requirements for our work, we define here the following concepts sequentially: Ideal space, local function, Kuratowski closure, dense, T*-dense, I-dense, codense, ?-operator, resolvable, I-open, pre-I-open, scattered set and Housdorff space. We start define the ideal space. Let (X, T) be a topological space with no separation properties assumed. The topic of ideal topological space has been considered by (Kuratowski, 1966) and (Vaidyanathaswamy, 1960). An ideal I on a topological space (X, T) is a nonempty collection of subsets of X which satisfies the following two condition:
(1) If A I and B A, then B I (heredity).
(2) If A I and B I, then A B I (finite additivity).
Moreover a ?-ideal on (X, T) is an ideal which settle (1), (2) and the following condition:
3.If {Ai : i = 1,2,3,….} I, then {Ai : i =1,2,3,….} I (countable additivity).
An ideal space is a topological space (X, T) with an ideal I on X and is denoted by (X, T, I).For a subset A X, A*( I)={x X : U A I for every U T (x)} is called the local function of A with respect to I and T (Kuratowski, 1933).We simply write A* instead of A*( I) in case there is no chance for disorder. It is familiar that Cl*(A) = A A* defines a Kuratowski closure operator for a topology T *(I) which is finer than T. During this paper, for a subset A X, Cl (A) and In (A) indicate the closure and the interior of A ,respectively. A subset A of an ideal space (X, T, I) is said to be dense (resp, T *-dense (Dontcher, Gansster and Rose, 1999), I –dense (Dontcher, Gansster and Rose,1999) if Cl (A) = X (resp Cl*(A) = X, A*= X).An ideal I on a space (X, T) is said to be codense (Devid, Sivaraj and Chelvam, 2005) if and only if T I = {?}.For an ideal space (X, T, I) and for any A X, where I is codense. Then: dense, T*-dense and I-dense are comparable (Jankovic and Hamlett, 1990). (Natkanies, 1986) used the idea of ideals to define another operator known as ?-operator elucidates as follow: For a subset A X, ? (A) = X-(X-A)*.Equivalently ?(A) = {M T: M-A I}.It is obvious that ?(A) for any A is a member of T. For an ideal space (X, T, I) and Y X then (Y, T y, I y) is an ideal space where T y = {U Y: U T} and I y = {U Y: U I} = {U I: U Y}. In 1943, Hewitt put forward the opinion of a resolvable space as follows: A nonempty topological space (X, T) is said to be resolvable (Hewitt, 1943) if X is the disjoint union of two dense subsets. Given a space (X, T) and A X, A is called I -open (Jankovic and Hamlett, 1990 )(resp per- I –open(Donkhev,1996)if A In (A*)(resp A In Cl*(A).A set A X is called scattered (Jankovic and Hamlett, 1990) i