The topology of θω-open sets

Abstract

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