Through this paper, we shall obtain common properties of local function and set operator  and introduce the approximations of local function and set operator . We also determined expansion of local function and set operator .

The study of two operators local function and the set operator on the ideal topological spaces are likely to be the same as the study of closure and interior operator of the topological spaces. However, they are not exactly equal to the interior and closure operator of the topological spaces. In this context, we introduce two new set operators on the ideal topological spaces. Detailed properties of these two operators are the part of this article. Furthermore, the operators interior (resp. ) and closure (local function) obey the relation Int(A) = X n Cl(X n A) (resp. (A) = X n (X n A) ). We search the general method of these relations, through this manuscript.

Through this paper we consider three operators in terms of operators $*$ and $\psi$ in an ideal topological space. Many properties of these operators have been discussed. Characterizations of Hayashi-Samuel spaces are obtained as applications of the properties.

In this paper, a collection of sets are investigated in such a way that the collection splits in the collection of preopen sets and the collection of b-open sets and also splits in the collection of semi-open sets and the collection of b-open sets. Functions in terms of this sets is a part of this paper and it will be considered that these collection remains invariant under homeomorphic image.