Through this chapter, we shall discuss the study of the joint operations of \(*\) and \(\varPsi \) of the ideal topological space \((X, \tau , \mathscr {I})\) with the help of the operators \(\varPsi ^*\) and \(*^{\varPsi }\) , where \( \begin{array}{c} \varPsi ^*(A)=(\varPsi (A))^*=\{ x\in X:\;U_x \cap \varPsi (A)\notin \mathscr {I}\}; \text {and}\\ *^{\varPsi }(A) = \varPsi (A^*) = \varPsi (\{x\in X:\; U_x \cap A \notin \mathscr {I} \}),\text { where }U_x\in \tau (x),\text { for }A \subseteq X. \end{array} \) From the operator \(\varPsi ^*\) , if \(x\in \varPsi ^*(A)\) , then no open set containing x belongs to \(\mathscr {I}\) . Using this fact, we characterize the Hayashi-Samuel spaces and obtain the value of joint operations of \(*\) (resp. \(\varPsi \) ) and \(\varPsi \) (resp. \(*\) ) on the continuous function, completely continuous function etc. Further, we determine the change of various types of generalized open sets under these operators. In another section, we also discuss the properties of the above two set operators in the Normed Linear Spaces which is a part of Functional Analysis.