Applied and Nonlinear Some Structures on Neutrosophic Topological Spaces

In this paper, we deﬁne boundary of neutrosophic soft set, neutrosophic soft dense set, neutrosophic soft basis and neutrosophic soft subspace topology on neutrosophic soft topological spaces. Furthermore, some important theorems are proved and interesting examples are given.


Introduction
The theory of fuzzy set was introduced by Zadeh in 1965 [19]. Fuzzy sets have been applied in many real life problems to handle uncertainty. After Zadeh, Smarandache introduced the theory of neutrosophic set [17]. This theory is the generalization of many theories such as; fuzzy set [19], intuitionistic fuzzy set [7]. In recent years, there have been many academic studies on the theory of neutrosophic set [3,4,8,9,13,14,16],. Many classical methods were not enough to solve problems related to uncertainties. Therefore Molodtsov introduced the soft set theory in 1999 [12]. The soft set theory is completely a new approach for dealing with uncertainties and vagueness. After Molodtsov, many different studies have been done on soft set theory. Also, many authors studied on different combination of fuzzy set, soft set, intuitionistic set, neutrosophic set, etc. [1-6, 8, 11, 15, 16, 18]. One of these combinations, neutrosophic soft set theory was first introduced by Maji [10]. Later, this theory was modified by Deli and Broumi [8]. Also, Bera presented neutrosophic soft topological spaces [4]. Recently, researchers have shown great interest in this theory. Operations on the neutrosophic soft set theory were re-defined as different from [4,8] by Ozturk T. Y. et. al [13]. They also studied some seperation axioms on neutrosophic soft topological spaces [9].
In this paper, considering these newly defined operations, unlike [13], boundary of neutrosophic soft set, neutrosophic soft basis, neutrosophic soft dense set, neutrosophic soft subspaces on neutrosophic soft topological spaces are defined. In addition, some important theorems together with proofs are given and study is supported by many different examples.
2 Preliminary Definition 1. [8] Let X be an initial universe set and E be a set of parameters. Let P(X) denote the set of all neutrosophic sets of X. Then, a neutrosophic soft set F, E over X is a set defined by a set valued function F representing a mapping F : E → P(X) where F is called approximate function of the neutrosophic soft set F, E . In other words, the neutrosophic soft set is a parameterized family of some elements of the set P(X) and therefore it can be written as a set of ordered pairs, 1], respectively called the truth-membership, indeterminacymembership, falsity-membership function of F(e). Since supremum of each T, I, F is 1 so the inequality [4] Let F, E be neutrosophic soft set over the universe set X. The complement of F, E is denoted by F, E c and is defined by: Obvious that, F, E c c = F, E .

Definition 3.
[10] Let F, E and G, E be two neutrosophic soft sets over the universe set X. F, E is said F, E is said to be neutrosophic soft equal to G, E if F, E is neutrosophic soft subset of G, E and G, E is neutrosophic soft subset of F, E . It is denoted by F, E = G, E .

Definition 4.
[13] Let F 1 , E and F 2 , E be two neutrosophic soft sets over the universe set X. Then their union is denoted by F 1 , E ∪ F 2 , E = F 3 , E and is defined by:

Definition 5.
[13] Let F 1 , E and F 2 , E be two neutrosophic soft sets over the universe set X. Then their intersection is denoted by F 1 , E ∩ F 2 , E = F 3 , E and is defined by:

Definition 6.
[13] Let F 1 , E and F 2 , E be two neutrosophic soft sets over the universe set X. Then

2.
A neutrosophic soft set F, E over the universe set X is said to be absolute neutrosophic soft set if T F(e) (x) = 1, I F(e) (x) = 1, F F(e) (x) = 0; ∀e ∈ E, ∀x ∈ X. It is denoted by 1 (X,E) .
[9] Let F, E be a neutrosophic soft set over the universe set X. We say that x e (α,β ,γ) ∈ F, E read as belongs to the neutrosophic soft set F, E , whenever α ≤ F F(e) (x), β ≤ I F(e) (x) and γ ≥ T F(e) (x).
It is clear that x e (α,β ,γ) and y e (α 1 ,β 1 ,γ 1 ) are distinct neutrosophic soft points if and only if x = y or e = e.  Clearly, F, E is the smallest neutrosophic soft closed set that containing F, E .

Some Structures on Neutrosophic Soft Topological Spaces
be a neutrosophic soft topology over X. Here, the neutrosophic soft sets ( F 1 , E), ( F 2 , E), ( F 3 , E) and ( F 4 , E) over X are defined as following; Suppose that the neutrosophic soft set ( F, E) over X is defined as; Then, let us find the boundary of the neutrosophic soft set ( F, E) : Theorem 1. Let (X, NSS τ , E) be a neutrosophic soft topological space over X and ( F 1 , E), ( F 2 , E) ∈ NSS(X, E). Then, From the condition-1, That is, ( F 1 , E) is a neutrosophic soft open set.   Proof. This is easily seen from the definition of neutrosophic soft basis.
and each ⇐ Suppose that the condition of theorem to be provided. Then, That is, NSS B is a neutrosophic soft basis for NSS τ .

Let
is a neutrosophic soft basis for neutrosophic soft topology NSS τ 1 over X, then Theorem 6. Let (X, NSS τ , E) be a neutrosophic soft topological space over X and ( F, E) ∈ NSS(X; E). Then the collection is a neutrosophic soft topology on ( F, E) and X ( F,E) , is a neutrosophic soft sub-topology on ( F, E) of the neutrosophic soft topology NSS τ . Here, the neutrosophic soft sets ( F 1 , E) , ( F 2 , E) , ( F 3 , E) and ( F 4 , E) over ( F, E) are defined as following:   E). Therefore, . Thus, ( G, E)∩ ( F, E) is a neutrosophic soft closure of ( G, E) in

Conclusion
In this study, we investigate some notions of neutrosophic soft topological space such as; boundary of neutrosophic soft set, neutrosophic soft dense set, neutrosophic soft basis and neutrosophic soft subspace topology. Furthermore we give some important theorems and many interesting examples. We hope that results of this paper will contribute to the studies on neutrosophic soft topological spaces.