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.
A Generalization of Tychonoff's Product Theorem
