We define  -closure operator as a new topological operator which lies between the -closure and the -closure. Some relationships between this new operator and each of -closure, -closure, and usual closure are obtained. Via  -closure operator, we introduce  -open sets as a new topology. Some mapping theorems related to the new topology are given. 2 topological spaces are characterized in terms of  -closure operator. Also, we use  -open sets to define  -regularity as a new separation axiom which lies strictly between -regularity and regularity. For a given topological space ( , ), we show that  -regularity is equivalent to the condition =  . Finally,  -continuity,  --continuity, weak  -continuity, and faint  -continuity are introduced and studied. Definition 2.1.[1] Let be a subset of a ts ( , ).a. The -closure of is denoted by ( ) and defined by