B\right)$ (resp. ${\text{Cl}}_{\theta }\left(B\right)={\text{Cl}}_{{\theta }_{\omega }}\left(B\right)$ and $D\left(B\right)=D_{{\theta }_{\omega }}\left(B\right)=D_{\theta }\left(B\right)$ ) for locally countable topological spaces (resp. antilocally countable topological spaces and regular topological spaces). In addition to these, we introduce several product theorems concerning metacompactness.