A look at soft neutrosophic spaces

1 Introduction

Fuzziness has modernized many areas such as mathematics, science, engineering and medicine. This concept was initiated by Zadeh [1]. Not only Zadeh discovered this concept, but he also developed the infrastructure of today’s popular forms of use such as relations of similarity, decision making, and fuzzy programming in a short time.

Pawlak [2] introduced a new concept of rough sets. Each of these theories has its own inherent difficulties. To finish out these difficulties, Molodtsov [3] introduced completely a new approach of soft set theory. Maji et al. [4] functionalized soft sets in multicriteria decision making problems by applying the technique of knowledge reduction to the information table induced by soft set. Maji et al. [5] introduced and studied several basic notions of soft set theory. Pei and Miao [6] and Chen [7] improved the work of Maji et al. Broumi et al. [8] extended the concept to the case of interval valued neutrosophic soft relation (IVNSS relation for short) which can be discussed as a generalization of soft relations, fuzzy soft relation, intuitionistic fuzzy soft relation, interval valued intuitionistic fuzzy soft relations and neutrosophic soft relations. Basic operations are presented and the various properties like reflexivity, symmetry, transitivity of IVNSS relations are also studied. Smarandache [9] for the first time floated the concept of neutrosophic set which is generalization of the intuitionistic fuzzy set (IFS), and intuitionistic set (NS). Some related examples are presented. Peculiarities between NS and IFS are underlined. Every element in the neutrosophic set has truth, indeterminacy and falsity values respectively and which are maps from universe of discourse to [0, 1] with the constraint that truth, indeterminacy and falsity values should not exceed 3 and not less than zero under addition. Many complex problems in statistics, in graph theory when relationship between the object have acceptance, rejection and also indeterminacy, physics, image processing, networking and in decision making which can’t be solved by existing classical methods. Wang et al. [10] thought that neutrosophic sets are difficult to be applied in some genuine issues on account of truth, indeterminacy and falsity membership degree.

So they jointly presented the idea of a single-valued neutrosophic set. The single-valued neutrosophic set can freely express truth-membership degree, indeterminacy-membership degree, and falsity-membership degree and manages inadequate, uncertain and conflicting data. All the aspects of the elements depicted by the single-valued neutrosophic set are entirely appropriate for human intuition because of the flaw of information that human gets or sees from the surrounding.

Smarandache [11] introduced a technique of extending soft set to Hyper soft set, and then to Plithogenic Hyper-soft set. Cagman et al. [12] defined the concept of soft topology on a soft set, and presented its related properties. The authors also discussed the foundations of the theory of soft topological spaces. Shabir and Naz [13] introduced soft topological spaces over an initial universe with a fixed set of parameters. The notions of soft open sets, soft closed sets, soft closure, soft interior points, soft neighborhood of a point, soft separation axioms and their basic properties were investigated. It is shown that a soft topological space gives a parametrized family of topological spaces. The authors discussed soft subspaces of a soft topological space and investigated characterization with respect to soft open and soft closed sets. Finally, soft T _i -spaces, notions of soft normal and soft regular spaces are addressed. Bayramov and Gunduz [14] investigated some basic notions of soft topological spaces by using soft point notion. Later the authors addressed T _i -soft spaces and the relationships between them. Lastly, the authors defined soft compactness and explored some of its significant characteristics.

Khattak et al. [15] introduced the concept of soft (α,β)-open sets and their characterization in soft single point topology. Atanassov [16] introduced the concept of ‘intuitionistic fuzzy set’ (IFS) which is an extension of the concept ‘fuzzy set’. The authors discussed various properties including operations and relations over sets. Bayramov and Gunduz [17] introduced some important properties of intuitionistic fuzzy soft topological spaces and defined the intuitionistic fuzzy soft closure and interior of an intuitionistic fuzzy soft set. Furthermore, intuitionistic fuzzy soft continuous mapping and structural characteristics were discussed in their study. Deli and Broumi [18] defined for the first define a relation on neutrosophic soft sets. The new concept allows the composition of two neutrosophic soft sets. It is devised to derive useful information through the composition of two neutrosophic soft sets. Finally a decision making method on neutrosophic soft sets is presented.

Bera and Mahapatra [19] introduced the concept of cartesian product and the relationship on neutrosophic soft sets with a new attempt. Some properties of this concept have been discussed and verified with appropriate real life examples. The neutrosophic soft composition has been defined and verified with the help of examples. Then, some basic properties neutrosophic set have been established. Further, the authors introduced neutrosophic soft function it’s properties have been introduced and some suitable examples. Injective, surjective, bijective, constant and identity neutrosophic soft functions have been defined. Finally, properties of inverse neutrosophic soft function have been discussed with proper examples.

Maji [20] stretched the study of Smarandache. The author used the concept of neutrosophic set in soft set and introduced neutrosophic soft set. Some definitions and related operations were introduced on neutrosophic soft set.

Akram et al. [21] introduced the notion of single-valued neutrosophic soft topological K-algebras. The authors discussed certain concepts, including interior, closure, connected, super connected, Compactness and Hausdorff in single-valued neutrosophic soft topological K-algebras. In continuation the authors illustrated these concepts with examples and investigate some of their related properties. The authors also studied image and pre-image of single-valued neutrosophic soft topological K-algebras.

Akram et al. [22] applied the notion of single-valued neutrosophic sets to K-algebras. Further, the authors studied certain properties, including connected, super connected introduced the notion of single-valued neutrosophic topological K-algebras and investigate some of their, compact and Hausdorff, of single-valued neutrosophic topological K-algebras. The authors also investigated the image and pre-image of single-valued neutrosophic topological K-algebras under homomorphism.

Akram et al. [23] leaked out that neutrosophic sets and soft sets are two different mathematical tools for representing vagueness and uncertainty. The authors applied these models in combination to study vagueness and uncertainty in K-algebras. Further, the authors introduced the notion of single-valued neutrosophic soft (SNS) K-algebras and investigated some of their properties. Lastly the authors established the notion of (∈, ∈ ∨q)-single-valued neutrosophic soft K-algebras and describe some of their related properties.

Kamal et al. [24] introduced the concept of multi-valued interval neutrosophic soft set which amalgamates multi-valued interval neutrosophic set and soft set. According to the authors the proposed set extended the notions of fuzzy set, intuitionistic fuzzy set, neutrosophic set, interval-valued neutrosophic set, multi-valued neutrosophic set, soft set and neutrosophic soft set. Further, the authors studied some basic operations such as complement, equality, inclusion, union, intersection, “AND” and “OR” for multi-valued interval neutrosophic soft elements and discuss its associated properties. Moreover, the derivation of its properties, related examples and some proofs on the propositions are included.

Karaaslan and Deli [25] introduced the concept of soft neutrosophic classical sets and its set theoretical operations such as; union, intersection, complement, AND-product and OR-product. In addition to these concept and operations, the authors defined four basic types of sets of degenerate elements in a soft neutrosophic classical set. Then the authors proposed a group decision making method based on soft neutrosophic classical sets, and give algorithm of proposed method. The authors also made an application of proposed method on a problem including soft neutrosophic classical data.

Karaaslan [26] studied the concept of single-valued neutrosophic refined soft set is defined as an extension of single-valued neutrosophic refined set. Also, set theoretical operations between two single-valued neutrosophic refined soft sets are defined and some basic properties of these operations are investigated. Furthermore, two methods to calculate correlation coefficient between two single-valued neutrosophic refined soft sets are proposed, and a clustering analysis application of one of proposed methods is given.

Karaaslan [27] introduced the concept of possibility neutrosophic soft set and defined some related concepts such as possibility neutrosophic soft subset, possibility neutrosophic soft null set, and possibility neutrosophic soft universal set. Then, based on definitions of n-norm and n-conorm, we define set theoretical operations of possibility neutrosophic soft sets such as union, intersection and complement, and investigate some properties of these operations. The author also introduced AND-product and OR-product operations between two possibility neutrosophic soft sets. The author proposed a decision making method called possibility neutrosophic soft decision making method (PNS-decision making method) which can be applied to the decision making problems involving uncertainty based on AND-product operation. The author finally gave a numerical example to display application of the method that can be successfully applied to the problems.

Karaaslan [28] introduced a similarity measure between possibility neutrosophic soft sets (PNS-set) is defined, and its properties are studied. A decision making method is established based on proposed similarity measure. Finally, an application of this similarity measure involving the real life problem is given.

Karaaslan [29] addressed firstly Maji’s definitions (Maji-2013) and verified that that some propositions are incorrect by a counterexample. The author then redefined concept of neutrosophic soft set and their operation based on Çağman (Çagman-2014} and investigated some properties of these operations. The author then constructed decision making method and group decision making which selects an optimum element from the alternatives by using weights of parameters. The author finally presented an example which shows that the method can be successfully applied to many problems that contain uncertainties.

Bera and Mahapatra [30] introduced the construction of topology on a neutrosophic soft set (NSS). The notion of neutrosophic soft interior, neutrosophic soft closure, neutrosophic soft neighborhood, neutrosophic soft boundary, regular NSS and their basic properties are studied in this study. The base for neutrosophic soft topology and subspace topology on NSS are defined with suitable examples. Some related properties were also established. Moreover, the concept of separation axioms on neutrosophic soft topological space have been introduced along with investigation of several structural characteristics. Hayat et al. [31] inaugurated some new operations on type-2 soft sets and discussed related properties. By means of this new definitions some certain laws are verified. Hayat et al. [32] established correspondence between a vertex and its neighbors has an essential role in the structure of a graph. Type-2 soft sets are also based on the correspondence of primary parameters and underlying parameters. The authors presented an application of type-2 soft sets in graph theory. In continuation, the authors introduced vertex-neighbors based type-2 soft sets over X (set of all vertices of a graph) and E (set of all edges of a graph). Moreover, the authors introduced some type-2 soft operations in graphs by presenting several examples to demonstrate these new concepts. Finally, the authors described an application of type-2 soft graphs in communication networks and present procedure as an algorithm. Hayat et al. [33] developed a promising framework based on soft sets, TOPSIS and the Shannon entropy. Customer’s preferences on incorporating design values are identified based on acceptable- and satisfactory-level needs, and these preferences are weighted through Shannon entropy. By performing AND operation on the soft set of level requirements of one customer with the soft set of requirements of another customer, several weighted tables of soft sets are obtained on the pair of design parameters values. To obtain the best concept on different levels of requirements, TOPSIS is performed which provides several integrated evaluations. An illustration is considered for the demonstration of the method, brings the best concept for two customers which is acceptable for both of the customers, satisfactory for both the customers and vise-versa. Finally, the comparisons are presented with recent major existing techniques.

Hayat et al. [34] introoduced some novel operations on neutrosophic matrices and addressed different results on these operations.

Khattak et al. [35] for the first time bounced up the idea of neutrosophic soft b-open set, neutrosophic soft b-closed sets and their properties. Also the idea of neutrosophic soft b-neighborhood and neutrosophic soft b-separation axioms in neutrosophic soft topological structures are reflected here. Later on the important results are discussed related to these newly defined concepts with respect to soft points. The concept of neutrosophic soft b-separation axioms of neutrosophic soft topological spaces is diffused in different results with respect to soft points. Furthermore, properties of neutrosophic soft b T _i -space (i = 0, 1, 2, 3, 4) and some associations between them are discussed. Mehmood et al. [35] leaked out the notion of neutrosophic soft p-open set, neutrosophic soft p-open set, neutrosophic soft p-close set and their important characteristics. Also the notion of neutrosophic soft p-neighborhood and neutrosophic soft p-separation axioms in neutrosophic soft topological spaces are developed.

These references [35–37] became source of motivation for my present work. In reference [35] the authors defined neutrosophic soft b-open set in neutrosophic soft topological spaces and then with the help of this definition, some sort of neutrosophic soft separation axioms are addressed. Same is done in reference [36] with respect to neutrosophic soft p-open sets. In reference [37] the authors defined neutrosophic soft * _b open sets in neutrosophic topological spaces. In our study, we proceed as follows:

In this article, the concept of neutrosophic soft **  _b open sets in neutrosophic topological spaces is initiated and its structural characteristics are attempted with respect to soft points. The rest of the article is pictured as follows. In section 2, some fundamental definitions, which are neutrosophic soft union, neutrosophic soft intersection, null neutrosophic soft set, absolute neutrosophic soft set, neutrosophic soft topology, neutrosophic soft and neutrosophic soft neighborhood are discussed. In section 3, eight new definitions are introduced with respect to soft points in neutrosophic soft topological spaces. Out of these eight definitions, neutrosophic soft **  _b open set is chosen to see its beauty in neutrosophic soft topological structures. Then with the application of this new definition some neutrosophic soft separation axioms and neutrosophic soft other separation axioms are generated with respect to soft points of the spaces. The inter-play among these neutrosophic soft separation axioms is also displayed relative to soft points of the spaces. These results are verified with examples. The second criteria of **  _{b 1} space is displayed. The engagement of neutrosophic soft **  _{b 1} space with NS **  _b -open set, neutrosophic soft **  _{b 3} spaces relation with neutrosophic soft closer with respect to soft points of the spaces.

In section 4, neutrosophic soft countability that neutrosophic soft first countability and neutrosophic soft second countability, the relation-ship between these results, neutrosophic soft sequences and their convergence in NS **  _b Hausdorff space and the cardinality results are discussed. In section 5, neutrosophic soft topological characteristics and inverse neutrosophic soft topological characteristics are addressed with respect to soft points. Neutrosophic soft product of neutrosophic soft structures is inaugurated. Characterization of Bolzano Weirstrass Property, neutrosophic soft compactness and related results and neutrosophic sequentially compactness are discussed. In section 6, some concluding remarks and future work is exhibited.

2 Preliminaries

In this section, we give some useful definitions for neutrosophic soft sets that are required for the up-coming sections. The contents of this section are mainly taken from [3, 35].

Definition 2.1. [9] A neutrosophic set A on the father set 〈 X 〉 is defined as: A = { 〈 x , T A ( x ) , I A ( x ) , F A ( x ) : x ∈ 〈 X 〉 〉 } where, T : 〈 X 〉 → ] 0 - , 1 + [ , I : 〈 X 〉 → ] 0 - , 1 + ] , F : 〈 X 〉 → ] 0 - , 1 + and 0 - ≼ T A ( x ) + I A ( x ) + F A ( x ) ≼ 3 + .

Definition 2.2. [3] Let 〈 X 〉 be a father set, θ be a set of all parameters, and L ( 〈 X 〉 ) denotes the power set of 〈 X 〉 . A pair ( f , θ ) is referred to as a soft set over 〈 X 〉 , where f is a map given by f : θ → L ( 〈 X 〉 ) . Then a NS set ( f ˜ , θ ) over 〈 X 〉 is a set defined by a set of valued functions signifying a mapping f ˜ : θ → L ( 〈 X 〉 ) , is referred to as the approximate NS set function ( f ˜ , θ ) . It can be written as a set of ordered pairs: ( f ˜ , θ ) = { ( ( n , x , T f ˜ ( n ) ( x ) , I f ˜ ( n ) ( x ) , F f ˜ ( n ) ( x ) : x ∈ 〈 X 〉 ) : n ∈ θ } .

Definition 2.3. [35] Let ( f ˜ , θ ) be a NSS over the father set 〈 X 〉 . The complement of ( f ˜ , θ ) is signified ( f ˜ , θ ) c and is defined as follows: ( f ˜ , θ ) c = { ( ( n , [ [ x , F f ˜ ( n ) ( x ) , 1 - I f ˜ ( n ) ( x ) , T f ˜ ( n ) ( x ) : x ∈ 〈 X 〉 ] ] ) : n ∈ θ } .

It’s clear that ( ( f ˜ , θ ) c ) c = ( f ˜ , θ ) .

Definition 2.4. [35] Let ( f ˜ , θ ) and ( ρ ˜ , θ ) two NSS over the father 〈 X 〉 . ( f ˜ , θ ) is supposed to be NSSS of ( ρ ˜ , θ ) if T f ˜ ( n ) ( x ) ≼ T ρ ˜ ( n ) ( x ) , I f ˜ ( n ) ( x ) ≼ I ρ ˜ ( n ) ( x ) , F f ˜ ( n ) ( x ) ≽ F ρ ˜ ( n ) ( x ) , ∀ n ∈ θ & ∀ x ∈ 〈 X 〉 . It is signifies as ( f ˜ , θ ) ⊑ ( ρ ˜ , θ ) . ( f ˜ , θ ) is said to be NS equal to ( ρ ˜ , θ ) if ( f ˜ , θ ) is NSSS of ( ρ ˜ , θ ) and ( ρ ˜ , θ ) is NSSS of if ( f ˜ , θ ) . It is symbolized as ( f ˜ , θ ) = ( ρ ˜ , θ ) .

Definition 2.5. [35] Let ( f ˜ 1 , θ ) and ( f ˜ 2 , θ ) be two NSSS over father set 〈 X 〉 such that ( f ˜ 1 , θ ) ≠ ( f ˜ 2 , θ ) . Then their union is signifies as ( f ˜ 1 , θ ) ∪ ˜ ( f ˜ 2 , θ ) = ( f ˜ 3 , θ ) & is defined as ( f ˜ 3 , θ ) = { ( ( n , [ [ x , T f ˜ 3 ( n ) ( x ) , I f ˜ 3 ( n ) ( x ) , F f ˜ 3 ( n ) ( x ) : n ∈ 〈 X 〉 ] ] ) : ∈θ}. where, { T f ˜ 3 ( n ) ( x ) = max [ T f ˜ 1 ( n ) ( x ) , T f ˜ 2 ( n ) ( x ) , I f ˜ 3 ( n ) ( x ) = max [ I f ˜ 1 ( n ) ( x ) , I f ˜ 2 ( n ) ( x ) , F f ˜ 3 ( n ) ( x ) = min [ F f ˜ 1 ( n ) ( x ) , F f ˜ 2 ( n ) ( x ) .

Definition 2.6. [35] Let ( f ˜ 1 , θ ) and ( f ˜ 2 , θ ) be two NSSS over the father set 〈 X 〉 such that ( f ˜ 1 , θ ) ≠ ( f ˜ 2 , θ ) . Then their intersection is signifies as ( f ˜ 1 , θ ) ⊓ ˜ ( f ˜ 2 , θ ) = ( f ˜ 3 , θ ) & is defined as ( f ˜ 3 , n ) = { ( ( n , n , T f ˜ 3 ( n ) ( x ) , I f ˜ 3 ( n ) ( x ) , F f ˜ 3 ( n ) ( x ) : 〈 X 〉 ) : ∈ θ } . where , where { T f ˜ 3 ( n ) ( x ) = min [ T f ˜ 1 ( n ) ( x ) , T f ˜ 2 ( n ) ( x ) , I f ˜ 3 ( n ) ( x ) = min [ I f ˜ 1 ( n ) ( x ) , I f ˜ 2 ( n ) ( x ) , F f ˜ 3 ( n ) ( x ) = max [ F f ˜ 1 ( n ) ( x ) , F f ˜ 2 ( n ) ( x ) .

Definition 2.7. [35] NSS ( f ˜ , θ ) be a NSS over the father set 〈 X 〉 is said to be a null neutrosophic soft set If T f ( n ) ( x ) = 0 , I f ( n ) ( x ) = 0 , F f ( n ) ( x ) = 1 ; , ∀ n ∈ θ & ∀ x ∈ 〈 X 〉 .

It is signifies as 0 ( 〈 X 〉 , θ ) .

Definition 2.8. [35] NSS ( f ˜ , θ ) over the father set 〈 X 〉 is an absolute NSS if T f ( n ) ( x ) = 1 , I f ( n ) ( x ) = 1 , F f ( n ) ( x ) = 0 ; , ∀ n ∈ θ & ∀ n ∈ 〈 X 〉 .

It is signified as 1 ( 〈 X 〉 , n ) . clearly, 0 ( 〈 X 〉 , n ) c = 1 ( 〈 X 〉 , n ) & 1 ( 〈 X 〉 , ) c = 0 ( 〈 X 〉 , n ) .

Definition 2.9. [35] Let NSS ( 〈 X ˜ 〉 , θ ) be the family of all NS soft sets and τ ⊂ NSS ( 〈 X ˜ 〉 , θ ) . then τ is said to be a NS soft topology on 〈 X ˜ 〉 if (1). 0 ( 〈 X 〉 , n ) , 1 ( 〈 X 〉 , n ) ∈ τ , ( 2 ) . The union of any number of NS soft sets in τ ∈ τ, (3). The intersection of a finite number of NSsoft sets in in τ ∈ τ. Then ( 〈 X ˜ 〉 , τ , θ ) is sai d to be a NSTS over 〈 X ˜ 〉 .

Definition 2.10. [35] Let NS be the family of all NS over 〈 X ˜ 〉 and ∈ 〈 X ˜ 〉 , then NS x ( a , b , c ) is supposed to be a N point, for 0 < a , b , c ⩽ 1 and is defined as follows: x ( a , b , c ) ( y ) = ⌊ { ( a , b , c ) if y = x } ( 0 , 0 , 1 ) if y ≠ x } ⌋

Definition 2.11. [35] Let NSS ( 〈 X 〉 ) ˜ be the family of all N soft sets over the father set 〈 X 〉 . ˜ Then NSS ( x ( a , b , c ) ) e is called a NS point, for every ∈ 〈 X 〉 , ˜ 0 ≺ { a , b , c } ≼ 1 , e ∝ θ , and is defined as follows: x ( a , b , c ) e ` ( y ) e = ( ( a , b , c ) if e ` = e ∧ y = x ( 0 , 0 , 1 ) if e ` ≠ e ∧ y ≠ x )

Definition 2.12. [35] Let ( f ˜ , θ ) be a NSS over the father set 〈 X ˜ 〉 . We say that e ( a , b , c ) ∈ ( f ˜ , θ ) read as belonging to the NSS ( f ˜ , θ ) whenever a ≼ T f ˜ ( x ) ( x ) , b ≼ I f ˜ ( x ) ( x ) , c ≽ F f ˜ ( x ) ( x ) .

Definition 2.13. [35] ( 〈 X ˜ 〉 , τ , θ ) be a NSTS over 〈 X ˜ 〉 & ( f ˜ , θ ) be a NSS over 〈 X ˜ 〉 . NSS ( f ˜ , θ ) in ( 〈 X ˜ 〉 , τ , θ ) is called a NS nbhd of the NS point x ( a , b , c ) e ∈ ( f ˜ , θ ) , if there exists a NS open set (g˜, θ) such that e ( a , b , c ) ∈ ( g ˜ , θ ) ⊏ ( f ˜ , θ ) .

3 Characterization of neutrosophic soft separation axioms in terms of neutrosophic soft points

In this section, eight new definitions are introduced with respect to crisp points in neutrosophic soft topological spaces. Among these definitions neutrosophic soft **  _b -open set is chosen to generate the neutrosophic soft structures with respect to soft pints. Suitable examples are also given to defend some results.

Definition 3.1. Let ( 〈 X ˜ 〉 , τ , θ ) be a NSTS over 〈 X ˜ 〉 & ( f ˜ , θ ) be a NS set over 〈 X ˜ 〉 then ( f ˜ , θ ) is called neutrosophic soft

(1):

Semi-open if ( f ˜ , θ ) ⊆ NScl ( NSint ( f ˜ , θ ) ) and neutrosophic soft Semi-close if ( f ˜ , θ ) ⊇ NSint ( NScl ( f ˜ , θ ) ) .

Pre-open if ( f ˜ , θ ) ⊆ NSint ( NScl ( f ˜ , θ ) ) and neutrosophic soft pre close if ( f ˜ , θ ) ⊇ NScl ( NSint ( f ˜ , θ ) ) .

α-open if ( f ˜ , θ ) ⊆ NSint ( NScl ( NSint ( f ˜ , θ ) ) ) and neutrosophic soft α close if ( f ˜ , θ ) ⊇ NScl ( NSint ( NScl ( f ˜ , θ ) ) ) .

β-open if ( f ˜ , θ ) ⊆ NScl ( NSint ( NScl ( f ˜ , θ ) ) ) and neutrosophic soft β close if ( f ˜ , θ ) ⊇ NSint ( NScl ( NSint ( f ˜ , θ ) ) ) .

b-open if ( f ˜ , θ ) ⊆ NScl ( NSint ( f ˜ , θ ) ) ∪ NSint ( NScl ( f ˜ , θ ) ) and neutrosophic soft b close if ( f ˜ , θ ) ⊇ NSint ( NScl ( f ˜ , θ ) ) ∩ NScl ( NSint ( f ˜ , θ ) ) .

* _b -open if ( f ˜ , θ ) ⊆ NScl ( NSint ( f ˜ , θ ) ) ∩ NSint ( NScl ( f ˜ , θ ) ) and neutrosophic soft * _b close if ( f ˜ , θ ) ⊇ NSint ( NScl ( f ˜ , θ ) ) ∪ NScl ( NSint ( f ˜ , θ ) ) .

b^**-open if ( f ˜ , θ ) ⊆ NSint ( NScl ( NSint ( f ˜ , θ ) ) ) ∪ NScl ( NSint ( NScl ( f ˜ , θ ) ) ) and neutrosophic soft b^**-close if ( f ˜ , θ ) ⊇ NScl ( NSint ( NScl ( f ˜ , θ ) ) ) ∩ NSint ( NScl ( NSint ( f ˜ , θ ) ) ) .

**  _b -open if ( f ˜ , θ ) ⊆ NSint ( NScl ( NSint ( f ˜ , θ ) ) ) ∩ NScl ( NSint ( NScl ( f ˜ , θ ) ) ) and neutrosophic soft **  _b -close if ( f ˜ , θ ) ⊇ NScl ( NSint ( NScl ( f ˜ , θ ) ) ) ∪ NSint ( NScl ( NSint ( f ˜ , θ ) ) ) .

Definition 3.2. Let ( 〈 X ˜ 〉 , τ , θ ) be a NSTS over 〈 X ˜ 〉 , x ( a , b , c ) e ≠ y ( a / , b / , c / ) e / are NS points. If there exist NS **  _b -open sets ( f ˜ , θ ) & ( g ˜ , θ ) such that

x ( a , b , c ) e ∈ ( f ˜ , θ ) , x ( a , b , c ) e ⊓ ( g ˜ , θ ) = 0 ( 〈 X ˜ 〉 , θ ) or y ( a / , b / , c / ) e / ∈ ( g ˜ , θ ) , y ( a / , b / , c / ) e / ⊓ ( f ˜ , θ ) = 0 ( 〈 X ˜ 〉 , θ ) , Then ( 〈 X ˜ 〉 , τ , θ ) is called a NS **  _{b 0} space.

Definition 3.3. Let ( 〈 X ˜ 〉 , τ , θ ) be a NSTS over 〈 X ˜ 〉 , x ( a , b , c ) e ≠ y ( a / , b / , c / ) e / are NS points. If there e xi st NS **  _b -open sets ( f ˜ , θ ) & ( g ˜ , θ ) such that x ( a , b , c ) e ∈ ( f ˜ , θ ) , x ( a , b , c ) e ⊓ ( g ˜ , θ ) = 0 ( X ˜ , θ ) and y ( a / , b / , c / ) e / ∈ ( g ˜ , θ ) , y ( a / , b / , c / ) e / ⊓ ( f ˜ , θ ) = 0 ( X ˜ , θ ) ,

Then ( 〈 X ˜ 〉 , τ , θ ) is called a NS **  _{b 1} space.

Definition 3.4. Let ( 〈 X ˜ 〉 , τ , θ ) be a NSTS over 〈 X 〉 , ˜ x ( a , b , c ) e ≠ y ( a / , b / , c / ) e / are NS points. If ∃NS **  _b open set ( f ˜ , θ ) & (g˜, θ) such that x ( a , b , c ) e ∈ ( f ˜ , θ ) and y ( a / , b / , c / ) e / ∈ ( g ˜ , θ ) and ( f ˜ , θ ) ⊓ ( g ˜ , θ ) = 0 ( X ˜ , θ ) . Then ( X ˜ , τ , θ ) is called a NS **  _{b 2} space.

Definition 3.5. Let ( 〈 X ˜ 〉 , τ , θ ) be a NSTS. ( f ˜ , θ ) be a NS **  _b closed set and ( x ( a , b , c ) e , θ ) ⊓ ( f ˜ , θ ) = 0 ( 〈 X ˜ 〉 , θ ) . If there exists NS **  _b -open sets (g˜₁, θ) & (g˜₂, θ) such that ( x ( a , b , c ) e , θ ) ∈ ( g ˜ 1 , θ ) , ( f ˜ , θ ) ⊂ ( g ˜ 2 , θ ) & ( x ( a , b , c ) e , θ ) ⊓ ( g ˜ 1 , θ ) = 0 ( 〈 X ˜ 〉 , θ ) , then ( 〈 X ˜ 〉 , τ , θ ) is called a NS **  _b -regular space. ( 〈 X ˜ 〉 , τ , θ ) is said to be NS **  _{b 3} space, if is both a NS regular and NS **  _{b 1} space.

Definition 3.6. Let ( 〈 X ˜ 〉 , τ , θ ) be a NSTS. This space is a NS **  _b normal space, if for every pair of disjoint NS **  _b close sets ( f ˜ 1 , θ ) and ( f ˜ 2 , θ ) , ∃ disjoint NS **  _b open sets (g˜₁, θ) and (g˜₂, θ) such that ( f ˜ 1 , θ ) ⊂ ( g ˜ 1 , θ ) & ( f ˜ 2 , θ ) ⊂ ( g ˜ 2 , θ ) .

( 〈 X ˜ 〉 , τ , θ ) is said to be a NS **  _{b 4} space if it is both a NS **  _b normal and NS **  _{b 1} space.

Example 3.7. Suppose that the father set 〈 X ˜ 〉 is assumed to be 〈 X ˜ 〉 = { x 1 , x 2 } & the set of conditions by θ ={ e₁, e₂ }. Let us consider NS set ( f ˜ , θ ) & x 1 ( 0.1 , 0.4 , 0.7 ) e 1 , x 1 ( 0.2 , 0.5 , 0.6 ) e 2 and x 2 ( 0.4 , 0.4 , 0.4 ) e 1 be NS points. Then the family τ = { 0 ( 〈 X ˜ 〉 , θ ) , 1 ( 〈 X ˜ 〉 , θ ) , ( f 1 ˜ , θ ) , ( f 2 ˜ , θ ) , ( f 3 ˜ , θ ) , ( f 4 ˜ , θ ) , ( f 5 ˜ , θ ) , ( f 6 ˜ , θ ) , ( f 7 ˜ , θ ) , ( f 8 ˜ , θ ) } , where ( f 1 ˜ , θ ) = x 1 ( 0.1 , 0.4 , 0.7 ) e 1 , ( f 2 ˜ , θ ) = x 1 ( 0.2 , 0.5 , 0.6 ) e 2 , ( f 3 ˜ , θ ) = x 2 ( 0.3 , 0.3 , 0.5 ) e 1 , ( f 4 ˜ , θ ) = ( f 1 ˜ , θ ) ∪ ( f 2 ˜ , θ ) , ( f 5 ˜ , θ ) = ( f 1 ˜ , θ ) ∪ ( f 3 ˜ , θ ) , ( f 6 ˜ , θ ) = ( f 2 ˜ , θ ) ∪ ( f 3 ˜ , θ ) , ( f 7 ˜ , θ ) = ( f 1 ˜ , θ ) ∪ ( f 2 ˜ , θ ) ∪ ( f 3 ˜ , θ ) , ( f 8 ˜ , θ ) = ‘ { x 1 ( 0.1 , 0.4 , 0.7 ) e 1 , x 1 ( 0.2 , 0.5 , 0.6 ) e 2 , x 2 ( 0.3 , 0.3 , 0.5 ) e 1 , x 2 ( 0.4 , 0.4 , 0.4 ) e 2 } is a NSTS. Thus ( 〈 X ˜ 〉 , τ, θ) be a NSTS. Also ( 〈 X ˜ 〉 , τ , θ ) is NS **  _{b 0} structure but it is not NS **  _{b 1} because for NS points x 1 ( 0.1 , 0.4 , 0.7 ) e 1 , x 2 ( 0.4 , 0.4 , 0.4 ) e 2 , ( 〈 X ˜ 〉 , τ , θ ) not NS **  _{b 1} .

Example 3.8. Suppose that the father set 〈 X ˜ 〉 is assumed to be 〈 X ˜ 〉 = { x 1 , x 2 } & the set of conditions by ^Fparameter ={ e₁, e₂ }. Let us consider NS set ( f ˜ , θ ) and x 1 ( 0.1 , 0.4 , 0.7 ) e 1 , x 1 ( 0.2 , 0.5 , 0.6 ) e 2 , x 2 ( 0.3 , 0.3 , 0.5 ) e 1 and x 2 ( 0.4 , 0.4 , 0.4 ) e 1 be NS points. Then the family τ = { 0 ( X ˜ , θ ) , 1 ( X ˜ , θ ) , ( f 1 ˜ , θ ) , ( f 2 ˜ , θ ) , ( f 3 ˜ , θ ) , , ( f 4 ˜ , θ ) , ( f 5 ˜ , θ ) , ( f 6 ˜ , θ ) , , ( f 7 ˜ , θ ) , ( f 8 ˜ , θ ) , … … … … ( f 15 ˜ , θ ) } , where ( f 1 ˜ , θ ) = x 1 ( 0.1 , 0.4 , 0.7 ) e 1 , ( f 2 ˜ , θ ) = x 1 ( 0.2 , 0.5 , 0.6 ) e 2 , ( f 3 ˜ , θ ) , x 1 ( 0.3 , 0.3 , 0.5 ) e 1 , ( f 4 ˜ , θ ) = x 1 ( 0.4 , 0.4 , 0.4 ) e 2 ( f 5 ˜ , θ ) = ( f 1 ˜ , θ ) ∪ ( f 2 ˜ , θ ) , ( f 6 ˜ , θ ) = ( f 1 ˜ , θ ) ∪ ( f 3 ˜ , θ ) , ( f 7 ˜ , θ ) = ( f 2 ˜ , θ ) ∪ ( f 4 ˜ , θ ) , ( f 8 ˜ , θ ) = ( f 2 ˜ , θ ) ∪ ( f 3 ˜ , θ ) , ( f 9 ˜ , θ ) = ( f 2 ˜ , θ ) ∪ ( f 4 ˜ , θ ) , ( f 10 ˜ , θ ) = ( f 3 ˜ , θ ) ∪ ( f 4 ˜ , θ ) , ( f 11 ˜ , θ ) = ( f 1 ˜ , θ ) ∪ ( f 2 ˜ , θ ) ∪ ( f 3 ˜ , θ ) , ( f 12 ˜ , θ ) = ( f 1 ˜ , θ ) ∪ ( f 2 ˜ , θ ) ∪ ( f 4 ˜ , θ ) , ( f 13 ˜ , θ ) = ( f 2 ˜ , θ ) ∪ ( f 3 ˜ , θ ) ∪ ( f 4 ˜ , θ ) , ( f 14 ˜ , θ ) = ( f 1 ˜ , θ ) ∪ ( f 3 ˜ , θ ) ∪ ( f 4 ˜ , θ ) , ( f 15 ˜ , θ ) = { x 1 ( 0.1 , 0.4 , 0.7 ) e 1 , x 1 ( 0.2 , 0.5 , 0.6 ) e 2 , x 2 ( 0.3 , 0.3 , 0.5 ) e 1 , x 2 ( 0.4 , 0.4 , 0.4 ) e 2 } , is a NSTS. Thus ( 〈 X ˜ 〉 , τ, θ) be a NSTS. Also ( 〈 X ˜ 〉 , τ, θ) is NS **  _{b 2} structure.

Theorem 3.9. Every sub space of a neutrosophic soft **  _{b 1} space is neutrosophic soft **  _{b 1} Space.

Proof. ( 〈 X ˜ 〉 , τ , θ ) be NS **  _{b 1} space and ( 〈 Y ˜ 〉 , τ Y ˜ , θ ) be an arbitrary NS sub-space of ( 〈 X ˜ 〉 , τ , θ ) . Suppose ( x ( a , b , c ) e 1 , θ ) 1 , ( x ( a / , b / , c / ) e 2 , θ ) 2 ∈ 〈 Y ˜ 〉 such that ( x ( a , b , c ) e 1 , θ ) 1 ≠ ( x ( a / , b / , c / ) e 2 , θ ) 2 . Since 〈 Y ˜ 〉 ⊆ 〈 X ˜ 〉 implies that ( x ( a , b , c ) e 1 , θ ) 1 , ( x ( a / , b / , c / ) e 2 , θ ) 2 ∈ 〈 X ˜ 〉 . But ( 〈 X ˜ 〉 , τ , θ ) is NS **  _{b 1} Space. So there exists NS **  _b open sets k₁, θ and k 2 , θ ∈ 〈 X ˜ 〉 , τ , θ ) such that ( x ( a , b , c ) e 1 , θ ) 1 ∈ k 1 , θ , ( x ( a / , b / , c / ) e 2 , θ ) 2 ∉ k 1 , θ and ( x ( a / , b / , c / ) e 2 , θ ) 2 ∈ 2 , θ , ( k ( a , b , c ) e 1 , θ ) 1 ∉ 2 , θ . Since k₁, θ and k 2 , θ ∈ 〈 X ˜ 〉 , τ , θ ) implies that 〈 Y ˜ 〉 ∩ k 1 , θ and 〈 Y ˜ 〉 ∩ k 2 , θ ∈ τ Y ˜ that is 〈 Y ˜ 〉 ∩ k 1 , θ and 〈 Y ˜ 〉 ∩ k 2 , θ are NS **  _b open in ( 〈 Y ˜ 〉 , τ Y ˜ , θ ) . Also ( x ( a , b , c ) e 1 , θ ) 1 ∈ 〈 Y ˜ 〉 and ( x ( a , b , c ) e 1 , θ ) 1 ∈ 〈 X ˜ 〉 , ( x ( a / , b / , c / ) e 2 , θ ) 2 ∉ 1 , θ implies that ( x ( a , b , c ) e 1 , θ ) 1 ∈ Y ˜ ∩ 1 , θ and ( x ( a a / , b / , c / ) e 2 , θ ) 2 ∉ Y ˜ ∩ 1 , θ and and ( x ( a / , b / , c / ) e 2 , θ ) 2 ∈ 〈 X ˜ 〉 , ( x ( a , b , c ) e 1 , θ ) 1 ∉ 2 , θ implies that ( x ( a / , b / , c / ) e 2 , θ ) 2 ∈ 〈 Y ˜ 〉 ( x ( a / , b / , c / ) e 2 , θ ) 2 ∈ Y ˜ ∩ k 2 , θ and ( x ( a , b , c ) e 1 , θ ) 1 ∉ 〈 Y ˜ 〉 ∩ 〈 k 2 , θ 〉 . Therefore, ( 〈 Y ˜ 〉 , τ 〈 Y ˜ 〉 , θ ) is NS **  _{b 1} space.

Theorem 3.10. Every sub space of a neutrosophic soft **  _{b 2} space is neutrosophic soft **  _{b 2} Space.

Proof. ( 〈 X ˜ 〉 , τ , θ ) be NS **  _{b 2} space and ( 〈 Y ˜ 〉 , τ Y ˜ , θ ) be an arbitrary NS sub-space of ( 〈 X ˜ 〉 , τ , θ ) . Suppose ( x ( a , b , c ) e 1 , θ ) 1 , ( n ( a / , b / , c / ) e 2 , θ ) 2 ∈ 〈 Y ˜ 〉 such that ( x ( a , b , c ) e 1 , θ ) 1 ≠ ( x ( a / , b / , c / ) e 2 , θ ) 2 . Since 〈 Y ˜ 〉 ⊆ 〈 X ˜ 〉 implies that ( x ( a , b , c ) e 1 , θ ) 1 , ( x ( a / , b / , c / ) e 2 , θ ) 2 ∈ 〈 X ˜ 〉 . But ( 〈 X ˜ 〉 , τ , θ ) is NS **  _{b 2} Space. So there exists NS **  _b open sets k₁, θ and k₂, θ such that ( x ( a , b , c ) e 1 , θ ) 1 ∈ k 1 , θ , ( x ( a / , b / , c / ) e 2 , θ ) 2 ∈ 1 , θ , 1 , θ ∩ 2 , θ = 0 ( 〈 X ˜ 〉 , θ ) . Since k₁, θ and k₂, θ ∈ τ. This implies that 〈 Y ˜ 〉 ∩ k 1 , θ and Y ˜ ∩ k 2 , θ ∈ τ Y ˜ that is Y ˜ ∩ k 1 , θ and Y ˜ ∩ k 2 , θ are NS **  _b open sets in Y ˜ . Now ( x ( a , b , c ) e 1 , θ ) 1 ∈ Y ˜ and ( x ( a , b , c ) e 1 , θ ) 1 ∈ k 1 , θ implies that ( x ( a , b , c ) e 1 , θ ) 1 ∈ Y ˜ ∩ k 1 , θ and ( x ( a / , b / , c / ) e 2 , θ ) 2 ∈ k 2 , θ implies that ( x ( a / , b / , c / ) e 2 , θ ) 2 ∈ Y ˜ ∩ k 2 , θ . Also Y ˜ ∩ k 1 , θ ∩ Y ˜ ∩ k 2 , θ = 0 ( X ˜ , θ ) = Y ˜ ∩ [ k 1 , θ ∩ k 2 , θ ] = Y ˜ ∩ 0 ( X ˜ , θ ) = 0 ( X ˜ , θ ) . This implies that ( Y ˜ , τ Y ˜ , θ ) is NS **  _{b 2} Space.

Theorem 3.11. Let ( 〈 X ˜ 〉 , τ , θ ) be a NSTS. Then ( 〈 X ˜ 〉 , τ , θ ) be a NS **  _{b 1} structure iff each NS point is a NS **  _b -close.

Proof. Let ( 〈 X ˜ 〉 , τ , θ ) be a NSTS over 〈 X 〉 . ˜ ( x ( a , b , c ) e , θ ) be an arbitrary NS point. We establish ( x ( a , b , c ) e , θ ) is a N soft **  _b -open set. Let ( y ( a / , b / , c / ) e / , θ ) ∈ ( x ( a , b , c ) e , θ ) . Then either ( y ( a / , b / , c / ) e / , θ ) ≻ ( x ( a , b , c ) e , θ ) or ( y ( a / , b / , c / ) e / , θ ) ≺ ( y ( a / , b / , c / ) e / , θ ) or ( y ( a / , b / , c / ) e / , θ ) ≻ ≻ ( x ( a , b , c ) e , θ ) or ( y ( a / , b / , c / ) e / , θ ) ≺ ≺ ( y ( a / , b / , c / ) e / , θ ) . This means that ( y ( a / , b / , c / ) e / , θ ) & ( x ( a , b , c ) e , θ ) are two are distinct NS points Thus x ≻ y or x ≺ y or e^/ ≻ e or e^/ ≺ e. or x ≻≻ y or x ≺≺ y or e^/ ≻≻ e or e^/ ≺≺ e. Since ( X ˜ , τ , θ ) be a NS **  _{b 1} structure, ∃ a NS **  _b -open set (g˜, θ) so that ( y ( a / , b / , c / ) e / , θ ) ∈ ( , ˜ θ ) & ( g ˜ ( a , b , c ) e , θ ) ⊓ ( , ˜ θ ) = 0 ( X ˜ , θ ) . Since, ( x ( a , b , c ) e , θ ) ⊓ ( g ˜ , θ ) = 0 ( X ˜ , θ ) . So ( y ( a / , b / , c / ) e / , θ ) ∈ ( g ˜ , θ ) ⊂ ( x ( a , b , c ) e , θ ) . Thus ( x ( a , b , c ) e , θ ) is a NS **  _b -open set, i.e., ( x ( a , b , c ) e , θ ) is a NS **  _b -closed set. Suppose that each NS point ( x ( a , b , c ) e , θ ) is a NS **  _b -closed. Then ( x ( a , b , c ) e , θ ) c is a NS **  _b -open set. Let ( x ( a , b , c ) e , θ ) ⊓ ( y ( a / , b / , c / ) e / , θ ) = 0 ( X ˜ , θ ) . Thus ( y ( a / , b / , c / ) e / , θ ) ∈ ( x ( a , b , c ) e , θ ) c & ( x ( a , b , c ) e , θ ) ⊓ ( x ( a , b , c ) e , θ ) c = 0 ( X ˜ , θ ) . So ( 〈 X ˜ 〉 , τ , θ ) be a NS-**  _{b 1} space.

Theorem 3.12. Let ( 〈 X ˜ 〉 , τ , θ ) be a NSTS over the father set 〈 X 〉 . ˜ Then ( 〈 X ˜ 〉 , τ , θ ) is NS-**  _{b 2} space if and only if for distinct NS points ( x ( a , b , c ) e , θ ) & ( y ( a / , b / , c / ) e / , θ ) , there exists a NS **  _b -open set ( f ˜ , θ ) containing there exists but not ( y ( a / , b / , c / ) e / , θ ) such that ( y ( a / , b / , c / ) e / , θ ) ∉ ( f ˜ , θ ) . ¯

Proof. Let ( n ( a , b , c ) e , θ ) ≻ ( y ( a / , b / , c / ) e / , θ ) be two NS points in NS **  _{b 2} space. Then there exists disjoint NS **  _b open sets ( f ˜ , θ ) & ( , ˜ θ ) such that ( x ( a , b , c ) e , θ ) ∈ ( f ˜ , θ ) & ( y ( a / , b / , c / ) e / , θ ) ∈ ( g ˜ , θ ) . Since ( x ( a , b , c ) e , θ ) ⊓ ( y ( a / , b / , c / ) e / , θ ) = 0 ( 〈 X ˜ 〉 , θ ) &

( f ˜ , θ ) ⊓ ( , ˜ θ ) = 0 ( 〈 X ˜ 〉 , θ ) . ( y ( a / , b / , c / ) e / , θ ) ∉ ( f ˜ , θ ) ( y ( a / , b / , c / ) e / , θ ) ∉ ( f ˜ , θ ) . ¯ Next suppose that, ( x ( a , b , c ) e , θ ) ≻ ( y ( a / , b / , c / ) e / , θ ) , there exists a NS **  _b open set ( f ˜ , θ ) containing ( x ( a , b , c ) e , θ ) but not ( y ( a / , b / , c / ) e / , θ ) s . t . ( y ( a / , b / , c / ) e / , θ ) ∉ ( f ˜ , θ ) ¯ c that is ( f ˜ , θ ) and ( f ˜ , θ ) ¯ c are mutually exclusive NS **  _b open sets supposing ( x ( a , b , c ) e , θ ) and ( y ( a / , b / , c / ) e / , θ ) in turn.

Theorem 3.13. Let ( 〈 X ˜ 〉 , τ , θ ) be a NSTS. Then ( 〈 X ˜ 〉 , τ , θ ) is NS **  _{b 1} space if every NS point ( x ( a , b , c ) e , θ ) ∈ ( f ˜ , θ ) ∈ ( 〈 X ˜ 〉 , τ , θ ) . if there exists a NS **  _b open set (g˜, θ) such that ( x ( a , b , c ) e , θ ) ∈ ( g ˜ , θ ) ⊂ ( g ˜ , θ ) ¯ ⊂ ( f ˜ , θ ) , Then ( 〈 X ˜ 〉 , τ , θ ) a NS **  _{b 2} space.

Proof. Suppose ( x ( a , b , c ) e , θ ) ⊓ ( y ( a / , b / , c / ) e / , θ ) = 0 ( 〈 X ˜ 〉 , θ ) . Since ( 〈 X ˜ 〉 , τ , θ ) is NS * _{b 1} space. ( x ( a , b , c ) e , θ ) & ( y ( a / , b / , c / ) e / , θ ) areNS * _b closed sets in ( 〈 X ˜ 〉 , τ , θ ) . Then ( x ( a , b , c ) e , θ ) ∈ ( ( y ( a / , b / , c / ) e / , θ ) ) c ∈ ( 〈 X ˜ 〉 , τ , θ ) . Thus there exists a NS **  _b open set ( g ˜ , θ ) ∈ ( 〈 X ˜ 〉 , τ , θ ) such that ( x ( a , b , c ) e , θ ) ∈ ( g ˜ , θ ) ⊂ ( , ˜ θ ) ¯ ⊂ ( ( y ( a / , b / , c / ) e / , θ ) ) c . So we have ( y ( a / , b / , c / ) e / , θ ) ∈ ( , ˜ θ ) and ( , ˜ θ ) ⊓ ( ( , ˜ θ ) ) c = 0 ( 〈 X ˜ 〉 , θ ) , that is ( 〈 X ˜ 〉 , τ, θ) is a NS soft **  _{b 2} space.

Theorem 3.14. Let ( 〈 X ˜ 〉 , τ , θ ) be a NSTS. ( 〈 X ˜ 〉 , τ , θ ) is soft **  _{b 3} space iff for every ( x ( a , b , c ) e , θ ) ∈ ( f ˜ , θ ) that is (g˜, θ)∈ ( 〈 X ˜ 〉 , τ , θ ) such that ( x ( a , b , c ) e , θ ) ∈ ( , ˜ θ ) ⊂ ( g ˜ , θ ) ¯ ⊂ ( g ˜ f , θ ) .

Proof. Let ( 〈 X ˜ 〉 , τ , θ ) is NS **  _{b 3} space & ( x ( a , b , c ) e , θ ) ∈ ( f ˜ , θ ) ∈ ( 〈 X ˜ 〉 , τ , θ ) . Since ( 〈 X ˜ 〉 , τ , θ ) is NS **  _{b 3} space for the NS point ( x ( a , b , c ) e , θ ) and NS **  _b closed set ( f ˜ , θ ) c , ∃ ( g ˜ 1 , θ ) and (g˜₂, θ) such that ( x ( a , b , c ) e , θ ) ∈ ( g ˜ 1 , θ ) , ( f ˜ , θ ) c ⊂ ( g ˜ 2 , θ ) and ( g ˜ 1 , θ ) ∩ ( g ˜ 2 , θ ) = 0 ( X ˜ , θ ) . Then we have ( x ( a , b , c ) e , θ ) ∈ ( g ˜ 1 , θ ) ⊂ ( g ˜ 2 , θ ) c ⊂ ( f ˜ , θ ) since (g˜₂, θ)  ^c NS **  _b closed set. ( g ˜ 1 , θ ) ¯ ⊂ ( g ˜ 2 , θ ) c . Conversely, let ( x ( a , b , c ) e , θ ) ∩ ( H ˜ , θ ) = 0 ( 〈 X ˜ 〉 , θ ) and (g˜, θ) be a NS **  _b closed set. ( x ( a , b , c ) e , θ ) ∝ ( H ˜ , θ ) c and from the condition of the theorem, we have ( x ( a , b , c ) e , θ ) ∈ ( g ˜ , θ ) ⊂ ( g ˜ , θ ) ¯ ⊂ ( g ˜ , θ ) c . Thus ( x ( a , b , c ) e , θ ) ∈ ( , ˜ θ ) , ( g ˜ , θ ) ⊂ ( g ˜ , θ ) ¯ c and ( g ˜ , θ ) ∩ ( g ˜ , θ ) ¯ c = 0 ( 〈 X ˜ 〉 , θ ) . So ( 〈 X ˜ 〉 , τ , θ ) is NS **  _{b 3} space.

Theorem 3.15. Let ( 〈 X ˜ 〉 , τ, θ) be a NSTS over the father set 〈 X 〉 . ˜ This space is a NS **  _{b 4} space if and only if for each NS **  _b closed set ( f ˜ , θ ) and NS **  _b open set (g˜, θ) with ( f ˜ , θ ) ⊂ ( g ˜ , θ ) , there exists a NS **  _b open set ( D ˜ , θ ) such tat ( f ˜ , θ ) ⊂ ( D ˜ , θ ) ⊂ ( D ˜ , θ ) ¯ ⊂ ( g ˜ , θ ) .

Proof. Let ( 〈 X ˜ 〉 , τ , θ ) be a NS **  _{b 4} over the father set 〈 X ˜ 〉 and let ( f ˜ , θ ) ⊂ ( g ˜ , θ ) . Then (g˜, θ)  ^c is a NS **  _b closed set and ( f ˜ , θ ) ∩ ( g ˜ , θ ) = 0 ( X ˜ , θ ) . Since ( 〈 X ˜ 〉 , τ, θ) be a NS **  _{b 4} space, ∃NS **  _b -open sets ( D 1 ˜ , θ ) and ( D 2 ˜ , θ ) such that ( f ˜ , θ ) ⊂ ( D 1 ˜ , θ ) , ( g ˜ , θ ) c ⊂ ( D 2 ˜ , θ ) & ( D 1 ˜ , θ ) ∩ ( D 2 ˜ , θ ) = 0 ( 〈 X ˜ 〉 , θ ) . Thus ( f ˜ , θ ) ⊂ ( D 1 ˜ , θ ) ⊂ ( D 2 ˜ , θ ) c ⊂ ( g ˜ , θ ) , ( D 2 ˜ , θ ) c is a NS **  _b closed set and ( D 1 ˜ , θ ) ¯ ⊂ ( D 2 ˜ , θ ) c . So ( f ˜ , θ ) ⊂ ( D 1 ˜ , θ ) ⊂ ( D 1 ˜ , θ ) ¯ ⊂ ( g ˜ , θ ) .

Conversely, ( f 1 ˜ , θ ) and ( f 2 ˜ , θ ) be two disjoint NS **  _b close sets. Then ( f 1 ˜ , θ ) ⊂ ( f 2 ˜ , θ ) c implies there exists NS **  _b open set ( D ˜ , θ ) such that ( f 1 ˜ , θ ) ⊂ ( D ˜ , θ ) ⊂ ( D 1 ˜ , θ ) ¯ ⊂ ( f 2 ˜ , θ ) c . Thus ( D ˜ , θ ) and ( D ˜ , θ ) c are NS **  _b open sets and ( f 1 ˜ , θ ) ⊂ ( f 2 ˜ , θ ) c , ( f 2 ˜ , θ ) ⊂ ( D ˜ , θ ) ¯ c and ( D ˜ , θ ) and ( D ˜ , θ ) ¯ c = 0 ( X ˜ , θ ) . ( X ˜ , τ, θ) be a NS **  _{b 4} space.

4 Main results concerning soft points