# Some Results in Neutrosophic Soft Topology Concerning Neutrosophic Soft∗b Open Sets

Authors: Arif Mehmood; Saleem Abdullah; Mohammed M. Al-Shomrani; Muhammad Imran Khan; Orawit Thinnukool
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

In this article, new generalised neutrosophic soft open known as neutrosophic soft∗b open set is introduced in neutrosophic soft topological spaces. Neutrosophic soft ∗bopen set is generated with the help of neutrosophic soft semiopen and neutrosophic soft preopen sets. Then, with the application of this new definition, some soft neutrosophical separation axioms, countability theorems, and countable space can be Hausdorff space under the subjection of neutrosophic soft sequence which is convergent, the cardinality of neutrosophic soft countable space, engagement of neutrosophic soft countable and uncountable spaces, neutrosophic soft topological features of the various spaces, soft neutrosophical continuity, the product of different soft neutrosophical spaces, and neutrosophic soft countably compact that has the characteristics of Bolzano Weierstrass Property (BVP) are studied. In addition to this, BVP shifting from one space to another through neutrosophic soft continuous functions, neutrosophic soft sequence convergence, and its marriage with neutrosophic soft compact space, sequentially compactness are addressed.


---

## Body

## 1. Introduction and Preliminaries

During the study towards possible applications in classical and nonclassical logic, fuzzy soft sets, vague soft set, and neutrosophic soft set are absolutely important. Nowadays, researchers daily deal with the complexities of modelling uncertain data in economics, engineering, environmental science, sociology, medical science, and many other fields. Classical methods are not always successful due to the reason that uncertainties appearing in these domains may be of various types. Zadeh [1] originated a new access of fuzzy set theory, which proved to be the most suitable agenda for dealing with uncertainties. While probability theory, rough sets [2], and other mathematical tools are considered as useful approaches to designate uncertainty. Each of these theories has its own inherent difficulties as pointed out by Molodtsov [3]. Molodtsov [4] suggested a completely new sophisticated approach of soft sets theory for modelling vagueness and uncertainty which is free from the complications affecting existing methods. In soft set theory the problem of setting the membership function, among other related problems, simply does not arise. Soft sets are considered as neighbourhood systems and are a special case of context-dependent fuzzy sets. Soft set theory has potential applications in many different fields, counting the smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability theory, and measurement theory. Maji et al. [5] functionalized soft sets in multicriteria decision-making problems by applying the technique of knowledge reduction to the information table induced by the soft set. In [6], they defined and studied several basic notions of soft set theory. In 2005, Pei and Miao [7] and Chen [8] improved the work of Maji et al. Cagman et al. [9] 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 [10] 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 neighbourhood 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 Ti-spaces and notions of soft normal and soft regular spaces are addressed. Bayramov and Gunduz [11] investigated some basic notions of soft topological spaces by using the soft point notion. Later, the authors addressed Ti-soft spaces and the relationships between them. Lastly, the author defined soft compactness and explored some of its significant characteristics.Khattak et al. [12] introduced the concept of soft (α, β) open sets and their characterization in soft single point topology. Zadeh [1] introduced the concept of a fuzzy set. The author described that a fuzzy set is a class of objects with a continuum of grades of membership. Further, the authors characterized the set by a membership function, assigning membership grade to each member of the set. The notions of inclusion, union, intersection, complement, relation, convexity, etc. were extended to such sets, and various properties of these notions in the context of fuzzy sets were established. In particular, a separation theorem for convex fuzzy sets was proved ignoring the prerequisites of mutually exclusive fuzzy sets.Atanassov [13] 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 [14] 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 [15] 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 [16] 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 its 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 [17] used the concept of the neutrosophic set in the soft set and introduced the neutrosophic soft set. Some definitions and related operations were introduced on the neutrosophic soft set.Kamal et al. [18] introduced the concept of multivalued interval neutrosophic soft set which amalgamates multivalued 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, multivalued 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 multivalued interval neutrosophic soft elements and discussed its associated properties. Moreover, the derivation of its properties, related examples, and some proofs on the propositions are included.Karaaslan and Deli [19] 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 concepts 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 gave the algorithm of the proposed method. The authors also made an application of the proposed method on a problem including soft neutrosophic classical data.Karaaslan [20] 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 the correlation coefficient between two single-valued neutrosophic refined soft sets are proposed, and a clustering analysis application of one of the proposed methods is given.Karaaslan [21] 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 the 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 the application of the method that can be successfully applied to the problems.Karaaslan [22] 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 the proposed similarity measure. Finally, an application of this similarity measure involving the real-life problem is given.Karaaslan [23] addressed firstly Maji’s definitions (Maji-2013) and verified that some propositions are incorrect by a counterexample. The author then redefined the 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 [24] introduced the construction of topology on a neutrosophic soft set (NSS). The notion of the neutrosophic soft interior, neutrosophic soft closure, neutrosophic soft neighbourhood, 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 has been introduced along with the investigation of several structural characteristics.Khattak et al. [25] for the first time bounced up the idea of the neutrosophic soft b open set, neutrosophic soft b closed sets and their properties. Also, the idea of neutrosophic soft b-neighbourhood 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 bTi-space (i=0,1,2,3,4) and some associations between them are discussed.In this article, the concept of vague soft topology is initiated, and its structural characteristics are attempted with new definitions. Some fundamental definitions are given which are necessary for the upcoming study. Neutrosophic soft union, neutrosophic soft intersection, null neutrosophic soft set, absolute neutrosophic soft set, neutrosophic soft topology, and neutrosophic soft neighbourhood are discussed. Three new definitions are introduced with respect to soft points in neutrosophic soft topological spaces. These three definitions are neutrosophic soft semiopen, neutrosophic soft preopen, and neutrosophic soft∗b open sets. Neutrosophic soft ∗b open set is generated with the help of neutrosophic soft semiopen and neutrosophic soft preopen sets. 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 interplay among these neutrosophic soft separation axioms is also displayed with respect to soft points of the spaces. These results are verified with examples. The second criteria of ∗b1 space are displayed. The engagement of neutrosophic soft ∗b1 space withNS∗b open set, neutrosophic soft ∗b3 spaces relation with neutrosophic soft closer with respect to soft points of the spaces.Neutrosophic soft countability, neutrosophic soft first countability and neutrosophic soft second countability, the relationship between these results, neutrosophic soft sequences and their convergence inNS∗b Hausdorff space, and the cardinality results are discussed. In continuation, 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 Weierstrass Property, neutrosophic soft compactness, related results, and neutrosophic sequentially compactness are discussed. In the end, some concluding remarks and future work are planted.Definition 1 (see [25]).
A neutrosophic set A on the father setX is defined as follows:
(1)A=x,TAx,IAx,FAx:x∈X,
whereT:X⟶]0−,1+,I:X⟶0−,1+,F:X⟶0−,1+[and0−≼TAx+IAx+FAx≼3+.Definition 2 (see [15]).
LetX be a father set, θ be a set of all parameters, and LX denotes the power set of X. A pair f,θ is referred to as a soft set over X, where fis a map given byf:θ⟶LX. Then, a NS set f~,θ over Xis a set defined by a set of valued functions signifying a mapping f~:θ⟶LX and is referred to as the approximateNS set functionf~,θ. It can be written as a set of ordered pairs:
(2)f~,θ=n,x,Tf~nx,If~nx,Ff~nx:x∈X:n∈θ.Definition 3 (see [24]).
Letf~,θ be aNSS over the father setX. The complement of f~,θ is signified f~,θc and is defined as follows:
(3)f~,θc=n,x,Tf~nx,1−If~nx,Ff~nx:x∈X:n∈θ.
It is clear that(4)f~,θcc=f~,θ.Definition 4 (see [17]).
Letf~,θ and ρ~,θ two NSS over the father X. f~,θ is supposed to be NSSS of ρ~,θ if Tf~nx≼Tρ~nx, If~nx≼Iρ~nx, Ff~nx≽Fρ~nx, ∀n∈θ&∀x∈X. It is signified 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 5 (see [25]).
Letf~1,θ&f~2,θ be two NSSS over father setXsuch that f~1,θ≠f~2,θ. Then, their union is signifies as f~1,θ⨆~f~2,θ=f~3,θ and is defined asf~3,θ=(n,x,Tf~3nx,If~3nx,Ff~3nx:x∈X:n∈θ, where
(5)Tf~3nx=maxTf~1nx,Tf~2nx,If~3nx=maxIf~1nx,If~2nx,Ff~3nx=minFf~1nx,Ff~2nx.Definition 6 (see [25]).
Letf~1,θ and f~2,θ be two NSSS over the fatherset Xs.t.f~1,θ≠f~2,θ. Then, their intersection is signified as f~1,θ⊓~f~2,θ=f~3,θ and is defined as f~3,n=(n,x,Tf~3nx,If~3nx,Ff~3nx:X:n∈θ,
where(6)Tf~3nx=minTf~1nx,Tf~2nx,If~3nx=minIf~1nx,If~2nx,Ff~3nx=maxFf~1nx,Ff~2nx.Definition 7 (see [25]).
Letf~,θ be a NSS over the father set X. The complement of f~,θ is signified f~,θc and is defined as follows:
(7)f~,θc=n,x,Ff~nx,1−If~nx,Tf~nx:x∈X:n∈θ.
It is clear that(8)f~,θcc=f~,θ.Definition 8 (see [25]).
NSSf~,θ be a NSS over the father setX is said to be a null neutrosophic soft set. IfTfnx=0,Ifnx=0,Ffnx=1;,∀n∈θ&∀x∈X.It is signified as0X,θ.Definition 9 (see [25]).
NSSf~,θ over the father set X is an absolute NSS if Tfnx=1,Ifnx=1,Ffnx=0;,∀n∈θ&∀x∈X.It is signified as1X,n. Clearly, 0X,nc=1X,n and 1X,nc=0X,n.Definition 10 (see [25]).
Let NSSX~,θ be the family of all NS soft sets and τ⊂NSSX~,θ. Then, τ is said to be a NS soft topology on X~ if (1). 0X,n,1X,n∈τ, (2). The union of any number of NS soft sets in τ∈τ, (3). The intersection of a finite number of NS soft sets in τ∈τ. Then, X~,τ,θ is said to be a NSTS over X~.Definition 11 (see [25]).
Let NS be the family of all NS overX~ and x∈X,~then NS xa,b,c is supposed to be a N point, for 0<a,b,c≤1 and is defined as follows:
(9)xa,b,cy=a,b,cprovidedy=x0,0,1providedy≠x.Definition 12 (see [25]).
Let NSS(X)~ be the family of all N soft sets over the father set X.~ Then, NSS xa,b,ceis called a NS point, for every x∈X,~0≺a,b,c≼1,e∝θ, and is defined as follows:
(10)xea,b,ce`y=a,b,cprovidede`=e∧y=x0,0,1providede`≠e∧y≠x.Definition 13 (see [25]).
Letf~,θ be a NSS over the father set X~. We say that xea,b,c∈f~,θ read as belonging to the NSS f~,θ whenever
(11)a≼Tf~xx,b≼If~xx,c≽Ff~xx.Definition 14 (see [25]).
X~,τ,θ be a NSTS over X~andf~,θ be a NSS over X~. NSS f~,θ in X~,τ,θ is called a NS nbhd of the NS point xea,b,c∈f~,θ, if there exists a NS open set ℊ~,θ such that xea,b,c∈ℊ~,θ⊏f~,θ.


## 2. Main Results

In this section, three new definitions are introduced with respect to soft points in neutrosophic soft topological spaces. These three definitions are neutrosophic soft semiopen, neutrosophic soft preopen, and neutrosophic soft∗b open sets. Neutrosophic soft ∗b open set is generated with the help of neutrosophic soft semiopen and neutrosophic soft preopen sets. Then, with the help of this new definition, some important results are discussed. Examples are also given for a better understanding of some results. Neutrosophic soft countability that neutrosophic soft first countability and neutrosophic soft second countability, the relationship between these results, neutrosophic soft sequences and their convergence in NS∗b Hausdorff space, and the cardinality results are discussed.Definition 15.
LetX~,τ,θ be a NSTS over X~&f~,θ be a NS set over X~. Then, f~,θ is as follows:
(1)
Neutrosophic soft semiopen iff~,θ⊆VSclVSintf~,θ(2)
Neutrosophic soft preopen iff~,θ⊆VSintVSclf~,θ(3)
Neutrosophic soft∗bopen if f~,θ⊆VSclVSintf~,θ⊔VSintVSclf~,θ and neutrosophic soft ∗bclose iff~,θ⊇VSclVSintf~,θ⊓VSintVSclf~,θDefinition 16.
LetX~,τ,θ be a NSTS over X~, xea,b,c≠ye/a/,b/,c/ are NS points. If there exist NS ∗b open sets f~,θandℊ~,θ such that xea,b,c∈f~,θ,xea,b,c⊓ℊ~,θ=0X~,θorye/a/,b/,c/∈ℊ~,θ,ye/a/,b/,c/⊓f~,θ=0X~,θ. Then, X~,τ,θ is called a NS ∗b0space.Definition 17.
LetX~,τ,θ be a NSTS over X~, xea,b,c≠ye/a/,b/,c/ are NS points. If there exist NS∗b open sets f~,θandℊ~,θ such that xea,b,c∈f~,θ,xea,b,c⊓ℊ~,θ=0X~,θandye/a/,b/,c/∈ℊ~,θ,ye/a/,b/,c/⊓f~,θ=0X~,θ. Then, X~,τ,θ is called a NS∗b1space.Definition 18.
LetX~,τ,θ be a NSTS overX,~xea,b,c≠ye/a/,b/,c/ are NS points. If ∃NS∗b open set f~,θ and ℊ~,θ such that xea,b,c∈f~,θ and ye/a/,b/,c/∈ℊ~,θ and f~,θ⊓ℊ~,θ=0X~,θ. Then, X~,τ,θ is called a NS∗∗b2 space.Definition 19.
LetX~,τ,θ be a NSTS.f~,θ be a NS ∗b closed set and xea,b,c,θ⊓f~,θ=0X~,θ. If there exists NS∗b open sets ℊ1~,θ and ℊ2~,θsuch that xea,b,c,θ∈ℊ1~,θ,f~,θ⊂ℊ2~,θ and xea,b,c,θ⊓ℊ1~,θ=0X~,θ, then X~,τ,θ is called a NS ∗b-regular space. X~,τ,θ is said to be NS∗b3 space, if is both a NS regular and NS ∗b1space.Definition 20.
LetX~,τ,θ be a NSTS. This space is a NS ∗∗bnormal space, if for every pair of disjoint NS ∗bclose sets f~1,θ and f~2,θ,∃ disjoint NS ∗∗b open sets ℊ~1,θ and ℊ~2,θs.t.f~1,θ⊂ℊ~1,θandf~2,θ⊂ℊ~2,θ.
X~,τ,θ is said to be a NS ∗b4space if it is both a NS ∗bnormal and NS ∗b1space.Example 21.
Suppose that the father setX~ is assumed to be X~=x1,x2 and the set of conditions byθ=e1,e2. Let us consider NS set f~,θ and xe110.1,0.4,0.7, xe210.2,0.5,0.6, xe120.3,0.3,0.5, and xe120.4,0.4,0.4 be NS points. Then, the family τ=0X~,θ,1X~,θ,f1~,θ,f2~,θ,f3~,θ,f4~,θ,f5~,θ,f6~,θ,f7~,θ,f8~,θ, where f1~,θ=xe110.1,0.4,0.7,f2~,θ=xe210.2,0.5,0.6, f3~,θ=xe120.3,0.3,0.5,f4~,θ=f1~,θ∪f2~,θ,f5~,θ=f1~,θ∪f3~,θ,f6~,θ=f2~,θ∪f3~,θ,f7~,θ=f1~,θ∪f2~,θ∪f3~,θ,f8~,θ=`xe110.1,0.4,0.7,xe210.2,0.5,0.6,xe120.3,0.3,0.5,xe220.4,0.4,0.4 is a NSTS. Thus, (X~,τ, Ⅎparameter) be a NSTS. Also, X~,τ,θ is NS∗b0 structure but it is not NS∗∗b1 because for NS points xe110.1,0.4,0.7,xe220.4,0.4,0.4,X~,τ,θ not NS∗b1.Example 22.
Suppose that the father setX~ is assumed to be X~=x1,x2 and the set of conditions by Ⅎparameter=e1,e2. Let us consider NS set f~,Ⅎparameter and xe110.1,0.4,0.7, xe210.2,0.5,0.6, xe120.3,0.3,0.5, and xe120.4,0.4,0.4 be NS points. Then, the family τ=0X~,θ,1X~,θ,f1~,θ,f2~,θ,f3~,θ,f4~,θ,f5~,θ,f6~,θ,f7~,θ,f8~,θ,⋯⋯⋯⋯f15~,θ,wheref1~,θ=xe110.1,0.4,0.7,f2~,θ=xe210.2,0.5,0.6,f3~,θ=xe120.3,0.3,0.5,f4~,θ=xe220.4,0.4,0.4f5~,θ=f1~,θ∪f2~,θ,f6~,θ=f1~,θ∪f3~,θ,f7~,θ=f2~,θ∪f4~,θ,f8~,θ=f2~,θ∪f3~,θ,f9~,θ=f2~,θ∪f4~,θ,f10~,θ=f3~,θ∪f4~,θ,f11~,θ=f1~,θ∪f2~,θ∪f3~,θ,f12~,θ=f1~,θ∪f2~,θ∪f4~,θ,f13~,θ=f2~,θ∪f3~,θ∪f4~,θ,f14~,θ=f1~,θ∪f3~,θ∪f4~,θ,f15~,θ=xe110.1,0.4,0.7,xe210.2,0.5,0.6,xe120.3,0.3,0.5,xe220.4,0.4,0.4 is a NSTS. Thus, X~,τ,θ be aNSTS. Also, X~,τ,θ is NS∗b2 structure.Theorem 23.
LetX~,τ,θ be a NSTS. Then, X~,τ,θ be a NS∗b1structure if each NS point is a NS∗b-close.Proof.
LetX~,τ,θ be a NSTS over X.~xea,b,c,θ be an arbitrary NS point. We establish xea,b,c,θ is a N soft ∗b open set. Let ye/a/,b/,c/,θ∈xea,b,c,θ. Then, either ye/a/,b/,c/,θ≻xea,b,c,Ⅎpθorye/a/,b/,c/,θ≺ye/a/,b/,c/,θorye/a/,b/,c/,θ≻≻xea,b,c,θorye/a/,b/,c/,θ≺≺ye/a/,b/,c/,θ. This means that ye/a/,b/,c/,θandxea,b,c,θ are two are distinct NSpoints. 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∗b1 structure, ∃a NS∗b open set ℊ~,θ so that ye/a/,b/,c/,θ∈ℊ~,θ and xea,b,c,θ⊓ℊ~,θ=0X~,θ. Since, xea,b,c,θ⊓ℊ~,θ=0X~,θ. So ye/a/,b/,c/,θ∈ℊ~,θ⊂xea,b,c,θ. Thus, xea,b,c,θ is a NS∗b open set, i.e., xea,b,c,θ is a NS∗b closed set. Suppose that each NS point xea,b,c,θ is a NS∗∗b closed set. Then, xea,b,c,θc is a NS∗b open set. Let xea,b,c,θ⊓ye/a/,b/,c/,θ=0X~,θ. Thus, ye/a/,b/,c/,θ∈xea,b,c,θc and xea,b,c,θ⊓xea,b,c,θc=0X~,θ. So X~,τ,θ be a NS-∗b1 space.☐Theorem 24.
LetX~,τ,θ be a NSTS over the father setX.~ Then, X~,τ,θ is NS-∗b2 space if and only if for distinct NS points xea,b,c,θ and ye/a/,b/,c/,θ, there exists a NS∗b open set f~,θ containing there exists but not ye/a/,b/,c/,θ such that ye/a/,b/,c/,θ∉f~,θ.¯Proof.
Letxea,b,c,θ≻ye/a/,b/,c/,θ be two NS points in NS∗b2 space. Then, there exist disjoint NS∗b open sets f~,θandℊ~,θ such that xea,b,c,θ∈f~,θandye/a/,b/,c/,θ∈ℊ~,θ. Since xea,b,c,θ⊓ye/a/,b/,c/,θ=0X~,θ and f~,θ⊓ℊ~,θ=0X~,θ.ye/a/,b/,c/,θ∉f~,θ⟹ye/a/,b/,c/,θ∉f~,θ.¯ Next, suppose that xea,b,c,θ≻ye/a/,b/,c/,θ, there exists a NS∗b open set f~,θ containing xea,b,c,θ but not ye/a/,b/,c/,θs.t.ye/a/,b/,c/,θ∉f~,θ¯c that is f~,θandf~,θ¯c are mutually exclusive NS∗b open sets supposing xea,b,c,θandye/a/,b/,c/,θ in turn.☐Theorem 25.
LetX~,τ,θ be a NSTS. Then, X~,τ,θ is NS∗b1space if every NS point xea,b,c,θ∈f~,θ∈X~,τ,θ. If there exists a NS∗b open set ℊ~,θ such that xea,b,c,θ∈ℊ~,θ⊂ℊ~,θ¯⊂f~,θ, then X~,τ,θ a NS∗b2 space.Proof.
Supposexea,b,c,θ⊓ye/a/,b/,c/,θ=0X~,θ. Since X~,τ,θ is NS∗b1 space. xea,b,c,θ and ye/a/,b/,c/,θ are NS∗b closed setsin X~,τ,θ. Then, xea,b,c,θ∈ye/a/,b/,c/,θc∈X~,τ,θ. Thus, there exists a NS∗b open set ℊ~,θ∈X~,τ,θ such that xea,b,c,θ∈ℊ~,θ⊂ℊ~,θ¯⊂ye/a/,b/,c/,θc. So we have ye/a/,b/,c/,θ∈ℊ~,θ and ℊ~,θ⊓ℊ~,θc=0X~,θ, that is X~,τ,θ is a NS soft ∗b2space.☐Theorem 26.
LetX~,τ,θ be a NSTS.X~,τ,θ is soft ∗b3space if for every xea,b,c,θ∈f~,θ that is ℊ~,θ∈(X~,τ,θ)such that xea,b,c,θ∈ℊ~,θ⊂ℊ~,θ¯⊂f~,θ.Proof.
LetX~,τ,θ is NS∗b3 space and xea,b,c,θ∈f~,θ∈X~,τ,θ. Since X~,τ,θ is NS∗b3 space for the NS pointxea,b,c,θ and NS∗b closed setf~,θc,∃ℊ1~,θ, and ℊ2~,θ such that xea,b,c,θ∈ℊ1~,θ,f~,θc⊂ℊ2~,θ and ℊ1~,θ∩ℊ2~,θ=0X~,θ. Then, we have xea,b,c,θ∈ℊ1~,θ⊂ℊ2~,θc⊂f~,θ. Since ℊ2~,θcNS∗b closed set. ℊ1~,θ¯⊂ℊ2~,θc. Conversely, let xea,b,c,θ∩H~,θ=0X~,θandH~,θ be a NS∗b closed set. xea,b,c,θ∝H~,θc and from the condition of the theorem, we havexea,b,c,θ∈ℊ~,θ⊂ℊ~,θ¯⊂H~,θc. Thus, xea,b,c,θ∈ℊ~,θ,H~,θ⊂ℊ~,θ¯c&ℊ~,θ∩ℊ~,θ¯c=0X~,θ. So X~,τ,θ is NS∗b3space.☐Theorem 27.
Let (X~,τ, θ) be a NSTS over the father set X.~ This space is a NS∗b4 space, if and only if for each NS∗b closed set f~,θand NS∗b open set ℊ~,θ with f~,θ⊂ℊ~,θ, there exists a NS∗b open set D~,θs.t.f~,θ⊂D~,θ⊂D~,θ¯⊂ℊ~,θ.Proof.
LetX~,τ,θ be a NS∗b4 over the father set X~and let f~,θ⊂ℊ~,θ. Then, ℊ~,θc is a NS∗b closed set and f~,θ∩ℊ~,θ=0X~,θ. Since (X~,τ, θ) be a NS∗b4space, ∃NS∗b open sets D~1,θandD~2,θs.t.f~,θ⊂D~1,θ,ℊ~,θc⊂D~2,θandD~1,θ∩D~2,θ=0X~,θ. Thus, f~,θ⊂D~1,θ⊂D~2,θc⊂ℊ~,θ,D~2,θc is a NS∗b closed set and D~1,θ¯⊂D~2,θc. So f~,θ⊂D~1,θ⊂D~1,θ¯⊂ℊ~,θ.
Conversely,f1~,θ and f2~,θ be two disjoint NS∗b closed sets. Then, f1~,θ⊂f2~,θc implies ∃NS∗b open set D~,θs.t.f1~,θ⊂D~,θ⊂D~1,θ¯⊂f2~,θc. Thus, D~,θandD~,θc are NS∗b open sets and f1~,θ⊂f2~,θc, f2~,θ⊂D~,θ¯c and D~,θ and D~,θ¯c=0X~,θ. (X~,τ, θ) be a NS∗b4space.☐Theorem 28.
Letxea,b,c,θ be a point in a NS first countable spaceX~,τ,θ and let W,θi:i=1,2,3,⋯. generates a NS countable ∗bopen base about the point ϰ; then, there exists an infinite soft subsequence V,θi:i=1,2,3,⋯. of the NS sequence W,θi:i=1,2,3,⋯., such that (i) for any NS∗b open set U,∂, containing xea,b,c,θ,∃ a suffix m such that V,θi⋐U,θ for all i≽m; and (ii) if X~,τ,θ be, in particular, a NS∗b1-space, then ∩~V,θi:i=1,2,3,⋯=xea,b,c,θ.Proof.
GivenW,θ1∩~W,θ2∩~W,θ3∩~⋯..∩~W,θk is NS∗b open sets, containing the point xea,b,c,θ. As the NS sets W,θi:i=1,2,3,⋯. form NS∗bopen base about xea,b,c,θ, there exists one among the NS sets W,∂i:i=1,2,3,⋯., which we shall denote by V,θk, such that xea,b,c,θ∈V,θk⋐(W,θ1∩~W,θ2∩~W,θ3∩~⋯..∩~W,θk),fork=1,2,3,⋯. The NS sequence V,θi:i=1,2,3,⋯, thus obtained, has the required properties. In fact, if U,θ is any NS∗∗b open set, containing xea,b,c,θ, then ∃a NS set W,θm say, belonging to the family W,θi:i=1,2,3,⋯., such that xea,b,c,θ∈W,θm⋐U,θ. Also, since V,θi⋐W,θm,foralli≽m. Next, let X~,τ,θ be NS∗b1-space, and let∩~V,θi:i=1,2,3,⋯=M,θ. As xea,b,c,θ is contained in each V,θi:i=1,2,3,⋯that is xea,b,c,θ∈V,θ1,xea,b,c,θ∈V,θ1,xea,b,c,θ∈V,θ2,xea,b,c,θ∈V,∂3,xea,b,c,θ∈V,θ4,⋯⋯..xea,b,c,θ∈V,θn, it follows that xea,b,c,θ∈M,θ. Letye/a/,b/,c/,θ be any point of X~,from xea,b,c,θ that is xea,b,c,θ≻ye/a/,b/,c/,θorxea,b,c,θ≺ye/a/,b/,c/,θ. By definition of NS∗b1-space, there exists NS∗b open set U,θ such that xea,b,c,θ∈U,θ and ye/a/,b/,c/,θ∉U,θ. There exists a suffix, s.t.V,θ1⋐U,θ,V,θ2⋐U,θ,V,θ3⋐U,θ,V,θ4⋐⋐U,θforalli≽m. Consequently, ye/a/,b/,c/,θ∉V,θiforalli≽m;hence, ye/a/,b/,c/,θ∉M,θ. Thus, M,θ consists of the point xea,b,c,θ only.Theorem 29.
NS second count-ability is NS first count-ability.Proof.
LetX~,τ,θ be a soft 2nd NS countable space. Then, this situation permits that there lives a NS countable base W for X~,τ,θ. In order to justify that X~,τ,θis NS, we proceed as, let xea,b,c,θ∈X~ be an arbitrary point. Let us assemble those members of W which absorbs xea,b,c,θ and named as Wxea,b,c,θ. If ℵ,∂ is soft n-hood of xea,b,c,θ, then this permit ∃NS∗b open set f,θ arresting xea,b,c,θ, in W and so in Wxea,b,c,θ such that xea,b,c,θ∈f,θ⋐ℵ,θ. This justifies that is a NS local base at xea,b,c,θ. One step more, Wxea,b,c,θ being a subfamily of a NS countable family W, it is therefore NS countable. Thus, every crisp point of X~ supposes a countable NSlocal base. This leads us to say X~,τ,θ is soft first N countable.Theorem 30.
ANS countable space in which every NS convergent sequence has a unique soft limit is a NS∗b Hausdorff space.Proof.
LetX~,τ,θ be NS Hausdorff space and let xea,b,c,θn~ be a soft convergent sequence in X~,τ,θ. We prove that the limit of this sequence is unique. We prove this result by contradiction. Suppose xea,b,c,θn~ converges to two soft points l~ and m~ such that l~≠m~. Then, by trichotomy law either l~<m~orl~>m~. Since it possess the NS∗b Hausdorff characteristics, there must happen two NS∗b open sets f,θandρ,θsuch that f,θ∩~ρ,θ=0X~,θ.
Now,xea,b,c,θn~ converges to l~ so there exists an integern1 such that xea,b,c,θn~∈f,∂∀n≥n1. Also, xea,b,c,θn~ converges to m~ so there exists an integer n2 such that xea,b,c,θn~∈ρ,θ∀n≥n2. We are interested todiscuss the maximum possibility, for that we must suppose the maximum of both the integers which will enable us to discuss the soft sequence for a single soft number. Maxn1,n2=n0, which leads to the situation xea,b,c,θn~∈f,θ∀n≥n0 and xea,b,c,θn~∈ρ,θ∀n≥n0. This implies that xea,b,c,θn~∈f,θ and xea,b,c,θn~∈ρ,θ∀n≥n0. This guarantees that xea,b,c,θn~∈f,∂∩~ρ,θ∀n≥n0, which beautifully contradict the fact that f,θ∩~ρ,θ=0X~,θ~. Hence, the limit of the NS sequence must be unique.Theorem 31.
The cardinality of allNS open sets in a NS second countable space is at most equal to ∁ (the power of the continuum).Proof.
LetX~,τ,θ be NSTS such that is NS second countable space. Let G,θ be any soft set of X~,τ,θ, then G,θ is the soft union of a certain soft subcollection of the NS countable collection W,θi:i=1,2,3,⋯.. Hence, the cardinality of the set of all soft ∗b open sets in X~,τ,θ is not greater than the cardinality of the soft set of all soft subcollections of the NS countable collection W,θi:i=1,2,3,⋯.. Thus, the cardinality of X~,τ,θ≺∁.Theorem 32.
Any collection of mutually exclusiveNS∗b open sets in a NS∗b second countable space is at most NS∗bcountable.Proof.
LetX~,τ,θ be NSTS such that it is NS second countable. Let ℶ,θ signifies any collection of mutually exclusive NS∗b open sets in X~,τ,θ. Let U,θ∈ℶ,θ, then ∃at least one soft set W,θm, belonging to the collection W,θi:i=1,2,3,⋯, such that W,θm⋐U,θ. Let n be the smallest suffix for which W,θn⋐U,θ. Since the soft sets U,θ in ℶ,θ are mutually disjoint, it follows that, for different soft sets U,θ∈ℶ,θ, there correspond soft sets W,θn with different suffices n. Hence, the soft sets in ℶ,θ are in a 1,1-correspondence with a soft subcollection of the NS countable collection W,θi:i=1,2,3,⋯; consequently, the cardinality of ℶ,θ is less than or equal to d.Theorem 33.
LetX~,τ,θ be NSTS such that it is NS second countabley NS∗b Hausdorff space. Then, set of all NS∗b open sets has the cardinality C.Proof.
LetX~,τ,θ be second countable NS∗b Hausdorff space. There exists in NSTSX~,τ,θ an infinite soft sequence of NS∗b open sets G,θ1,G,θ2,G,θ3,G,θ4,G,θ5,⋯.,G,θn,⋯⋯ such that G,θ1≠0X~,θ,G,θ2≠0X~,θ,G,θ3≠0X~,θ,G,θ4≠0X~,θ,⋯.G,θn≠0X~,θ⋯⋯
G,θ1∩~G,θ2∩~G,θ3∩~G,θ4∩~⋯.∩~G,θn∩~⋯=0X~,θ. Different soft subsequences of the sequences G,θ1,G,θ2,G,θ3,G,θ4,G,θ5,⋯.,G,θn will determine, as their unions, different NS∗b open sets. But since the soft set of all soft subsets of a countable soft set has the cardinality C, it follows that soft set of all the NS∗b open sets in X~,τ,θ has the cardinality ≽C. Again, the cardinality of the soft set of all NS∗b open sets in X~,τ,θ≼C. Consequently, the cardinality of all NS∗b open sets in X~,τ,θ is exactly equal to C.Theorem 34.
LetX~,τ,θ be NSST such that it is NS second countable ∗b1 space; the soft set of all NS points in this space has the cardinality at most equal to C (i.e., possess the power of the continuum at most).Proof.
Given thatX~,τ,θ is NS second countable ∗b1 space, for which the NS sets W,θ1,W,θ2,W,θ3,W,θ4,W,θ5,⋯. Forms a soft countable NS∗b open base of X~,τ,θ. Then, for any given point xea,b,c,θ∈X~,W,θς1,W,θς2,W,θς3,W,θς4,W,θς5,⋯. generate countable NS∗b open base about the point ς in NSSTX~,τ,θ,where ςi,i=1,2,3,⋯.., is a soft subsequence of the soft sequence W,θn:n=1,2,3,⋯, consisting of all those W,θn which contain the point ς. Since a second NS countable space is necessary first NS countable. So corresponding to the point ς, there exists an infinite soft sequence of NS∗b open sets b,∂i:i=i,2,3,⋯, which is a soft subsequence of the soft sequence W,θpn:n=1,2,3,.. and, therefore, also a soft subsequence of the soft sequence W,θn:n=1,2,3,.., such that ∩~b,θi:i=i,2,3,⋯=xea,b,c,θ. Thus, to each point xea,b,c,θ in X~, there corresponds a soft subsequence b,θi:i=i,2,3,⋯ of the soft sequence W,θn:n=1,2,3,.., and two in different points, there correspond two such different soft subspaces b,θi:i=i,2,3,⋯. Hence, the soft set of all points in X~, i.e., the crisp set X~, has the same cardinality as that of a certain soft subcollection of the soft collection of all soft subspace of the sequence W,θn:n=1,2,3,... Thus, the cardinality of X~ is less than or equal to ∁. In other words, the crisp set X~ has the power of the continuum at most.Theorem 35.
LetX~,τ,θ be NSTS such that it is NS second countable. Every soft uncountable soft subset contains a point of condensation.Proof.
SinceX~,τ,θ is NS second countable space and W,θi:i=1,2,3,.. be a soft countable NS∗b open base of X~,τ,θ. Let f,θ be a NS subset of X~ such that f,θ does not contain any point of condensation. For each point xea,b,c,θ∈f,∂, xea,b,c,θ is not the point of condensation of f,θ. Hence, there exists NS open ∗b set G,∂, containing xea,b,c,θ such that G,∂∩~f,θ is soft countable at most. There exists a suffix nxea,b,c,θ, such that xea,b,c,θ∈W,θnp⋐G,θ, and then, f,θ∩~W,θnp is also NS countable at most. But we can express f,∂ in the form f,θ=∪~xea,b,c,θ:xea,b,c,θ∈f,θ∪~f,θ∩~W,θnxea,b,c,θ:xea,b,c,θ∈f,θ, and there can be at most a NS countable number of different suffices. So, f,θ is at most a NS union of NS uncountable soft subset, that is, f,θ is at most a NS countable soft subset of X. Consequently, if f,θ is a soft uncountable soft subset, then it must possess a point of condensation.Theorem 36.
X~,τ,θ beNSTS such that it is NS second countable. If ψ,θ is uncountable NS subset of X~,τ,θ, then the soft subset ϖ,θ, consisting of all those N points of ψ,θ which are not points of condensation of ψ,θ, is at most NS countable.Proof.
SinceX~,τ,θ is NS second countable space, W,θi:i=1,2,3,.. be a countable NS∗b open base of X~,τ,θ. Let V,θi:i=1,2,3,⋯ be a soft subsequence of the soft sequence W,θi:i=1,2,3,.., consisting of all those soft sets W,θj for which W,θj∩~ψ,θ is at most soft countable. Then, V,θi∩~ψ,θ:i=1,2,3,⋯ is at most NS countable. We shall establish that ϖ,θ=ψ,θ∩~V,θ1∪~V,θ2∪~,⋯. xea,b,c,θ∈ϖ,θ, there is not a point of condensation of ψ,θ; hence, there exists a soft nbhd U,∂ of xea,b,c,θ, such that U,θ∩~ψ,θ is at most NS countable. Also, there exists a soft set W,θj, belonging to the soft sequence W,θi:i=1,2,3,.., satisfying xea,b,c,θ∈W,θj⋐U,θ. Then, W,θj∩~ψ,θ is at most NS countable, and so W,θj must be one of the soft sets V,θi and therefore xea,b,c,θ∈V,θi. Again, since xea,b,c,θ∈ψ,∂, it follows xea,b,c,θ∈ψ,θ∩~V,θ1∪~V,∂2∪~,⋯. Next, let ye/a/,b/,c/,θ∈ψ,θ∩~V,θ1∪~V,θ2∪~,⋯, then ye/a/,b/,c/,θ belongs to some V,θi. Now, as V,θi is a soft nbd of ye/a/,b/,c/,θ, and V,θi∩~ψ,θ is at most NS countable. ye/a/,b/,c/,θ cannot be a point of condensation of ψ,∂. Also, as ye/a/,b/,c/,θ∈ψ,θ. It follows that ye/a/,b/,c/,θ∈ϖ,θ. Thus, ϖ,θ=ψ,θ∩~V,θ1∪~V,θ2∪~,⋯=ψ,θ∩~V,θ1∪~ψ,θ∩~V,θ2∪~,⋯) and since each ψ,θ∪~V,θi is at most NS countable. It follows that ϖ,∂ is also at most NS countable.☐


## 3. Main More Results

In this section, 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 Weierstrass Property, neutrosophic soft compactness and related results, and neutrosophic sequentially compactness are discussed.Theorem 37.
LetX~,τ,θ be NSTS such that it is NS∗b Hausdorff space and Y~,F,θ be any NSST. Let f,∂:X~,τ,θ⟶Y~,F,θbe a soft function such that it is soft monotone and continuous. Then, Y~,F,θ is also of characteristics of NS∗b Hausdorffness.Proof.
Supposexea,b,c,θ1,xea,b,c,θ1∈X~ such that either xea,b,c,θ1≻xea,b,c,θ2orxea,b,c,θ1≺xea,b,c,θ2. Since f,θ is soft monotone. Let us suppose the monotonically increasing case. So, xea,b,c,θ1≻fxea,b,c,θ2orxea,b,c,θ1≺fxea,b,c,θ2 implies that fxea,b,c,θ1≻fx2orfxea,b,c,θ1≺fxea,b,c,θ2, respectively. Suppose ye/a/,b/,c/,θ1,ye/a/,b/,c/,θ2∈Y~such that ye/a/,b/,c/,θ1≻ye/a/,b/,c/,θ2orye/a/,b/,c/,θ1≺ye/a/,b/,c/,θ2, so ye/a/,b/,c/,θ1≻ye/a/,b/,c/,θ2orye/a/,b/,c/,θ1≺ye/a/,b/,c/,θ2, respectively, such that ye/a/,b/,c/,θ=fxea,b,c,θ1,ye/a/,b/,c/,θ2=fxea,b,c,θ2. Since, X~,τ,θ)is NS∗b Hausdorff space so there exists mutually disjoint NS∗b open sets k1,θ and k2,θ∈X~,τ,θ)⟹fk1,θ andfk2,θ∈Y~,F,θ). We claim that fk1,θ∩~fk2,θ=0(X~,θ~. Otherwise, fk1,θ∩~fk2,θ≠0X~,θ~. Suppose ∃te//a//,b//,c//,θ1∈fk1,θ∩~fk2,θ⟹te//a//,b//,c//,θ1∈fk1,θ&te//a//,b//,c//,θ1∈fk2,θ,te//a//,b//,c//,θ1∈fk1,θ,f is soft one−one∃te//a//,b//,c//,θ2∈k1,θ such that te//a//,b//,c//,θ1=fte//a//,b//,c//,θ2,te//a//,b//,c//,θ1∈fk2,∂⟹∃te//a//,b//,c//,θ3∈k2,∂ such that fte//a//,b//,c//,θ3⟹fte//a//,b//,c//,θ2=fte//a//,b//,c//,θ3. Since, f is soft one-one ⟹te//a//,b//,c//,θ2=te//a//,b//,c//,θ3 implies that te//a//,b//,c//,θ2∈fk1,∂,te//a//,b//,c//,θ2∈fk2,∂ implies that te//a//,b//,c//,θ2∈fk1,∂∩~fk2,∂. This is a contradiction becausek1,∂∩~k2,∂=0X~,θ. Therefore, fk1,∂∩~fk2,∂=0X~,θ. Finally, xea,b,c,θ1≻xea,b,c,θ2orxea,b,c,θ1≺xea,b,c,θ2⟹xea,b,c,θ1≠xea,b,c,θ2⟹fxea,b,c,θ1≻fxea,b,c,θ2orfxea,b,c,θ1≺fxea,b,c,θ2⟹fxea,b,c,θ1≠fxea,b,c,θ2. Given a pair of points ye/a/,b/,c/,θ1,ye/a/,b/,c/,θ2∈Y~∋ye/a/,b/,c/,θ1≠ye/a/,b/,c/,θ2. We are able to find out mutually exclusive NS∗b open sets fk1,θ,fk2,θ∈Y~,F,θs.t.ye/a/,b/,c/,θ1∈fk1,∂,ye/a/,b/,c/,θ2∈fk2,θ. This proves that Y~,F,θ is NS∗b Husdorff space.☐Theorem 38.
LetX~,τ,θ)be NSTS and Y~,F,θ be another NSTS which satisfies one more condition of NS∗b Hausdorffness. Let f,θ:X~,τ,θ⟶Y~,F,θ a soft function s.t. it is soft monotone and continuous. Then, (X~,τ,θ) is also of characteristics of NS∗b Hausdorfness.Proof.
Supposexea,b,c,θ1,xea,b,c,θ2∈X~ such that either xea,b,c,θ1≻xea,b,c,θ2orxea,b,c,θ1≺xea,b,c,θ2. Let us suppose the NS monotonically increasing case. So, xea,b,c,θ1≻xea,b,c,θ2orxea,b,c,θ1≺xea,b,c,θ2 implies that fxea,b,c,θ1≻fxea,b,c,θ2orfxea,b,c,θ1≺fxea,b,c,θ2, respectively. Suppose ye/a/,b/,c/,θ1,ye/a/,b/,c/,θ2∈Y~ such that ye/a/,b/,c/,θ1≻ye/a/,b/,c/,θ2orye/a/,b/,c/,θ1≺ye/a/,b/,c/,θ2. So, ye/a/,b/,c/,θ1≻ye/a/,b/,c/,θ2orye/a/,b/,c/,θ1≺ye/a/,b/,c/,θ2, respectively, such that ye/a/,b/,c/,θ1=fxea,b,c,θ1,ye/a/,b/,c/,θ2=fxea,b,c,θ2 such thatxea,b,c,θ1=f−1y1 and xea,b,c,θ2=f−1ye/a/,b/,c/,θ2. Since ye/a/,b/,c/,θ1,ye/a/,b/,c/,θ2∈Y~ but Y~,F,θ is NS∗b Hausdorff space. So according to definition ye/a/,b/,c/,θ1≻ye/a/,b/,c/,θ2 or ye/a/,b/,c/,θ1≺ye/a/,b/,c/,θ2. There definitely exist NS∗b open sets k1,θ and k2,θ∈Y~,F,θ such that ye/a/,b/,c/,θ1∈k1,θ and ye/a/,b/,c/,θ2∈k2,θand these two NS∗b open sets are disjoint. Since f−1k1,θ andf−1k2,∂ are NS∗b open in X~,τ,θ). Now, f−1k1,θ∩~f−1k1,θ=f−1k1,θ∩~k2,θ=f−1∅~=0X~,∂~ and ye/a/,b/,c/,θ1∈k1,θ⇒f−1ye/a/,b/,c/,θ1∈f−1k1,θ⇒xea,b,c,θ1∈k1,θ,ye/a/,b/,c/,θ2∈k2,∂⇒f−1ye/a/,b/,c/,θ2∈f−1k2,θ implies that xea,b,c,θ2∈k2,θ. We see that it has been shown for every pair of distinct points of X, there suppose disjoint NS∗b open sets, namely, f−1k1,θ and f−1k2,θ belong to X~,τ,θ) such that xea,b,c,θ1∈f−1k1,θ and xea,b,c,θ2∈f−1k2,θ. Accordingly, NSTS is NS∗b Hausdorff space.Theorem 39.
LetX~,τ,θ be NSST and Y~,F,θ be another NSTS. Let f,∂:X~,τ,θ)⟶Y~,F,θ be a soft mapping such that it is continuous mapping. Let Y~,F,θ is NS∗b Hausdorff space, then it is guaranteed that xea,b,c,θ,ye/a/,b/,c/,θ:fxea,b,c,θ=fye/a/,b/,c/,θ is a NS∗b closed subset of X~,τ,θ×Y~,F,θ.Proof.
Given thatX~,τ,θ) be NSTS and Y~,F,θ be another NSTS. Let f,∂:X~,τ,θ)⟶Y~,F,θ be a soft mapping such that it is continuous mapping. Y~,F,θ is NS∗b Hausdorff space. Then, we will prove that xea,b,c,θ`,ye/a/,b/,c/,θ:fxea,b,c,θ=fye/a/,b/,c/,θ is a NS∗b closed subset of X~,τ,θ)×Y~,F,θ. Equivalently, we will prove that xea,b,c,θ,ye/a/,b/,c/,θ:fxea,b,c,θ=ye/a/,b/,c/,θc is NS∗b open subset of X~,τ,θ)×Y~,F,θ. Let xea,b,c,θ,ye/a/,b/,c/,θ∈xea,b,c,θ,ye/a/,b/,c/,θwithxea,b,c,θ≻ye/a/,b/,c/,θ:fxea,b,c,θ≻fye/a/,b/,c/,θcorxea,b,c,θ,ye/a/,b/,c/,θ∈xea,b,c,θ,ye/a/,b/,c/,θwithxea,b,c,θ≺ye/a/,b/,c/,θ:fxea,b,c,θ≺fye/a/,b/,c/,θc. Then, fxea,b,c,θ≻fye/a/,b/,c/,θ or ≺fxea,b,c,θ≺fye/a/,b/,c/,θ accordingly. Since, Y~,F,θ is NS∗b Hausdorff space. Certainly, fxea,b,c,θ,fye/a/,b/,c/,θ are points of Y~,F,θ; there exists NS∗b open sets G,∂,k,∂∈Y~,F,θ such that fxea,b,c,θ∈G,∂ and fxea,b,c,θ∈k,∂ provided G,∂∩~k,∂=0X~,θY~. Since f,θ is soft continuous, f−1(G,θ and f−1(k,θ are NS∗b open sets in X~,τ,θ)supposing xea,b,c,θ and ye/a/,b/,c/,θ, respectively, and so f−1(G,∂×f−1(H,∂ is basic NS∗b open set in X~,τ,θ)×Y~,F,θ containing xea,b,c,θ,ye/a/,b/,c/,θ. Since G,θ∩~k,θ=0θY~, it is clear by the definition of xea,b,c,θ,ye/a/,b/,c/,θ:fxea,b,c,θ=fye/a/,b/,c/,θ that is f−1(G,θ&f−1(k,θ∩~xea,b,c,θ,ye/a/,b/,c/,θ:fx=fye/a/,b/,c/,θ=0X~,θ, that is f−1(G,∂×f−1(k,θ⋐xea,b,c,θ,ye/a/,b/,c/,θ:fxea,b,c,θ=fye/a/,b/,c/,θc. Hence, xea,b,c,θ,ye/a/,b/,c/,θ:fxea,b,c,θ=fye/a/,b/,c/,θc implies that xea,b,c,θ,ye/a/,b/,c/,θ:fxea,b,c,θ=fye/a/,b/,c/,θ is NS∗b closed.Theorem 40.
LetX~,τ,θ) be NSTS and Y~,F,θ be another NSSTS. Let f,∂:X~,τ,θ)⟶Y~,F,θ be a NS∗b open mapping such that it is onto. If the soft set xea,b,c,θ,ye/a/,b/,c/,θ:fxea,b,c,θ=fye/a/,b/,c/,θ is NS∗b closed in X~,τ,θ)×Y~,F,θ, then X~,τ,θ)will behave as NS∗b Hausdorff space.Proof.
Supposefxea,b,c,θ,fye/a/,b/,c/,θ be two points of Y~ such that either fxea,b,c,θ≻fye/a/,b/,c/,θ or ≺fxea,b,c,θ≺fye/a/,b/,c/,θ. Then, xea,b,c,θ,ye/a/,b/,c/,θ∉x,ye/a/,b/,c/,θwithxea,b,c,θ≻ye/a/,b/,c/,θ:fxea,b,c,θ≻fye/a/,b/,c/,θorxea,b,c,θ,ye/a/,b/,c/,θ∉xea,b,c,θ,ye/a/,b/,c/,θwithxea,b,c,θ≺ye/a/,b/,c/,θ:fxea,b,c,θ≺fye/a/,b/,c/,θ, that is xea,b,c,θ,ye/a/,b/,c/,θ∈xea,b,c,θ,ye/a/,b/,c/,θwithxea,b,c,θ≻ye/a/,b/,c/,θ:fxea,b,c,θ≻fye/a/,b/,c/,θc or xea,b,c,θ,ye/a/,b/,c/,θ∈xea,b,c,θ,ye/a/,b/,c/,θwithxea,b,c,θ≺ye/a/,b/,c/,θ:fxea,b,c,θ≺fye/a/,b/,c/,θc. Since, xea,b,c,θ,ye/a/,b/,c/,θ∈xea,b,c,θ,ye/a/,b/,c/,θwithxea,b,c,θ≻ye/a/,b/,c/,θ:fxea,b,c,θ≻fye/a/,b/,c/,θc or xea,b,c,θ,ye/a/,b/,c/,θ∈xea,b,c,θ,ye/a/,b/,c/,θwithxea,b,c,θ≺ye/a/,b/,c/,θ:fxea,b,c,θ≺fye/a/,b/,c/,θc is soft in X~,τ,θ)×Y~,F,θ, then there exists NS∗∗b open sets G,θ and k,θ in X~,τ,θ) such that xea,b,c,θ,ye/a/,b/,c/,θye/a/,b/,c/,θ∈G,θ×k,θ⋐xea,b,c,θ,ye/a/,b/,c/,θwithxea,b,c,θ≻ye/a/,b/,c/,θ:fxea,b,c,θ≻fye/a/,b/,c/,θcorxea,b,c,θ,ye/a/,b/,c/,θ∈G,∂×k,θ⋐xea,b,c,θ,ye/a/,b/,c/,θwithxea,b,c,θ≺ye/a/,b/,c/,θ:fxea,b,c,θ≺fye/a/,b/,c/,θc. Then, since f isNS∗b open, f(G,θ) and fk,θ are NS∗b open sets in Y~,F,θ containing fxea,b,c,θ and fye/a/,b/,c/,θ, respectively, and f(G,θ)∩~fk,θ=0X~,θ~ otherwise fG,θ×fk,θ∩~xea,b,c,θ,withxea,b,c,θ≻ye/a/,b/,c/,θ:fxea,b,c,θ≻fy or xea,b,c,θ,ye/a/,b/,c/,θ∉xea,b,c,θ`,ye/a/,b/,c/,θwithxea,b,c,θ≺ye/a/,b/,c/,θ:fxea,b,c,θ≺fye/a/,b/,c/,θ=0X~,θ~. It follows that Y~,F,θ is NS∗b Hausdorff space.Theorem 41.
LetX~,τ,θ be a NS second countable space then it is guaranteed that every family of nonempty disjoint NS∗b open subsets of a NS second countable space X~,τ,θ) is NS countable.Proof.
Given thatX~,τ,θ be a NS second countable space.
Then,∃ a NS countable base W=B1,B2,B3,B4,⋯..Bn:n∈ℕforX~,τ,θ). Let C,θ be a family of nonvacuous mutually exclusive NS∗b open subsets of X~,τ,θ. Then, for each f,θ of in C,θ, there exists a soft Bn∈W in such a way that Bn⋐f,θ. Let us attach with f,θ, the smallest positive intern such that Bn⋐f,θ. Since the candidates of C,θ are mutually exclusive because of this behaviour distinct candidates will be associated with distinct positive integers. Now, if we put the elements of C,θ in order so that the order is increasing relative to the positive integers associated with them, we obtain a sequence of candidates of C,θ. This verifies that C,θ is NS countable.☐Theorem 42.
LetX~,τ,θ be a NS second countable space and let f,θ be NS uncountable subset of X~,τ,θ. Then, at least one point of f,θ is a soft limit point off,θ.Proof.
LetW=B1,B2,B3,B4,⋯..Bn:n∈ℕforX~,τ,θ.
Let, if possible, no point off,θ be a soft limit point of f,θ. Then, for each xea,b,c,θ∈f,θ, there exists NS∗b open set ρ,θxea,b,c,θ such that xea,b,c,θ∈ρ,θxea,b,c,θ and ρ,θxea,b,c,θ∩~f,θ=xea,b,c,θ. Since W is a soft base ∃Bnxea,b,c,θ∈W such that xea,b,c,θ∈Bnxea,b,c,θ⋐ρ,θxea,b,c,θ. Therefore, Bnxea,b,c,θ∩~f,θ⋐ρ,θxea,b,c,θ∩~f,θ=xea,b,c,θ. Moreover, if xea,b,c,θ1 and xea,b,c,θ2 be any two NS points such that xea,b,c,θ1≠xea,b,c,θ2 which means either xea,b,c,θ1≻xea,b,c,θ2 or xea,b,c,θ1≺xea,b,c,θ2, then there exists Bnxea,b,c,θ1 and Bnxea,b,c,θ2 in W such that Bnxea,b,c,θ1∩~f,θ=xea,b,c,θ1 and Bnxea,b,c,θ2∩~f,θ=xea,b,c,θ2. Now, xea,b,c,θ1≠xea,b,c,θ2 which guarantees that xea,b,c,θ1≠xea,b,c,θ2 which implies that
Bnxea,b,c,θ1∩~f,θ≠Bnxea,b,c,θ2∩~f,θ which implies Bnxea,b,c,θ1≠Bnxea,b,c,θ2. Thus, there exists a one to one soft correspondence of f,θ on to Bnxea,b,c,θ:xea,b,c,θ∈f,θ. Now, f,θ being NS uncountable, it follows that Bnxea,b,c,θ:xea,b,c,θ∈f,θ is NS uncountable. But, this is purely a contradiction, since Bnxea,b,c,θ:xea,b,c,θ∈f,θ being a NS subfamily of the NS countable collection W. This contradiction is taking birth that on point of f,θ is a soft limit point of f,θ, so at least one point of f,θ is a soft limit point of f,θ.☐Theorem 43.
LetX~,τ,θNSTS such that is is NS countably compact then this space has the characteristics of Bolzano Weierstrass PropertyBWP.Proof.
LetX~,τ,θ be a NS countably compact space and suppose, if possible, that it negates the Bolzano Weierstrass Property (BWP). Then, there must exists an infinite NS set f,∂ supposing no soft limit point. Further, suppose ρ,θ be soft countability infinite soft subset f,θ that is ρ,θ⋐f,θ. Then, this guarantees ρ,θ has no soft limit point. It follows that ρ,θ is NS∗b closed set. Also for each xea,b,c,θn~∈ρ,θ,xea,b,c,θn~ is not a soft limit point of ρ,k,θ. Hence, there exists NS∗∗b open set Gn,θ, such that xea,b,c,θn~∈Gn,k,θ and Gn,θ∩~ρ,θ=xea,b,c,θn~. Then, the collection Gn,k,θ:n∈N∩~ρ,k,θc is countable NS∗b open cover of X~,τ,θ. This soft cover has no finite subcover. For this, we exhaust a single Gn,θ, it would not be a soft cover of X~,τ,θ since then xea,b,c,θn~ would be covered. Result in X~,τ,θ is not NS countably compact.Theorem 44.
LetX~,τ,θ and Y~,F,θ be two NSTS and suppose f,θ be a NS continuous function such that f,∂:X~,τ,θ⟶Y~,F,θ is NS continuous function and let L,θ⋐X~,τ,θ supposes the B.V.P. then safely fL,θ has the B.V.P.Proof.
SupposeL,θ be an infinite NS subset of f,θ, so that L,θ contains an enumerable NS set xea,b,c,θn:n∈N then there exists enumerable NS set ye/a/,b/,c/,θn:n∈N⋐L,θs.t.fye/a/,b/,c/,θn=xea,b,c,θn. L,θ has B.V.P implies that every infinite soft subset of L,θ supposes soft accumulation point belonging to L,θ this implies that ye/a/,b/,c/,θn:n∈N has a soft neutrosophic limit point, say, ye/a/,b/,c/,θ0∈L,θ implies that the limit of the soft sequence ye/a/,b/,c/,θn:n∈N is ye/a/,b/,c/,θ0∈L,∂⟹ye/a/,b/,c/,θn⟶ye/a/,b/,c/,θ0∈L,θ. f is soft continuous implies thatit is soft continuous. Furthermore, ye/a/,b/,c/,θn⟶ye/a/,b/,c/,θ0∈L,θ⟹fye/a/,b/,c/,θn⟶fye/a/,b/,c/,θ0∈fL,θ⟹xea,b,c,θn⟶fye/a/,b/,c/,θ0∈fL,θ implies that limit of a soft sequence xea,b,c,θn:n∈N is fye/a/,b/,c/,θ0∈fL,θ implies that limit of a soft sequence xea,b,c,θn:n∈Nisfye/a/,b/,c/,θ0∈f,θL,θ. Finally, we have shown that there exists an infinite soft subset xea,b,c,θn:n∈N of fL,θ containing a limit point fye/a/,b/,c/,θ0∈fL,θ. This guarantees that fL,θ has B.V.P.☐Theorem 45.
LetX~,τ,θ be a NSTS and f,T,θ be NS subspace of X~,τ,θ. The necessary and sufficient condition for f,θ to be NS∗b compact relative to f,T,θ is that f,θ is NS∗b compact relative to X~,τ,θ.Proof.
First, we prove thatf,θ relative to X~,τ,θ).
Letk,θi:i∈I that is k,θ1,k,θ2,k,θ3,k,θ4,⋯ be X~,τ,θ)−NS∗b open cover of f,θ, then f,θ⋐∪i~k,∂i.k,∂i∈X~,τ,θ⟹∃ℊ,∂i∈X~,τ,θ such that k,θi=ℊ,θi∩~f,θ⋐ℊ,θi implies that there exists ℊ,θi∈X~,τ,θ) such that k,θi⋐ℊ,θi⟹∪i~k,θi⋐∪i~ but f,θ⋐k,θi. So that f,θ⋐∪i~k,θi. This guarantees that ℊ,θi:i∈I is a X~,τ,θ−NS∗b open cover of f,θ which is known to be NS∗b compact relative X~,τ,θ, and hence, the soft cover ℊ,θi:ixea,b,c,θ must be freezable to a finite soft cub cover, say,ℊ,θir:r=1,2,3,4,⋯.,n. Then, f,θ⋐⋃r=1nG,θir~⟹f,θ∩~f,θ⋐fcrip,∂∩~⋃r=1nG,θir~=⋃r=1n(~f,θ∩~ℊ,θir=⋃r=1nk,θir~orf,θ⋐⋃r=1nk,θir~⟹k,θir:1≤r≤n is a X~,τ,θ−NS∗b open cover of f,θ. Stepping from an arbitrary X~,τ,θ−∗bopen cover of f,θ, we are able to show that the NS∗b cover is freezable to a finite soft subcover k,θir:1≤r≤n of k,θ, meaning there by f,θ is X~,τ,θ−NS∗b compact. The condition is sufficient: suppose f,T,θ be soft subspace of X~,τ,θ and also f,θ is X~,τ,θ)−NS compact. We have to prove that f,θis X~,τ,θ−NS∗b compact. Let k,θ1,k,θ2,k,θ3,k,θ4,⋯ be soft X~,τ,θ−NS∗b open cover of f,θ, so that f,θ⋐∪i~ℊ,θi from which f,θ∩~f,θ⋐f,θ∩~∪i~ℊ,θior,f,θ⋐∪i~f,θ∩~ℊ,θi. On taking k,θi=ℊ,θi∩~f,θ, we get f,θ⋐∪~k,θi,ℊ,θi∈X~,τ,θ⟹k,θi=ℊ,θi∩~f,θ∈k,H,θ⋯.1. Now from 1 it is clear that k,θ1,k,θ2,k,θ3,k,θ4,⋯ is f,T,θ−NS∗bopen soft cover of f,θ which is known to be f,T,θ−NS∗bcompact; hence, this soft cover must be reducible to a finite soft subcover say, k,θir:1≤r≤n. This⟹f,θ⋐⋃r=1nk,θir~=⋃r=1nℊ,θir∩~f,θ∈X~,τ,θ, or
(12)f,θ⋐⋃r=1nℊ,θir∩~fcrip,θ⋐⋃r=1nℊ,θir,~orf,θ⋃r=1nℊ,θir.
This proves thatℊ,θir:1≤r≤n is a finite soft subcover to the soft cover ℊ,θi. Starting from an arbitrary X~,τ,θ−NS∗b open soft cover of f,θ, we are able to show that this soft neutrosophic open cover is freezable to a finite soft subcover, showing there by f,θ is X~,τ,θ−N∗bS compact.Theorem 46.
LetX~,τ,θNSTS and let xea,b,c,θn~ be a NS sequence in X~,τ,θ such that it converges to a point xea,b,c,θ0, then the soft set ℊ,θ consisting of the points xea,b,c,θn0 and xea,b,c,θnn=1,2,3,⋯⋯ is soft NS∗b compact.Proof.
GivenX~,τ,θNSTS and let xea,b,c,θn~ be a NS sequence in X~,τ,θ such that it converges to a point xea,b,c,θn0 that is xea,b,c,θn~⟶xea,b,c,θn0∈X~. Let ℊ,θ=xea,b,c,θ1~,xea,b,c,θ2~,xea,b,c,θ3~,xea,b,c,θ4~,xea,b,c,θ5~,xea,b,c,θ7~,⋯. Let S,θα:α∈∆ be NS∗b open covering of ℊ,θ so that ℊ,θ⋐∪~S,θα:α∈∆,xea,b,c,θn0∈ℊ,θ implies that ∃α0∈∆ such that xea,b,c,θn0∈S,θα0. According to the definition of soft convergence, xea,b,c,θn0∈S,θα0∈X~,τ,θ implies thatthere exists n0∈Ns.t.n≥n0 and xea,b,c,θn∈S,θα0. Evidently, S,θα0 contains the points xea,b,c,θn0,(13)xea,b,c,θn0+1,xea,b,c,θn0+2,xea,b,c,θn0+3,xea,b,c,θ,⋯.
xea,b,c,θn0+n,⋯⋯ Look carefully at the points and train them in a way as, xea,b,c,θ1,xea,b,c,θ2,xea,b,c,θ3,xea,b,c,θ4,⋯.
xea,b,c,θ,⋯⋯ generating a finite soft set. Let 1≤n0−1. Then, xea,b,c,θi∈ℊ,θ. For this i,xea,b,c,θi∈ℊ,θ. Hence, there exists αi∈∆ such that xea,b,c,θi∈S,θαi. Evidently, ℊ,θ⋐⋃r=0n0−1S,θαi~. This shows that S,θαi:0≤n0−1 is NS∗b open cover of ℊ,θ. Thus, an arbitrary NS∗b open cover S,θα:α∈∆ of ℊ,θ is reducible to a finite NS cub-cover S,θαi:i=0,1,2,3,⋯n0−1, it follows that ℊ,θ is soft NS∗b compact.☐Theorem 47.
IfX~,τ,θNSTS such that it has the characteristics of NS∗b sequentially compactness. Then,X~,τ,θ is safely NS∗b countably compact.Proof.
LetX~,τ,θNSTS and let ρ,θ be finite soft subset of X~,τ,θ. Let
(14)xea,b,c,θ1~,xea,b,c,θ2~,xea,b,c,θ3~,xea,b,c,θ4~,xea,b,c,θ5~,xea,b,c,θ6~,xea,b,c,θ7~,⋯be a soft sequence of soft points of ρ,θ. Then, ρ,θ being finite, at least one of the elements. In ρ,∂, say xea,b,c,θ0~ must be duplicated an infinite number of times in the NS sequence. Hence,
(15)xea,b,c,θ0~,xea,b,c,θ0~,xea,b,c,θ0~,xea,b,c,θ0~,xea,b,c,θ0~,xea,b,c,θ0~,xea,b,c,θ0~,⋯is a soft subsequence of xea,b,c,θn~ such that it is soft constant sequence and repeatedly constructed by a single soft number xea,b,c,θ0~ and we know that a soft constant sequence converges on its self. So it converges to xea,b,c,θ0~ which belongs to ρ,θ. Hence, ρ,θ is soft sequentially NS∗b compact.Theorem 48.
LetX~,τ,θNSTS and Y~,T,θ be anotherNSTS. Let f,θ be a soft continuous mapping of a soft neutrosophic sequentially compactNS∗b space X~,τ,θ into Y~,T,θ. Then, f,θX~,τ,θ is NS∗b sequentially compact.Proof.
GivenX~,τ,θNSTS and Y~,T,θ be anotherNSTS. Let f,θ be a soft continuous mapping of a NS sequentially compact space X~,τ,θ into Y~,T,θ. Then, we have to prove f,θX~,τ,θNS sequentially. For this, we proceed. Let
(16)ye/a/,b/,c/,θ1~,ye/a/,b/,c/,θ2~,,ye/a/,b/,c/,θ5~,ye/a/,b/,c/,θ6~,ye/a/,b/,c/,θ7~,⋯ye/a/,b/,c/,θn~,⋯.be a soft sequence of NS points in f,θX~,τ,θ. Then, for each n∈N, there exists
(17)xea,b,c,θ1~,xea,b,c,θ2~,xea,b,c,θ4~,xea,b,c,θ5~,xea,b,c,θ7~,⋯xea,b,c,θn~,⋯.∈X~,τ,θsuch that
(18)f,∂xea,b,c,θ1~,xea,b,c,θ2~,xea,b,c,θ3~,xea,b,c,θ7~,⋯xea,b,c,θn~,⋯.=ye/a/,b/,c/,θ1~,ye/a/,b/,c/,θ2~,ye/a/,b/,c/,θ3~,ye/a/,b/,c/,θ4~,ye/a/,b/,c/,θ6~,ye/a/,b/,c/,θ7~,⋯ye/a/,b/,c/,θn~,⋯..
Thus, we obtain a soft sequence(19)xea,b,c,θ1~,xea,b,c,θ2~,xea,b,c,θ3~,xea,b,c,θ4~,xea,b,c,θ~,xea,b,c,θ6~,xea,b,c,θ7~,⋯xea,b,c,θn~,⋯.
inX~,τ,θ. But X~,τ,θ being soft sequentially NS∗b compact, there is a NS subsequence xea,b,c,θni~ of xea,b,c,θn~ such that xea,b,c,θni~⟶xea,b,c,θ~∈X~,τ,θ. So, by NS∗b continuity of f,θ,xea,b,c,θni~⟶xea,b,c,θ~ implies that f,θxea,b,c,θni~⟶f,θxea,b,c,θn~∈f,θX~,τ,θ. Thus, f,∂xea,b,c,θni~ is a soft subsequence of
(20)ye/a/,b/,c/,θ1~,ye/a/,b/,c/,θ2~,ye/a/,b/,c/,θ,xea,b,c,θ4~,ye/a/,b/,c/,θ5~,ye/a/,b/,c/,θ6~,ye/a/,b/,c/θ7~,⋯ye/a/,b/,c/,θn~,⋯.converges to f,θϰ~ in f,θX~,τ,θ. Hence, f,θX~,τ,θ is NS∗b sequentially compact.☐Theorem 49.
LetX~,τ,θ be a NS∗b1 space and xea,b,c,θ,ye/a/,b/,c/,θ∈X~ such that xea,b,c,θ>ye/a/,b/,c/,θ or xea,b,c,θ<ye/a/,b/,c/,θ. If Wxea,b,c,θ is a NS local base at xea,b,c,θ, then there exists at least one member of Wxea,b,c,θ which does not suppose ye/a/,b/,c/,θ.Proof.
SinceX~,τ,θ be a NS∗b1 space and xea,b,c,θ>ye/a/,b/,c/,θ or xea,b,c,θ<ye/a/,b/,c/,θ,∃NS∗b open sets G,θ and H,θ such that xea,b,c,θ∈G,θ but ye/a/,b/,c/,θ∉G,∂ and ye/a/,b/,c/,θ∈H,∂ but xea,b,c,θ∉H,θ. Since, Wxea,b,c,θ is NS local base at xea,b,c,θ; there exists xea,b,c,θ∈B⋐G,∂. Since ye/a/,b/,c/,θ∉G,θ and B⋐G,θ, so ye/a/,b/,c/,θ∉B. Thus, B∈Wxea,b,c,θ such that ye/a/,b/,c/,θ∉B.Theorem 50.
LetX~,τ,θNSTS and suppose f,θ,ℊ,θ be two NS continuous functions on a NSTSX~,τ,θ in to a NSTSY~,F,θ which is NS∗b Hausdorff. Then, soft set xea,b,c,θ∈X~:fxea,b,c,θ=ℊ is NS∗b closed of X~,τ,θ.Proof.
Let ifxea,b,c,θ∈X~:fxea,b,c,θ=ℊxea,b,c,θ is a NS set of function. If xea,b,c,θ∈X~:fxea,b,c,θ=ℊϰc=∅~, it is clearly NS∗b open, and therefore, xea,b,c,θ∈X~:fxea,b,c,θ=ℊxea,b,c,θ is NS∗bclosed, which is nothing is proved in this case. Let us consider the case when xea,b,c,θ∈X~:fxea,b,c,θ=ℊxea,b,c,θc≠xea,b,c,θ. And let ρ∈xea,b,c,θ∈Xcrip:fxea,b,c,θ=ℊxea,b,c,θc. Then, ρ does not belong xea,b,c,θ∈X~:fxea,b,c,θ=ℊϰ. Result in fρ≠ℊρ. Now, Ycrip,F,∂ being NS∗b Hausdorff space so there exists NS∗b open sets ℊ,θ and H,θ of fρ and ℊρ, respectively, such that ℊ,θ and H,θ such that these NS sets such that the possibility of one rules out the possibility of other. By soft continuity of f,θ,ℊ,θ,f,θ−1 as well as ℊ,θ−1is NS∗∗b open nhd of ρ, and therefore, so is f,θ−1∩~ℊ,θ−1 is contained in xea,b,c,θ∈X~:fxea,b,c,θ=ℊxea,b,c,θ, for, xea,b,c,θ∈f,∂−1∩~ℊ,θ−1⟹fxea,b,c,θ∈ℊ,θ and ℊfxea,b,c,θ≠ℊxea,b,c,θ because ℊ,∂ and H,θ are mutually exclusive. This implies thatx does not belong to xea,b,c,θ∈X~:fxea,b,c,θ=ℊxea,b,c,θ. Therefore, ρ∈f−1ℊ,θ∩~ℊ−1ℊ,θ⋐xea,b,c,θ∈X~:fxea,b,c,θ=ℊxea,b,c,θc. This shows that xea,b,c,θ∈X~:fxea,b,c,θ=ℊxea,b,c,θcis nhd of each of its points. So, xea,b,c,θ∈X~:fxea,b,c,θ=ℊxea,b,c,θcNS∗b open, and hence, xea,b,c,θ∈X~:fxea,b,c,θ=ℊxea,b,c,θ is NS∗b closed.Theorem 51.
LetX~,τ,θNSTS such that it is NS∗b Hausdorff space and let f be a soft continuous function of X~,τ,θ into itself. Then, the NS set of fixed points under f is a NS∗b closed set.Proof.
Letδ=fxea,b,c,θ=xea,b,c,θ. If δc=∅~, then is NS∗b open, and therefore, fxea,b,c,θ=xea,b,c,θNS∗bclosed. So, let fxea,b,c,θ=xea,b,c,θc≠∅~ and let ye/a/,b/,c/,θ∈fxea,b,c,θ=xea,b,c,θc. Then, ye/a/,b/,c/,θ does not belong to fxea,b,c,θ=xea,b,c,θ, and therefore, fye/a/,b/,c/,θ≠ye/a/,b/,c/,θ. Now, ye/a/,b/,c/,θ and fye/a/,b/,c/,θ being two distinct points of the NS∗b Hausdorff space X~,τ,θ, so there exists NS∗b open sets ℊ,∂ and H,θ such that ye/a/,b/,c/,θ∈ℊ,θ,fye/a/,b/,c/,θ∈H,∂ and ℊ,θ,H,θ are disjoint. Also, by the NS continuity of f, f−1(H,θ is NS∗b open set containing y. We pretend that ℊ,θ∩~f−1H,θ⋐fxea,b,c,θ=xea,b,c,θc. Since μ∈ℊ,θ∩~f−1H,θ⟹μ∈ℊ,θ&μ∈f−1⟹μ∈ℊ,θ&fμ∈H,θ⟹μ≠fμ. As ℊ,θ∩~H,θ=∅~ implies that μ does not belong to fxea,b,c,θ=xea,b,c,θ⟹μ∈fxea,b,c,θ=xea,b,c,θc. Therefore, ye/a/,b/,c/,θ∈ℊ,∂∩~f−1H,θ⋐fxea,b,c,θ=xea,b,c,θc. Thus, fxea,b,c,θ=xea,b,c,θc is the NS nhd of each of its points. So, fxea,b,c,θ=xea,b,c,θc is NS∗b open, and hence, fxea,b,c,θ=xea,b,c,θ is NS∗b closed.☐


## 4. Conclusion

Crisp topology is such an important branch of mathematics which is used as applied mathematics as well as pure mathematics. Soft topology is an extension of crisp topology. It actually discusses the behaviour of the subsets of the crisp set with the help of parameters. Fuzzy soft topology only discusses the membership value, and it has nothing to do with the nonmembership value. This is the drawback of fuzzy soft topology. The intuitionistic fuzzy soft topology is an extension of fuzzy soft topology. It addresses both the degree of membership as well as the degree of nonmembership. Intuitionistic fuzzy soft topology is failed to address the indeterminacy case. This deficiency is filled by neutrosophic soft topology. This neutrosophic soft topology addresses acceptance, rejection, and also indeterminacy case. Characterization of neutrosophic soft points, neutrosophic soft separation axioms, countability theorems, and countable space can be Hausdorff space under the restriction of neutrosophic soft sequence which is convergent, cardinality of neutrosophic soft countable space, engagement of neutrosophic soft countable and uncountable spaces, neutrosophic soft topological characteristics of different spaces, neutrosophic soft continuity, product of different neutrosophic soft spaces, and neutrosophic soft countably compact has the characteristics of Bolzano Weierstrass Property are studied. BVP shifting from one space to another through neutrosophic soft continuous functions, neutrosophic soft sequence convergence and its linking with neutrosophic soft compact space, sequentially compactness are addressed.



---

*Source: 5544319-2021-05-25.xml*