δ ω -Continuity and some Results on δ ω -Closure Operator

Abstract

Al-Jarrah et al. defined a new topological operator, namely, ${\delta }_{\omega }$ -closure operator, and proved that it lies between the $\delta$ -closure operator and the usual closure operator. Al-Ghour et al. defined ${\theta }_{\omega }$ -closure operator and discussed its properties. In this paper, it is proved that the ${\delta }_{\omega }$ -closure operator lies between the ${\theta }_{\omega }$ -closure operator and the usual closure operator. Also, sufficient conditions are given for the equivalence between the ${\delta }_{\omega }$ -closure operator and the ${\theta }_{\omega }$ -closure operator. Moreover, we define three new types of continuity, namely, ${\delta }_{\omega }$ -continuity, $\omega$ - $\delta$ -continuity, and almost ${\delta }_{\omega }$ -continuity, and discuss their properties. It is proved that the concepts of usual continuity and ${\delta }_{\omega }$ -continuity are independent of each other. In addition, the relationships between different types of continuity have been investigated. Further, some examples and counter examples are given.