The first intervention in the current study is the concept of ⋉ -operator, which the researcher defined using the concept of Φʆ-operator and Γʆ-operator in grill topological spaces, in addition this concept has important properties and it has impact on the tool with which it is defined it. On the other hand, through the definition of the phantom set with its mathematical properties, the necessary and sufficient condition is obtained for the closed of Γʆ-operator under the influence of finite unions.