The Topology of Theta Omega Open Sets

Abstract

We define θ_{ω}-closure operator as a new topological operator. We show that θ_{ω}-closure of a subset of a topological space is strictly between its usual closure and its θ-closure. Moreover, we give several sufficient conditions for the equivalence between θ_{ω}-closure and usual closure operators, and between θ_{ω}-closure and θ-closure operators. Also, we use θ_{ω}-closure operator to introduce θ_{ω}-open sets as a new class of sets and we prove that this class of sets lies strictly between the class of open sets and the class of θ_{ω}-open sets. We investigate θ_{ω}-open sets, in particular, we obtain a product theorem and several mapping theorems. Moreover, we introduce ω-T₂ as a new separation axioms by utilizing ω-open sets, we prove that the class of ω-T₂ is strictly between the class of T₂ topological spaces and the class of T₁ topological spaces. We study relationship between ω-T₂ and ω-regularity. As main results of this chapter, we give a chaterization of ω-T₂ via θ_{ω}-closure and we give characterizations of ω-regularity via θ_{ω}-closure and via θ_{ω}-open sets.