A new class between theta open sets and theta omega open sets

Abstract

We define ${\theta }_N$ -closure operator as a new topological operator which lies between the θ-closure and the ${\theta }_{\omega }$ -closure. Some relationships between this new operator and each of θ-closure, ${\theta }_{\omega }$ -closure, and usual closure are obtained. Via ${\theta }_N$ -closure operator, we introduce ${\theta }_N$ -open sets as a new topology. Some mapping theorems related to the new topology are given. $T_2$ topological spaces are characterized in terms of ${\theta }_N$ -closure operator. Also, we use $N$ -open sets to define $N$ -regularity as a new separation axiom which lies strictly between ω-regularity and regularity. For a given topological space $(Y,\sigma )$ , we show that $N$ -regularity is equivalent to the condition $\sigma ={\sigma }_{{\theta }_N}$ . Finally, ${\theta }_N$ -continuity, $N$ -θ-continuity, weak ${\theta }_N$ -continuity, and faint ${\theta }_N$ -continuity are introduced and studied.

Keywords: θ ω -continuity; θ ω -open sets; N -open sets; θ-continuity; θ-open sets; ω-open sets.