Neutrosophic Soft Bitopological Spaces

: In this paper, we built bitopological space on the concept of neutrosophic soft set, we defined the basic topological concepts of this spaces which are N 3 -(bi) * -open set, N 3 -(bi) * -closed set, (bi) * -neutrosophic soft interior, (bi) * -neutrosophic soft closure , (bi) * -neutrosophic soft boundary , (bi) * -neutrosophic soft exterior and we introduced their properties. In addition, we investigated the relations of these basic topological concepts with their counterparts in neutrosophic soft topological spaces and we introduced many examples.


Introduction-
The concept of soft set is defined by Molodtsov [1] as follows: Let M be an initial universe set and E be a set of parameters. Let P (M) denotes the set of all the subsets-of-M. Consider B≠ ∅‚ B ⊆E. The collection (β‚B) is termed to be the soft set, where β is a mapping by β:B→P (M), and later this concept has been redefined by Naim Cagman [20]. Smarandache [2] introduced neutrosophic set as a generalization of fuzzy set [3] and intuitionistic fuzzy set [4]. P. K. Maji [5] defended the concept of neutrosophic soft set by combining the concept of neutrosophic set and soft set. This the concept is defined as follows: let M be an initial universe set and E be a set of parameters. Let P(M) denote the set of all the neutrosohpic sets of M. Consider B≠ ∅‚ B ⊆E. The collection (β‚B) is termed to be the soft neutrosophic set, where β is a mapping by β:B→P (M). This concept has been modified by [6,7].The concept of neutrosohpic soft topological space was introduced by Bera [8]. Taha et al. [9] redefined the neutrosop-hic soft topological spaces differently from the study [8]. Other theoretical studies on these concepts were presented by a number of researchers, for example, Narmada, Georgiou, Cageman, Al-Nafee, Evanzalin and Salama, (see [10,11,22,13,14,15,16,17,18,19,20]). "Kelly, [21] introduced the concept of bitopological space. This concept is introduced as an extension of topological space. This concept has been introduced with interest in fuzzy set, soft set and neutrosophic set (see [22,23,24,25]). Therefore, we find it important and necessary to build a bitopological spaces on the concept of neutrosophic soft set. In this paper, bitopological space on the concept of neutrosophic soft set is built, the basic topological concepts of this spaces which are N3-(bi) * -open set, N3-(bi) * -closed set, (bi) *neutrosophic soft interior, (bi) * -neutrosophic soft closure, (bi) * -neutrosophic soft boundary, (bi) * -neutrosophic soft exterior are defined, the relations of these basic topological concepts with their counterparts in neutrosophic soft topological spaces are investigated and many examples on this concepts are given.

Preliminary
In this section, we will refer to the basic definitions required in our work.
From philosophical point of view the neutrosophic set takes the value from real standard or non-standard subsets of ] − 0,+1["."But in real life application in scientific and engineering problems it is difficult to use a neutrosophic set with value from real standard or non-standard subset of ] − 0,+1["."Hence we consider the neutrosophic set which takes the value from the subset of [0, 1]".
Firstly, neutrosophic set defined by Maji, [5],and later this concept and its operations have been redefined by [7]. Our work in this research is based on the definition below:

Definintion [7]
Let M be an initial universe set and B be a set of parameters. Let P(M) denote the set of all the neutronsohpic sets of M. Then, a neutrosophic soft set β over M is a set defined by a set valued function β representing a mapping from B to P (M), where e β is called approximate function of the neutrosophic soft set β . In other words, β is a parameterized family of,some elements of the set P(M) and therefore it can be written as a set of ordered pairs,

3."Neutrosophic soft bitopological space"
In this section, we defined the neutrosophic soft bitopological space or (N3-Bi-Top for short) on the concept of neutrosophic soft set and the basic topological concepts of this spaces which are N3-biopen and N3biclosed.  (1) Clearly that ∅ ‚ M ∈ (T1∩T2 ).

Remark
If we take the operation of union instead of the operation of intersection, then the above theorem is not generally correct.

Remark
In above theorem, it is not necessary the converse of (a) and (d) be true.

Theorem
Let (