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.

#### Keywords

- neutrosophic soft set
- neutrosophic soft topological spaces
- boundary of neutrosophic soft set
- neutrosophic soft dense set
- neutrosophic soft basis
- neutrosophic soft subspace topology

#### MSC 2010

- 54A40
- 54E55
- 54D10

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. [1,2,3,4,5,6, 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.

_{F̃}_{(e)}(_{F̃}_{(e)}(_{F̃}_{(e)}(_{F̃}_{(e)}(_{F̃}_{(e)}(_{F̃}_{(e)} (

^{c} and is defined by:

^{c}^{c}

_{F̃}_{(e)} (_{G̃}_{(e)} (_{F̃}_{(e)} (_{G̃}_{(e)}(_{F̃}_{(e)} (_{G̃}_{(e)} (

(

_{1},_{2},_{1},_{2},_{3},

_{1},_{2},_{1},_{2},_{3},

_{1},_{2},_{1},_{2},_{1},_{2},_{3},_{3},_{1},_{2},^{c} as follows:

_{F̃}_{(e)} (_{F̃}_{(e)} (_{F̃}_{(e)} (_{(X,E)}.

_{F̃}_{(e)}(_{F̃}_{(e)}(_{F̃}_{(e)} (_{(X,E)}.

_{1},_{2},_{3},

(_{1},_{2},_{3},_{1},_{2},_{3},_{1},_{2},_{3},_{1},_{2},_{3},

(_{1},_{2},_{3},_{1},_{2},_{1},_{3},_{1},_{2},_{3},_{1},_{2},_{1},_{3},

(_{1},_{(X,E)} = (_{1},_{1},_{(X,E)} = 0_{(X,E)};

(_{1},_{(X,E)} = 1_{(X,E)} _{1},_{(X,E)} = (_{1},

_{1},_{2},

[(_{1},_{2},^{c}_{1},^{c}_{2},^{c}

[(_{1},_{2},^{c}_{1},^{c}_{2},^{c}

_{F̃}_{(e)}(_{F̃}_{(e)} (_{F̃}_{(e)}(

It is clear that
^{′}

0_{(X,E)} _{(X,E)}

_{1},_{2},_{3}} _{1},_{2}} _{1},_{2},_{3},_{4},

_{1},_{2},

(_{1},_{1},_{1},

_{1},_{2},_{1},_{2},

_{1},^{c}_{1},

1_{(X,E)} = (_{1},_{1},_{(X,E)}_{1},

_{1},_{1},

(_{1},

(_{1},_{1},_{1},_{1},

It is clear.

It is clear.

It is clear.

Suppose that (_{1},_{1},^{c}

From the condition-1,

That is, (_{1},

(

(

(

(_{(X,E)}

(_{(X,E)}_{(X,E)}

(_{(X,E)} _{(X,E)} ≠ (

Straightforward.

This is easily seen from the definition of neutrosophic soft basis.

1.

⇐ Suppose that the condition of theorem to be provided. Then,

That is,

2. Let
_{i1},_{i2},

⇒ Suppose that

⇐ Conversely, assume that the hypothesis holds. Let

Since 0_{(X̃,E)} ∩ (_{0(F,E)} and 1_{(X,E)} ∩ (_{(F̃,E)} and

_{1},^{′}_{2},^{′}_{3},^{′} and_{4},^{′} over

1. Since

2. We first show that if (

Let (_{i},E^{c}_{i},E^{c}^{c}^{c}

Conversely, suppose that (^{c}

3.
_{i},E_{i},E_{i},E_{i},E_{i},E

_{1},_{1},

Suppose that (_{1},_{1},_{1},

Conversely, assume that (_{1},_{1},_{1},_{1},_{1},

_{1},_{1},

Suppose that (_{1},_{1},_{1},_{1},

Conversely, assume that (_{1},_{1},_{1},_{1},_{1},

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.

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