operator Proximity in i-Topological Space

In 1909, the researcher Riesz[1] presented the idea of proximity spaces in his theory "theory of enchainment", but this idea did not receive attention at that time. Then this idea was presented and developed by the Russian scientist Efremovic [3] and presented by the name of infinitesmma1 spaces in a series of research, and then he generalized the concept of proximity spaces by using the meaning of proximity neighborhood After that, the proximity spaces witnessed a clear development through the many and varied researches included the concept like [4,15,19,20]. As for the concept of ideal and ideal topology, it was introduced by the scientist K. Kuratowski in 1933[2], and this topic has witnessed many different researches that dealt with various aspects of this topic such as [23,13,12]. The itopology, it is another form of topological spaces defined by the use of the family T with the ideal I , as it was defined by Irina Zvina in 2006[6]. The Ψ-operator was defined by T. Natkaniec [5] which is defined as the complement of the local function in ideal topological


INTRODUCTION
In 1909, the researcher Riesz [1] presented the idea of proximity spaces in his theory "theory of enchainment", but this idea did not receive attention at that time. Then this idea was presented and developed by the Russian scientist Efremovic [3] and presented by the name of infinitesmma1 spaces in a series of research, and then he generalized the concept of proximity spaces by using the meaning of proximity neighborhood After that, the proximity spaces witnessed a clear development through the many and varied researches included the concept like [4,15,19,20]. As for the concept of ideal and ideal topology, it was introduced by the scientist K. Kuratowski in 1933[2], and this topic has witnessed many different researches that dealt with various aspects of this topic such as [23,13,12]. The i-topology, it is another form of topological spaces defined by the use of the family T with the ideal I , as it was defined by Irina Zvina in 2006 [6]. The Ψ-operator was defined by T. Natkaniec [5] which is defined as the complement of the local function in ideal topological spaces , where different types and studies wer presented of -operator and enrich this topic in the ideal topological spaces, and we will shed light on some of the researchers who worked in the fuzzy and soft topological spaces and proximity spaces And of them [9,10,17] In this paper, we will study the effects that can have on the -operator in the i-topological space using proximity spaces and using a set of principles that were previously studied

proposition (3-3) :
let is itopological space and a proximity space , all of the Suffix phrase are Verified : 1. .

. Proposition (3-4) :
Let is itopological space and is a proximity space then and the converse is not true.

Proof :
Let so there exist , if possible that so u such that A ,hence A c ,and this is contradiction , Then .

Proposition (3-5) :
Let is itopological space , and is a proximity space . Then for each A of I .

There are several properties of -operator as in the following theorem Theorem (3-6) :
Let is itopological space and is a proximity space then each of the following are holds :

2.
For each

8.
for any subset A of X.

12.
If A is iclose set then .

Proposition (3-7) :
Let is itopological space and is a proximity space if then .

Proof :
Since or B , let and Then and by proposition (3-10) (10) we have so .

conclusion :
1. In this paper the definition of the operator in the topological space was presented and it became clear to us the clear effect of proximity spaces on some characteristics and anthologies related to the operator. 2. This definition can be applied to a group of subjects presented by a group of researchers such as [7,8,11,14,18].