1. bookVolume 6 (2021): Issue 1 (January 2021)
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Some Structures on Neutrosophic Topological Spaces

Published Online: 31 Dec 2020
Page range: 467 - 478
Received: 19 Jul 2019
Accepted: 03 Oct 2019
Journal Details
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Format
Journal
First Published
01 Jan 2016
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2 times per year
Languages
English
Abstract

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

MSC 2010

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

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 (F̃,E) over X is a set defined by a set valued function F̃ representing a mapping F̃ : EP(X) where F̃ is called approximate function of the neutrosophic soft set (F̃,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, (F˜,E)={(e,x,TF˜(e)(x),IF˜(e)(x),FF˜(e)(x):xX):eE} \left( {\widetilde F,E} \right) = \left\{ {\left( {e,\left\langle {x,{T_{\widetilde F(e)}}(x),{I_{\widetilde F(e)}}(x),{F_{\widetilde F(e)}}(x)} \right\rangle :x \in X} \right):e \in E} \right\} where T(e)(x), I(e)(x), F(e)(x) ∈ [0,1], respectively called the truth-membership, indeterminacy-membership, falsity-membership function of F̃(e). Since supremum of each T, I, F is 1 so the inequality 0 ≤ T(e)(x) + I(e)(x) + F(e) (x) ≤ 3 is obvious.

Definition 2

[4] Let (F̃,E) be neutrosophic soft set over the universe set X. The complement of (F̃,E) is denoted by (F̃,E)c and is defined by: (F˜,E)c={(e,x,FF˜(e)(x),1IF˜(e)(x),TF˜(e)(x):xX):eE}. {\left( {\widetilde F,E} \right)^c} = \left\{ {\left( {e,\left\langle {x,{F_{\widetilde F(e)}}(x),1 - {I_{\widetilde F(e)}}(x),{T_{\widetilde F(e)}}(x)} \right\rangle :x \in X} \right):e \in E} \right\}.

Obvious that, ((F̃,E)c)c = (F̃,E).

Definition 3

[10] Let (F̃,E) and (G̃,E) be two neutrosophic soft sets over the universe set X. (F̃,E) is said to be neutrosophic soft subset of (G̃,E) if T(e) (x) ≤ T(e) (x), I(e) (x) ≤ I(e)(x), F(e) (x) ≥ F(e) (x), ∀eE,xX. It is denoted by (F̃,E) ⊆ (G̃,E).

(F̃,E)is said to be neutrosophic soft equal to (G̃,E) if (F̃,E) is neutrosophic soft subset of (G̃,E) and (G̃,E) is neutrosophic soft subset of (F̃,E). It is denoted by (F̃,E) = (G̃,E).

Definition 4

[13] Let (1,E) and (2,E) be two neutrosophic soft sets over the universe set X. Then their union is denoted by (1,E) ∪ (2,E) = (3,E) and is defined by: (F˜3,E)={(e,x,TF˜3(e)(x),IF˜3(e)(x),FF˜3(e)(x):xX):eE} \left( {{{\widetilde F}_3},E} \right) = \left\{ {\left( {e,\left\langle {x,{T_{{{\widetilde F}_3}(e)}}(x),{I_{{{\widetilde F}_3}(e)}}(x),{F_{{{\widetilde F}_3}(e)}}(x)} \right\rangle :x \in X} \right):e \in E} \right\} where TF˜3(e)(x)=max{TF˜1(e)(x),TF˜2(e)(x)},IF˜3(e)(x)=max{IF˜1(e)(x),IF˜2(e)(x)},FF˜3(e)(x)=min{FF˜1(e)(x),FF˜2(e)(x)}. \matrix{ {{T_{{{\widetilde F}_3}(e)}}(x) = \max \left\{ {{T_{{{\widetilde F}_1}(e)}}(x),{T_{{{\widetilde F}_2}(e)}}(x)} \right\},} \hfill \cr {{I_{{{\widetilde F}_3}(e)}}(x) = \max \left\{ {{I_{{{\widetilde F}_1}(e)}}(x),{I_{{{\widetilde F}_2}(e)}}(x)} \right\},} \hfill \cr {{F_{{{\widetilde F}_3}(e)}}(x) = \min \left\{ {{F_{{{\widetilde F}_1}(e)}}(x),{F_{{{\widetilde F}_2}(e)}}(x)} \right\}.} \hfill \cr }

Definition 5

[13] Let (1,E) and (2,E) be two neutrosophic soft sets over the universe set X. Then their intersection is denoted by (1,E) ∩ (2,E) = (3,E) and is defined by: (F˜3,E)={(e,x,TF˜3(e)(x),IF˜3(e)(x),FF˜3(e)(x):xX):eE} \left( {{{\widetilde F}_3},E} \right) = \left\{ {\left( {e,\left\langle {x,{T_{{{\widetilde F}_3}(e)}}(x),{I_{{{\widetilde F}_3}(e)}}(x),{F_{{{\widetilde F}_3}(e)}}(x)} \right\rangle :x \in X} \right):e \in E} \right\} where TF˜3(e)(x)=min{TF˜1(e)(x),TF˜2(e)(x)},IF˜3(e)(x)=min{IF˜1(e)(x),IF˜2(e)(x)},FF˜3(e)(x)=max{FF˜1(e)(x),FF˜2(e)(x)}. \matrix{ {{T_{{{\widetilde F}_3}(e)}}(x) = \min \left\{ {{T_{{{\widetilde F}_1}(e)}}(x),{T_{{{\widetilde F}_2}(e)}}(x)} \right\},} \hfill \cr {{I_{{{\widetilde F}_3}(e)}}(x) = \min \left\{ {{I_{{{\widetilde F}_1}(e)}}(x),{I_{{{\widetilde F}_2}(e)}}(x)} \right\},} \hfill \cr {{F_{{{\widetilde F}_3}(e)}}(x) = \max \left\{ {{F_{{{\widetilde F}_1}(e)}}(x),{F_{{{\widetilde F}_2}(e)}}(x)} \right\}.} \hfill \cr }

Definition 6

[13] Let (1,E) and (2,E) be two neutrosophic soft sets over the universe set X. Then “(1,E) difference (2,E)” operation on them is denoted by (1,E) \ (2,E) = (3,E) and is defined by (3,E) = (1,E) ∩ (2,E)c as follows: (F˜3,E)={(e,x,TF˜3(e)(x),IF˜3(e)(x),FF˜3(e)(x):xX):eE} \left( {{{\widetilde F}_3},E} \right) = \left\{ {\left( {e,\left\langle {x,{T_{{{\widetilde F}_3}(e)}}(x),{I_{{{\widetilde F}_3}(e)}}(x),{F_{{{\widetilde F}_3}(e)}}(x)} \right\rangle :x \in X} \right):e \in E} \right\} where TF˜3(e)(x)=min{TF˜1(e)(x),FF˜2(e)(x)},IF˜3(e)(x)=min{IF˜1(e)(x),1IF˜2(e)(x)},FF˜3(e)(x)=max{FF˜1(e)(x),TF˜2(e)(x)}. \matrix{ {{T_{{{\widetilde F}_3}(e)}}(x) = \min \left\{ {{T_{{{\widetilde F}_1}(e)}}(x),{F_{{{\widetilde F}_2}(e)}}(x)} \right\},} \hfill \cr {{I_{{{\widetilde F}_3}(e)}}(x) = \min \left\{ {{I_{{{\widetilde F}_1}(e)}}(x),1 - {I_{{{\widetilde F}_2}(e)}}(x)} \right\},} \hfill \cr {{F_{{{\widetilde F}_3}(e)}}(x) = \max \left\{ {{F_{{{\widetilde F}_1}(e)}}(x),{T_{{{\widetilde F}_2}(e)}}(x)} \right\}.} \hfill \cr }

Definition 7

[13] 1. A neutrosophic soft set (F̃,E) over the universe set X is said to be null neutrosophic soft set if T(e) (x) = 0, I(e) (x) = 0, F(e) (x) = 1; ∀eE,xX. It is denoted by 0(X,E).

2. A neutrosophic soft set (F̃,E) over the universe set X is said to be absolute neutrosophic soft set if T(e)(x) = 1, I(e)(x) = 1, F(e) (x) = 0; ∀eE,xX. It is denoted by 1(X,E).

Clearly, 0(X,E)c=1(X,E) 0_{(X,E)}^c{ = 1_{(X,E)}} and 1(X,E)c=0(X,E) 1_{(X,E)}^c{ = 0_{(X,E)}} .

Proposition 1

[13] Let (1,E), (2,E) and (3,E) be neutrosophic soft sets over the universe set X. Then,

(1,E) ∪ [(2,E) ∪ (3,E)] = [(1,E) ∪ (2,E)] ∪ (3,E) and (1,E) ∩ [(2,E) ∩ (3,E)] = [(1,E) ∩ (2,E)] ∩ (3,E);

(1,E) ∪ [(2,E) ∩ (3,E)] = [(1,E) ∪ (2,E)] ∩ [(1,E) ∪ (3,E)] and (1,E) ∩ [(2,E) ∪ (3,E)] = [(1,E) ∩ (2,E)] ∪ [(1,E) ∩ (3,E)];

(1,E) ∪ 0(X,E) = (1,E) and (1,E) ∩ 0(X,E) = 0(X,E);

(1,E) ∪ 1(X,E) = 1(X,E) and (1,E) ∩ 1(X,E) = (1,E).

Proposition 2

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

[(1,E) ∪ (2,E)]c = (1,E)c ∩ (2,E)c;

[(1,E) ∩ (2,E)]c = (1,E)c ∪ (2,E)c.

Definition 8

[9] Let NSS(X,E) be the family of all neutrosophic soft sets over the universe set X. Then neutrosophic soft set x(α,β,γ)e x_{\left( {\alpha ,\beta ,\gamma } \right)}^e is called a neutrosophic soft point, for every xX,0 < α,β, γ ≤ 1, eE, and defined as follows: x(α,β,γ)e(e)(y)={(α,β,γ)ife=eandy=x,(0,0,1)ifeeoryx. x_{\left( {\alpha ,\beta ,\gamma } \right)}^e\left( {{e^\prime}} \right)\left( y \right) = \left\{ {\matrix{ {\left( {\alpha ,\beta ,\gamma } \right)\;{\textit{if}}\;{e^\prime} = \textit{e and y} = x,} \cr {\left( {0,0,1} \right)\,{\textit{if}}\;{e^\prime} \ne {\textit e\; or\; y} \ne x.} \cr } } \right.

Definition 9

[9] Let (F̃,E) be a neutrosophic soft set over the universe set X. We say that x(α,β,γ)e(F˜,E) x_{\left( {\alpha ,\beta ,\gamma } \right)}^e \in \left( {\widetilde F,E} \right) read as belongs to the neutrosophic soft set (F̃,E), whenever αF(e)(x),βI(e) (x) and γT(e)(x).

Definition 10

[9] Let (X,τNSS,E) \left( {X,\mathop \tau \limits^{NSS} ,E} \right) be a neutrosophic soft topological space over X. A neutrosophic soft set (F̃,E) in (X,τNSS,E) \left( {X,\mathop \tau \limits^{NSS} ,E} \right) is called a neutrosophic soft neighborhood of the neutrosophic soft point x(α,β,γ)e(F˜,E) x_{\left( {\alpha ,\beta ,\gamma } \right)}^e \in \left( {\widetilde F,E} \right) , if there exists a neutrosophic soft open set (G̃,E) such that x(α,β,γ)e(G˜,E)(F˜,E) x_{\left( {\alpha ,\beta ,\gamma } \right)}^e \in \left( {\widetilde G,E} \right) \subset \left( {\widetilde F,E} \right) .

Definition 11

[9] Let x(α,β,γ)e x_{\left( {\alpha ,\beta ,\gamma } \right)}^e and y(α,β,γ)e y_{\left( {{\alpha ^\prime},{\beta ^\prime},{\gamma ^\prime}} \right)}^{{e^\prime}} be two neutrosophic soft points. For the neutrosophic soft points x(α,β,γ)e x_{\left( {\alpha ,\beta ,\gamma } \right)}^e and y(α,β,γ)e y_{\left( {{\alpha ^\prime},{\beta ^\prime},{\gamma ^\prime}} \right)}^{{e^\prime}} over a common universe X, we say that the neutrosophic soft points are distinct points if x(α,β,γ)ey(α,β,γ)e=0(X,E) x_{\left( {\alpha ,\beta ,\gamma } \right)}^e \cap y_{\left( {{\alpha ^\prime},{\beta ^\prime},{\gamma ^\prime}} \right)}^{{e^\prime}}{ = 0_{(X,E)}} .

It is clear that x(α,β,γ)e x_{\left( {\alpha ,\beta ,\gamma } \right)}^e and y(α1,β1,γ1)e y_{\left( {{\alpha _1},{\beta _1},{\gamma _1}} \right)}^{{e^\prime}} are distinct neutrosophic soft points if and only if xy or ee.

Definition 12

[13] Let NSS(X,E) be the family of all neutrosophic soft sets over the universe set X and τNSSNSS(X,E) \mathop \tau \limits^{NSS} \subset NSS(X,E) . Then τNSS \mathop \tau \limits^{NSS} is said to be a neutrosophic soft topology on X if

0(X,E) and 1(X,E) belongs to τNSS \mathop \tau \limits^{NSS}

the union of any number of neutrosophic soft sets in τNSS \mathop \tau \limits^{NSS} belongs to τNSS \mathop \tau \limits^{NSS}

the intersection of finite number of neutrosophic soft sets in τNSS \mathop \tau \limits^{NSS} belongs to τNSS \mathop \tau \limits^{NSS} .

Then (X,τNSS,E) \left( {X,\mathop \tau \limits^{NSS} ,E} \right) is said to be a neutrosophic soft topological space over X. Each members of τNSS \mathop \tau \limits^{NSS} is said to be neutrosophic soft open set.

Definition 13

[13] Let (X,τNSS,E) \left( {X,\mathop \tau \limits^{NSS} ,E} \right) be a neutrosophic soft topological space over X and (F̃,E) be a neutrosophic soft set over X. Then (F̃,E) is said to be neutrosophic soft closed set iff its complement is a neutrosophic soft open set.

Definition 14

[13] Let (X,τNSS,E) \left( {X,\mathop \tau \limits^{NSS} ,E} \right) be a neutrosophic soft topological space over X and (F̃,E) ∈ NSS(X,E) be a neutrosophic soft set. Then, the neutrosophic soft interior of (F̃,E), denoted (F̃,E)°, is defined as the neutrosophic soft union of all neutrosophic soft open subsets of (F̃,E).

Clearly, (F̃,Eis the biggest neutrosophic soft open set that is contained by (F̃,E).

Definition 15

[13] Let (X,τNSS,E) \left( {X,\mathop \tau \limits^{NSS} ,E} \right) be a neutrosophic soft topological space over X and (F̃,E) ∈ NSS(X,E) be a neutrosophic soft set. Then, the neutrosophic soft closure of (F̃,E), denoted (F˜,E)¯ \overline {\left( {\widetilde F,E} \right)} , is defined as the neutrosophic soft intersection of all neutrosophic soft closed supersets of (F̃,E).

Clearly, (F˜,E)¯ \overline {\left( {\widetilde F,E} \right)} is the smallest neutrosophic soft closed set that containing (F̃,E).

Some Structures on Neutrosophic Soft Topological Spaces
Definition 16

Let (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) be a neutrosophic soft topological space over X and (F̃,E) ∈ NSS(X,E) be a neutrosophic soft set over X. If Fr(F˜,E)=(F˜,E)¯((F˜,E)c)¯ Fr(\widetilde F,E) = \overline {(\widetilde F,E)} \cap \overline {((\widetilde F,E{)^c})} , then Fr(F̃,E) is said to be boundary of the neutrosophic soft set (F̃,E).

Example 1

Let X = {x1,x2,x3} be an initial universe set, E = {e1,e2} be a set of parameters and τNSS={0(X,E),1(X,E),(F˜1,E),(F˜2,E),(F˜3,E),(F˜4,E)} \mathop \tau \limits^{NSS} = \left\{ {{0_{(X,E)}}{{,1}_{(X,E)}},({{\widetilde F}_1},E),({{\widetilde F}_2},E),({{\widetilde F}_3},E),({{\widetilde F}_4},E)} \right\} be a neutrosophic soft topology over X. Here, the neutrosophic soft sets (1,E), (2,E), (3,E) and (4,E) over X are defined as following;

(F˜1,E)={e1,{x1,0.6,0.4,0.7,x2,0.3,0.5,0.2,x3,0.4,0.6,0.9}e2,{x1,0.3,0.7,0.4,x2,0.1,0.2,0.5,x3,0.7,0.8,0.9}}, ({\widetilde F_1},E) = \left\{ {\matrix{ {{e_1},\left\{ {\langle {x_1},0.6,0.4,0.7\rangle ,\langle {x_2},0.3,0.5,0.2\rangle ,\langle {x_3},0.4,0.6,0.9\rangle } \right\}} \cr {{e_2},\left\{ {\langle {x_1},0.3,0.7,0.4\rangle ,\langle {x_2},0.1,0.2,0.5\rangle ,\langle {x_3},0.7,0.8,0.9\rangle } \right\}} \cr } } \right\},

(F˜2,E)={e1,{x1,0.2,0.7,0.6,x2,0.3,0.7,0.9,x3,0.5,0.8,0.2}e2,{x1,0.1,0.9,0.5,x2,0.4,0.6,0.8,x3,0.4,0.3,0.7}}, ({\widetilde F_2},E) = \left\{ {\matrix{ {{e_1},\left\{ {\langle {x_1},0.2,0.7,0.6\rangle ,\langle {x_2},0.3,0.7,0.9\rangle ,\langle {x_3},0.5,0.8,0.2\rangle } \right\}} \cr {{e_2},\left\{ {\langle {x_1},0.1,0.9,0.5\rangle ,\langle {x_2},0.4,0.6,0.8\rangle ,\langle {x_3},0.4,0.3,0.7\rangle } \right\}} \cr } } \right\},

(F˜3,E)={e1,{x1,0.2,0.4,0.7,x2,0.3,0.5,0.9,x3,0.4,0.6,0.9}e2,{x1,0.1,0.7,0.5,x2,0.1,0.2,0.8,x3,0.4,0.3,0.9}}, ({\widetilde F_3},E) = \left\{ {\matrix{ {{e_1},\left\{ {\langle {x_1},0.2,0.4,0.7\rangle ,\langle {x_2},0.3,0.5,0.9\rangle ,\langle {x_3},0.4,0.6,0.9\rangle } \right\}} \cr {{e_2},\left\{ {\langle {x_1},0.1,0.7,0.5\rangle ,\langle {x_2},0.1,0.2,0.8\rangle ,\langle {x_3},0.4,0.3,0.9\rangle } \right\}} \cr } } \right\},

(F˜4,E)={e1,{x1,0.6,0.7,0.6,x2,0.3,0.7,0.2,x3,0.5,0.8,0.2}e2,{x1,0.3,0.9,0.5,x2,0.4,0.6,0.5,x3,0.7,0.8,0.7}}. ({\widetilde F_4},E) = \left\{ {\matrix{ {{e_1},\left\{ {\langle {x_1},0.6,0.7,0.6\rangle ,\langle {x_2},0.3,0.7,0.2\rangle ,\langle {x_3},0.5,0.8,0.2\rangle } \right\}} \cr {{e_2},\left\{ {\langle {x_1},0.3,0.9,0.5\rangle ,\langle {x_2},0.4,0.6,0.5\rangle ,\langle {x_3},0.7,0.8,0.7\rangle } \right\}} \cr } } \right\}.

Suppose that the neutrosophic soft set (F̃,E) over X is defined as;

(F˜,E)={e1,{x1,0.7,0.6,0.3,x2,0.5,0.7,0.1,x3,0.8,0.8,0.6}e2,{x1,0.4,0.9,0.2,x2,0.3,0.4,0.4,x3,0.9,0.8,0.7}}. (\widetilde F,E) = \left\{ {\matrix{ {{e_1},\left\{ {\langle {x_1},0.7,0.6,0.3\rangle ,\langle {x_2},0.5,0.7,0.1\rangle ,\langle {x_3},0.8,0.8,0.6\rangle } \right\}} \cr {{e_2},\left\{ {\langle {x_1},0.4,0.9,0.2\rangle ,\langle {x_2},0.3,0.4,0.4\rangle ,\langle {x_3},0.9,0.8,0.7\rangle } \right\}} \cr } } \right\}.

Then, let us find the boundary of the neutrosophic soft set (F̃,E) :

(F˜,E)¯=1(X,E) \overline {(\widetilde F,E)} = { 1_{(X,E)}} and ((F˜,E)c¯)=(F˜1,E)c(F˜3,E)c \left( {\overline {{{(\widetilde F,E)}^c}} } \right) = ({\widetilde F_1},E{)^c} \cap {({\widetilde F_3},E)^c}

Therefore,

Fr(F˜,E)=(F˜,E)¯((F˜,E)c¯)=1(X,E)(F˜1,E)c(F˜3,E)c={e1,{x1,0.7,0.7,0.6,x2,0.2,0.5,0.3,x3,0.9,0.9,0.4}e2,{x1,0.4,0.3,0.3,x2,0.5,0.8,0.1,x3,0.9,0.2,0.7}}. \matrix{ {Fr(\widetilde F,E) = \overline {(\widetilde F,E)} \cap \left( {\overline {{{(\widetilde F,E)}^c}} } \right){{ = 1}_{(X,E)}} \cap {{({{\widetilde F}_1},E)}^c} \cap {{({{\widetilde F}_3},E)}^c}} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\; = \left\{ {\matrix{ {{e_1},\left\{ {\langle {x_1},0.7,0.7,0.6\rangle ,\langle {x_2},0.2,0.5,0.3\rangle ,\langle {x_3},0.9,0.9,0.4\rangle } \right\}} \cr {{e_2},\left\{ {\langle {x_1},0.4,0.3,0.3\rangle ,\langle {x_2},0.5,0.8,0.1\rangle ,\langle {x_3},0.9,0.2,0.7\rangle } \right\}} \cr } } \right\}.} \hfill \cr }

Theorem 1

Let (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) be a neutrosophic soft topological space over X and (1,E), (2,E) ∈ NSS(X,E). Then,

(1,E)°= (1,E)\Fr(1,E),

(F˜1,E)¯=(F˜1,E)Fr(F˜1,E) \overline {({{\widetilde F}_1},E)} = ({\widetilde F_1},E) \cup Fr({\widetilde F_1},E) ,

Fr((1,E) ∪ (2,E)) ⊆ Fr(1,E) ∪ Fr(2,E),

Fr((1,E)c) = Fr(1,E),

1(X,E) = (1,E)° ∪ Fr(1,E)∪ (1(X,E)\(1,E))°,

Fr((F˜1,E)¯)Fr(F˜1,E) Fr\left( {\overline {({{\widetilde F}_1},E)} } \right) \subseteq Fr({\widetilde F_1},E) ,

Fr((1,E)°) ⊆ Fr(1,E),

(1,E) is a neutrosophic soft open set Fr(F˜1,E)=(F˜1,E)¯\(F˜1,E) Fr({\widetilde F_1},E) = \overline {({{\widetilde F}_1},E)} \backslash ({\widetilde F_1},E) ,

(1,E) is a neutrosophic soft closed setFr(1,E) = (1,E)\(1,E)°.

Proof

(F˜1,E)\Fr(F˜1,E)=(F˜1,E)\((F˜1,E)¯(1(X,E)\(F˜1,E))¯)=(F˜1,E)((F˜1,E)¯(1(X,E)\(F˜1,E))¯)c=((F˜1,E)((F˜1,E)¯)c)((F˜1,E)((1(X,E)\(F˜1,E))¯)c)=((F˜1,E)\(F˜1,E)¯)((F˜1,E)\((1(X,E)\(F˜1,E))¯))=(F˜1,E)\(1(X,E)\(F˜1,E))¯=(F˜1,E)((1(X,E)\(F˜1,E))¯)c=(F˜1,E)(F˜1,E)o=(F˜1,E)o. \matrix{ {({{\widetilde F}_1},E)\backslash Fr({{\widetilde F}_1},E)} \hfill & = \hfill & {({{\widetilde F}_1},E)\backslash \left( {\overline {({{\widetilde F}_1},E)} \cap \overline {\left( {{1_{(X,E)}}\backslash ({{\widetilde F}_1},E)} \right)} } \right)} \hfill \cr {} \hfill & = \hfill & {({{\widetilde F}_1},E) \cap {{\left( {\overline {({{\widetilde F}_1},E)} \cap \overline {\left( {{1_{(X,E)}}\backslash ({{\widetilde F}_1},E)} \right)} } \right)}^c}} \hfill \cr {} \hfill & = \hfill & {\left( {({{\widetilde F}_1},E) \cap {{\left( {\overline {({{\widetilde F}_1},E)} } \right)}^c}} \right) \cup \left( {({{\widetilde F}_1},E) \cap {{\left( {\overline {\left( {{1_{(X,E)}}\backslash ({{\widetilde F}_1},E)} \right)} } \right)}^c}} \right)} \hfill \cr {} \hfill & = \hfill & {\left( {({{\widetilde F}_1},E)\backslash \overline {({{\widetilde F}_1},E)} } \right) \cup \left( {({{\widetilde F}_1},E)\backslash \left( {\overline {\left( {{1_{(X,E)}}\backslash ({{\widetilde F}_1},E)} \right)} } \right)} \right)} \hfill \cr {} \hfill & = \hfill & {({{\widetilde F}_1},E)\backslash \overline {\left( {{1_{(X,E)}}\backslash ({{\widetilde F}_1},E)} \right)} = ({{\widetilde F}_1},E) \cap {{\left( {\overline {\left( {{1_{(X,E)}}\backslash ({{\widetilde F}_1},E)} \right)} } \right)}^c}} \hfill \cr {} \hfill & = \hfill & {({{\widetilde F}_1},E) \cap {{({{\widetilde F}_1},E)}^o} = ({{\widetilde F}_1},E{)^o}.} \hfill \cr }

It is clear.

Fr((F˜1,E)(F˜2,E))=((F˜1,E)(F˜2,E)¯)(1(X,E)\((F˜1,E)(F˜2,E))¯)=((F˜1,E)¯(F˜2,E)¯)((1(X,E)\(F˜1,E))(1(X,E)\(F˜2,E))¯)((F˜1,E)¯(F˜2,E)¯)(1(X,E)\(F˜1,E)¯)(1(X,E)\(F˜2,E)¯)((F˜1,E)¯(1(X,E)\(F˜1,E))¯)((F˜2,E)¯(1(X,E)\(F˜2,E))¯)=Fr(F˜1,E)Fr(F˜2,E) \matrix{ {Fr\left( {({{\widetilde F}_1},E) \cup ({{\widetilde F}_2},E)} \right)} \hfill & = \hfill & {\left( {\overline {({{\widetilde F}_1},E) \cup ({{\widetilde F}_2},E)} } \right) \cap \left( {\overline {{1_{(X,E)}}\backslash \left( {({{\widetilde F}_1},E) \cup ({{\widetilde F}_2},E)} \right)} } \right)} \hfill \cr {} \hfill & = \hfill & {\left( {\overline {({{\widetilde F}_1},E)} \cup \overline {({{\widetilde F}_2},E)} } \right) \cap \left( {\overline {{{(1}_{(X,E)}}\backslash ({{\widetilde F}_1},E)) \cap \left( {{1_{(X,E)}}\backslash ({{\widetilde F}_2},E)} \right)} } \right)} \hfill \cr {} \hfill & \subseteq \hfill & {\left( {\overline {({{\widetilde F}_1},E)} \cup \overline {({{\widetilde F}_2},E)} } \right) \cap \left( {\overline {{1_{(X,E)}}\backslash ({{\widetilde F}_1},E)} } \right) \cap \left( {\overline {{1_{(X,E)}}\backslash ({{\widetilde F}_2},E)} } \right)} \hfill \cr {} \hfill & \subseteq \hfill & {\left( {\overline {({{\widetilde F}_1},E)} \cap \overline {\left( {{1_{(X,E)}}\backslash ({{\widetilde F}_1},E)} \right)} } \right) \cup \left( {\overline {({{\widetilde F}_2},E)} \cap \overline {\left( {{1_{(X,E)}}\backslash ({{\widetilde F}_2},E)} \right)} } \right)} \hfill \cr {} \hfill & = \hfill & {Fr({{\widetilde F}_1},E) \cup Fr({{\widetilde F}_2},E)} \hfill \cr }

Fr((F˜1,E)c)=((F˜1,E)c¯)(((F˜1,E)c)c¯)=((F˜1,E)c¯)(F˜1,E)¯=Fr(F˜1,E) \matrix{ {Fr\left( {{{({{\widetilde F}_1},E)}^c}} \right)} \hfill & = \hfill & {\left( {\overline {{{({{\widetilde F}_1},E)}^c}} } \right) \cap \left( {\overline {{{\left( {{{({{\widetilde F}_1},E)}^c}} \right)}^c}} } \right) = \left( {\overline {{{({{\widetilde F}_1},E)}^c}} } \right) \cap \overline {({{\widetilde F}_1},E)} } \hfill \cr {} \hfill & = \hfill & {Fr({{\widetilde F}_1},E)} \hfill \cr }

It is clear.

Fr((F˜1,E)¯)=((F˜1,E)¯)¯(((F˜1,E)¯)c)¯(F˜1,E)¯((F˜1,E)c)¯=Fr(F˜1,E) Fr\left( {\overline {({{\widetilde F}_1},E)} } \right) = \overline {\left( {\overline {({{\widetilde F}_1},E)} } \right)} \cap \overline {\left( {{{\left( {\overline {({{\widetilde F}_1},E)} } \right)}^c}} \right)} \subseteq \overline {({{\widetilde F}_1},E)} \cap \overline {\left( {{{({{\widetilde F}_1},E)}^c}} \right)} = Fr({\widetilde F_1},E)

It is clear.

Suppose that (1,E) is a neutrosophic soft open set. Then (1,E)c is a neutrosophic soft closed set and ((F˜1,E)c)¯=(F˜1,E)c \overline {\left( {{{({{\widetilde F}_1},E)}^c}} \right)} = ({\widetilde F_1},E{)^c} . In here, Fr(F˜1,E)=(F˜1,E)¯((F˜1,E)c)¯=(F˜1,E)¯(F˜1,E)c=(F˜1,E)¯\(F˜1,E). Fr({\widetilde F_1},E) = \overline {({{\widetilde F}_1},E)} \cap \overline {\left( {{{({{\widetilde F}_1},E)}^c}} \right)} = \overline {({{\widetilde F}_1},E)} \cap {({\widetilde F_1},E)^c} = \overline {({{\widetilde F}_1},E)} \backslash ({\widetilde F_1},E).

From the condition-1, (F˜1,E)o=(F˜1,E)\Fr(F˜1,E)=(F˜1,E)\((F˜1,E)¯\(F˜1,E))=(F˜1,E)((F˜1,E)¯(F˜1,E)c)c=(F˜1,E)(((F˜1,E)¯)c((F˜1,E)c)c)=((F˜1,E)((F˜1,E)¯)c)((F˜1,E)(F˜1,E))=((F˜1,E)((F˜1,E)¯)c)(F˜1,E)=(F˜1,E) \matrix{ {{{({{\widetilde F}_1},E)}^o}} \hfill & = \hfill & {({{\widetilde F}_1},E)\backslash Fr({{\widetilde F}_1},E) = ({{\widetilde F}_1},E)\backslash \left( {\overline {({{\widetilde F}_1},E)} \backslash ({{\widetilde F}_1},E)} \right) = ({{\widetilde F}_1},E) \cap {{\left( {\overline {({{\widetilde F}_1},E)} \cap {{({{\widetilde F}_1},E)}^c}} \right)}^c}} \hfill \cr {} \hfill & = \hfill & {({{\widetilde F}_1},E) \cap \left( {{{\left( {\overline {({{\widetilde F}_1},E)} } \right)}^c} \cup {{\left( {{{({{\widetilde F}_1},E)}^c}} \right)}^c}} \right)} \hfill \cr {} \hfill & = \hfill & {\left( {({{\widetilde F}_1},E) \cap {{\left( {\overline {({{\widetilde F}_1},E)} } \right)}^c}} \right) \cup \left( {({{\widetilde F}_1},E) \cap ({{\widetilde F}_1},E)} \right)} \hfill \cr {} \hfill & = \hfill & {\left( {({{\widetilde F}_1},E) \cap {{\left( {\overline {({{\widetilde F}_1},E)} } \right)}^c}} \right) \cup ({{\widetilde F}_1},E) = ({{\widetilde F}_1},E)} \hfill \cr }

That is, (1,E) is a neutrosophic soft open set.

Definition 17

Let (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) be a neutrosophic soft topological space over X and (F̃,E) ∈ NSS(X,E).

(F̃,E) is said to be a neutrosophic soft dense set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) if (F˜,E)¯=1(X,E) \;\overline {(\widetilde F,E)} { = 1_{(X,E)}} ,

(F̃,E) is said to be a neutrosophic soft co-dense set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) if (1(X,E)\(F˜,E))¯=1(X,E) \overline {\left( {{1_{(X,E)}}\backslash (\widetilde F,E)} \right)} { = 1_{(X,E)}} ,

(F̃,E) is said to be a neutrosophic soft not-dense set in any part of (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) if (F˜,E)¯ \;\overline {(\widetilde F,E)} is a neutrosophic soft dense set over (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) .

Theorem 2

Let (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) be a neutrosophic soft topological space over X and (F̃,E) ∈ NSS(X,E). Then,

(F̃,E) is a neutrosophic soft dense set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) iff (F̃,E) ∩ (Ũ,E) ≠ 0(X,E) for each 0(X,E)(U˜,E)τNSS {0_{(X,E)}} \ne (\widetilde U,E) \in \mathop \tau \limits^{NSS} ,

(F̃,E) is a neutrosophic soft co-dense set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) iff (1(X,E)\(F̃,E)) ∩ (Ũ,E) ≠ 0(X,E) for each 0(X,E)(U˜,E)τNSS {0_{(X,E)}} \ne (\widetilde U,E) \in \mathop \tau \limits^{NSS} ,

(F̃,E) is a neutrosophic soft not-dense set in any part of (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) iff there is a neutrosophic soft open set (V˜,E)τNSS (\widetilde V,E) \in \mathop \tau \limits^{NSS} such that (Ṽ,E) ∩(F̃,E) = 0(X,E) and 0(X,E) ≠ (Ṽ,E) ∩ (Ũ,E) for each 0(X,E)(U˜,E)τNSS {0_{(X,E)}} \ne (\widetilde U,E) \in \mathop \tau \limits^{NSS} .

Proof

Straightforward.

Definition 18

Let (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) be a neutrosophic soft topological space over X and BNSS \mathop B\limits^{NSS} be a sub-family of τNSS \mathop \tau \limits^{NSS} . BNSS \mathop B\limits^{NSS} is said to be a neutrosophic soft basis for the neutrosophic soft topology τNSS \mathop \tau \limits^{NSS} if every element of τNSS \mathop \tau \limits^{NSS} can be written as the neutrosophic soft union of elements of BNSS \mathop B\limits^{NSS} .

Theorem 3

Let (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) be a neutrosophic soft topological space over X and BNSS \mathop B\limits^{NSS} be a neutrosophic soft basis for τNSS \mathop \tau \limits^{NSS} . Then, τNSS \mathop \tau \limits^{NSS} equals to the collection of all neutrosophic soft unions of elements of BNSS \mathop B\limits^{NSS} .

Proof

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

Example 2

Let us consider the neutrosophic soft topology τNSS \mathop \tau \limits^{NSS} given in Example-1. Then, the family BNSS={0(X,E),1(X,E),(F˜1,E),(F˜2,E),(F˜3,E)} \mathop B\limits^{NSS} = \left\{ {{0_{(X,E)}}{{,1}_{(X,E)}},({{\widetilde F}_1},E),({{\widetilde F}_2},E),({{\widetilde F}_3},E)} \right\} is a neutrosophic soft basis for the neutrosophic soft topology τNSS \mathop \tau \limits^{NSS} .

Theorem 4

Let (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) be a neutrosophic soft topological space over X and BNSS \mathop B\limits^{NSS} be a sub-family of τNSS \mathop \tau \limits^{NSS} . Then,

The family BNSS \mathop B\limits^{NSS} is a neutrosophic soft basis of the neutrosophic soft topology τNSS \mathop \tau \limits^{NSS} iff there exist a neutrosophic soft set (B˜,E)BNSS (\widetilde B,E) \in \mathop B\limits^{NSS} such that x(α,β,γ)e(B˜,E)(F˜,E) x_{(\alpha ,\beta ,\gamma )}^e \in (\widetilde B,E) \subseteq (\widetilde F,E) for each (F˜,E)τNSS (\widetilde F,E) \in \mathop \tau \limits^{NSS} and x(α,β,γ)e(F˜,E) x_{(\alpha ,\beta ,\gamma )}^e \in (\widetilde F,E) .

If the family BNSS={(B˜i,E)}iI \mathop B\limits^{NSS} = {\left\{ {({{\widetilde B}_i},E)} \right\}_{i \in I}} is a neutrosophic soft basis for τNSS \mathop \tau \limits^{NSS} , then there exist a neutrosophic soft set (B˜i3,E)BNSS ({\widetilde B_{{i_3}}},E) \in \mathop B\limits^{NSS} such that x(α,β,γ)e(B˜i3,E)(B˜i1,E)(B˜i2,E) x_{(\alpha ,\beta ,\gamma )}^e \in ({\widetilde B_{{i_3}}},E) \subseteq ({\widetilde B_{{i_1}}},E) \cap ({\widetilde B_{{i_2}}},E) for each (B˜i1,E),(B˜i2,E)BNSS ({\widetilde B_{{i_1}}},E),({\widetilde B_{{i_2}}},E) \in \mathop B\limits^{NSS} and each x(α,β,γ)e(B˜i1,E)(B˜i2,E) x_{(\alpha ,\beta ,\gamma )}^e \in ({\widetilde B_{{i_1}}},E) \cap ({\widetilde B_{{i_2}}},E) .

Proof

1. Suppose that BNSS \mathop B\limits^{NSS} is a neutrosophic soft basis of the neutrosophic soft topology τNSS \mathop \tau \limits^{NSS} , (F˜,E)τNSS (\widetilde F,E) \in \mathop \tau \limits^{NSS} and x(α,β,γ)e(F˜,E) x_{(\alpha ,\beta ,\gamma )}^e \in (\widetilde F,E) . Then (F˜,E)=(B˜,E)BNSS(B˜,E). \left( {\widetilde F,E} \right) = \bigcup\limits_{(\widetilde B,E) \in \mathop B\limits^{NSS} } {\left( {\widetilde B,E} \right)} . Therefore x(α,β,γ)e(B˜,E)(F˜,E) x_{(\alpha ,\beta ,\gamma )}^e \in (\widetilde B,E) \subseteq (\widetilde F,E) from x(α,β,γ)e(F˜,E)=(B˜,E)BNSS(B˜,E) x_{(\alpha ,\beta ,\gamma )}^e \in \left( {\widetilde F,E} \right) = \bigcup\limits_{(\widetilde B,E) \in \mathop B\limits^{NSS} } {\left( {\widetilde B,E} \right)} for x(α,β,γ)e(B˜,E)(F˜,E) x_{(\alpha ,\beta ,\gamma )}^e \in (\widetilde B,E) \subseteq (\widetilde F,E) .

⇐ Suppose that the condition of theorem to be provided. Then, (F˜,E)=x(α,β,γ)e(F˜,E){x(α,β,γ)e}x(α,β,γ)e(F˜,E)(B˜,E)(F˜,E)(F˜,E)=x(α,β,γ)e(F˜,E)(B˜,E). \matrix{ {(\widetilde F,E)} \hfill & = \hfill & {\bigcup\limits_{x_{(\alpha ,\beta ,\gamma )}^e \in (\widetilde F,E)} \left\{ {x_{(\alpha ,\beta ,\gamma )}^e} \right\} \subseteq \bigcup\limits_{x_{(\alpha ,\beta ,\gamma )}^e \in (\widetilde F,E)} (\widetilde B,E) \subseteq (\widetilde F,E)} \hfill \cr {} \hfill & \Rightarrow \hfill & {(\widetilde F,E) = \bigcup\limits_{x_{(\alpha ,\beta ,\gamma )}^e \in (\widetilde F,E)} (\widetilde B,E).} \hfill \cr }

That is, BNSS \mathop B\limits^{NSS} is a neutrosophic soft basis for τNSS \mathop \tau \limits^{NSS} .

2. Let (B˜i1,E),(B˜i2,E)BNSS ({\widetilde B_{{i_1}}},E),({\widetilde B_{{i_2}}},E) \in \mathop B\limits^{NSS} and x(α,β,γ)e(B˜i1,E)(B˜i2,E) x_{(\alpha ,\beta ,\gamma )}^e \in ({\widetilde B_{{i_1}}},E) \cap ({\widetilde B_{{i_2}}},E) . Since (i1,E) ∩ (i2,E) is a neutrosophic soft open set and BNSS \mathop B\limits^{NSS} is a neutrosophic soft basis for τNSS \mathop \tau \limits^{NSS} , then (B˜i1,E)(B˜i2,E)=j(B˜j,E)x(α,β,γ)e(B˜i1,E)(B˜i2,E)=j(B˜j,E)(B˜i3,E) ({\widetilde B_{{i_1}}},E) \cap ({\widetilde B_{{i_2}}},E) = \bigcup\limits_j ({\widetilde B_j},E) \Rightarrow x_{(\alpha ,\beta ,\gamma )}^e \in ({\widetilde B_{{i_1}}},E) \cap ({\widetilde B_{{i_2}}},E) = \bigcup\limits_j ({\widetilde B_j},E) \Rightarrow \exists ({\widetilde B_{{i_3}}},E) , x(α,β,γ)e(B˜i3,E)(B˜i1,E)(B˜i2,E) x_{(\alpha ,\beta ,\gamma )}^e \in ({\widetilde B_{{i_3}}},E) \subseteq ({\widetilde B_{{i_1}}},E) \cap ({\widetilde B_{{i_2}}},E) .

Theorem 5

Let τ1NSS \mathop {{\tau _1}}\limits^{NSS} and τ2NSS \mathop {{\tau _2}}\limits^{NSS} be two neutrosophic soft topologies over X generated by the neutrosophic soft bases B1NSS \mathop {{B_1}}\limits^{NSS} and B2NSS, \mathop {{B_2}}\limits^{NSS} , respectively. Then τ1NSSτ2NSS \mathop {{\tau _1}}\limits^{NSS} \subseteq \mathop {{\tau _2}}\limits^{NSS} iff for each x(α,β,γ)eNSS(X,E) x_{(\alpha ,\beta ,\gamma )}^e \in NSS(X,E) and for each (B˜1,E)B1NSS ({\widetilde B_1},E) \in \mathop {{B_1}}\limits^{NSS} containing x(α,β,γ)e x_{(\alpha ,\beta ,\gamma )}^e , there exists (B˜2,E)B2NSS ({\widetilde B_2},E) \in \mathop {{B_2}}\limits^{NSS} such that x(α,β,γ)e(B˜2,E)(B˜1,E) x_{(\alpha ,\beta ,\gamma )}^e \in ({\widetilde B_2},E) \subseteq ({\widetilde B_1},E) .

Proof

⇒ Suppose that τ1NSSτ2NSS \mathop {{\tau _1}}\limits^{NSS} \subseteq \mathop {{\tau _2}}\limits^{NSS} and x(α,β,γ)eNSS(X,E) x_{(\alpha ,\beta ,\gamma )}^e \in NSS(X,E) , (B˜1,E)B1NSS ({\widetilde B_1},E) \in \mathop {{B_1}}\limits^{NSS} such that x(α,β,γ)e(B˜1,E) x_{(\alpha ,\beta ,\gamma )}^e \in ({\widetilde B_1},E) . Since B1NSS \mathop {{B_1}}\limits^{NSS} is a neutrosophic soft basis for neutrosophic soft topology τ1NSS \mathop {{\tau _1}}\limits^{NSS} over X, then B1NSSτ1NSSx(α,β,γ)e(B˜1,E)B1NSSτ1NSSτ2NSS \mathop {{B_1}}\limits^{NSS} \subseteq \mathop {{\tau _1}}\limits^{NSS} \Rightarrow x_{(\alpha ,\beta ,\gamma )}^e \in ({\widetilde B_1},E) \in \mathop {{B_1}}\limits^{NSS} \subseteq \mathop {{\tau _1}}\limits^{NSS} \subseteq \mathop {{\tau _2}}\limits^{NSS} i.e., x(α,β,γ)e(B˜1,E)τ2NSS x_{(\alpha ,\beta ,\gamma )}^e \in ({\widetilde B_1},E) \in \mathop {{\tau _2}}\limits^{NSS} . Since B2NSS \mathop {{B_2}}\limits^{NSS} is a neutrosophic soft basis for τ2NSS \mathop {{\tau _2}}\limits^{NSS} , so for (B˜2,E)B2NSS ({\widetilde B_2},E) \in \mathop {{B_2}}\limits^{NSS} we have x(α,β,γ)e(B˜2,E)(B˜1,E) x_{(\alpha ,\beta ,\gamma )}^e \in ({\widetilde B_2},E) \subseteq ({\widetilde B_1},E) .

⇐ Conversely, assume that the hypothesis holds. Let (F˜,E)τ1NSS (\widetilde F,E) \in \mathop {{\tau _1}}\limits^{NSS} . Since B1NSS \mathop {{B_1}}\limits^{NSS} is a neutrosophic soft basis for neutrosophic soft topology τ1NSS \mathop {{\tau _1}}\limits^{NSS} , then for x(α,β,γ)e(F˜,E) x_{(\alpha ,\beta ,\gamma )}^e \in (\widetilde F,E) there exist (B˜1,E)B1NSS ({\widetilde B_1},E) \in \mathop {{B_1}}\limits^{NSS} such that x(α,β,γ)e(B˜1,E)(F˜,E) x_{(\alpha ,\beta ,\gamma )}^e \in ({\widetilde B_1},E) \subseteq (\widetilde F,E) . No by hypothesis, there exist (B˜2,E)B2NSS ({\widetilde B_2},E) \in \mathop {{B_2}}\limits^{NSS} such that (B˜2,E)(B˜1,E)(B˜2,E)(B˜1,E)(F˜,E)(B˜2,E)(F˜,E)(F˜,E)τ2NSS ({\widetilde B_2},E) \subseteq ({\widetilde B_1},E) \Rightarrow ({\widetilde B_2},E) \subseteq ({\widetilde B_1},E) \subseteq (\widetilde F,E) \Rightarrow ({\widetilde B_2},E) \subseteq (\widetilde F,E) \Rightarrow (\widetilde F,E) \in \mathop {{\tau _2}}\limits^{NSS} . This show that τ1NSSτ2NSS \mathop {{\tau _1}}\limits^{NSS} \subseteq \mathop {{\tau _2}}\limits^{NSS} .

Theorem 6

Let (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) be a neutrosophic soft topological space over X and (F̃,E) ∈ NSS(X;E). Then the collection τNSS(F˜,E)={(F˜,E)(F˜i,E):(F˜i,E)τNSSforiI} {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} = \left\{ {(\widetilde F,E) \cap ({{\widetilde F}_i},E):({{\widetilde F}_i},E) \in \mathop \tau \limits^{NSS} {{for}}\,i \in I} \right\} is a neutrosophic soft topology on (F̃,E) and (X(F˜,E),τNSS(F˜,E),E) \left( {{X_{(\widetilde F,E)}},{{\mathop \tau \limits^{NSS} }_{(\widetilde F,E)}},E} \right) is a neutrosophic soft topological space.

Proof

Since 0(X̃,E) ∩ (F̃,E) = 0(F,E) and 1(X,E) ∩ (F̃,E) = (F̃,E), then 0(F̃,E) and (F˜,E)τNSS(F˜,E) (\widetilde F,E) \in {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} . Moreover, i=1n((F˜i,E)(F˜,E))=(i=1n(F˜i,E))(F˜,E) \bigcap\limits_{i = 1}^n \left( {({{\widetilde F}_i},E) \cap (\widetilde F,E)} \right) = \left( {\bigcap\limits_{i = 1}^n ({{\widetilde F}_i},E)} \right) \cap (\widetilde F,E) and iI((F˜i,E)(F˜,E))=(iI(F˜i,E))(F˜,E) \bigcup\limits_{i \in I} \left( {({{\widetilde F}_i},E) \cup (\widetilde F,E)} \right) = \left( {\bigcup\limits_{i \in I} ({{\widetilde F}_i},E)} \right) \cup (\widetilde F,E) for τNSS={(F˜i,E):iI} \mathop \tau \limits^{NSS} = \left\{ {({{\widetilde F}_i},E):i \in I} \right\} . Therefore τNSS(F˜,E) {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} is a neutrosophic soft topology over (F̃,E).

Definition 19

Let (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) be a neutrosophic soft topological space over X and (F̃,E) ∈ NSS(X,E). Then the collection τNSS(F˜,E)={(F˜,E)(F˜i,E):(F˜i,E)τNSSforiI} {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} = \left\{ {(\widetilde F,E) \cap ({{\widetilde F}_i},E):({{\widetilde F}_i},E) \in \mathop \tau \limits^{NSS}\, {{for}}\,i \in I} \right\} is called a neutrosophic soft subspace topology on (F̃,E) and (X(F˜,E),τNSS(F˜,E),E) \left( {{X_{(\widetilde F,E)}},{{\mathop \tau \limits^{NSS} }_{(\widetilde F,E)}},E} \right) is called a neutrosophic soft topological subspace of (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) .

Example 3

Let us consider the neutrosophic soft topology τNSS \mathop \tau \limits^{NSS} and the neutrosophic soft set (F̃,E) given in Example-1. Then the collection τNSS(F˜,E)={0(X˜,E),(F˜,E),(F˜1,E),(F˜2,E),(F˜3,E),(F˜4,E)} {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} = \left\{ {{0_{(\widetilde X,E)}},(\widetilde F,E),({{\widetilde F}_1},E{)^\prime},({{\widetilde F}_2},E{)^\prime},({{\widetilde F}_3},E{)^\prime},({{\widetilde F}_4},E{)^\prime}} \right\} is a neutrosophic soft sub-topology on (F̃,E) of the neutrosophic soft topology τNSS \mathop \tau \limits^{NSS} . Here, the neutrosophic soft sets (1,E), (2,E), (3,E) and (4,E) over (F̃,E) are defined as following:

(F˜1,E)=(F˜,E)(F˜1,E)={e1,{x1,0.6,0.4,0.7,x2,0.3,0.5,0.2,x3,0.4,0.6,0.9}e2,{x1,0.3,0.7,0.4,x2,0.1,0.2,0.5,x3,0.7,0.8,0.9}}, {({\widetilde F_1},E)^\prime} = (\widetilde F,E) \cap ({\widetilde F_1},E) = \left\{ {\matrix{ {{e_1},\left\{ {\langle {x_1},0.6,0.4,0.7\rangle ,\langle {x_2},0.3,0.5,0.2\rangle ,\langle {x_3},0.4,0.6,0.9\rangle } \right\}} \cr {{e_2},\left\{ {\langle {x_1},0.3,0.7,0.4\rangle ,\langle {x_2},0.1,0.2,0.5\rangle ,\langle {x_3},0.7,0.8,0.9\rangle } \right\}} \cr } } \right\},

(F˜2,E)=(F˜,E)(F˜2,E)={e1,{x1,0.2,0.6,0.6,x2,0.3,0.7,0.9,x3,0.5,0.8,0.6}e2,{x1,0.1,0.9,0.5,x2,0.3,0.4,0.8,x3,0.4,0.3,0.7}}, {({\widetilde F_2},E)^\prime} = (\widetilde F,E) \cap ({\widetilde F_2},E) = \left\{ {\matrix{ {{e_1},\left\{ {\langle {x_1},0.2,0.6,0.6\rangle ,\langle {x_2},0.3,0.7,0.9\rangle ,\langle {x_3},0.5,0.8,0.6\rangle } \right\}} \cr {{e_2},\left\{ {\langle {x_1},0.1,0.9,0.5\rangle ,\langle {x_2},0.3,0.4,0.8\rangle ,\langle {x_3},0.4,0.3,0.7\rangle } \right\}} \cr } } \right\},

(F˜3,E)=(F˜,E)(F˜3,E)={e1,{x1,0.2,0.4,0.7,x2,0.3,0.5,0.9,x3,0.4,0.6,0.9}e2,{x1,0.1,0.7,0.5,x2,0.1,0.2,0.8,x3,0.4,0.3,0.9}}, {({\widetilde F_3},E)^\prime} = (\widetilde F,E) \cap ({\widetilde F_3},E) = \left\{ {\matrix{ {{e_1},\left\{ {\langle {x_1},0.2,0.4,0.7\rangle ,\langle {x_2},0.3,0.5,0.9\rangle ,\langle {x_3},0.4,0.6,0.9\rangle } \right\}} \cr {{e_2},\left\{ {\langle {x_1},0.1,0.7,0.5\rangle ,\langle {x_2},0.1,0.2,0.8\rangle ,\langle {x_3},0.4,0.3,0.9\rangle } \right\}} \cr } } \right\},

(F˜4,E)=(F˜,E)(F˜4,E)={e1,{x1,0.6,0.7,0.6,x2,0.2,0.5,0.3,x3,0.5,0.8,0.4}e2,{x1,0.3,0.3,0.5,x2,0.4,0.6,0.5,x3,0.7,0.2,0.7}}. {({\widetilde F_4},E)^\prime} = (\widetilde F,E) \cap ({\widetilde F_4},E) = \left\{ {\matrix{ {{e_1},\left\{ {\langle {x_1},0.6,0.7,0.6\rangle ,\langle {x_2},0.2,0.5,0.3\rangle ,\langle {x_3},0.5,0.8,0.4\rangle } \right\}} \cr {{e_2},\left\{ {\langle {x_1},0.3,0.3,0.5\rangle ,\langle {x_2},0.4,0.6,0.5\rangle ,\langle {x_3},0.7,0.2,0.7\rangle } \right\}} \cr } } \right\}.

In addition, ((F˜,E),τNSS(F˜,E),E) \left( {(\widetilde F,E),{{\mathop \tau \limits^{NSS} }_{(\widetilde F,E)}},E} \right) is a neutrosophic soft topological subspace of (X,τNSS,E) \;(X,\mathop \tau \limits^{NSS} ,E) .

Theorem 7

Let (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) be a neutrosophic soft topological space over X and (F̃,E), (K̃,E) ∈ NSS(X,E). Then,

If BNSS \mathop B\limits^{NSS} is a neutrosophic soft base for τNSS \mathop \tau \limits^{NSS} , then BNSS(F˜,E)={(B˜,E)(F˜,E):(B˜,E)BNSS} {{\mathop B\limits^{NSS}}_{(\widetilde F,E)}} = \left\{ {(\widetilde B,E) \cap (\widetilde F,E):(\widetilde B,E) \in \mathop B\limits^{NSS} } \right\} is a neutrosophic soft base for the neutrosophic soft sub-topology τNSS(F˜,E) {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} ,

If (G̃,E) is a neutrosophic soft closed set in τNSS(F˜,E) {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} and (F̃,E) is a neutrosophic soft closed set in τNSS(K,E) {{\mathop \tau \limits^{NSS}}_{(K,E)}} , then (G̃,E) is a neutrosophic soft closed set in τNSS(K,E) {{\mathop \tau \limits^{NSS}}_{(K,E)}} ,

Let (G̃,E) ⊆ (F̃,E). If (G˜,E)¯ \overline {(\widetilde G,E)} is the neutrosophic soft closure in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) , then (G˜,E)¯(F˜,E) \overline {(\widetilde G,E)} \cap (\widetilde F,E) is the neutrosophic soft closure in (X(F˜,E),τNSS(F˜,E),E) \left( {{X_{(\widetilde F,E)}},{{\mathop \tau \limits^{NSS} }_{(\widetilde F,E)}},E} \right) .

Proof

1. Since BNSS \mathop B\limits^{NSS} is a neutrosophic soft base for τNSS \mathop \tau \limits^{NSS} so for arbitrary (U˜,E)τNSS (\widetilde U,E) \in \mathop \tau \limits^{NSS} , we have (U˜,E)=(B˜,E)BNSS(B˜,E) (\widetilde U,E) = \bigcup\limits_{(\widetilde B,E) \in \mathop B\limits^{NSS} } (\widetilde B,E) . In case, (U˜,E)(F˜,E)=((B˜,E)BNSS(B˜,E))(F˜,E)=(B˜,E)BNSS((B˜,E)(F˜,E)) (\widetilde U,E) \cap (\widetilde F,E) = \left( {\bigcup\limits_{(\widetilde B,E) \in \mathop B\limits^{NSS} } (\widetilde B,E)} \right) \cap (\widetilde F,E) = \bigcup\limits_{(\widetilde B,E) \in \mathop B\limits^{NSS} } \left( {(\widetilde B,E) \cap (\widetilde F,E)} \right) for (U˜,E)(F˜,E)τNSS(F˜,E) (\widetilde U,E) \cap (\widetilde F,E) \in {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} . Since arbitrary member of τNSS(F˜,E) {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} can be expressed as the union of members of BNSS(F˜,E) {{\mathop B\limits^{NSS}}_{(\widetilde F,E)}} , hence the theorem is completed.

2. We first show that if (G̃,E) is a neutrosophic soft closed set in τNSS(F˜,E) {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} then there exist a closed set (Ṽ,E) ⊆ (K̃,E) i.e., (V˜,E)τNSS (\widetilde V,E) \notin \mathop \tau \limits^{NSS} such that (G̃,E) = (Ṽ,E) ∩ (F̃,E).

Let (G̃,E) be a closed in τNSS(F˜,E) {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} . Then (i,E)c is a neutrosophic soft open set in τNSS(F˜,E) {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} i.e., (i,E)c can be put as (G̃,E)c = (Ũ,E) ∩ (F̃,E) for (U˜,E)τNSS((G˜,E)c)c=(F˜,E)((U˜,E)(F˜,E))c=(U˜,E)c(F˜,E) (\widetilde U,E) \in \mathop \tau \limits^{NSS} \Rightarrow {\left( {{{(\widetilde G,E)}^c}} \right)^c} = (\widetilde F,E) \cap {\left( {(\widetilde U,E) \cap (\widetilde F,E)} \right)^c} = (\widetilde U,E{)^c} \cap (\widetilde F,E) . Here (U˜,E)cτNSS {(\widetilde U,E)^c} \notin \mathop \tau \limits^{NSS} i.e., (Ũ,E)c is a closed in τNSS \mathop \tau \limits^{NSS} . So here acts as (Ṽ,E) ⊆ (K̃,E).

Conversely, suppose that (G̃,E) = (Ṽ, E) ∩ (F̃,E) where (F̃,E) ⊆ (K̃,E) and (Ṽ, E) is closed in τNSS(K,E) {{\mathop \tau \limits^{NSS}}_{(K,E)}} . Clearly, (V˜,E)cτNSS {(\widetilde V,E)^c} \in \mathop \tau \limits^{NSS} so that (V˜,E)c(F˜,E)τNSS(F˜,E) {(\widetilde V,E)^c} \cap (\widetilde F,E) \in {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} . Now, (Ṽ,E)c ∩ (F̃,E) = ((K̃,E) \ (Ṽ,E)) ∩ (F̃,E) = ((K̃,E) ∩ (F̃,E)) \ ((Ṽ,E) ∩ (F̃,E)) = (F̃,E)\(G̃,E). This implies (F̃,E)\(G̃,E) is a neutrosophic soft open set in (F̃,E) i.e., (G̃,E) is a neutrosophic soft closed set in τNSS(F˜,E) {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} .

3. (G˜,E)¯={(G˜i,E):(G˜i,E)isclosedand(G˜i,E)(G˜,E)} \overline {(\widetilde G,E)} = \bigcap\nolimits \left\{ {({{\widetilde G}_i},E):({{\widetilde G}_i},E)\;{\rm{is}}\;{\rm{closed}}\;{\rm{and}}\;({{\widetilde G}_i},E) \supseteq (\widetilde G,E)} \right\} is a neutrosophic soft closure of (G̃,E) and so (G˜,E)¯ \overline {(\widetilde G,E)} is a neutrosophic soft closed set. Now, (G˜,E)¯(F˜,E)={(G˜i,E):(G˜i,E)isclosedand(G˜i,E)(G˜,E)}(F˜,E)=((G˜i,E)(F˜,E)) \overline {(\widetilde G,E)} \cap (\widetilde F,E) = \bigcap\nolimits \left\{ {({{\widetilde G}_i},E):({{\widetilde G}_i},E)\;{\rm{is}}\;{\rm{closed}}\;{\rm{and}}\;({{\widetilde G}_i},E) \supseteq (\widetilde G,E)} \right\} \cap (\widetilde F,E) = \bigcap\nolimits_ \left( {({{\widetilde G}_i},E) \cap (\widetilde F,E)} \right) . Since each (i,E) is closed, then each (i,E) ∩ (F̃,E) is closed in τNSS(F˜,E) {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} by Theorem-5. Now (G,E) ⊆ (i,E) and (G,E) ⊆ (F̃,E). So ((G̃,E) ∩ (F̃,E)) ⊆ ((i,E) ∩ (F̃,E)) ⇒ (G̃,E) ⊆ (i,E) ∩ (F̃,E). Therefore, (G˜,E)¯(F˜,E)={((G˜i,E)(F˜,E)):((G˜i,E)(F˜,E))isclosedand((G˜i,E)(F˜,E))(G˜,E)} \overline {(\widetilde G,E)} \cap (\widetilde F,E) = \bigcap \left\{ {\left( {({{\widetilde G}_i},E) \cap (\widetilde F,E)} \right):\left( {({{\widetilde G}_i},E) \cap (\widetilde F,E)} \right)\;{\rm{is}}\;{\rm{closed}}\;{\rm{and}}\;\left( {({{\widetilde G}_i},E) \cap (\widetilde F,E)} \right) \supseteq (\widetilde G,E)} \right\} . Thus, (G˜,E)¯(F˜,E) \overline {(\widetilde G,E)} \cap (\widetilde F,E) is a neutrosophic soft closure of (G̃,E) in τNSS(F˜,E) {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} .

Theorem 8

Let (X(F˜,E),τNSS(F˜,E),E) \left( {{X_{(\widetilde F,E)}},{{\mathop \tau \limits^{NSS} }_{(\widetilde F,E)}},E} \right) be a neutrosophic soft subspace of a neutrosophic soft topological space (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) over X. If (F̃,E) is a neutrosophic soft open set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) , then a neutrosophic soft set (1,E) ⊆ (F̃,E) is neutrosophic soft open set in (X(F˜,E),τNSS(F˜,E),E) \left( {{X_{(\widetilde F,E)}},{{\mathop \tau \limits^{NSS} }_{(\widetilde F,E)}},E} \right) iff (1,E) is a neutrosophic soft open set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) .

Proof

Suppose that (F̃,E) is a neutrosophic soft open set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) such that a neutrosophic soft subset (1,E) of (F̃,E) is open set in (X(F˜,E),τNSS(F˜,E),E) \left( {{X_{(\widetilde F,E)}},{{\mathop \tau \limits^{NSS} }_{(\widetilde F,E)}},E} \right) . Then (F˜1,E)τNSS(F˜,E) ({\widetilde F_1},E) \in {{\mathop \tau \limits^{NSS}}_{(\widetilde F,E)}} and so (1,E) = (Ũ,E) ∩ (F̃,E) for (U˜,E)τNSS (\widetilde U,E) \in \mathop \tau \limits^{NSS} . But (1,E) is a neutrosophic soft open set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) as (Ũ,E) and (F̃,E) both are neutrosophic soft open set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) .

Conversely, assume that (1,E) is a neutrosophic soft open set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) when (F̃,E) is a neutrosophic soft open set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) and (1,E) ⊆ (F̃,E). Then (F˜1,E)τNSS ({\widetilde F_1},E) \in \mathop \tau \limits^{NSS} . But (1,E) ∩ (F̃,E) = (1,E) and so (1,E) is a neutrosophic soft open set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) . Therefore, the first part is proved.

Theorem 9

Let (X(K˜,E),τNSS(K˜,E),E) \left( {{X_{(\widetilde K,E)}},{{\mathop \tau \limits^{NSS} }_{(\widetilde K,E)}},E} \right) be a neutrosophic soft subspace of a neutrosophic soft topological space (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) over X. If (K̃,E) is a neutrosophic soft closed set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) , then a neutrosophic soft set (1,E) ⊆ (K̃,E) is a neutrosophic soft closed set in (X(K˜,E),τNSS(K˜,E),E) \left( {{X_{(\widetilde K,E)}},{{\mathop \tau \limits^{NSS} }_{(\widetilde K,E)}},E} \right) iff (1,E) is a neutrosophic soft closed set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) .

Proof

Suppose that (K̃,E) is a neutrosophic soft closed set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) such that a neutrosophic soft subset (1,E) of (K̃,E) is neutrosophic soft closed set in (X(K˜,E),τNSS(K˜,E),E) \left( {{X_{(\widetilde K,E)}},{{\mathop \tau \limits^{NSS} }_{(\widetilde K,E)}},E} \right) . Since (1,E) is closed in (X(K˜,E),τNSS(K˜,E),E) \left( {{X_{(\widetilde K,E)}},{{\mathop \tau \limits^{NSS} }_{(\widetilde K,E)}},E} \right) and so (1,E) = (Ṽ, E) ∩ (K̃,E) for (Ṽ, E) being neutrosophic soft closed set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) . But (1,E) is a neutrosophic soft closed set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) as (Ṽ, E) and (K̃,E) both are neutrosophic soft closed sets in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) .

Conversely, assume that (1,E) is a neutrosophic soft closed set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) when (K̃,E) is neutrosophic soft closed set in (X,τNSS,E) (X,\mathop \tau \limits^{NSS} ,E) and (1,E) ⊆ (K̃,E). Then (1,E) ∩ (K̃,E) = (1,E) and so (1,E) is a neutrosophic soft closed in (X(K˜,E),τNSS(K˜,E),E) \left( {{X_{(\widetilde K,E)}},{{\mathop \tau \limits^{NSS} }_{(\widetilde K,E)}},E} \right) . Hence the first part is proved.

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.

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