# Neutrosophic Soft Quad Structures Concerning Neutrosophic Soft Points

Authors: Arif Mehmood; Farkhanda Afzal; Muhammad Imran Khan; Saleem Abdullah; Saeed Gul
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this article, a new concept of neutrosophic soft quad-topological structures is introduced. Generalised neutrosophic soft∗b open sets in neutrosophic soft quad-topological structures concerning soft points of the space are introduced. Different results are addressed in neutrosophic soft quad-topological structure on the basis of these new neutrosophic soft ∗b open sets. Neutrosophic soft separation axioms and other separation axioms are addressed in neutrosophic soft quad-topological structures. The engagement of these axioms is switched over with different results with respect to soft points. Neutrosophic soft topological properties of some results are also addressed in neutrosophic soft quad-topological structure. To secure the results, examples are constructed. The nonvalidity of some results are justified with examples. The understanding of some complicated problems is secured with simple techniques.


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## Body

## 1. Introduction

Zadeh [1] introduced fuzzy set theory. It addresses vagueness and incomplete data used in various fields of science. Atanassov [2] addressed the short comings in fuzzy set theory with cultured way and opened a door to a new idea with new title intuitionistic fuzzy set theory. Molodtsov [3] inaugurated the concept of soft set theory to address the uncertainty. Soft set theory hugged with many applications in many fields, like smoothness of function, Riemann integration, measurement theory, and game theory. Molodtsov [4] continuously work on soft set theory. Alshehri et al. [5] applied the concept of soft set theory to K-algebra and traced some characteristics of Abelian soft K-algebras. In addition, they introduced the notion of intersection K-algebras and addressed some of their properties. Maji et al. [6] initiated first practical application of soft sets in decision-making problems. Feng et al. [7] for the first time considered the combination of soft sets, fuzzy sets, and rough sets. Using soft sets as the granulation structures, Feng et al. [8] defined soft approximation spaces, soft rough approximations, and soft rough sets, which are generalizations of Pawlak’s rough set model based on soft sets. It has been proved that, in some cases, Feng’s soft rough set model could provide better approximations than classical rough sets.Xu et al. [9] unveiled the concept of vague soft set theory which is an extension to soft set theory. Huang et al. [10] deeply studied [9] and pointed out some incorrect results. They verified the incorrect results with examples and gave some more new definitions. Chang Wang and Yaya Li [11] initiated the concept of vague soft topological structures with title topological structure of vague soft sets. The authors discussed the basic concepts related to vague soft topological space and studied the results in vague soft topology. The soft set to the hyper-soft set was generalized by Smarandache [12]. In addition to this, the author introduced the hybrids of crisp, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic hyper-soft set. Bera and Mahapatra [13] introduced the concept of neutrosophic soft topology and discussed some fundamental results. Ozturk in [14] introduced new concepts in neutrosophic soft topological spaces. These concepts are boundary, dense set, and neutrosophic soft basis. In addition, the concept of soft subspace on neutrosophic soft topological spaces. Some interesting results are addressed with respect to soft points. Some complicated results are secured with best examples.Yolcu et al. redefined neutrosophic soft mapping and studied the images and inverse images of neutrosophic soft sets. The authors continued to trace the basic operations and other related properties of neutrosophic soft mapping. The authors beautifully applied neutrosophic soft mapping to application in decision-making problems.Ozturk et al. [14] are pioneers of new operations on neutrosophic soft sets. Ozturk et al. [15] unveiled the concept of neutrosophic soft mapping, neutrosophic soft open mapping, and neutrosophic soft homeomorphism on the basis of operation defined in [14]. Some results are secured with best understandable examples. Gunduz et al. [16] introduced new concepts of neutrosophic soft sets. They defined new separation axioms in neutrosophic soft topological spaces with respect to soft points. In continuation, the relationship among these neutrosophic soft axioms has been addressed. Interior and closer of neutrosophic soft sets are also addressed. On the basis of these concepts, some other structures are also discussed. Most of the difficult results are secured with best examples.AL-Nafee [17] introduced new family of neutrosophic soft sets. The author defined new operations on the neutrosophic set. These operations are union and intersection. On the basis of these new operations, the author defined neutrosophic soft topological space. AL-Nafee et al. [18] continued their work and extended the neutrosophic soft topological space to neutrosophic soft bitopological space on the basis of operations defined in [18]. The authors regenerated all the fundamental results of NSBTS on these basic operations. Dadas and Demiralp [19] inaugurated NSBTS on the basis of the operations defined in [13]. The authors introduced pairwise neutrosophic soft (closed) sets in NSBTS. These references [13–19] became source of motivation for my new research.In our study, we have worked with the operations given in references [14–16] which are entirely different from references [13, 17]. In Section 2, some basic recipes are inaugurated. Originality begins in Section 3. In this section, new concepts of neutrosophic soft quad-topological spaces are addressed with examples. Some results, union and intersection are also studied in neutrosophic soft quad-topological spaces. In Section 4, some important definitions of generalized neutrosophic soft open sets in neutrosophic soft quad topological spaces are introduced. These definitions are semiopen, preopen, and ∗b open sets, respectively. These definitions became source of generation of different neutrosophical soft separation axioms and neutrosophical soft other separation axioms in neutrosophic soft quad topological spaces with respect to soft points of the second space. Neutrosophical soft quad homeomorphism which is a safe carriage for different structure from one space to another is defined. Soft closer attachment with neutrosophical soft separation axioms and neutrosophical soft other separation axioms in neutrosophic soft quad topological spaces with respect to soft points of the second space are addressed. In Section 5, more main results are addressed. Among neutrosophical soft separation axioms, Hausdorff space is considered to be the most important separation axioms. Important things should be given serious attention. So, this section has almost been engaged with the study of Hausdorff space. Through neutrosophical soft function, the characteristics of one space can be migrated to another space if the neutrosophical soft function is satisfying conditions of neutrosophical soft bijections and bicontinuousness. A soft function satisfying these conditions is known as neutrosophical soft homeomorphism. Neutrosophical soft homeomorphism is giving birth to neutrosophical soft topological property. Some neutrosophical soft topological properties of Hausdorff space are addressed with respect to soft points. Sequentially, compactness and countably compact in neutrosophic soft quad topological spaces with respect to soft points of the second space are addressed. Engagement of Hausdorff space with closed sets in neutrosophic soft quad topological spaces is addressed. In Section 6, some concluding remarks and future work are given.


## 2. Fundamental Concepts

In this section, fundamental concepts are addressed. These fundamental concepts are neutrosophic set, soft set, neutrosophic soft set, neutrosophic soft complement, neutrosophic soft subsets, neutrosophic soft union, neutrosophic soft intersection, null neutrosophic soft, neutrosophic soft absolute set, neutrosophic soft points, and neutrosophic soft topological spaces.Definition 1.
(see [20]). A neutrosophic set (NS) symbolized by A on the key set L is defined as: A=£,TA£,IA£,FA£:£∈L, where T:L0−,1+I:L0−,1+,F:L0−,1+0−°TA〈£〉+I〈£〉+FA〈£〉°3+.,Definition 2.
(see [3]). Let L be the key set, ∂ be a set of parameters, and ℒL symbolizes the power set of L. A pair η˜,∂ is referred to as a soft set (SS) over L, where η˜ is a map given by η˜:∂⟶ℒL.Definition 3.
(see [21, 22]). Let L be the key set and ∂ set of parameters. Let ℒL signifies power set of all neutrosophic sets on L. Then, a neutrosophic soft set η˜,∂ over L is a set defined by a set valued function η˜ representing a mapping η˜:∂⟶ℒL, where η˜ is called approximate function of the neutrosophic soft set 〈η˜,∂〉 It can be written as a set of ordered pairs: η˜,∂=s,£,Tf˜s£,If˜s£,Ff˜s£:£∈L:s∈∂.Definition 4.
(see [13]). Let η˜,∂ be a neutrosophic soft set over key set L, the complement of η˜,∂ is signified η˜,∂c and is defined as follows:
η˜,∂c=s,£,Tf˜s£,1−If˜s£,Ff˜s£:£∈L:s∈∂. It is clear that(1)η˜,∂cc=η˜,∂.Definition 5.
(see [23, 24]). Let 〈η˜,∂〉 and 〈ρ˜,∂〉 are two neutrosophic soft set over key set L. Then, 〈η˜,∂〉 is said to be neutrosophic soft subset of 〈ρ˜,∂〉 if Tf˜s£≼Tρ˜s£, If˜s£≼Iρ˜s£, Ff˜s£≽Fρ˜s£, ∀s∈∂, and ∀£∈L. It is denoted by 〈η˜,∂〉⋐ρ˜,θ. Neutrosophic soft 〈η˜,∂〉 is said to be neutrosophic soft equal to 〈ρ˜,∂〉, if 〈η˜,∂〉 is neutrosophic soft subset of 〈ρ˜,∂〉 and 〈ρ˜,∂〉 neutrosophic soft subset of 〈η˜,∂〉 It is dented by 〈η˜,∂〉=〈η˜,∂〉.Definition 6.
(see [16]). Let 〈η1˜,∂〉,〈η2˜,∂〉 be two neutrosophic soft subsets over key set L so that 〈η1˜,∂〉≠〈η2˜,∂〉, then their union is denoted by 〈η1˜,∂〉∪˜〈η2˜,∂〉=〈η3˜,∂〉 and is defined as 〈η3˜,∂〉=s,£,Tf˜3s£,If˜3s£,Ff˜3s£:£∈L:s∈∂, where(2)Tf˜3s£=maxTf˜1s£,Tf˜2s£,If˜3s£=maxIf˜1s£,If˜2s£,f˜3s£=minFf˜1s£,Ff˜2s£..Definition 7.
(see [16]). Let 〈η1˜,∂〉 and 〈η2˜,∂〉 be two neutrosophic soft subsets over key set L such that 〈η1˜,∂〉≠〈η2˜,∂〉, then their intersection is defined as 〈η1˜,∂〉∩˜〈η2˜,∂〉=〈η3˜,∂〉 is given as 〈η3˜,∂〉=s,£,Tf˜3s£,If˜3s£,Ff˜3s£:£∈L:s∈∂ where(3)Tf˜3s£=minTf˜1s£,Tf˜2s£,If˜3s£=minIf˜1s£,If˜2s£,Ff˜3s£=maxFf˜1s£,Ff˜2s£.Definition 8.
(see [14]). Let η˜,∂ be a neutrosophic soft set over a key set L, then it is said to be a null neutrosophic soft set if Tfs£=0,Ifs£=0,Ffs£=1;,∀s∈∂,∀£∈L.
It is signified as0L,∂.Definition 9.
(see [14]). Let η˜,∂ be a neutrosophic soft set over a key set L, then it is said to be an absolute soft set if Tfs£=1,Ifs£=1,Ffs£=0;,∀s∈∂ and for all £∈L.
It is signified as1L,s; clearly, 0L,∂c=1L,s,1L,sc=0L,∂c.Definition 10.
(see [14]). Let NSS L˜,∂ be the family of all neutrosophic soft sets and τ⊂˜NSSL˜,∂, then τ is said to be a neutrosophic soft topology on L˜ if 1:0L,∂,1L,∂∈τ,2: the union of any number of neutrosophic soft sets in τ∈τ,3 : the intersection of a finite number of neutrosophic soft sets in in τ∈τ; then, L˜,τ,∂) is said to be an NSTS over L˜.Definition 11.
(see [14]). Let NS be the family of all neutrosophic sets over L˜, £∈L˜, then neutrosophic set £⋎1,⋎2,⋎3y is neutrosophic point, for 0<⋎1,⋎2,⋎3≤1 and is defined as follows:(4)£⋎1,⋎2,⋎3y=⋎1,⋎2,⋎3ify=£0,0,1ify≠£.Definition 12.
(see [14]). Let NSSL˜ be the family of all neutrosophic soft sets over KS L Then, neutrosophic soft set £⋎1,⋎2,⋎3s is called a neutrosophic soft point if for every £∈L˜,0≺⋎1,⋎2,⋎3≼1,s∈∂ and is defined as follows:(5)£s=⋎1,⋎2,⋎3ifs`=s,theny=£0,0,1ifs≠s,theny≠£.Definition 13.
(see [14]). Let η˜,∂ (be a neutrosophic soft sets over KS L, then £s∈NSSη˜,∂) if(6)⋎1≼Tf˜s£,⋎2≼If˜s£,⋎3≽Ff˜s£.Definition 14.
(see [14]). Let L˜,τ,∂ be a soft neutrosophic topological space over L. Let η˜,∂ be a neutrosophic soft set. Then, η˜,∂ is called a neutrosophic soft neighborhood of the neutrosophic soft point £s, if there exists a neutrosophic soft open set U,∂ such that £s∈U,∂⊂˜η˜,∂.


## 3. Characterisation of Neutrosophic Soft Quad-Topological Structures

In this section, the concept of neutrosophic soft quad topological spaceNSQTS is defined. Furthermore, new types of open and closed sets have been introduced in neutrosophic soft quad topological spaces.


### 3.1. Definition

IfL,τ1,∂,L,τ2,∂,L,τ3,∂,andL,τ4,∂ are four neutrosophic soft topological space (NSTS), then L,τ1,τ2,τ3,τ4,∂ is called neutrosophic soft quad topological space. A neutrosophic soft subset 〈η˜,∂〉 is said to be neutrosophic soft quad open in L,τ1,τ2,τ3,τ4,∂ if there exists a neutrosophic soft open set 〈η1˜,∂〉 in τ1, neutrosophic soft open set 〈η2˜,∂〉 in τ2, neutrosophic soft open set 〈η3˜,∂〉 in τ3 and neutrosophic soft open 〈η4˜,∂〉 in τ4 such that 〈η˜,∂〉=〈η1˜,∂〉∪˜〈η2˜,∂〉∪˜〈η3˜,∂〉∪˜〈η4˜,∂〉.


### 3.2. Example

LetL=£1,£2,£3, ∂=s1,s2 and τ1=0L,∂,1L,∂,p1,∂,p2,∂, τ2=0L,∂,1L,∂,J1,∂,J2,∂, τ3=0L,∂,1L,∂,Q1,∂,Q2,∂, and τ4=0L,∂,1L,∂where p1,∂,p2,∂,J1,∂, J2,∂ and Q1,∂,Q2,∂ being NSSs as follows:(7)fp1,∂s1=<£1,02×10−1,03×10−1,08×10−1>,<£2,02×10−4,03×10−4,08×10−4>,£3,02×10−1,04×10−1,03×10−1(8)fp1,∂s2=<£1,03×10−1,02×10−1,06×10−1>,<£2,01×10−1,05×10−1,05×10−1>,£3,04×10−1,03×10−1,04×10−1(9)fp2,∂s1=<£1,04×10−1,03×10−1,06×10−1>,<£2,04×10−1,05×10−1,03×10−1>,£3,03×10−1,05×10−1,02×10−1(10)fp2,∂s2=<£1,03×10−1,04×10−1,05×10−1>,<£2,02×10−1,06×10−1,04×10−1>,£3,04×10−1,06×10−1,03×10−1(11)fJ1,∂s1=<£1,05×10−1,04×10−1,04×10−1>,<£2,06×10−1,06×10−1,02×10−1>,£3,04×10−1,06×10−1,01×10−1(12)fJ1,∂s2=<£1,04×10−1,06×10−1,03×10−1>,<£2,03×10−1,07×10−1,03×10−1>,£3,05×10−1,07×10−1,01×10−1(13)JfJ2,∂s1=<£1,01×10−1,02×10−1,07×10−1>,<£2,03×10−1,03×10−1,03×10−1>,£3,01×10−1,02×10−1,02×10−1(14)fJ2,∂s2=<£1,01×10−1,02×10−1,07×10−1>,<£2,03×10−1,03×10−1,03×10−1>,£3,01×10−1,02×10−1,02×10−1(15)fQ1,∂s1=<£1,06×10−1,05×10−1,02×10−1>,<£2,04×10−1,05×10−1,02×10−1>,£3,04×10−1,05×10−1,01×10−1(16)fQ1,∂s2=<£1,03×10−1,04×10−1,04×10−1>,<£2,03×10−1,06×10−1,02×10−1>,£3,04×10−1,06×10−1,01×10−1(17)fQ2,∂s1=<£1,05×10−1,04×10−1,02×10−1>,<£2,04×10−1,05×10−1,02×10−1>,£3,03×10−1,05×10−1,01×10−1(18)fQ2,∂s2=<£1,03×10−1,04×10−1,05×10−1>,<£2,02×10−1,06×10−1,02×10−1>,£3,04×10−1,05×10−1,01×10−1Therefore,τ1, τ2,τ3,andτ4 are NSQTS on L and so L,τ1,τ2,τ3,τ4,∂ is a NSQTS.


### 3.3. Theorem

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS. Then, τ1∩˜τ2∩˜τ3∩˜τ4 is a NSQTS on L.Proof.
For this, we have to verify all the three conditions of neutrosophic soft quad topological space. Conditions (1) and (3) are obvious; for condition (2), letpi,∂;i∈I∈τ1∩˜τ2∩˜τ3∩˜τ4, then pi,∂∈τ1, pi,∂∈τ2, pi,∂∈τ3, and pi,∂∈τ4Asτ1, τ3, and τ4 are NSTS on L, then ∪ipi,∂∈τ1, ∪ivi,∂∈τ2,∪ipi,∂∈τ3, and ∪ipi,∂∈τ4. Therefore, ∪ipi,∂∈τ1∩˜τ2∩˜τ3∩˜τ4.Remark 1.
LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS, then τ1∪˜τ2∪˜τ3∪˜τ4 need not be a NSQTS on L.


### 3.4. Example

LetL=£1,£2,£3, ∂=s1,s2 and τ1=0L,∂,1L,∂,p1,∂,p2,∂,p3,∂, τ2=0L,∂,1L,∂,J1,∂,J2,∂, τ3=0L,∂,1L,∂,Q1,∂,Q2,∂, and τ4=0L,∂,1L,∂, where p1,∂,p2,∂,J1,∂, J2,∂ and Q1,∂,Q2,∂ being NSSs as follows:(19)fp1,∂s1=<£1,02×10−1,03×10−1,08×10−1>,<£2,04×10−1,04×10−1,04×10−1>,£3,02×10−1,04×10−1,03×10−1(20)fp1,∂s2=<£1,03×10−1,02×10−1,06×10−1>,<£2,01×10−1,05×10−1,05×10−1>,£3,04×10−1,03×10−1,05×10−1(21)fp2,∂s1=<£1,04×10−1,03×10−1,06×10−1>,<£2,04×10−1,05×10−1,03×10−1>,£3,03×10−1,05×10−1,02×10−1(22)fp2,∂s2=<£1,03×10−1,04×10−1,05×10−1>,<£2,02×10−1,06×10−1,04×10−1>,£3,04×10−1,06×10−1,03×10−1(23)fp3,∂s1=<£1,05×10−1,04×10−1,04×10−1>,<£2,06×10−1,06×10−1,02×10−1>,£3,04×10−1,06×10−1,00×10−1(24)fp3,∂s2=<£1,04×10−1,06×10−1,03×10−1>,<£2,03×10−1,07×10−1,03×10−1>,£3,05×10−1,07×10−1,01×10−1(25)fJ1,∂s1=<£1,05×10−1,04×10−1,04×10−1>,<£2,06×10−1,06×10−1,02×10−1>,£3,04×10−1,06×10−1,01×10−1(26)fJ1,∂s2=<£1,04×10−1,06×10−1,03×10−1>,<£2,03×10−1,07×10−1,03×10−1>,£3,05×10−1,07×10−1,01×10−1(27)fJ2,∂s1=<£1,01×10−1,02×10−1,07×10−1>,<£2,03×10−1,03×10−1,03×10−1>,£3,01×10−1,02×10−1,02×10−1(28)fJ2,∂s2=<£1,01×10−1,02×10−1,07×10−1>,<£2,03×10−1,03×10−1,03×10−1>,£3,01×10−1,02×10−1,02×10−1(29)fQ1,∂s1=<£1,06×10−1,05×10−1,02×10−1>,<£2,04×10−1,05×10−1,02×10−1>,£3,04×10−1,05×10−1,01×10−1(30)fQ1,∂s2=<£1,03×10−1,04×10−1,04×10−1>,<£2,03×10−1,06×10−1,02×10−1>,£3,04×10−1,06×10−1,01×10−1(31)fQ2,∂s1=<£1,03×10−1,04×10−1,04×10−1>,<£2,03×10−1,06×10−1,02×10−1>,£3,04×10−1,06×10−1,01×10−1(32)fQ2,∂s2=<£1,04×10−1,05×10−1,02×10−1>,<£2,04×10−1,05×10−1,02×10−1>,£3,03×10−1,05×10−1,01×10−1Here,τ1∪˜τ2∪˜τ3∪˜τ4=0L,∂,1L,∂,p1,∂,p2,∂,p3,∂,J1,∂,J2,∂,Q1,∂,Q2,∂ is not a NSQTS on. L.


### 3.5. Definition

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS. Then, an NSS(33)J,∂=s,£,TJs£,IJs£,FJs£:£∈L,s∈∂,is called as a pairwise NSQ open set PNSOS if there exists a NSOS J1,∂ in τ1,NSOSJ2,∂ in τ2,NSOSJ3,∂ in τ3 and NSOSJ4,∂ in τ4 such that for all £∈L such that(34)J,∂=J1,∂∪˜J2,∂∪˜J3,∂∪˜J4,∂=s,<£,maxTJs£,TJs£,maxIJs£,IJs£,minFJs£,FJs£>:s∈∂.


### 3.6. Definition

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS. Then, an NSS(35)J,∂=s,£,TJs£,IJs£,FJs£:£∈L,s∈∂is called as a PNSOS if there exist a NSOS J1,∂ in τ1 and a,NSOSJ2,∂ in τ2,NSOSJ3,∂ in τ3, and NSOSJ4,∂ in τ4 such that, for all £∈L,(36)J,∂=J1,∂∪˜J2,∂∪˜J3,∂∪˜J4,∂=s,<£,maxTJs£,TJs£,maxIJs£,IJs£,minFJs£,FJs£>:s∈∂.The set of all pairwise neutrosophic soft quad open sets inL,τ1,τ2,τ3,τ4,∂ is denoted by PNSO L,τ1,τ2,τ3,τ4,∂.


### 3.7. Definition

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS. Then, a NSS(37)J,∂=s,£,TJs£,IJs£,FJs£:£∈L,s∈∂,is called as a pairwise neutrosophic soft quad closed set (PNSC) if J,∂c is a PNSO. It is clear that J,∂ is a PNSC set if there exists a NSOCJ1,∂ in τ1, NSOC J2,∂ in τ2, NSOSJ3,∂ in τ3, and NSOSJ4,∂ in τ4 such that, for all £∈L,(38)J,∂=J1,∂∪˜J2,∂∪˜J3,∂∪˜J4,∂=s,<£,maxTJs£,TJs£,maxIJs£,IJs£,minFJs£,FJs£>:s∈∂.The set of allPNSC in L,τ1,τ2,τ3,τ4,∂ is denoted by. PNSCL,τ1,τ2,τ3,τ4,∂.


### 3.8. Theorem

LetL,τ1,τ2,τ3,τ4,∂ be a NSBTS. In this case,(1)
0L,∂,1L,∂∈PNSOL,τ1,τ2,τ3,τ4,∂(2)
IfJi,∂|i∈I⋐PNSOL,τ1,τ2,τ3,τ4,∂, then Ui∈IJi,∂∈PNSOL,τ1,τ2,τ3,τ4,∂(3)
IfUi,∂|i∈I⋐PNSCL,τ1,τ2,τ3,τ4,∂ then Ui∈IUi,∂∈PNSCL,τ1,τ2,τ3,τ4,∂Proof.
.(1)
Since0L,∂∪0L,∂=0L,θ and 1L,∂∪1L,θ=1L,θ, then 0L,∂ and 1L,∂ are PNSC.(2)
SinceJi,∂∈PNSOL,τ1,τ2,τ3,τ4,∂, there exist Ji1,∂∈τ1, Ji2,∂∈τ2, Ji3,∂∈τ3, and Ji4,∂∈τ4 such that Ji,∂=Ji1,∂∪˜Ji2,∂∪˜Ji3,∂∪˜Ji4,∂ for all i∈I. Then,(39)Ui∈IJi,∂=Ui∈IJi,∂=Ji1,∂∪˜Ji2,∂∪˜Ji3,∂∪˜Ji4,∂=Ui∈IJi1,∂∪˜Ui∈IJi2,∂∪˜Ui∈IJi3,∂Ui∈IJi4,∂.As τ1,τ2,τ3andτ4 are NSTS on Ui∈IJi1,∂∈τ1,Ui∈IJi2,∂∈τ2,Ui∈IJi3,∂∈τ3,andUi∈IJi4,∂∈τ4. Therefore, Ui∈IJi,∂∈PNSOL,τ1,τ2,τ3,τ4,∂.(3)
SinceUi1,∂∈PNSCL,τ1,τ2,τ3,τ4,∂, there exist Ui1,∂c∈τ1,Ui2,∂c∈τ2,Ui3,∂c∈τ3, and Ui4,∂c∈τ4 such that Ui,∂=Ui1,∂∩˜Ui2,∂∩˜Ui3,∂∩˜Ui4,∂, for all i∈I. Then,(40)Ui∈IUi,∂=∩i∈IUi1,∂∩˜Ui2,∂∩˜Ui3,∂∩˜Ui4,∂=∩i∈IUi1,∂∩˜∩i∈IUi2,∂∩˜Ui3,∂∩˜Ui4,∂.
Then,∩i∈IUi,∂∈PNSCL,τ1,τ2,τ3,τ4,∂ as ∩i∈IUi1,∂c∈τ1, ∩i∈IUi2,∂c∈τ2, ∩i∈IUi3,∂c∈τ3, and ∩i∈IUi4,∂c∈τ4.


### 3.9. Definition

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS,J,∂∈NSSL. The PNS closure of J,∂, denoted by clPNSSJ,∂, is the intersection of all PNSC containing J,∂, i.e.,(41)clPNSSJ,∂=ω,∂∈PNSCL|J,∂⋐ω,∂.It is clear thatclPNSSJ,∂ is the smallest PNSCS containing J,∂.


### 3.10. Theorem

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS and J,∂,Q,∂∈NSSL. Then,(1)clPNSS0L,∂=0L,∂ and clPNSS1L,∂=1L,∂(2)J,∂⋐clPNSSJ,∂(3)J,∂ is a PNSCS if clPNSSJ,∂=J,∂(4)clPNSSJ,∂⋐clPNSSQ,∂ if J,∂⋐Q,∂(5)clPNSSJ,∂∪˜clPNSSQ,∂⋐clPNSSJ,∂∪˜Q,∂(6)clPNSSclPNSSJ,∂=clPNSSJ,∂, i.e., clPNSSJ,∂ is a PNSCSProof.
It is obvious.


### 3.11. Theorem

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS,⋌,∂∈NSSL. Then, £s∈clPNSSJ,∂ if and only if for all U£s∈τ1234, where U£s any is PNSOS contains £s and τ1234£s is the family of all PNSOS contains £s,U£s∩˜J,∂≠0L,∂.Proof.
Let£s∈clPNSSJ,∂ and suppose that there exists U£s∈τ1234£s such that U£s∩˜J,∂=0L,∂. Then, J,∂⋐U£sc. Thus, clPNSSJ,∂⋐clPNSSU£sc=U£sc which implies clPNSSJ,∂∩˜U£s=0L,∂, a contradiction.
Conversely, that£s∉clPNSSJ,∂, then £s∈clPNSSJ,∂c∈τ1234£s. Therefore, by hypothesis clPNSSJ,∂c∩˜J,∂≠0L,∂, a contradiction.


## 3.1. Definition

IfL,τ1,∂,L,τ2,∂,L,τ3,∂,andL,τ4,∂ are four neutrosophic soft topological space (NSTS), then L,τ1,τ2,τ3,τ4,∂ is called neutrosophic soft quad topological space. A neutrosophic soft subset 〈η˜,∂〉 is said to be neutrosophic soft quad open in L,τ1,τ2,τ3,τ4,∂ if there exists a neutrosophic soft open set 〈η1˜,∂〉 in τ1, neutrosophic soft open set 〈η2˜,∂〉 in τ2, neutrosophic soft open set 〈η3˜,∂〉 in τ3 and neutrosophic soft open 〈η4˜,∂〉 in τ4 such that 〈η˜,∂〉=〈η1˜,∂〉∪˜〈η2˜,∂〉∪˜〈η3˜,∂〉∪˜〈η4˜,∂〉.


## 3.2. Example

LetL=£1,£2,£3, ∂=s1,s2 and τ1=0L,∂,1L,∂,p1,∂,p2,∂, τ2=0L,∂,1L,∂,J1,∂,J2,∂, τ3=0L,∂,1L,∂,Q1,∂,Q2,∂, and τ4=0L,∂,1L,∂where p1,∂,p2,∂,J1,∂, J2,∂ and Q1,∂,Q2,∂ being NSSs as follows:(7)fp1,∂s1=<£1,02×10−1,03×10−1,08×10−1>,<£2,02×10−4,03×10−4,08×10−4>,£3,02×10−1,04×10−1,03×10−1(8)fp1,∂s2=<£1,03×10−1,02×10−1,06×10−1>,<£2,01×10−1,05×10−1,05×10−1>,£3,04×10−1,03×10−1,04×10−1(9)fp2,∂s1=<£1,04×10−1,03×10−1,06×10−1>,<£2,04×10−1,05×10−1,03×10−1>,£3,03×10−1,05×10−1,02×10−1(10)fp2,∂s2=<£1,03×10−1,04×10−1,05×10−1>,<£2,02×10−1,06×10−1,04×10−1>,£3,04×10−1,06×10−1,03×10−1(11)fJ1,∂s1=<£1,05×10−1,04×10−1,04×10−1>,<£2,06×10−1,06×10−1,02×10−1>,£3,04×10−1,06×10−1,01×10−1(12)fJ1,∂s2=<£1,04×10−1,06×10−1,03×10−1>,<£2,03×10−1,07×10−1,03×10−1>,£3,05×10−1,07×10−1,01×10−1(13)JfJ2,∂s1=<£1,01×10−1,02×10−1,07×10−1>,<£2,03×10−1,03×10−1,03×10−1>,£3,01×10−1,02×10−1,02×10−1(14)fJ2,∂s2=<£1,01×10−1,02×10−1,07×10−1>,<£2,03×10−1,03×10−1,03×10−1>,£3,01×10−1,02×10−1,02×10−1(15)fQ1,∂s1=<£1,06×10−1,05×10−1,02×10−1>,<£2,04×10−1,05×10−1,02×10−1>,£3,04×10−1,05×10−1,01×10−1(16)fQ1,∂s2=<£1,03×10−1,04×10−1,04×10−1>,<£2,03×10−1,06×10−1,02×10−1>,£3,04×10−1,06×10−1,01×10−1(17)fQ2,∂s1=<£1,05×10−1,04×10−1,02×10−1>,<£2,04×10−1,05×10−1,02×10−1>,£3,03×10−1,05×10−1,01×10−1(18)fQ2,∂s2=<£1,03×10−1,04×10−1,05×10−1>,<£2,02×10−1,06×10−1,02×10−1>,£3,04×10−1,05×10−1,01×10−1Therefore,τ1, τ2,τ3,andτ4 are NSQTS on L and so L,τ1,τ2,τ3,τ4,∂ is a NSQTS.


## 3.3. Theorem

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS. Then, τ1∩˜τ2∩˜τ3∩˜τ4 is a NSQTS on L.Proof.
For this, we have to verify all the three conditions of neutrosophic soft quad topological space. Conditions (1) and (3) are obvious; for condition (2), letpi,∂;i∈I∈τ1∩˜τ2∩˜τ3∩˜τ4, then pi,∂∈τ1, pi,∂∈τ2, pi,∂∈τ3, and pi,∂∈τ4Asτ1, τ3, and τ4 are NSTS on L, then ∪ipi,∂∈τ1, ∪ivi,∂∈τ2,∪ipi,∂∈τ3, and ∪ipi,∂∈τ4. Therefore, ∪ipi,∂∈τ1∩˜τ2∩˜τ3∩˜τ4.Remark 1.
LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS, then τ1∪˜τ2∪˜τ3∪˜τ4 need not be a NSQTS on L.


## 3.4. Example

LetL=£1,£2,£3, ∂=s1,s2 and τ1=0L,∂,1L,∂,p1,∂,p2,∂,p3,∂, τ2=0L,∂,1L,∂,J1,∂,J2,∂, τ3=0L,∂,1L,∂,Q1,∂,Q2,∂, and τ4=0L,∂,1L,∂, where p1,∂,p2,∂,J1,∂, J2,∂ and Q1,∂,Q2,∂ being NSSs as follows:(19)fp1,∂s1=<£1,02×10−1,03×10−1,08×10−1>,<£2,04×10−1,04×10−1,04×10−1>,£3,02×10−1,04×10−1,03×10−1(20)fp1,∂s2=<£1,03×10−1,02×10−1,06×10−1>,<£2,01×10−1,05×10−1,05×10−1>,£3,04×10−1,03×10−1,05×10−1(21)fp2,∂s1=<£1,04×10−1,03×10−1,06×10−1>,<£2,04×10−1,05×10−1,03×10−1>,£3,03×10−1,05×10−1,02×10−1(22)fp2,∂s2=<£1,03×10−1,04×10−1,05×10−1>,<£2,02×10−1,06×10−1,04×10−1>,£3,04×10−1,06×10−1,03×10−1(23)fp3,∂s1=<£1,05×10−1,04×10−1,04×10−1>,<£2,06×10−1,06×10−1,02×10−1>,£3,04×10−1,06×10−1,00×10−1(24)fp3,∂s2=<£1,04×10−1,06×10−1,03×10−1>,<£2,03×10−1,07×10−1,03×10−1>,£3,05×10−1,07×10−1,01×10−1(25)fJ1,∂s1=<£1,05×10−1,04×10−1,04×10−1>,<£2,06×10−1,06×10−1,02×10−1>,£3,04×10−1,06×10−1,01×10−1(26)fJ1,∂s2=<£1,04×10−1,06×10−1,03×10−1>,<£2,03×10−1,07×10−1,03×10−1>,£3,05×10−1,07×10−1,01×10−1(27)fJ2,∂s1=<£1,01×10−1,02×10−1,07×10−1>,<£2,03×10−1,03×10−1,03×10−1>,£3,01×10−1,02×10−1,02×10−1(28)fJ2,∂s2=<£1,01×10−1,02×10−1,07×10−1>,<£2,03×10−1,03×10−1,03×10−1>,£3,01×10−1,02×10−1,02×10−1(29)fQ1,∂s1=<£1,06×10−1,05×10−1,02×10−1>,<£2,04×10−1,05×10−1,02×10−1>,£3,04×10−1,05×10−1,01×10−1(30)fQ1,∂s2=<£1,03×10−1,04×10−1,04×10−1>,<£2,03×10−1,06×10−1,02×10−1>,£3,04×10−1,06×10−1,01×10−1(31)fQ2,∂s1=<£1,03×10−1,04×10−1,04×10−1>,<£2,03×10−1,06×10−1,02×10−1>,£3,04×10−1,06×10−1,01×10−1(32)fQ2,∂s2=<£1,04×10−1,05×10−1,02×10−1>,<£2,04×10−1,05×10−1,02×10−1>,£3,03×10−1,05×10−1,01×10−1Here,τ1∪˜τ2∪˜τ3∪˜τ4=0L,∂,1L,∂,p1,∂,p2,∂,p3,∂,J1,∂,J2,∂,Q1,∂,Q2,∂ is not a NSQTS on. L.


## 3.5. Definition

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS. Then, an NSS(33)J,∂=s,£,TJs£,IJs£,FJs£:£∈L,s∈∂,is called as a pairwise NSQ open set PNSOS if there exists a NSOS J1,∂ in τ1,NSOSJ2,∂ in τ2,NSOSJ3,∂ in τ3 and NSOSJ4,∂ in τ4 such that for all £∈L such that(34)J,∂=J1,∂∪˜J2,∂∪˜J3,∂∪˜J4,∂=s,<£,maxTJs£,TJs£,maxIJs£,IJs£,minFJs£,FJs£>:s∈∂.


## 3.6. Definition

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS. Then, an NSS(35)J,∂=s,£,TJs£,IJs£,FJs£:£∈L,s∈∂is called as a PNSOS if there exist a NSOS J1,∂ in τ1 and a,NSOSJ2,∂ in τ2,NSOSJ3,∂ in τ3, and NSOSJ4,∂ in τ4 such that, for all £∈L,(36)J,∂=J1,∂∪˜J2,∂∪˜J3,∂∪˜J4,∂=s,<£,maxTJs£,TJs£,maxIJs£,IJs£,minFJs£,FJs£>:s∈∂.The set of all pairwise neutrosophic soft quad open sets inL,τ1,τ2,τ3,τ4,∂ is denoted by PNSO L,τ1,τ2,τ3,τ4,∂.


## 3.7. Definition

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS. Then, a NSS(37)J,∂=s,£,TJs£,IJs£,FJs£:£∈L,s∈∂,is called as a pairwise neutrosophic soft quad closed set (PNSC) if J,∂c is a PNSO. It is clear that J,∂ is a PNSC set if there exists a NSOCJ1,∂ in τ1, NSOC J2,∂ in τ2, NSOSJ3,∂ in τ3, and NSOSJ4,∂ in τ4 such that, for all £∈L,(38)J,∂=J1,∂∪˜J2,∂∪˜J3,∂∪˜J4,∂=s,<£,maxTJs£,TJs£,maxIJs£,IJs£,minFJs£,FJs£>:s∈∂.The set of allPNSC in L,τ1,τ2,τ3,τ4,∂ is denoted by. PNSCL,τ1,τ2,τ3,τ4,∂.


## 3.8. Theorem

LetL,τ1,τ2,τ3,τ4,∂ be a NSBTS. In this case,(1)
0L,∂,1L,∂∈PNSOL,τ1,τ2,τ3,τ4,∂(2)
IfJi,∂|i∈I⋐PNSOL,τ1,τ2,τ3,τ4,∂, then Ui∈IJi,∂∈PNSOL,τ1,τ2,τ3,τ4,∂(3)
IfUi,∂|i∈I⋐PNSCL,τ1,τ2,τ3,τ4,∂ then Ui∈IUi,∂∈PNSCL,τ1,τ2,τ3,τ4,∂Proof.
.(1)
Since0L,∂∪0L,∂=0L,θ and 1L,∂∪1L,θ=1L,θ, then 0L,∂ and 1L,∂ are PNSC.(2)
SinceJi,∂∈PNSOL,τ1,τ2,τ3,τ4,∂, there exist Ji1,∂∈τ1, Ji2,∂∈τ2, Ji3,∂∈τ3, and Ji4,∂∈τ4 such that Ji,∂=Ji1,∂∪˜Ji2,∂∪˜Ji3,∂∪˜Ji4,∂ for all i∈I. Then,(39)Ui∈IJi,∂=Ui∈IJi,∂=Ji1,∂∪˜Ji2,∂∪˜Ji3,∂∪˜Ji4,∂=Ui∈IJi1,∂∪˜Ui∈IJi2,∂∪˜Ui∈IJi3,∂Ui∈IJi4,∂.As τ1,τ2,τ3andτ4 are NSTS on Ui∈IJi1,∂∈τ1,Ui∈IJi2,∂∈τ2,Ui∈IJi3,∂∈τ3,andUi∈IJi4,∂∈τ4. Therefore, Ui∈IJi,∂∈PNSOL,τ1,τ2,τ3,τ4,∂.(3)
SinceUi1,∂∈PNSCL,τ1,τ2,τ3,τ4,∂, there exist Ui1,∂c∈τ1,Ui2,∂c∈τ2,Ui3,∂c∈τ3, and Ui4,∂c∈τ4 such that Ui,∂=Ui1,∂∩˜Ui2,∂∩˜Ui3,∂∩˜Ui4,∂, for all i∈I. Then,(40)Ui∈IUi,∂=∩i∈IUi1,∂∩˜Ui2,∂∩˜Ui3,∂∩˜Ui4,∂=∩i∈IUi1,∂∩˜∩i∈IUi2,∂∩˜Ui3,∂∩˜Ui4,∂.
Then,∩i∈IUi,∂∈PNSCL,τ1,τ2,τ3,τ4,∂ as ∩i∈IUi1,∂c∈τ1, ∩i∈IUi2,∂c∈τ2, ∩i∈IUi3,∂c∈τ3, and ∩i∈IUi4,∂c∈τ4.


## 3.9. Definition

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS,J,∂∈NSSL. The PNS closure of J,∂, denoted by clPNSSJ,∂, is the intersection of all PNSC containing J,∂, i.e.,(41)clPNSSJ,∂=ω,∂∈PNSCL|J,∂⋐ω,∂.It is clear thatclPNSSJ,∂ is the smallest PNSCS containing J,∂.


## 3.10. Theorem

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS and J,∂,Q,∂∈NSSL. Then,(1)clPNSS0L,∂=0L,∂ and clPNSS1L,∂=1L,∂(2)J,∂⋐clPNSSJ,∂(3)J,∂ is a PNSCS if clPNSSJ,∂=J,∂(4)clPNSSJ,∂⋐clPNSSQ,∂ if J,∂⋐Q,∂(5)clPNSSJ,∂∪˜clPNSSQ,∂⋐clPNSSJ,∂∪˜Q,∂(6)clPNSSclPNSSJ,∂=clPNSSJ,∂, i.e., clPNSSJ,∂ is a PNSCSProof.
It is obvious.


## 3.11. Theorem

LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS,⋌,∂∈NSSL. Then, £s∈clPNSSJ,∂ if and only if for all U£s∈τ1234, where U£s any is PNSOS contains £s and τ1234£s is the family of all PNSOS contains £s,U£s∩˜J,∂≠0L,∂.Proof.
Let£s∈clPNSSJ,∂ and suppose that there exists U£s∈τ1234£s such that U£s∩˜J,∂=0L,∂. Then, J,∂⋐U£sc. Thus, clPNSSJ,∂⋐clPNSSU£sc=U£sc which implies clPNSSJ,∂∩˜U£s=0L,∂, a contradiction.
Conversely, that£s∉clPNSSJ,∂, then £s∈clPNSSJ,∂c∈τ1234£s. Therefore, by hypothesis clPNSSJ,∂c∩˜J,∂≠0L,∂, a contradiction.


## 4. Exhabitation of Some Definitions and Main Results

In this section, some important definitions of generalized neutrosophic soft open sets in neutrosophic soft quad topological spaces are introduced which pave the way to our new results. These definitions are semiopen, preopen, and∗b open sets, respectively. These definitions became source of generation of different neutrosophical soft separation axioms and neutrosophical soft other separation axioms in neutrosophic soft quad topological spaces with respect to soft points of the second space. Neutrosophical soft quad homeomorphism which is a safe carriage for different structure from one space to another is defined. Soft closer attachment with neutrosophical soft separation axioms and neutrosophical soft other separation axioms in neutrosophic soft quad topological spaces with respect to soft points of the second space are addressed.Definition 15.
LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS over L,η˜,∂ be a NS set over L. Then, 〈η˜,∂〉 is(1)
Neutrosophic soft quad semiopen ifη˜,∂⋐VSclVSintη˜,∂(2)
: Neutrosophic soft quad preopen ifη˜,∂⋐VSintVSclη˜,∂(3)
: Neutrosophic soft quad∗b open if η˜,∂⋐VSclVSintη˜,∂∪˜VSintVSclη˜,∂ and neutrosophic soft quad ∗b close if η˜,∂⋑VSclVSintη˜,∂∩˜VSintVSclη˜,∂Definition 16.
LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS over L, £1⋎1,⋎2,⋎3≠£2⋎1,⋎2,⋎3 are NSQ points if there exist NSQ∗b-open sets η˜,∂ and U,∂ such that
£1⋎1,⋎2,⋎3∈η˜,∂,£1⋎1,⋎2,⋎3∩˜U,∂=0L,∂or£2⋎1,⋎2,⋎3∈U,∂,£2⋎1,⋎2,⋎3∩˜η˜,∂=0L,∂. Then, L,τ1,τ2,τ3,τ4,∂ is called a NSQ∗b space.Definition 17.
LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS over L, £1⋎1,⋎2,⋎3≠£2⋎1,⋎2,⋎3 are NSQ points. If there exist NSQ∗b-open sets η˜,∂,U,∂ such that
£1⋎1,⋎2,⋎3∈η˜,∂,£1⋎1,⋎2,⋎3∩˜U,∂=0L˜,∂,£2⋎1,⋎2,⋎3∈U,∂,£2⋎1,⋎2,⋎3∩˜η˜,∂=0L˜,∂, then L,τ1,τ2,τ3,τ4,∂ is called a NSQ∗b space.Definition 18.
LetL,τ1,τ2,τ3,τ4,∂NSQTS over L,£1⋎1,⋎2,⋎3≠£2⋎1,⋎2,⋎3 are NSQ points. If there exists NSQ∗b open set η˜,∂,U,∂ such that £1⋎1,⋎2,⋎3∈η˜,∂, £2⋎1,⋎2,⋎3∈U,∂ and η˜,∂∩˜U,∂=0L˜,∂, then L,τ1,τ2,τ3,τ4,θ is called a NSQ∗b space.Definition 19.
LetL,τ1,τ2,τ3,τ4,θ and Y˜,F1,F2,F3,F4,∂ be two NSQTS. A NS function η˜,∂: L,τ1,τ2,τ3,τ4,∂⟶Y˜,F1,F2,F3,F4,∂ is said to NSQ homeomorphism if (i) η˜,∂ is NSQ bijective; (ii) η˜,∂ is NSQ continuous; (iii) η˜,∂−1 is NSQ continuous or η˜,∂ is NSO or η˜,∂ is NSC.Theorem 1.
LetL,τ1,τ2,τ3,τ4,θ be a NSTQS over the father set L, then is NS Q ∗b space if and only if for distinct NS points £1⋎1,⋎2,⋎3 and £2⋎1,⋎2,⋎3, there exists an NSQ∗b-open set η˜,∂ containing but not £2⋎1,⋎2,⋎3 such that £2⋎1,⋎2,⋎3∉η˜,∂¯.Proof.
Let£1⋎1,⋎2,⋎3≠£2⋎1,⋎2,⋎3 be two NS points in NSQ∗b space, then there exists disjoint NSQ∗b open sets η˜,∂,U,∂ such that £1⋎1,⋎2,⋎3∈η˜,∂ and £2⋎1,⋎2,⋎3∈U,∂, since £1⋎1,⋎2,⋎3∩˜£2⋎1,⋎2,⋎3=0L,∂ and η˜,∂∩˜U,∂=0L,∂.£2⋎1,⋎2,⋎3∉η˜,∂⇒£2⋎1,⋎2,⋎3∉η˜,∂.¯ Next suppose that, £1⋎1,⋎2,⋎3≠£2⋎1,⋎2,⋎3 then there exists a NSQ∗b open set η˜,∂ containing £1⋎1,⋎2,⋎3 but not £2⋎1,⋎2,⋎3 such that £2⋎1,⋎2,⋎3∉η˜,∂¯c that is and η˜,∂¯c are mutually exclusive NSQ∗b open sets supposing £1⋎1,⋎2,⋎3 and £2⋎1,⋎2,⋎3 in turn.Theorem 2.
LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS. Then, L,τ1,τ2,τ3,τ4,θ is NSQ∗b space if every NSQ point £1⋎1,⋎2,⋎3∈η˜,∂∈L,τ1,τ2,τ3,τ4,∂ if there exists an NSQ∗b open set U,∂ such that £1⋎1,⋎2,⋎3∈U,∂⋐U,∂¯⋐η˜,∂, then L,τ1,τ2,τ3,τ4,θ an NSQ∗b space.Proof.
Suppose£1⋎1,⋎2,⋎3∩˜£2⋎1,⋎2,⋎3=0L,∂. Since L,τ1,τ2,τ3,τ4,∂ is NSQ∗b space. £1⋎1,⋎2,⋎3 and £2⋎1,⋎2,⋎3 are NSQ∗b closed sets in L,τ1,τ2,τ3,τ4,∂; then, £1⋎1,⋎2,⋎3∈£2⋎1,⋎2,⋎3∈L,τ1,τ2,τ3,τ4,∂. Thus, there exists a NSQ∗b open set U,∂∈L,τ1,τ2,τ3,τ4,∂ such that £1⋎1,⋎2,⋎3∈U,∂⋐U,∂¯⋐£2⋎1,⋎2,⋎3c. So, we have £2⋎1,⋎2,⋎3∈U,∂ and U,∂∩˜U,∂c=0L,,∂, that is, L,τ1,τ2,τ3,τ4,∂ is a NSQ∗b2 space.Definition 20.
LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS. η˜,∂ be a NSQ∗b closed set and £1⋎1,⋎2,⋎3∩˜η˜,∂=0L,∂. If there exists NSQ∗b open sets U1,∂ and U2,∂ such that £1⋎1,⋎2,⋎3∈U1,∂,η˜,∂⋐U2,∂ and £1⋎1,⋎2,⋎3∩˜U1,∂=0L,∂, then L,τ1,τ2,τ3,τ4,∂ is called a NSQ∗b-regular space. L,τ1,τ2,τ3,τ4,∂ is said to be NSQ∗b3 space, if is both a NSQ regular and NSQ∗b space.Theorem 3.
LetL,τ1,τ2,τ3,τ4,∂ be a NSQTS.L,τ1,τ2,τ3,τ4,∂ is NSQ∗b3 space iff for every £1⋎1,⋎2,⋎3∈η˜,∂, that is, U,∂∈ (L,τ1,τ2,τ3,τ4,∂ such that £1⋎1,⋎2,⋎3∈U,∂⋐U,∂¯⋐η˜,∂.Proof.
LetL,τ1,τ2,τ3,τ4,∂ is NSQ∗b3 space and £1⋎1,⋎2,⋎3∈η,∂∈L,τ1,τ2,τ3,τ4,∂. Since L,τ1,τ2,τ3,τ4,∂ is NSQ∗b3 space for the NSQ point £1⋎1,⋎2,⋎3 and NSQ∗b closed set η˜,∂c, there exists U1,∂ and U2,∂ such that £1⋎1,⋎2,⋎3∈U1,∂,η,∂c⋐U2,∂ and U1,∂∩˜U2,∂=0L˜,∂. Then, we have £1s∈U1,∂⋐U2,∂c⋐η˜,∂. Since U2,∂cNSQ∗b closed set. U1,∂¯⋐U2,∂c. Conversely, let £1s∩˜h,∂=0L˜,∂ and h,∂ be a NSQ∗b closed set. £1s∈h,∂c and we have £1s∈U,∂⋐U,∂¯⋐h,∂c. Thus, £1s∈U,∂,h,∂⋐U,∂¯c and U,∂∩˜U,∂¯c=0L˜,∂. So, L,τ1,τ2,τ3,τ4,∂ is NSQ∗b3 space.


## 5. Characterization of More Results Concerning New Definition

In this section, more main results are addressed. Among neutrosophical soft quad separation axioms, Hausdorff space is considered to be the most important separation axioms. Important things should be given serious attention. So, this section has almost been engaged with the study of Hausdorff space. Through neutrosophical soft quad function, the characteristics of one space can be migrated to another space if the neutrosophical soft quad function is satisfying conditions of neutrosophical soft quad bijections and bicontinuousness. A soft quad function satisfying these conditions is known as neutrosophical soft quad homeomorphism. Neutrosophical soft quad homeomorphism is giving birth to neutrosophical soft quad topological property. Some neutrosophical soft quad topological properties of Hausdorff space are addressed with respect to soft points. Sequentially compactness and countably compact in neutrosophic soft quad topological spaces with respect to soft points of the second space are addressed. Engagement of Hausdorff space with closed sets in neutrosophic soft quad topological spaces is addressed.Theorem 4.
LetL,τ1,τ2,τ3,τ4,∂ be NSQTS such that it is NSQ∗b Hausdorff space and Y˜,F1,F2,F3,F4,∂ be another NSQTS such that let f,θ: L,τ1,τ2,τ3,τ4,∂⟶andY˜,F1,F2,F3,F4,∂ be NSQ homeomorphism. Then, Y˜,F1,F2,F3,F4,∂ is also of the same behavior of NSQ∗b Hausdorffness.Proof.
Letℶ1⋎1,⋎2,⋎3,ℶ2⋎1,⋎2,⋎3∈Y˜ such that ℶ1⋎1,⋎2,⋎3≠ℶ2⋎1,⋎2,⋎3. Since f,∂:L,τ1,τ2,τ3,τ4,∂⟶Y˜,F1,F2,F3,F4,∂: L,τ1,τ2,τ3,τ4,∂⟶Y˜,F1,F2,F3,F4,∂ be NSQ homeomorphism. So, there exists £1⋎1,⋎2,⋎3,£2⋎1,⋎2,⋎3∈L˜ such that £1⋎1,⋎2,⋎3≠£2⋎1,⋎2,⋎3. Since L,τ1,τ2,τ3,τ4,∂ is NSQ∗b Hausdorff space, so there exist h,∂ and U,∂ in NSQ∗b Hausdorff space such that £1⋎1,⋎2,⋎3∈h,∂ and £2⋎1,⋎2,⋎3∈U,∂ such that £1⋎1,⋎2,⋎3∩˜U,∂=0L,∂,£2⋎1,⋎2,⋎3∩˜h,∂=0L,∂. Hence, f£1⋎1,⋎2,⋎3∈fh,∂,f£2⋎1,⋎2,⋎3∈fU,∂f£1⋎1,⋎2,⋎3∈fh,∂,£2⋎1,⋎2,⋎3∈U,∂. Since f,∂:L,τ1,τ2,τ3,τ4,∂⟶Y˜,F1,F2,F3,F4,∂ is NSQ homeomorphism, so f,θ is NSQ ∗b open. Also, fh,∂∩˜fU,∂=fh,∂∩˜U,∂=f0L,∂=0Y,∂. This proves Y˜,F1,F2,F3,F4,∂ is also of characteristics of NSQ∗b Hausdorffness.Theorem 5.
LetL,τ1,τ2,τ3,τ4,∂ be NSQTS and Y˜,F1,F2,F3,F4,∂ be another NSQTS which satisfies one more condition of NSQ∗b Hausdorffness. Let f,θ: L,τ1,τ2,τ3,τ4,∂⟶Y˜,F1,F2,F3,F4,∂ a NSQ function such that it is NSQ homeomorphism, then L,τ1,τ2,τ3,τ4,∂ is also of characteristics of NSQ∗b Hausdorffness.Proof.
Suppose£1s///,£2s///∈L˜ such that £1s///≠£2s///. Since f,∂ NSQ homeomorphism, so f£1s///≠£2s///. Let ℶ1s,ℶ2s∈Y˜ such that ℶ1s≠ℶ2s/.
Letf£1s///=ℶ1s,f£2s///=ℶ2s/f⇒£1⋎1,⋎2,⋎3=f−1ℶ1sand£1s///=f−1ℶ2s/
Letℶ1s,ℶ2s/∈Y˜ such that ℶ1s≠ℶ2s/. Since Y˜,F1,F2,F3,F4,θa is NSQ∗b Hausdorff space. There definitely exists NSQ∗b open sets h1,θ and h2,∂ such that ℶ1s∈h1,∂ and ℶ2s/∈h2,∂ with h1,∂≠h2,∂ .
f−1ℶ1s,ℶ2s/ are NSQ∗b open in L,τ1,τ2,τ3,τ4,∂ Now, f−1ℶ1s∩˜f−1ℶ2s/=f−1ℶ1s∩˜ℶ2s/⇒f−10L,∂=0L,∂. So is NSQTS is ∗b Hausdorff space.Theorem 6.
LetL,τ1,τ2,τ3,τ4,∂ be NSQTS and Y˜,F1,F2,F3,F4,∂ be another NSQTS. Let f,∂: L,τ1,τ2,τ3,τ4,θ⟶Y˜,F1,F2,F3,F4,∂ be a NSQ mapping. Let Y˜,F1,F2,F3,F4,∂a is NSQ∗b Hausdorff space, then it is guaranteed that ℳ,θ=£1⋎1,⋎2,⋎3,£2⋎1,⋎2,⋎3:f£1⋎1,⋎2,⋎3=f£2⋎1,⋎2,⋎3 is a NSQ∗b closed subset of L˜×˜Y˜ .Proof.
Given thatL,τ1,τ2,τ3,τ4,θ be NSQTS and Y˜,F1,F2,F3,F4,∂ be another NSQTS. Let L,τ1,τ2,τ3,τ4,θ⟶Y˜,F1,F2,F3,F4,∂ be a NSQ mapping such that it is NSQ continuous mapping Y˜,F1,F2,F3,F4,∂ is NSQ∗b Hausdorff space Then, we will prove that ℳ,∂ is NSQ close subset of L˜×˜L˜. Equivalently, we will prove that ℳ,∂c is NSQ∗b an open subset of L˜×˜L˜. Let £1s,£2s/∈ℳ,∂c. Then, £1s≠£2s/⇒f£1s≻f£2s/ or ≺f£1s≺f£2s/ accordingly. Since, Y˜,F1,F2,F3,F4,∂ is NSQ ∗b Hausdorff space. Certainly, f£1s≠£2s/,f£2s/ are NS points of Y˜, so there exists NSQ∗b open sets G,∂,h,∂∈Y˜ such that f£1s∈G,∂, f£1s∈f,∂ provided G,∂∩˜h,∂=0L,∂˜. Since, f,∂ is soft continuous, f−1G,θ×f−1h,∂ are NSQ∗b open sets in L˜. So, f−1G,θ×f−1h,∂ is NSQ∗b open in L˜×˜L˜. Now, we show that f−1G,θ×f−1h,∂ For this, let £1⋎1,⋎2,⋎3,£2⋎1,⋎2,⋎3∈⇒f−1G,∂×f−1f,∂⟶£1⋎1,⋎2,⋎3∈f−1G,∂ and £2⋎1,⋎2,⋎3∈f−1h,∂⇒f£1s=G,∂ and £2s/=h,∂. Since G,∂∩˜h,∂=0π˜˜,∂˜⇒£1s≠£2s/⇒£1s∈ℳ,∂c⇒f−1G,∂×˜f−1h,∂⊆˜ℳ,∂c. Thus, f−1G,∂×˜f−1h,∂ is NSQ∗b open in L˜×˜L˜. So, this implies that £1s,£2s/ is NSQ interior point of ℳ,∂c. So, every point of ℳ,∂c is NSQ interior point of ℳ,∂c. So ℳ,∂c is ∗b open in L˜×˜L˜.Theorem 7.
LetL,τ1,τ2,τ3,τ4,∂ be NSQTS and Y˜,F1,F2,F3,F4,∂ be another NSQTS. Let h,∂: L,τ1,τ2,τ3,τ4,∂⟶Y˜,F1,F2,F3,F4,θ be an NSQ∗b open mapping such that it is onto. If the soft set ℳ,∂=£1s,£2s/:f£1s=f£2s/ is NSQ∗b closed in L˜×˜L˜, then Y˜ is NSQ∗b Hausdorff space.Proof.
Supposef£1s,f£2s/ be two NSQ points of Y˜ such f£1s≠f£2s/⟶f£1s≻f£2s/ or ≺£1s≺f£2s/. Then, £1⋎1,⋎2,⋎3,£2⋎1,⋎2,⋎3∉ℳ,∂, that is, £1⋎1,⋎2,⋎3,£2⋎1,⋎2,⋎3∈ℳ,∂c. Since ℳ,∂cNSQ∗b open in L˜×˜L˜, there exist NSQ∗b open sets G,∂ and h,∂ in π˜ such that £1s,£2s/∈G,∂×˜h,∂⋐ℳ,∂c. Then, since NSQ∗b is open, G,∂,fh,∂ are NSQ∗b open in Y˜ containing f£1s,f£2s/, and G,∂ as h,∂ disjoint. It follows that Y˜ is NSQ∗b Hausdorff space.Theorem 8.
LetL,τ1,τ2,τ3,τ4,∂ be a NSQ second countable space and let h,∂ be NSQ uncountable subset of L,τ1,τ2,τ3,τ4,∂, then at least one point of h,∂ is a NSQ limit point of h,∂.Proof.
LetW=ℬ1,ℬ2,ℬ3,ℬ4,…,ℬn:n∈ℕ for L,τ1,τ2,τ3,τ4,∂.
Let, if possible, no point off,∂ be a NSQ limit point of f,∂. Then, for each £1s∈f,∂, there exists NSQ∗b open set ρ,∂£1s such that £1s∈ρ,∂£1s and ρ,∂£1s∩˜f,∂=£1s. Since W is soft base, there exists ℬn∈W such that £1s∈ℬn⋐ρ,∂£1s. Therefore, ℬn∩˜f,∂⋐ρ,∂£1s∩˜f,∂=£1s. Moreover, if £1s1 and £1s2 be any two NSQ points such that £1s1≠£1s2 which means either £1s1≻£1s2 or £1s1≺£1s2, then there exists ℬn and ℬn in W such that ℬn∩˜f,∂=£1s1 and ℬn∩˜f,∂=£1s2. Now, £1s1≠£1s2 which guarantees that £1s1≠£1s2 which implies that ℬn∩˜f,∂≠ℬn∩˜f,∂ which implies ℬn≠ℬn. Thus, there exists a one to one NSQ correspondence of f,∂ on to ℬn:£1s∈f,∂. Now, f,∂ being NSQ uncountable, it follows that ℬn:£1s∈f,∂ is NSQ uncountable. This is a contradiction.Theorem 9.
LetL,τ1,τ2,τ3,τ4,∂ and Y˜,F1,F2,F3,F4,∂ be two NSQTS and suppose f,θ be a NSQ continuous function such that f,∂:L,τ1,τ2,τ3,τ4,∂⟶Y˜,F1,F2,F3,F4,∂ is NSQ continuous function and let ℒ,∂⋐L,τ1,τ2,τ3,τ4,∂ supposes the B.V.P. Then, safely, fℒ,∂ has the B.V.P.Proof.
Supposeℒ,∂ be an infinite NSQ subset of f,∂, so that ℒ,∂ contains an enumerable NS set £1sn:n∈N, then there exists enumerable NSQ set £2s/n:n∈N⋐ℒ,∂ such that f£2s/n=£1sn.ℒ,∂ has B.V.P implying that every infinite soft subset ℒ,∂ supposes NSQ accumulation point belonging to ℒ,∂, and this implies that £2s/n:n∈N has NSQ limit point, say, £2s/0∈ℒ,∂ implies that the limit of NSQ sequence £2s/n:n∈N is £2s/0∈ℒ,∂⟶£2s/n⟶£2s/0∈ℒ,∂.f is NSQ continuous. Furthermore, £2s/n£2s/0∈ℒ,∂⟶f£2⋎1,⋎2,⋎3nf£2s/0∈fℒ,∂⟶£1snf£2s/0∈fℒ,∂ implies that NSQ limit of a NSQ sequence £1sn:n∈N is f£2s/0∈fℒ,∂ implying that NSQ limit of a NSQ sequence £1sn:n∈Nisf£2s/0∈f,∂ℒ,∂. Finally, we have shown that there exists infinite soft subset £1sn:n∈N of fℒ,∂ containing a limit point f£2s/0∈fℒ,∂; this guarantees that fℒ,∂ has B.V.P.Theorem 10.
LetL,τ1,τ2,τ3,τ4,∂NSQTS and let £1sn˜ be a NSQ sequence in L,τ1,τ2,τ3,τ4,∂ such that it converges to NSQ a point £1s0, then the NSQ set f,∂ consisting of NSQ points £1sn0 and £1snn=1,2,3,…… is soft NSQ compact.Proof.
GivenL,τ1,τ2,τ3,τ4,∂NSQTS and let £1sn˜ be an NSQ sequence in π˜,τ1,τ2,τ3,τ4,∂ such that it converges to a point £1sn0, that is £1sn˜⟶£1sn0∈L˜. Let g,∂=£1s1˜,£1s2˜,£1s3˜,£1s4˜,£1s5˜,£1s6˜,£1s7˜,…. Let S,∂α:α∈Δ be NSQ∗b open covering of g,∂ so that g,∂⋐∪˜S,∂α:α∈Δ,£1sn0∈g,∂ implies that there exists α0∈Δ such that £1sn0∈S,∂α0. By NSQ convergence, £1sn0∈S,∂α0∈L,τ1,τ2,τ3,τ4,∂ implies that there exists n0∈Ns.t.n≥n0 and £1sn∈S,∂α0. Evidently, S,∂α0 contains the NSQ points £1sn0,£1sn0+1,£1sn0+2,£1sn0+3,£1s,….,£1sn0+n,…. Look carefully at the points and train them in a way as £1s1,£1s2,£1s3,£1s4,….£1sn generating a finite soft set. Let 1≤n0−1. Then, £1⋎1,⋎2,⋎3i∈g,∂. For this i,£1si∈g,∂. Hence, there exists αi∈Δ such that £1si∈S,θαi. Evidently, g,θ⋐Ur=0n0−1S,θαi˜. This shows that S,θαi:0≤n0−1 is NSQ∗b open cover of g,∂. Thus, an arbitrary NS Q∗b open cover S,∂α:α∈Δ of g,∂ is reducible to a finite NSQ subcover S,∂αi:i=0,1,2,3,…n0−1, and it follows that g,θ is NSQ∗b compact.Theorem 11.
IfL,τ1,τ2,τ3,τ4,∂NSQTS such that it has the characteristics of NSQ ∗b sequentially compactness. Then, L,τ1,τ2,τ3,τ4,∂ is NSQ∗b countably compact.Proof.
LetL,τ1,τ2,τ3,τ4,∂NSQTS and let ρ,∂ be finite soft subset of L,τ1,τ2,τ3,τ4,∂. Let £1s1˜,£1s2˜,£1s3˜,£1s4˜,£1s5˜,£1s6˜,£1s7˜,… be an NSQ sequence of NSQ points of ρ,∂. Then, ρ,∂ being finite, at least one of the elements in ρ,∂ say £1s0˜ must be duplicated an infinite number of times in the NSQ sequence. Hence, £1s0˜,£1s0˜,£1s0˜,£1s0˜,£1s0˜,£1s0˜,£1s0˜,… is soft NSQ subsequence of £1sn˜ such that it is NSQ constant sequence and repeatedly constructed by single soft number £1s0˜, and we know that a soft constant sequence converges on its self. So, it converges to £1s0˜ which belongs to ρ,∂. Hence, ρ,∂ is soft sequentially NSQ∗b compact.Theorem 12.
LetL,τ1,τ2,τ3,τ4,∂NSQTS and Y˜,F1,F2,F3,F4,∂ be another NSQTS. Let f,∂be a soft continuous mapping of an NSQ sequentially compact NSQ∗b space L,τ1,τ2,τ3,τ4,∂ into Y˜,F1,F2,F3,F4,∂, then f,∂L,τ1,τ2,τ3,τ4,∂ is NSQ∗b sequentially compact.Proof.
GivenL,τ1,τ2,τ3,τ4,∂NSQTS and Y˜,F1,F2,F3,F4,∂ be another NSQTS. Let f,∂ be a NSQ continuous mapping of a NSB sequentially compact space L,τ1,τ2,τ3,τ4,∂ into Y˜,F1,F2,F3,F4,∂, then we have to prove L,τ1,τ2,τ3,τ4,∂NSQ sequentially. For this, we proceed as follows. Let £2s/1˜,£2s/2˜,£2s/3˜,£2s/4˜,…£2s/7˜,…£2s/n˜,…. be a NSQ sequence of NSQ points. Then, for each n∈N, there exists £1s1˜,£1s2˜,£1s4˜,£1s5˜,..£1s7˜,…£1sn˜,….∈L,τ1,τ2,τ3,τ4,θ. Thus, we obtain an NSQ sequence £1s1˜,£1s2˜,£1s3˜,£1s4˜,..£1s5,£1s6˜,£1s7˜,…£1sn˜,…. in L,τ1,τ2,τ3,τ4,∂. But L,τ1,τ2,τ3,τ4,∂ being soft sequentially NSQ∗b compact, there is an NSQ subsequence £1sni˜ of
£1sn˜ such that £1sni˜⟶£1s˜∈L˜,τ1,τ2,τ3,∂. So, by NSQ∗b continuity of f,£1sni˜⟶£1s˜ implies that f£1sni˜f⟶£1sn˜∈fL,τ1,τ2,τ3,τ4,∂. Thus, f£2⋎1,⋎2,⋎3ni˜ is a soft subsequence of £2s/1˜,£2s/2˜,£2s/3,£2s/5˜,£2s/5˜,£2s/6˜,£2s/7˜,…£2s/n˜,…. converges to fκ1˜ in L˜,τ1,τ2,τ3,τ4,∂. Hence, L˜,τ1,τ2,τ3,τ4,∂NSQ∗b sequentially compact.Theorem 13.
LetL˜,τ1,τ2,τ3,τ4,∂NSQTS and suppose f,∂,g,∂ be two NSQ continuous function on an NSQTS L˜,τ1,τ2,τ3,τ4,∂ in to an NSQTS Y˜,F1,F2,F3,F4,∂ which is NSQ∗b Hausdorff. Then, soft set £1s∈L˜:f£1s=g£1s is NSQ∗b closed of L,τ1,τ2,τ3,τ4,∂.Proof.
Let if£1s∈L˜:f£1s=g£1s is an NSQ set of function. If £1s∈L˜:f£1s=g£1s=0L,∂, it is clearly NSQ∗b open, and therefore, £1s∈L˜:f£1s=g£1s is NSQ∗b closed, that is, nothing is proved in this case. Let us consider the case when £1s∈L˜:f£1s=g£1s and let ρ∈£1s∈L˜:f£1s=g£1s. Then, ρ does not belong £1s∈L˜:f£1s=gk1. Result in fρ≠gρ. Now, Y˜,F1,F2,F3,∂ being NSQ∗b Hausdorff space, so there exists NSQ∗b open sets f,∂,ℌ,∂ of fρ and gρ, respectively, such that f,∂ and ℌ,∂ such that these NSQ sets such that the possibility of one rules out the possibility of other. By soft continuity of f,∂,g,∂,f,∂−1 as well as is NSQ∗b open nhd of ρ, and therefore, f,∂−1∩˜g,∂−1 is contained in £1s∈L˜:f£1s=g£1s for, £1s∈f,∂−1∩˜g,∂−1g£1⋎1,⋎2,⋎3∈g,∂ and gf£1s≠g£1s because f,∂ and ℌ,∂ are mutually exclusive. This implies that κ1 does not belong to £1s∈L˜:f£1s=h£1s. Therefore, this shows that £1s∈L˜:f£1s=g£1s is nhd of each of its points. So, £1s∈L˜:f£1s=g£1sNSQ∗b open, and hence £1s∈L˜:f£1s=g£1s is NSQ∗b closed.Theorem 14.
LetL,τ1,τ2,τ3,τ4,∂NSQTS such that it is NSQ∗b Hausdorff space and let f be soft continuous function of L,τ1,τ2,τ3,τ4,∂ into itself. Then, the NSQ set of fixed points under g is a NSQ∗b closed set.Proof.
LetS=g£1s=£1s. If Sc=0L,∂, Then, NSQ∗b is open, and therefore, S=g£1s=£1s closed. So, let S=g£1s=£1s and let £2s/∈g£1s=£1sc.Then, £2s/ does not belong to g£1s=£1s, and therefore, g£1s=£1s. Now, £2s/, g£2s/ being two distinct points of the NSQ∗b Hausdorff space L,τ1,τ2,τ3,τ4,∂, so there exist NSQ∗b open sets g,∂,ℌ,∂ such that £2s/∈g,∂,h£2s/∈ℌ,∂ and g,∂,ℌ,∂ are disjoint. Also, by the NSQ continuity, NSQ∗b is an open set containing £2s/, we pretend that g,∂∩˜f−1ℌ,∂⋐g£1s=£1sc. Since μ∈g,∂∩˜f−1ℌ,∂⇒μ∈g,∂,μ∈h−1⇒μ∈g,∂,hμ∈ℌ,∂⇒μ≠gμ. As g,∂∩˜ℌ,∂=0L,∂ implies that μ does not belong to f£1s=£1s⇒μ∈g£1s=£1sc.Therefore, £1s∈g,0L,∂∩˜g−1ℌ,0L,∂⋐fκ1⋎1,⋎2,⋎3=κ1⋎1,⋎2,⋎3c. Thus, f£1s=£1sc is the NSQ neighborhood of each of its points. So, g£1s=£1sc is NSQ∗b open, and hence, g£1s=£1s is NSQ∗b closed.


## 6. Conclusion

Neutrosophic soft topology (NST) is extension of vague soft topology (VST) and VST is extension of fuzzy soft topology (FST). VST gives two type of informations. One is true, and second one is false. It does not give informations about the indeterminacy (doubtful) case. NST is dominant over VST because it supposes all the three informations that is true, false, and indeterminacy at the same time. NST has narrow domain as compared to neutrosophic soft bitopology (NSBT). Some problems are very hard to discuss in NST, so need of NSBT is felt. Still NSTS is unable to afford problems of large number of domain. So, extension is needed to neutrosophic soft tritopology (NSTT) and neutrosophic soft quad-topology (NSQT). In our work, we regenerated some structures in NSQTS with new definition that is∗b open sets relative to soft points. We worked with the operations given in references [14–16] which are entirely different from references [13, 17]. In future, we will develop neutrosophic soft penta-topology and neutrosophic soft hexa-topology relative to soft points of the space under more generalized neutrosophic soft open sets. We will try to develop neutrosophic soft separation axioms and their engagement with each other. In addition to this, neutrosophic soft other separation axioms will also be given special attention. After doing this, on the basis of reference [12], we will try to build the same structures with respect to hyper-soft sets and then with respect to plithogenic hyper-soft sets.



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*Source: 8483632-2022-01-19.xml*