A study on Soft Pre-Open Sets using γ Operation

The concept of strong soft γ pre-open set was initiated by Biswas and Parsanann.We utilize this notion to study several characterizations and properties of this set. We investigate the relationships between this set and other types of soft open sets. Moreover, the properties of the strong soft γ pre-interior and closure are discussed. Furthermore, we define a new concept by using strong soft γ pre-closed that we denote as locally strong soft γ pre-closed, in which several results are obtained. We establish a new type of soft pre-open set, namely soft γ pre-open. Also, we continue to study preγ soft open set and discuss the relationships among all these sets. Some counter examples are given to show some relationships obtained in this work.


Introduction
Moldstov [1] investigated the soft set theory, as a new approach for uncertainties, and the vague set theory. Also, he presented many uses of soft sets in some directions, such as game theory, Perron integration, and probability.

ISSN: 0067-2904
Jamil Iraqi Journal of Science, 2021, Vol. 62, No. 4, pp: 1276-1283 2366 Kasahara [2] initiated the notion of operation on topological space and explored several important properties. Later, Maji and others [3] studied several operations on soft sets in more details and proved many propositions about these operations. Jamil [4] established new open sets by using the idea of operation and provided several characterizations Moreover, Ogata [5] utilized the idea of operation to introduce new types of open sets, called open and pre-open sets. Also, he mentioned several properties of each set. Recently, Biswas and Prasannan [6] introduced and discussed various types of soft open sets by using the notions and . In this paper, we continue studying such many forms of soft pre-open sets involving operation. Finally, Al-shami [7][8][9][10][11] studied a new type of soft open sets and investigated the properties of its separation axioms.

Preliminaries and basic results
Definition 2.1 [1]. Let be the set of the universe and be a set of parameters, then the pair ( ) is named a soft set over space such that is mapping from to the family of all subsets of which is denoted by ( ). Definition 2.2 [3]. Let [3]. A soft set ( ) of is named a null soft set, which is represented by ̃ if for any , ( ) . Definition 2.5 [3]. A soft set ( ) of is named an absolute soft set that is represented by ̃ if for any , ( ) . Definition 2.6 [1]. Let ( ) and ( ) be two soft sets over , then [10]. Relative complement for any soft set ( ) is defined by ( ) ( ), such that ( ) is a function given by ( ) ( ) for each Definition 2.8 [12]. The family of soft sets is in the universe set , and is a set of parameters, then is called soft topology on if the following conditions hold 1) ̃, ̃ belong to 2) If *( ) ) being a soft topological space. The family of all soft open sets is stated by ( ) Definition 2.9 [13]. The soft closure of a soft subset ( ) over a space ( ) is the intersection of each soft closed supersets of ( ) which is stated by ( ) Definition 2.10 [13]. The soft interior of a soft set of ( ) in space ( ) is the union of all soft open subsets of ( ) which is stated by ( ) Definition 2.11 [2]. A soft space ( ). An operation is a function from soft topology into ( ) that is ( ) ̃( ) for all ( ) , in which ( ) represents the value of at ( ). For simplicity, we use the notation ̃ to indicate the soft space ( ) in which is an operation defined on . Definition 2.12 [6]

Some properties of strong soft pre open set
The family of all strong soft pre-open sets over ̃ is stated by    Proof:

4) ( ) is strong soft preclosed if and only if
Proof: we are going to prove only (7). Proof: since ( ) is locally strong soft pre-closed, then there exists ( ). Note that ( ) ( ̃ ) and since ( ) is strong soft pre-closed, so ( ) is locally strong soft pre-closed.

5-Conclusions
Soft sets were initiated by Molodtstove in 1999 and, since then, many researchers defined and investigated several types of soft sets. Some of these studies have real applications such as solving problems for medical diagnosis, determining educational obstacles, and taking right decisions about them. In this paper, we defined and studied new soft sets and named them soft pre-open sets. Also, we provided several properties and characterizations about pre soft open and strong soft pre-open sets. Also, the relationships among these sets were discussed.