Neutrosophic Soft alpha-Open Set in Neutrosophic Soft Topological Spaces

journal homepage: http://jac.ut.ac.ir

Neutrosophic Soft α−Open Set in Neutrosophic Soft

Topological Spaces

Arif Mehmood

, Fawad Nadeem

, Choonkil Park

, Gorge Nordo

,

Humaira Kalsoom

, Muhammad Rahim Khan

and Naeem Abbas

Islamabad, Pakistan.

2_{Department of Mathematics, University of Science and Technology, Bannu, Khyber}

Pakhtunkhwa, Pakistan.

3_{Department of Mathematics, Research Institute for Natural Sciences, Hanyang University}

Seoul 133-791, South Korea.

4_{MIFT – Dipartimento di Scienze Matematiche e Informatiche, scienze Fisiche e scienze della}

Terra, Messina University, Messina, Italy.

5_{School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China.}

ABSTRACT ARTICLE INFO

In this paper, the notion of generalized neutrosophic soft open set (GNSOS) in neutrosophic soft open set (GN-SOS) in neutrosophic soft topological structures relative to neutrosophic soft points is introduced.The concept of generalized neutrosophic soft separation axioms in neu-trosophic soft topological spaces with respect to

Article history:

Received 17, September 2019 Received in revised form 11, April 2020

Accepted 15 May 2020

Available online 01, June 2020

Keyword: neutrosophic soft set; neutrosophic soft point.

AMS subject Classification: 54F05.

∗_{Corresponding author: Arif Mehmood. Email: mehdaniyal@gmail.com} †_{fawadnadeem2@gmail.com}

‡_{baak@hanyang.ac.kr} §_{giorgio.nordo@unime.it} ¶_{humaira87@zju.edu.cn}

1

Abstract continued

soft points. Several related properties, structural characteristics have been investigated. Then the convergence of sequence in neutrosophic soft topological space is defined and its uniqueness in generalized neutrosophic soft Hausdorff space (GNSHS) relative to soft points is examined. Neutrosophic monotonous soft function and its characteristics are switched over to different results. Lastly, generalized neutrosophic soft product spaces with respect to crisp points have been addressed.

Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of in-clusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint. Atanassov [19] introduced the concept of intuitionistic fuzzy set (IFS) which is generalization of the concept fuzzy set. Various properties are proved, which are connected to the operations and relations over sets, and with modal and topological operators, defined over the set of IFS’s. Bayramov and Gunduz [2] introduced some important properties of intuitionistic fuzzy soft topological spaces and define the intuitionistic fuzzy soft closure and interior of an intuitionistic fuzzy soft set. Furthermore, intuitionistic fuzzy soft continuous mapping are given and structural char-acteristics are discussed and studied. Deli and Broumi [6] defined for the first define a relation on neutrosophic soft sets which allows to compose two neutrosophic soft sets. It is devised to derive useful information through the composition of two neutrosophic soft sets. The authors then, examined symmetric, transitive and reflexive neutrosophic soft relations and many related concepts such as equivalent neutrosophic soft set relation, partition of neutrosophic soft sets, equivalence classes, quotient neutrosophic soft sets, neutrosophic soft composition are given and their propositions are discussed. Finally a decision making method on neutrosophic soft sets is presented. Bera and Mahapatra [3] Introduced the concept of Cartesian product and the relations on neutrosophic soft sets in a new approach. Some properties of this concept have been discussed and verified with suitable real life examples. The neutrosophic soft composition has been defined and verified with the help of example. Then, some basic properties to it have been established. After that the concept of neutrosophic soft function along with some of its basic properties have been introduced and verified by suitable examples. Injective, sur-jective, bisur-jective, constant and identity neutrosophic soft functions have been defined. Finally, properties of inverse neutrosophic soft function have been discussed with proper example. Smarandache [17] for the first time initiated the concept of neutrosophic set which is generalization of the intuitionistic fuzzy set (IFS), para-consistent set, and

in-tuitionistic set to the neutrosophic set (NS). many examples are presented. Peculiarities between NS and IFS are underlined.Maji [16] studied the concept of neutrosophic set of Smarandache. The author introduced this concept in soft sets and defined neutrosophic soft set. Some definitions and operations have been introduced on neutrosophic soft set. Some properties of this concept have been established. Bera and Mahapatra [4] introduced how to construct a topology on a neutrosophic soft set (NSS). the notion of neutrosophic soft interior, neutrosophic soft closure, neutrosophic soft neighborhood, and neutrosophic soft boundary, regular NSS are introduced and some of their basic properties are studied in this paper. Then the base for neutrosophic soft topology and subspace topology on NSS have been defined with suitable examples. Some related properties have been developed, too. Moreover, the concept of separation axioms on neutrosophic soft topological space have been introduced along with investigation of several structural characteristics.

Khattak et al. [9, 10] continued work on soft bi-topological structures. The authors discussed weak separation axioms and other separation axioms in soft bi-topological space with respect to crisp points and soft points of the spaces respectively.

Mehmood et al. [13] discussed soft α-connectedness, soft α-dis-connectedness and soft α-compact spaces in bi-polar soft topological spaces with respect to ordinary points. For better understanding the authors provided suitable examples.

Khattak et al. [11] introduced for the first time application soft semi open sets in binary topology. An important outcome of this work is a formal framework for the study of informations associated with ordered pairs of soft sets. Moreover the authors discussed five main results concerning binary soft topological spaces in the same article. Khattak et al.[8] for the first time bounced up the idea of NS b-open set, NS b-closed sets and their properties. Also the idea of neutrosophic soft b-neighborhood and neu-trosophic soft b-separation axioms in NSTS 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 NSTS is diffused in different results with respect to soft points. Furthermore, properties of neutrosophic soft b-Ti

-space (i = 0,1,2,3,4) and some associations between them are discussed.

Khattak et al. [12, 14, 15] continued has work on another structures known as neutrosophic soft topological structures. The authors introduced generalized and most generalized neutrosophic soft open sets in neutrosophic soft topological structures. They discussed different results with respect to soft point of the space.

The first aim of this paper is to reintroduce the concept of GNS-open set and develop a neutrosophic soft topology on GNS-open set with respect to soft points. Later the notions of neutrosophic soft point and their characteristics are addressed in connection

with GNS-neighborhood. The concept of generalized separation axioms of neutrosophic soft topological spaces and related results are studied with respect to soft points. Study is slowly extended to neutrosophic soft Countability, NS sequences and their conver-gence in Hausdorff spaces. Furthermore, neutrosophic soft monotonous function and neutrosophic soft product spaces are addressed in connection with neutrosophic soft generalized open sets with respect to soft points. We hope that these results will be useful for future study on neutrosophic soft topology to carry out a general framework for most real-world applications.

2

Preliminaries

In this section we now state certain useful definitions, theorems, and several existing results for neutrosophic soft sets that we require in the next sections.

Definition 2.1. [8] N SS on father set hXi is characterized as Aneutrosophic _{= [dx,}

T_AntophichXi I_Antophic_{hXie: x ∝ hXi]. T : (x) →]0}−_{, 1}+ _{[I : hXi →]0}−_{, 1}+ _{[F : hXi →}

]0−, 1+_{[ So that’s it o}−

4 {T + I + F 4 3+_}

Definition 2.2. [8] let hXi be a fatherset, `paramater _{be a set of all conditions, and}

£(hXi) denote the efficiency set of hXi. A pair (f, `paramater_{) is referred to as a soft}

set over hXi where f is a map given by F : `paramater<sub>) → £(hXi) . For n ∝ `</sub>paramater_,

f (n) may be viewed as the set of n -softset elements (f, `paramater_{) or as a set of n −}

estimatedthe soft set components, i.e.(f, `paramater<sub>) = {n, f (n) : n ∝ `</sub>paramater_{, F :}

`paramater) → £(hXi)}

Definition 2.3. [8] let hXi be a fatherset, `paramater be a set of all conditions, and £(hXi) denote the efficiency set of hXi. A pair (f, `paramater) is referred to as a soft set over hXi where f is a map given by F : `paramater) → £(hXi) . Then a NS set ( ˜<sub>f , `</sub>paramater) over hXi is a set defined by a set of valued functions signifying a mapping ˜_{f : `</sub>paramater → £(hXi) is referred to as the approximate NSset function ( ˜<sub>f : `}paramater) In other words, the NSset is a group of conditions of certain elements of the set £(hXi) so it can be written as a set of ordered pairs:( ˜<sub>f : `</sub>paramater) = {((n, [x, I<sub>f</sub>˜(x)(x), I<sub>f</sub>˜(x)(x), F<sub>f</sub>˜(x)(x), n :∝ `paramater} T_f˜(x)(x), I_f˜(x)(x), F_f˜(x)(x) [0, 1] are

membership of truth, membership of indeterminacy and membership of falsehood ˜f (n) Since the supremum of eachT, I, f is 1 , the inequality that 0  {Tf(x)(x)+  {If(x)(x)+ 

{Ff(x)(x) +3+} is obvious

Definition 2.4. [8] Let ( ˜_{f : `</sub>paramater) be a NSS over the father set hXi. The comple-ment of ( ˜<sub>f : `}paramater) is signified ( ˜_{f : `}paramater)c and is defined as follows:

( ˜<sub>f : `</sub>paramater)c= {((n, [x, I<sub>f</sub>˜(x)(x), I<sub>f</sub>˜(x)(x), F<sub>f</sub>˜(x)(x), n :∝ `paramater} T_f˜(x)(x), I_f˜(x)(x), F_f˜(x)(x)

It’s clear that (( ˜_{f : `</sub>paramater)c)c= ( ˜<sub>f : `}paramater).

Definition 2.5. [8] Let ( ˜f , n) and ( ˜ρ, n) two N SS over the father set (X) ( ˜f , n) is supposed to be N SSS of ( ˜ρ, n) if T_f˜(x)(x)  Tρ˜(x)(x), I_f˜(x)(x)  Tρ˜(x)(x), I_f˜(x)(x) 

Fρ˜(x)(x), n ∝ `paramaterandx ∝ (x) It is signifies as ( ˜f , n) v ( ˜ρ, n) .( ˜f , n) is is said to

be N S equal to ( ˜ρ, n) if ( ˜f , n) is N SSS of ( ˜ρ, n) and ( ˜ρ, n) is N SSS of if ( ˜f , n). It is symbolized as ( ˜f , n) = ( ˜ρ, n)

3

Neutrosophic soft points and their properties

Definition 3.1. [8] Let ( ˜f1, n) and ( ˜f2, n) be two N SSS over father set hXi s.t. ( ˜f1, n)

6= ( ˜f2, n) and then their union is signifies as ( ˜f1, n) ˜S( ˜f2, n)=( ˜f3, n) and is defined as

( ˜f3, n) = {((n, [x, T_f˜₃(x)(x), I_f˜₃(x)(x), F_f˜₃(x)(x), n :∝ `paramater}

Where,

T_f˜₃(x)(x)= max[T_f˜₁(x)(x), T_f˜₂(x)(x)I_f˜₃(x)(x) = max[I_f˜₁(x)(x), I_f˜₂(x)(x), F_f˜₃(x)(x) = min[I_f˜₁(x)(x), I_f˜₂(x)(x)

Definition 3.2. [8] Let ( ˜f1, n) and ( ˜f2, n) be two N SSS over father set hXi s.t. ( ˜f1, n)

6= ( ˜f2, n) and. Then their union is signifies as ( ˜f1, n) ˜T( ˜f2, n)=( ˜f3, n) and is defined as

( ˜f3, n) = {((n, [x, T_f˜₃(x)(x), I_f˜₃(x)(x), F_f˜₃(x)(x), n :∝ `paramater}

Where,

T_f˜₃(x)(x)= max[T_f˜₁(x)(x), T_f˜₂(x)(x)I_f˜₃(x)(x) = max[I_f˜₁(x)(x), I_f˜₂(x)(x), F_f˜₃(x)(x) = min[I_f˜₁(x)(x), I_f˜₂(x)(x)

Definition 3.3. [8] N S ( ˜f1, n) be a N SS over the father set hXi is said to be a vacuous

set if Tf(x)(x) =0,Tf(x)(x) =0,Tf(x)(x)=1,ge ∝ nand g x ∝ (x) It is signifies as 0((x),n) .

Definition 3.4. [8] N S ( ˜f1, n) be a N SS over the father set X It is said to be an

absolute neutrosophical softness if Tf(x)(x) =0,Tf(x)(x) =0,Tf(x)(x)=1,ge ∝ nand g x ∝

(x)Itissignif iesas1((x),n) clearly, 0((x),n)c= 1((x).n)and1((x).n)c= 0((x),n)

Definition 3.5. [8] Let N SS (h˜_{xi, `</sub>parameter) be the family of all N S sets over the father set ( ˜X ) and τ ⊂ NSS(h˜<sub>xi, `}parameter). Then τ is said to be a N ST on ( ˜X) if: (1) 0(hXi,n).1((x).n) ∝ τ

(2) The union of any number of N SS in τ belongs to τ

(3) The intersection of a finite number of N SS in τ belongs to τ Then N SS (h˜_{xi, τ `}parameter) is said to be a N ST S over ( ˜X) . Each member of τ is said to be a NSopen set.

Definition 3.6. [8] Let (h˜_{xi, τ `</sub>parameter) be a N ST S over ( ˜X ) and ( ˜<sub>f : `}paramater) be a neutrosophic soft set over ( ˜X ) Then ( ˜_{f : `}paramater) is supposed to be a N S closed set if f its complement is a N Sopen set..

Definition 3.7. [8] Let (h˜_{xi, τ `</sub>parameter) be a N ST S over ( ˜X ) and ( ˜<sub>f : `}paramater) be a neutrosophic soft set over ( ˜X ) Then ( ˜_{f : `</sub>paramater) is supposed to be a α − open if. ( ˜<sub>f : `}paramater) ⊆ int(cl(int(( ˜_{f : `</sub>paramater))) and N S α − close if ( ˜<sub>f : `}paramater) ⊇ cl(int(cl(( ˜_{f : `}paramater)))

Definition 3.8. [8] Let N S be the family of all N S over father set ( ˜_{X ) and x ∝ ( ˜}X) The NS x(a,b,c) is supposed to be a neutrosophic point, for 0 < a, b, c ≤ 1, and is defined

as follows: x(a,b,c)y = $( (a, b, c) provided y = x} (0, 0, c) provided y 6= x} % (1)

Example 3.9. Suppose that ( ˜X) = {x1, x2} Then N set A = {< x1,0.1, 0.3, 0.5 >, <

x2, 0.5, 0.4, 0.7 >} is the union of N points x1(0.1,0.3,0.5) and x2(0.5,0.4,0.7) Now we define the

concept of NSpoints for NSsets. .

Definition 3.10. Let N SS(hXi)be the family of all N soft sets over the father set˜ ˜

(X) Then NSS (X(a,b,c))e is called a NSpoint, for every x ∝ ˜hXi, 0 ≺ {a, b, c  1}, e ∝

`parameter , and is defined as follows:

xe_(a,b,c)(e)y =

(a, b, c) provided e = e ∧ y = x} (a, b, c) provided e 6= e ∧ y 6= x}

!

. (2)

Definition 3.11. Suppose that the father set ˜(X) is assumed to be ( ˜X) = {x1, x2} and

the set of conditions by `parameter = {e1, e2} Let us consider NSS ( ˜f , `paramater) over the

father set(X) as follows:˜

( ˜_{f , `}paramater) = $(( (a, b, c) provided y = x} (0, 0, c) provided y 6= x} )% (3)

It is clear that ( ˜_{f , `}paramater_{) is the union of its NSpoints X}e11_{(0.3,0.7,0.6)}

Xe21(0.4,0.6,0.8), Xe12(0.4,0.3,0.8)AN D _Xe12(0.3,0.7,0.2). Xe11(0.3,0.7,0.6) ₌ $( e1 = {< x1, 0.3, 0.7, 0.6 >, < X2, 0, 0, 1 >} e2 = {< x1, , 0, 0, 1 >, < X2, 0, 0, 1 >} % (4) Xe21(0.4,0.6,0.8) ₌ $( e1 = {< x1, 0, 0, 1 >, < X2, 0, 0, 1 >} e2 = {< x1, 0.4, 0.6, 0.8 >, < X2, 0, 0, 1 >} % (5) Xe22(0.3,0.7,0.2) ₌ $( e1 = {< x1, 0, 0, 1 >, < X2, 0, 0, 1 >} e2 = {< x1, 0, 0, 1 >, < X2, 0.3, 0.7, 0.2 >} % (6)

Definition 3.12. Let ( ˜_{F , `</sub>parameter) be a NSS over the father set < ˜x > . we say that xe<sub>(a,b,c) ∝ ( ˜F , `}parameter) read as belonging to the NSS ( ˜_{F , `}parameter) whenever

a  T_{f (x)}˜ (x),b  I_{f (x)}˜ (x),c  F_{f (x)}˜ (x)

Definition 3.13. (h˜_{xi, τ, `</sub>parameter) be a NSTS over ( ˜X) ( ˜<sub>f , `}parameter) be a NSS over ( ˜X) NSS ( ˜_{f , `</sub>parameter) in (h˜<sub>xi, τ, `}parameter) is called a Nnbhd of the NS point xe_(a,b,c) ∝ ( ˜f , `parameter) if ∃ a NS α open set (˜<sub>g, `</sub>parameter) such that xe_{(a,b,c) ∝ (˜g, `</sub>parameter<sub>) @</sub> ( ˜<sub>f , `}parameter)

Theorem 3.14. (h˜_{xi, τ, `</sub>parameter) be a NSTS over and ( ˜<sub>f , `}parameter) Then ( ˜_{f , `</sub>parameter) is a NS α open set ⇔ ( ˜<sub>f , `}parameter) is a NS nbhd of its NS points.

Proof. . Let ( ˜_{f , `}parameter_{) be a NS open set and x}e

(a,b,c) ∝ ( ˜f , `

parameter_{) @ is a NS nbhd}

of xe

(a,b,c). conversely, let ( ˜f , `

parameter_{) be a NS nbhd of its NS points. Let x}e

(a,b,c) ∝

( ˜_{f , `</sub>parameter<sub>) since ( ˜f , `}parameter_{) is a NS nbhd of the NS pointx}e

(a,b,c) ∃ (˜g, `

parameter₎

∝ (h˜xi, τ, `parameter_{) s.t. x}e

(a,b,c) ∝ (˜g, `

parameter_{) @ ( ˜f , `</sub>parameter<sub>) Since ( ˜f , `}parameter₎

= txe

(a,b,c) such that x e

(a,b,c) ∝ ( ˜f , `

parameter_{) it follows}

That ( ˜_{f , `</sub>parameter<sub>) is a union of NS α open sets and hence ( ˜f , `}parameter_{) is a NS α}

open set. The nbhd system of a NS point xe

(a,b,c), denoted by t x e

(a,b,c) ,`

parameter_{) , is}

the family of all its nbhds.

Theorem 3.15. The nbhd system t xe (a,b,c),`

parameter_{) at x}e

(a,b,c)in a NSTS (h˜xi, τ, `

parameter₎

has the following properties:                                   

(1)if ( ˜_{f , `</sub>parameter<sub>) ∝ tx</sub>e<sub>(a,b,c), `}parameter), thenxe_{(a,b,c) ,`</sub>parameter) (2)if ( ˜<sub>f , `}parameter_{) ∝ tx}e

(a,b,c), `

parameter_{), ( ˜f , `</sub>parameter<sub>) @ (˜h, `}parameter_)then

(˜_{h, `</sub>parameter<sub>) ∝ tx</sub>e<sub>(a,b,c), `}parameter) : (3)if ( ˜_{f , `</sub>parameter<sub>), (˜g, `}parameter_{) ∝ tx}e

(a,b,c), `

parameter_)then

( ˜_{f , `</sub>parameter) u (˜<sub>g, `}parameter_{) ∝ tx}e_{(a,b,c), `</sub>parameter) (4)if ( ˜<sub>f , `}parameter_{) ∝ tx}e (a,b,c), ` parameter<sub>)then</sub> ∃if (˜<sub>g, `</sub>parameter ) ∝ txe(a,b,c), ` parameter<sub>)s.t</sub> if (˜<sub>g, `</sub>parameter_{) ∝ tx}e0

(a0_,b0_,c0₎, `parameter), f orf oreach

xe0

(a0_,b0_,c0₎, `parameter)( ˜f , `parameter) u (˜g, `parameter) ∝ xe<sub>(a,b,c)</sub>, `parameter)

(7)

Proof. The proof of 1),2),and 3) is obvious from Definition 3.13.

if ( ˜_{f , `</sub>parameter<sub>) ∝ ∪x</sub>e<sub>(a,b,c), `}parameter) then ∃ NS α open set (˜_{g, `</sub>parameter) s.t xe<sub>(a,b,c), `}parameter_{) ∝} (˜_{g, `</sub>parameter<sub>) @ ( ˜f , `}parameter) From Proposition 3.14,(˜_{g, `</sub>parameter<sub>) ∝ ∪x</sub>e<sub>(a,b,c), `}parameter)

so for each xe_(a00_,b0_,c0₎, ∝ (˜g, `parameter), (˜g, `parameter) ∝ (˜g, `parameter) ∝ ∪xe 0

(a0_,b0_,c0₎, `parameter)

Definition 3.16. Let xe_(a,b,c)and xe_(a00_,b0_,c0₎be two NSpoints. For the NSpoints Over father

set ( ˜X) we say that the NSpoints are distinct points xe_(a,b,c)u xe0

(a0_,b0_,c0₎ = 0, (˜x, `parameter)

It is clear that xe_(a,b,c) and xe_(a00_,b0_,c0₎ are distinct NS points if and only if x  y or X ≺ Y

or e0  e or e0 _{≺ e}

4

Neutrosophic soft α separation axioms

In this section, we define neutrosophical soft α separation in neutrosophical soft topo-logical space with respect to NS points.

Definition 4.1. Let (h˜_{xi, τ, `}parameter_{) be a NSTS over( ˜X) , and}

     xe_(a,b,c)  xe0 (a0_,b0_,c0₎ or xe (a,b,c) ≺ x e0 (a0_,b0_,c0₎ (8)

NSpoints. If there exist NS α-open sets ( ˜_{f , `</sub>parameter<sub>) and (˜g, `}parameter_{) such that}

xe (a,b,c) ∝ ( ˜f , ` parameter<sub>) x</sub>e (a,b,c)u( ˜f , ` parameter_{) = 0} ( ˜X,,`parameter<sub>)</sub>or ye 0 (a0<sub>,b</sub>0<sub>,c</sub>0<sub>)</sub> ∝ (˜g, `parameter), ye0

(a0_,b0_,c0₎u (˜g, `parameter) = 0<sub>( ˜X,,`</sub>parameter₎ Then (h˜xi, τ, `parameter) is called a NS α0

Definition 4.2. Let (h˜_{xi, τ, `}parameter_{) be a NSTS over( ˜X) , and}

     xe_(a,b,c)  xe0 (a0_,b0_,c0₎ or xe (a,b,c) ≺ x e0 (a0_,b0_,c0₎ (9)

NSpoints. If there exist NS α-open sets ( ˜_{f , `</sub>parameter<sub>) and (˜g, `}parameter_{) such that}

xe (a,b,c) ∝ ( ˜f , ` parameter<sub>) x</sub>e (a,b,c)u( ˜f , ` parameter_{) = 0} ( ˜X,,`parameter<sub>)</sub>or ye 0 (a0<sub>,b</sub>0<sub>,c</sub>0<sub>)</sub> ∝ (˜g, `parameter), ye0

(a0_,b0_,c0₎u (˜g, `parameter) = 0<sub>( ˜X,,`</sub>parameter₎ Then (< ˜X >, τ, `parameter) is called a NS α1

Definition 4.3. Let (h˜_{xi, τ, `}parameter_{) be a NSTS over( ˜X) , and}

     xe (a,b,c)  xe 0 (a0_,b0_,c0₎ or xe (a,b,c) ≺ x e0 (a0_,b0_,c0₎ (10)

NSpoints. If there exist NS α-open sets ( ˜_{f , `</sub>parameter<sub>) and (˜g, `}parameter_{) such that}

xe (a,b,c) ∝ ( ˜f , ` parameter<sub>) x</sub>e (a,b,c)u( ˜f , ` parameter_{) = 0} ( ˜X,,`parameter<sub>)</sub>or ye 0 (a0<sub>,b</sub>0<sub>,c</sub>0<sub>)</sub> ∝ (˜g, `parameter), ye0

Example 4.4. Suppose that the father set ( ˜X) is assumed to be

( ˜X) = {x1, x2} and the set of conditions by `parameter = {e1, e2}.Let us consider NSset

( ˜_{f , `}parameter) over the father set ( ˜X) and xe11(0.1,0.4,0.7)_{, x}e21(0.2,0.5,0.6)_{, x}e12(0.3,0.3,0.5) _and

xe12(0.4,0.4,0.4) _{NSpoints. Then the family}

τ = {0_{( ˜X,,`</sub>parameter<sub>)</sub>, 1<sub>( ˜X,,`}parameter₎, ( ˜f₁, `parameter),( ˜f<sub>2</sub>, `parameter),( ˜f₃, `parameter),( ˜f₄,

`parameter), ( ˜f5, `parameter),( ˜f6, `parameter),( ˜f7, `parameter),( ˜f8, `parameter)}.

where ( ˜f1, `parameter) = x e<sub>11</sub> (0.1,0.4,0.7)<sub>( ˜f</sub> 2, `parameter) = x e₂₁ (0.2,0.5,0.6)_{, ( ˜f} 3, `parameter) = x e<sub>12</sub> (0.3,0.3,0.5)<sub>, ( ˜f</sub> 4, `parameter)

= ( ˜f1, `parameter)∪( ˜f2, `parameter), ( ˜f5, `parameter) = ( ˜f1, `parameter)∪( ˜f3, `parameter), ( ˜f6, `parameter)

= ( ˜f2, `parameter) ∪ ( ˜f3, `parameter), ( ˜f7, `parameter) = ( ˜f1, `parameter) ∪ ( ˜f2, `parameter) ∪

( ˜f3, `parameter),

( ˜f8, `parameter)} = {x e₁₁

(0.1,0.4,0.7)_{, x}e21(0.2,0.5,0.6)_{, x}e12(0.3,0.3,0.5) _{and x}e12(0.4,0.4,0.4) _{is a NSTSover}

the father set ( ˜X) . Thus (< ˜_{X >, τ, `</sub>parameter) be a NSTSover the father set ( ˜X) . Also (< ˜<sub>X >, τ, `}parameter) is NS α0 structure but it is not NS α1 because for NSpoints

xe11(0.1,0.4,0.7)_,xe22(0.4,0.4,0.4)_{,(< ˜X >, τ, `}parameter_{) not soft neutrosophic α}

1

Example 4.5. Suppose that the father set ( ˜X) is assumed to be

( ˜X) = {x1, x2} and the set of conditions by `parameter = {e1, e2}.Let us consider NSset

( ˜_{f , `}parameter) over the father set ( ˜X) and xe11(0.1,0.4,0.7)_{, x}e21(0.2,0.5,0.6)_{, x}e12(0.3,0.3,0.5) _and

xe12(0.4,0.4,0.4) _{NSpoints. Then the family}

τ = {0_{( ˜X,,`</sub>parameter<sub>)</sub>, 1<sub>( ˜X,,`}parameter₎, ( ˜f₁, `parameter),( ˜f<sub>2</sub>, `parameter),( ˜f₃, `parameter),( ˜f₄,

`parameter), ( ˜f5, `parameter),( ˜f6, `parameter),( ˜f7, `parameter),( ˜f8, `parameter)}...( ˜f15, `parameter).

where ( ˜f1, `parameter) = x e<sub>11</sub> (0.1,0.4,0.7)<sub>( ˜f</sub> 2, `parameter) = x e₂₁ (0.2,0.5,0.6)_{, ( ˜f} 3, `parameter) = x e<sub>12</sub> (0.3,0.3,0.5)<sub>, ( ˜f</sub> 4, `parameter) xe22(0.4,0.4,0.4) _{( ˜f}

3, `parameter) = ( ˜f3, `parameter)∪( ˜f2, `parameter), ( ˜f6, `parameter) = ( ˜f1, `parameter)∪

( ˜f3, `parameter), ( ˜f7, `parameter) = ( ˜f2, `parameter)∪( ˜f4, `parameter), ( ˜f8, `parameter) = ( ˜f2, `parameter)∪

( ˜f3, `parameter), ( ˜f9, `parameter) = ( ˜f2, `parameter)∪( ˜f4, `parameter), ( ˜f10, `parameter) = ( ˜f3, `parameter)∪

( ˜f4, `parameter), ( ˜f11, `parameter) = ( ˜f1, `parameter)∪( ˜f2, `parameter), ( ˜f3, `parameter), ( ˜f12, `parameter) =

( ˜f1, `parameter)∪( ˜f2, `parameter)∪( ˜f4, `parameter), ( ˜f13, `parameter) = ( ˜f2, `parameter)∪( ˜f3, `parameter)∪

( ˜f4, `parameter), ( ˜f14, `parameter) = ( ˜f1, `parameter)∪( ˜f3, `parameter)∪( ˜f4, `parameter), ( ˜f15, `parameter)

= {xe11(0.1,0.4,0.7)_{, x}e21(0.2,0.5,0.6)_{, x}e12(0.3,0.3,0.5) _{and x}e12(0.4,0.4,0.4) _{is a NSTSover the father}

set ( ˜X) . Thus (< ˜_{X >, τ, `</sub>parameter) be a NSTSover the father set ( ˜X) . Also (< ˜<sub>X >, τ, `}parameter) is NS α1 because for NSpoints x

e₁₁

(0.1,0.4,0.7)_,xe22(0.4,0.4,0.4)_{,(< ˜X >}

, τ, `parameter) not soft neutrosophic α2

Theorem 4.6. Let (h˜_{xi, τ, `</sub>parameter) be a NSTS over the father set ( ˜X).Then (h˜<sub>xi, τ, `}parameter) be a N ST1 structure iffeach NSpointis a NS α − closed set.

Proof. Let (h˜<sub>xi, τ, `</sub>parameter<sub>) be a NSTS over the father set ( ˜X). (x</sub>e (a,b,c), `

be an arbitrary NSpoint.We establish (xe<sub>(a,b,c), `</sub>parameter) parameter is a NS α − open set. Let (y<sub>(a</sub>e00<sub>,b</sub>0<sub>,c</sub>0<sub>)</sub>, `parameter) ∝ (xe_(a,b,c), `parameter) . Then either (ye

0

(a0_,b0_,c0₎, `parameter) 

(xe<sub>(a,b,c), `</sub>parameter) or (y<sub>(a</sub>e00<sub>,b</sub>0<sub>,c</sub>0<sub>)</sub>, `parameter) ≺ (xe_(a,b,c), `parameter)

or (y_(ae00_,b0_,c0₎, `parameter)  (xe<sub>(a,b,c)</sub>, `parameter) or (ye 0

(a0_,b0_,c0₎, `parameter) ≺≺ (xe<sub>(a,b,c)</sub>, `parameter)

. This means that (ye_(a00_,b0_,c0₎, `parameter) and (xe<sub>(a,b,c)</sub>, `parameter) are two are distinct

NS-points. Thus x  y or X ≺ Y or e0  e or e0 _{≺ e,x  y or X ≺≺ Y or e}0 _{e or}

e0 ≺≺ e Since (h˜_{xi, τ, `}parameter_{) be a N ST}

1 structure, ?a NS α − open set (˜g, `parameter)

so that (ye_(a00_,b0_,c0₎, `parameter) ∝ (˜g, `parameter) and (xe_(a,b,c), `parameter) u (˜g, `parameter) =

0_{( ˜X,,`</sub>parameter<sub>)</sub> since (xe<sub>(a,b,c)</sub>, `parameter) u (˜g, `parameter) = 0<sub>( ˜X,,`}parameter₎ so

(y_(ae00_,b0_,c0₎, `parameter) ∝ (˜g, `parameter) ⊂ (xe_(a,b,c), `parameter) Thus (xe<sub>(a,b,c)</sub>, `parameter) is a

NS α−open set, i.e.(xe_{(a,b,c), `</sub>parameter) is a NS α−closed set. Suppose that each NSpoint (xe<sub>(a,b,c), `}parameter)is a NSα − closed set.Then (xe_{(a,b,c), `</sub>parameter)Cis a NS α − open set. Let (xe<sub>(a,b,c), `}parameter) ∩ (ye_(a00_,b0_,c0₎, `parameter) = 0<sub>( ˜X,,`</sub>parameter₎

Thus

(y_(ae00_,b0_,c0₎, `parameter) ∝ (xe<sub>(a,b,c)</sub>, `parameter)C and

(xe_{(a,b,c), `</sub>parameter)∩(xe<sub>(a,b,c), `}parameter)C = 0<sub>( ˜X,,`</sub>parameter<sub>)</sub>So (h˜xi, τ, `parameter) be a N S −

α1 space.

Theorem 4.7. Let (h˜_{xi, τ, `</sub>parameter) be a NSTS over the father set ( ˜X).Then (h˜<sub>xi, τ, `}parameter) be a N ST2 spaceiff for distinct NSpoints . (xe(a,b,c), `

parameter_{) and (y}e0

(a0_,b0_,c0₎, `parameter)

, there exists a NS α − open set ( ˜<sub>f `</sub>parameter) containing ∃ but not (y<sub>(a</sub>e00<sub>,b</sub>0<sub>,c</sub>0<sub>)</sub>, `parameter)

s.t.(ye_(a00_,b0_,c0₎, `parameter) 3 ( ˜f `parameter)

Proof. (xe<sub>(a,b,c), `</sub>parameter)  (ye<sub>(a</sub>00<sub>,b</sub>0<sub>,c</sub>0<sub>)</sub>, `parameter) be two NSpoints in N S2 space. Then

∃ disjoint NS α open sets ( ˜_{f `}parameter_{) and (˜}

g`parameter) such (xe<sub>(a,b,c), `</sub>parameter_{) ∝} ( ˜_{f `}parameter)

and

(y_(ae00_,b0_,c0₎, `parameter) ∝ (˜g`parameter).

since

(xe<sub>(a,b,c), `</sub>parameter) u (y<sub>(a</sub>e00<sub>,b</sub>0<sub>,c</sub>0<sub>)</sub>, `parameter) = 0_{( ˜X,,`}parameter₎

and

( ˜_{f , `</sub>parameter) u (˜<sub>g, `}parameter) = 0_{( ˜X,,`}parameter₎ (ye 0

(a0_,b0_,c0₎, `parameter) 3 ( ˜f , `parameter) ⇒

(y_(ae00_,b0_,c0₎, `parameter) 3 ( ˜f `parameter). Next suppose that, (xe_(a,b,c), `parameter)  (ye 0

(a0_,b0_,c0₎, `parameter),

∃ a NS α open set ( ˜<sub>f , `</sub>parameter<sub>) containing (x</sub>e (a,b,c), `

parameter_{) but not (y}e0

(a0_,b0_,c0₎, `parameter)

s.t. (y_(ae00_,b0_,c0₎, `parameter) 3 ( ˜f `parameter)cthat is ( ˜f , `parameter) and ( ˜f `parameter)c are

mu-tually exclusive NS α open sets supposing (xe<sub>(a,b,c), `</sub>parameter) and (ye<sub>(a</sub>00<sub>,b</sub>0<sub>,c</sub>0<sub>)</sub>, `parameter) in

turn.

is N S1 space if every NS point (xe(a,b,c), `

parameter

) ∝ ( ˜f , `parameter) ∝ (h˜xi, τ, `parameter). If there exists a NS α open set (˜_{g, `</sub>parameter) s.t. (xe<sub>(a,b,c), `}parameter_{) ∝ (h˜xi, τ, `}parameter) a N S2space.

Proof. Suppose(xe<sub>(a,b,c), `</sub>parameter)u (y<sub>(a</sub>e00<sub>,b</sub>0<sub>,c</sub>0<sub>)</sub>, `parameter) = 0<sub>( ˜X,,`</sub>parameter<sub>)</sub>. Since (h˜xi, τ, `parameter)

is N S1space. (xe_(a,b,c), `parameter) and (ye

0

(a0_,b0_,c0₎, `parameter) are NS α − closed sets in

(h˜_{xi, τ, `</sub>parameter) Then (xe<sub>(a,b,c), `}parameter_{) ∝ ((y(a}e00_,b0_,c0₎, `parameter))c∝ (h˜xi, τ, `parameter)

. Thus ∃ a NS (˜_{g, `</sub>parameter<sub>) ∝ (h˜xi, τ, `}parameter) s.t. (xe_{(a,b,c), `</sub>parameter<sub>) ∝ (˜g, `}parameter) ⊂ (˜_{g, `}parameter_{) ⊂ ((y}e0

(a0_,b0_,c0₎, `parameter))c. So we have ((ye 0

(a0_,b0_,c0₎, `parameter))c∝ (˜g, `parameter)

and (((˜_{g, `</sub>parameter) u ((˜<sub>g, `}parameter)c = 0<sub>( ˜X,,`</sub>parameter<sub>)</sub>,i.e. (h˜xi, τ, `parameter) is a N S₂

space. .

Definition 4.9. . Let (h˜_{xi, τ, `</sub>parameter<sub>) be a NSTS over the father set ( ˜X) (h˜xi, τ, `}parameter₎

be a NS α − closed set and (xe (a,b,c), `

parameter_{) ∝ ( ˜f , `}parameter_{) = 0}

( ˜X,,`parameter₎. If ∃ NS

α open sets (( ˜g_{1, `}parameter_{) and(( ˜g}

2, ` parameter<sub>) s.t. (x</sub>e (a,b,c), ` parameter_{) ∝ (( ˜g} 2, ` parameter<sub>),</sub> ( ˜<sub>f , `</sub>parameter_{) ⊂ (( ˜g} 2, ` parameter<sub>) and (x</sub>e (a,b,c), ` parameter_{)u(( ˜g} 1, ` parameter<sub>) = 0</sub> ( ˜X,,`parameter₎

, then (< ˜_{X >, τ, `</sub>parameter<sub>) is called a NS α − regular space. (h˜xi, τ, `}parameter_{) is said}

to be N S3 space, if is both a a NS α − regular and N S1 space.

Theorem 4.10. Let (h˜_{xi, τ, `</sub>parameter<sub>) be a NSTS over the father set ( ˜X) (h˜xi, τ, `}parameter₎

is soft α space iff for every (xe (a,b,c), `

parameter_{) ∝ ( ˜f , `</sub>parameter<sub>) ,i.e, ( ˜f , `}parameter_{) ∝ (<}

˜

X >, τ, `parameter<sub>) s.t. (x</sub>e (a,b,c), `

parameter_{) ∝ (˜g, `</sub>parameter<sub>) ⊂ (˜g, `}parameter_{), ( ˜f , `}parameter_).

Proof. Let (h˜_{xi, τ, `}parameter_{) is N S −α}

3 space and (xe_(a,b,c), `parameter) ∝ ( ˜f , `parameter) ∝

(h˜_{xi, τ, `</sub>parameter<sub>). Since (h˜xi, τ, `}parameter_{) isN S−α}

3space for the NS point (xe_(a,b,c), `parameter)

and a NS α−closed set( ˜_{f , `}parameter₎c_{∃( ˜g}

1, `parameter) and ( ˜g2, `parameter) s.t. (xe_(a,b,c), `parameter) ∝

( ˜g1, `parameter), ( ˜f , `parameter)c ⊂ ( ˜g2, `parameter) and ( ˜g1, `parameter) ∩ ( ˜g2, `parameter) =

0_{( ˜X,,`</sub>parameter<sub>)</sub> Conversely, let (xe<sub>(a,b,c)</sub>, `parameter) ∩ ( ˜H, `parameter) = 0<sub>( ˜X,,`}parameter₎ and

( ˜<sub>H, `</sub>parameter<sub>) be a NS α − closed set. (x</sub>e (a,b,c), `

parameter_{) ∝ ( ˜H, `}parameter₎c

and from the condition of the theorem, we have (xe

(a,b,c), `

parameter_{) ∝ (˜g, `</sub>parameter<sub>) ⊂ (˜g, `}parameter_{)( ˜H, `}parameter₎c_.Thus

(xe (a,b,c), `

parameter_{) ∝ (˜g, `</sub>parameter<sub>) ⊂ ( ˜H, `}parameter_{)(˜g, `}parameter₎c

and

(˜_{g, `</sub>parameter<sub>) ∩ (˜g, `}parameter₎c _{= 0}

( ˜X,,`parameter<sub>)</sub>. So (< ˜X >, τ, `parameter) is N S − α3

space.

Definition 4.11. Let (h˜<sub>xi, τ, `</sub>parameter) be a NSTS over the father set ( ˜X) This space is a NS α − normal space, if for every pair of disjoint a NS α closed sets ( ˜f1, `parameter)

and ( ˜f2, `parameter), disjoint a NS α − open sets (( ˜g1, `parameter) and ( ˜g2, `parameter)

s.t.( ˜f1, `parameter) ⊂ ( ˜g1, `parameter) and ( ˜f2, `parameter) ⊂ ( ˜g2, `parameter). (h˜xi, τ, `parameter)

Theorem 4.12. Let (h˜<sub>xi, τ, `</sub>parameter) be a NSTS over the father set ( ˜X) This space is a N S − α4 space ⇔ for each NS α − closed set ( ˜f , `parameter)and NS α − open set

(˜_{g, `</sub>parameter) with ( ˜<sub>f , `}parameter) ⊂ (˜_{g, `</sub>parameter), ∃ a NS α − open set ( ˜<sub>D, `}parameter)s.t. ( ˜_{f , `</sub>parameter) ⊂ ( ˜<sub>D, `}parameter) ⊂ ( ˜_{D, `</sub>parameter<sub>) ⊂ (˜g, `}parameter_).

Proof. Let (h˜_{xi, τ, `}parameter_{) be a N S−α}

4over the father set ( ˜X). and let ( ˜f , `parameter) ⊂

(˜_{g, `</sub>parameter<sub>). Then (˜g, `}parameter₎c_{is a NS α−closed set and ( ˜f , `</sub>parameter<sub>)∩(˜g, `}parameter_{) =}

0(h˜xi,`parameter<sub>)</sub>. Since (h˜xi, τ, `parameter) be a N S−α₄space,∃ NS α open sets ( ˜D₁, `parameter)

and ( ˜D2, `parameter) s.t. ( ˜f , `parameter) ⊂ ( ˜D1, `parameter) ,(˜g, `parameter)c⊂ ( ˜D2, `parameter)and( ˜D1, `parameter)∩

( ˜D2, `parameter) = 0(h˜xi,`parameter₎. Thus ( ˜f , `parameter) ⊂ ( ˜D<sub>1</sub>, `parameter) ⊂ ( ˜D₂, `parameter)c⊂

(˜_{g, `}parameter_{), ( ˜D}

2, `parameter)cis a NS b−closed set and ( ˜D1, `parameter) ⊂ ( ˜D2, `parameter)c

. So ( ˜_{f , `}parameter_{) ⊂ ( ˜D}

1, `parameter) ⊂ ( ˜D1, `parameter) ⊂ (˜g, `parameter)

Conversely, let ( ˜f1, `parameter) and ( ˜f2, `parameter), disjoint a NS α−closed. Then( ˜f1, `parameter) ⊂

( ˜f2, `parameter)c. From the condition of theorem, there exists a NS α−open set ( ˜D, `parameter)

s.t. ( ˜f1, `parameter) ⊂ ( ˜D, `parameter) ⊂ ( ˜D1, `parameter) ⊂ ( ˜f2, `parameter)cThus ( ˜D, `parameter)

and ( ˜_{D, `}parameter₎c _{are NS ? -openα − open sets and( ˜f}

1, `parameter) ⊂ ( ˜f2, `parameter)c,

( ˜f2, `parameter) ⊂ ( ˜D, `parameter) and ( ˜D, `parameter) and (( ˜D, `parameter) = 0(h˜xi,`parameter₎.

(h˜_{xi, τ, `}parameter_{) be a N S − α}

4 space.

5

Neutrosophic soft Countability:

Theorem 5.1. Let (xe (a,b,c), `

parameter_{) be a point in a first NS countable space hx}crip_{, =, ∂i}

and let {hW, ∂ii : i = 1, 2, 3....} generates a NS countable α − open base about the point

(xe (a,b,c), `

parameter_{), then there exists an infinite soft sub-sequence {hV, ∂i}

i : i = 1, 2, 3....}

of the NS sequence {hW, ∂ii : i = 1, 2, 3....},such that (i) for any NS α − open set

hU, ∂i,containing (xe (a,b,c), `

parameter_{) ,∃ a suffix m such that {hV, ∂i}

i b hU, ∂i for all

i < m; and (ii) if hxcrip_{, =, ∂i be, in particular , a NS α}

1 − space, then ∩{hV, ∂ii : i =

1, 2, 3....} = {(xe (a,b,c), `

parameter_)}.

Proof. Given hW, ∂i1∩hW, ∂ie 2∩hW, ∂ie 3∩....hW, ∂ie kis NSopen sets, contains the point(xe(a,b,c), `parameter)

As the NS sets {hW, ∂ii : i = 1, 2, 3....} Forms NS α−open base about (xe(a,b,c), `parameter)

∃ one among theNSsets {hW, ∂ii : i = 1, 2, 3....} , which we shall denote by hV, ∂ik,

such that (xe

(a,b,c), `parameter) ∝ hV, ∂ik b hW, ∂i1∩hW, ∂ie 2∩hW, ∂ie 3∩....hW, ∂ie k for k =

1, 2, 3... The NSsequence {hV, ∂ii : i = 1, 2, 3....} Thus obtained, has the required

properties. In fact, if hU, ∂i is any NS α−open set, containing (xe

(a,b,c), `parameter) then ∃ a

NS set hW, ∂imsay, belonging to the family {hW, ∂ii : i = 1, 2, 3....}, s.t.(xe(a,b,c), `parameter) ∝

α1 − space and let∩{hV, ∂ii : i = 1, 2, 3....} = {hM, ∂i As{(xe(a,b,c), `

parameter_{)} is}

con-tained in each {hV, ∂ii : i = 1, 2, 3....} that is (xe(a,b,c), `

parameter ) ∝ hV, ∂i1, (xe(a,b,c), ` parameter ) ∝ hV, ∂i2, (xe(a,b,c), ` parameter ) ∝ hV, ∂i3, (xe(a,b,c), ` parameter<sub>)hV, ∂i</sub> 4...(xe(a,b,c), ` parameter ) ∝ hV, ∂init follows that (xe(a,b,c), `

parameter

) ∝ hM, ∂i, Let (y(ae00_,b0_,c0), `parameter) be any point

of xcrip from x that is (xe<sub>(a,b,c), `</sub>parameter)  (y<sub>(a</sub>e00<sub>,b</sub>0<sub>,c</sub>0<sub>)</sub>), `parameter) or(xe_(a,b,c), `parameter) ≺

(y_(ae00_,b0_,c0₎, `parameter) By definition of NS α<sub>1</sub>−space,∃ NS α−open sethU, ∂i ,s.t. (xe<sub>(a,b,c)</sub>, `parameter) ∝

hU, ∂i and (ye0

(a0_,b0_,c0₎), `parameter) 3 hU, ∂i .There exists a suffix m ,s.t. hV, ∂i₁ b hU, ∂i, hV, ∂i₂ b

hU, ∂i, hV, ∂i3 b hU, ∂i, hV, ∂i4, hU, ∂i...hV, ∂ii b hU, ∂i for all i  m. consequently,(ye

0

(a0_,b0_,c0₎, `parameter) 3

hV, ∂ii for all i  m ; hence(ye

0

(a0_,b0_,c0₎, `parameter) 3 hM, ∂i_i. Thus hM, ∂i_i cosists of the

point (xe_{(a,b,c), `}parameter) only.

Theorem 5.2. Soft second NS countability is first NS countability.

Proof. Let hxcrip_{, =, ∂i be a soft 2nd NS countable space. Then this situation permit}

that there live a NS countable base < for hxcrip_{, =, ∂i In order to justify that hx}crip_{, =, ∂i}

isNS, we proceed as, let(xe (a,b,c), `

parameter_{) ∝ X}crip _{be arbitrary point. Let us assemble}

those members of < which absorbs (xe<sub>(a,b,c), `</sub>parameter) and named as <(xe(a,b,c),`

parameter₎

. If hℵ, ∂i is soft n − hood of(xe_{(a,b,c), `}parameter), then this permit ∃ NS open seth℘, ∂i arresting (xe

(a,b,c), `parameter) in < and so in < (xe

(a,b,c),`

parameter₎

.s.t. (xe

(a,b,c), `parameter) ∝

h℘, ∂i b hℵ, ∂i. This justifies that is a NS local base at (xe

(a,b,c), `parameter).One step

more,<(xe(a,b,c),`

parameter₎

being a sub-family of a NScountable family <, it is therefore NS countable. Thus every crisp point of xcrip _{supposes a countable NS local base. This}

leads us to say hxcrip_{, =, ∂i is soft first N countable.}

Theorem 5.3. A NS α countable space in which every NSconvergent sequence has a unique soft limit is a N Sα2 space.

Proof. Let hxcrip, =, ∂i be N Sα2space and let h(xe(a,b,c),`parameter^ )ni be a soft convergent

sequence in hxcrip, =, ∂i .We prove that the limit of this sequence is unique. We prove this result by contradiction. Suppose h(xe_(a,b,c),_`parameter^ ₎

ni converges to two soft points

˜

I and ˜m such that ˜I 6= ˜m. Then by trichotomy law either ˜I < ˜m or ˜I > ˜m. Since the possess the N Sα2f characteristics, there must happen two NS α open sets h℘, ∂i and

hρ, ∂i such that h℘, ∂ie∩hρ, ∂i = 0(hexi,∂parameter₎.That is, the possibility of one rules out

the possibility of other.Now, h(xe

(a,b,c),`parameter^ )ni converges to ˜I so ∃ an interger n1

s.t.h(xe_(a,b,c),_`parameter^ ₎

ni ∝ h℘, ∂i∀n ≥ n1. Also, h(xe_(a,b,c),`parameter^ )ni converges to ˜m

so there exists an interger n2 such thath(xe(a,b,c),`parameter^ )ni ∝ hρ, ∂i∀n ≥ n1. We are

interested to discuss the maximum possibility, fot that we must suppose maximum of both the integers which will enable us to discuss the soft sequence for single soft number. Max(n1, n2) = n0. Which leads to the situation h(x(a,b,c)e ,`parameter^ )ni ∝ h℘, ∂i∀n ≥ n0

and h(xe

(a,b,c),`parameter^ )ni ∝ hρ, ∂i∀n ≥ n0. This implies that h(x e

(a,b,c),`parameter^ )ni ∝

h℘, ∂i and h(xe

(a,b,c),`parameter^ )ni ∝ hρ, ∂i, ∀n ≥ n0.This guarantees that (x e

(a,b,c),`parameter^ )n ∝

(h℘, ∂i∩hρ, ∂i)∀n ≥ ne 0 .Which beautifully contradict the fact that h℘, ∂i∩hρ, ∂i =e 0(hxi,∂e

parameter₎. Hence, the limit of the NSsequence must be unique. Let hxcrip, =, ∂i

be N Sα2 be the NS first axiom space in which every NSconvergent sequence has a

unique soft limit. We prove the space is N S2 space, suppose if possible hxcrip, =, ∂i

be N Sα2 is not N S2 space. Then there must exists two points (xe_(a,b,c), `parameter) 6=

(ye0

(a0_,b0_,c0₎, `parameter),that is (xe<sub>(a,b,c)</sub>, `parameter)  (ye 0

(a0_,b0_,c0₎, `parameter) or (xe<sub>(a,b,c)</sub>, `parameter) ≺

(ye0

(a0_,b0_,c0₎, `parameter) such that every soft set containinig (xe<sub>(a,b,c)</sub>, `parameter) has a

non-empty intersection with every NS α open set containing (ye0

(a0_,b0_,c0₎, `parameter). Since the

space is NSfirst countable, there exist nested monotone decreasing NS local bases at x and (ye0

(a0_,b0_,c0₎, `parameter). Let <(xe<sub>(a,b,c)</sub>, `parameter) = hB1(xe_(a,b,c), `parameter), ...Bn((xe<sub>(a,b,c)</sub>, `parameter)) :

n ∝ Ni and <(ye0 (a0_,b0_,c0₎, `parameter) = hB1(ye 0 (a0<sub>,b</sub>0<sub>,c</sub>0<sub>)</sub>, `parameter), ...Bn(ye 0 (a0_,b0_,c0₎, `parameter) : n ∝ N i

be the nested NS local bases at(xe (a,b,c), `

parameter_{) and (y}e0

(a0_,b0_,c0₎, `parameter) .

respec-tively.Then, we nust have Bn(xe_(a,b,c), `parameter)∩Be n((y

e0

(a0_,b0_,c0₎, `parameter)) 6= eO∀n ∝ N

and so ∃(xe (a,b,c), `

parameter₎

n∝ hxcrip, =, ∂i s.t. (xe_(a,b,c), `parameter) ∝ Bn(xe<sub>(a,b,c)</sub>, `parameter)∩Be n((y

e0 (a0_,b0_,c0₎, `parameter))∀n ∝ N . Therefore, (xe (a,b,c), ` parameter₎

n ∝ Bn(x) and (xe_(a,b,c), `parameter)n ∝ Bn((ye

0

(a0_,b0_,c0₎, `parameter))∀n ∝

N .Let h℘, ∂i and hρ, ∂i be arbitrary NS α open set such that (xe (a,b,c), `

parameter_{) ∝ h℘, ∂i}

and(ye0

(a0_,b0_,c0₎, `parameter) ∝ hρ, ∂i. Then by definition of soft nested base, there exists an

integer no such that Bn((xe_(a,b,c), `parameter)) b h℘, ∂i and Bn((ye

0

(a0_,b0_,c0₎, `parameter)) b

hρ, ∂in > n0. This means that (xe_(a,b,c), `parameter)n∝ h℘, ∂i and (ye

0

(a0_,b0_,c0₎, `parameter)n ∝

hρ, ∂in > n0. It allows that (xe_(a,b,c), `parameter)n→ (xe<sub>(a,b,c)</sub>, `parameter) and (ye

0

(a0_,b0_,c0₎, `parameter)n→

(ye0

(a0_,b0_,c0₎, `parameter. But this is contradiction to the given that every soft convergent

se-quence in hxcrip_{, =, ∂i has unique soft limit. Hence hx}crip_{, =, ∂i is NS α}

2 space.

Theorem 5.4. The cardinality of all NSopen sets in a neutrosophic second countable space is at most equal to C (the power of the continuum).

Proof. Let hxcrip_{, =, ∂i be NSSTsuch that is NSsecond countable space. Let hζ, ∂i be}

any soft set of hxcrip_{, =, ∂i . Then hζ, ∂i is the soft union of a certain soft sub-collection}

of the NScountable collection {hW, ∂ii : i = 1, 2, 3....}. Hence the cardinality of the set

of all NS? open sets in hxcrip_{, =, ∂i is not greater than the cardinality of the soft set of}

all soft sub-collections of the NScountable collection {hW, ∂ii : i = 1, 2, 3....}. thus the

cardinality of hxcrip_{, =, ∂i ≺ C .}

Theorem 5.5. Any collection of mutually exclusive NSopen sets in a NSsecond count-able space is at most NScountcount-able.

Proof. hxcrip_{, =, ∂i be NSSTsuch that it is NSsecond countable. Let hi, ∂i signifies any} collection of mutually exclusive NS α open sets in hxcrip_{, =, ∂i. . Let hU, ∂i ∝ hi, ∂i,} then ∃ at least one soft set hW, ∂im ,belonging to the collection {hW, ∂ii : i = 1, 2, 3....}

such thathW, ∂im b hU, ∂i. Let n be the smallest suffix for which hW, ∂in b hU, ∂i.

Since the soft sets hU, ∂i in hi, ∂i are mutually disjoint, it follows that, for different soft sets hU, ∂i ∝ hi, ∂i,there correspond soft setshW, ∂in with different suffices n. Hence

the soft sets in hζ, ∂i are in a (1,1)-correspondence with a soft sub-collection of the NScountable collection {hW, ∂ii : i = 1, 2, 3....}; consequently,the cardinality of hi, ∂i is

less than or equal to d.

Theorem 5.6. Let hxcrip, =, ∂i be NSST such that it is α2 space.Then this space posses

an infinite soft sequence of non NS α open sets which are mutually exclusive

Proof. Given hxcrip_{, =, ∂i be NSST.If the space hx}crip_{, =, ∂i is soft discrete, then}

defi-nitely every one-pointic soft set is NS α open ; and these NS α open sets constitute an in-finite soft set of mutually dis-joint non-null NS α open subsets of hxcrip_{, =, ∂i,and an}

infinite soft sequence can be picked from this infinite soft set. Next, hxcrip_{, =, ∂i be not}

a soft discrete space; then ?a point (ye0

(a0_,b0_,c0₎, `parameter) in xcrip which is not a limiting

point of xcrip_{. Let (x}e (a,b,c), `

parameter₎

1 be a poit in xcrip which may be at least one

of the possibilities (ye0

(a0_,b0_,c0₎, `parameter)  (xe<sub>(a,b,c)</sub>, `parameter)1 or (xe_(a,b,c), `parameter)1 ≺

(ye0

(a0_,b0_,c0₎, `parameter) then by NS α2 space,∃ two NS α open sets hU1, ∂i and hV1, ∂i such

that (xe (a,b,c), `

parameter₎

1 ∝ hU1, ∂i, q ∝ hV1, ∂i and hU1, ∂i∩hVe 1, ∂i = 0(h ˜Xi,,`parameter₎

. Again, since (ye0

(a0_,b0_,c0₎, `parameter) ∝ hV1, ∂i and q is limiting point of xcrip ∃ a point

(xe (a,b,c), `

parameter₎

2 ∝ hV1, ∂i which behaves as (xe_(a,b,c), `parameter)2  q or (xe<sub>(a,b,c)</sub>, `parameter)2 ≺

(ye0

(a0_,b0_,c0₎, `parameter). This implies that these are points of NS α2 space. So by

defini-tion ∃ two NS α open sets hU2, ∂i and hV2, ∂i such that (xe_(a,b,c), `parameter)2 ∝ hU2, ∂i

,(ye0

(a0_,b0_,c0₎, `parameter) ∝ hV2, ∂i and hU1, ∂i∩hVe 1, ∂i = o_{(h exi,∂}parameter_{^ )}Again, since (y

e0

(a0_,b0_,c0₎, `parameter) ∝

hV1, ∂i and q is limiting point of

xcrip_(xe (a,b,c), `

parameter₎

2 ∝ hV1, ∂i which behaves as (xe_(a,b,c), `parameter)2  q or (xe<sub>(a,b,c)</sub>, `parameter)2 ≺

(ye0

(a0_,b0_,c0₎, `parameter). This implies that these are points of NS α2space. So by definition ∃

two NS α open sets hU2, ∂i and hV2, ∂i such that (xe_(a,b,c), `parameter)2 ∝ hU2, ∂i, (ye

0

(a0_,b0_,c0₎, `parameter) ∝

hV2, ∂i and hU2, ∂i∩hVe 2, ∂i = o_{(h exi,∂}parameter_{^ )} ashU2, ∂i b hV1, ∂i and hV2, ∂i b hV1, ∂i ,

for otherwise we could pick hU2, ∂ie∩hV1, ∂i and hV2, ∂ie∩hV1, ∂i in place of hU2, ∂i and

hV2, ∂i respectively. Keeping continue the above process repeatedly and determing the

points (xe<sub>(a,b,c), `</sub>parameter)1, (xe(a,b,c), `

parameter₎

2, , , (xe(a,b,c), `

parameter₎

n which may behave

as

(xe<sub>(a,b,c), `</sub>parameter)1 , (xe(a,b,c), `

parameter₎ 2 , (xe(a,b,c), ` parameter<sub>)</sub> 3 , (xe(a,b,c), ` parameter₎ 4  , (xe_(a,b,c),

`parameter)5 , (xe(a,b,c), ` parameter₎ 6  (xe(a,b,c), ` parameter<sub>)</sub> 7  ...  (xe(a,b,c), ` parameter₎ n or

(xe<sub>(a,b,c), `</sub>parameter)1 ≺, (xe(a,b,c), `

parameter₎ 2 ≺, (xe(a,b,c), ` parameter<sub>)</sub> 3 ≺, (xe(a,b,c), ` parameter₎ 4 ≺ , (xe_(a,b,c), `parameter)5 ≺, (xe(a,b,c), ` parameter₎ 6 ≺ (xe(a,b,c), ` parameter<sub>)</sub> 7 ≺ ... ≺ (xe(a,b,c), ` parameter₎ n and

accordingly the NSsoft open sets

hU1, ∂i, hU2, ∂i, hU3, ∂i, hU4, ∂i, hU5, ∂i...hUn, ∂i and hV1, ∂i, hV2, ∂i, hV3, ∂i, hV4, ∂i, hV5, ∂i...hVn, ∂i

such that for k = 1, 2, 3, 4, ., n, (xe_{(a,b,c), `}parameter)k ∝ hUk, ∂i, b hVk−1, ∂i, (ye

0

(a0_,b0_,c0₎, `parameter) ∝

hVk, ∂i b hVk−1, ∂i and hUk, ∂ie∩hVk, ∂i = o_{(h exi,∂}parameter_{^ )} Now, since (y

e0

(a0_,b0_,c0₎, `parameter)

is a limiting point of Xcrip _{and hV}

n, ∂i NS α open set containing (ye

0

(a0_,b0_,c0₎, `parameter),

there exists a point (xe

(a,b,c), `parameter)k+1 in Xcrip, different from q and lying within

hVn, ∂i . Then by NS α2 space, ∃ two NSopen sets hUn+1, ∂i and hVn+1, ∂i satisfying

the conditions (xe

(a,b,c), `parameter)k+1 ∝ hUn+1, ∂i , (ye

0

(a0_,b0_,c0₎, `parameter) ∝ hVn+1, ∂i and

hUn+1, ∂ie∩hVn+1, ∂i = o_{(h exi,∂}parameter_{^ )}. As above it may be so disciplinized as hUn+1, ∂i b

hVn, ∂i and hVn+1, ∂i b hVn, ∂i.The infinite soft sequence of NS α open sets hU1, ∂i, hU2, ∂i, hU3, ∂i, hU4, ∂i, hU5, ∂i...

thus defined by induction, the presence of one rules out the possibility of other. Since hUn+1, ∂i b hVn, ∂i, hUk, ∂ie∩hVk, ∂i = o_{(h exi,∂}parameter_{^ )} and hVn, ∂i b hVn+1, ∂i for n =

1, 2, 3....

Theorem 5.7. Let hxcrip_{, =, ∂i be NSSTsuch that it is NSsecond countabley NS α} 2

space. Then set of all NS α open sets has the cardinalityC. Proof. Let hxcrip_{, =, ∂i be second countable NS α}

2.Then, by theorem 2.6, there exists in

NSST hxcrip_{, =, ∂i an infinite soft sequence of NS α open sets hζ, ∂i}

1, hζ, ∂i2, hζ, ∂i3, hζ, ∂i4hζ, ∂i5, ..., hζ, ∂in...suchthathζ, ∂i1 6=

o

(h exi,∂parameter^ ₎, hζ, ∂i2 6= o_{(h exi,∂}parameter_{^ )}, hζ, ∂i3 6= o_{(h exi,∂}parameter_{^ )}, hζ, ∂i4 6= o_{(h exi,∂}parameter_{^ )}, ...hζ, ∂in 6=

o

(h exi,∂parameter^ ₎...

hζ, ∂i1∩hζ, ∂ie 2∩hζ, ∂ie 3∩hζ, ∂ie 4∩...ee ∩hζ, ∂in∩... = oe _{(h exi,∂}parameter_{^ )}. sub-sequences of the

se-quences

hζ, ∂i1, hζ, ∂i2, hζ, ∂i3, hζ, ∂i4hζ, ∂i5, ..., hζ, ∂in nwill determine, as their unions, different

NS α open sets. But since the soft set of all soft sub-sets of a countable soft set has the cardinality C, it follows that soft set of all the NS α open sets in hxcrip_{, =, ∂i has}

the cardinality C. Again by the theorem 2.5, the cardinality of the soft set of all NS open sets in hxcrip_{, =, ∂i  C. consequently, the cardinality of all NS α open sets in}

hxcrip_{, =, ∂i is exactly equal to C.}

Theorem 5.8. Let hxcrip_{, =, ∂i be NSSTsuch that it is NS second countable α}

1 space

the the soft set of all NSpoints in this space has the cardinality at most equal to C(i.e, posses the power of the continuum at most).

Proof. Given that hxcrip, =, ∂i is NS second countable T1 space, for which the NS sets

hW, ∂i1, hW, ∂i2, hW, ∂i3, hW, ∂i4hW, ∂i5, . Forms a soft countable NS α open base of

hxcrip_{, =, ∂i . Then, for any given point (x}e (a,b,c), `

parameter

) ∝ xcrip, hW, ∂iζ1, hW, ∂iζ2, hW, ∂iζ3, hW, ∂iζ4hW, ∂iζ5, ....

generate countable NS open base about the point ζ in NSST hxcrip, =, ∂i , where ζi, i = 1, 2, 3, ..., is a soft sub-sequence of the soft sequence hW, ∂in : n = 1, 2, 3, ...,

consisting of all those hW, ∂i4 which contain the point ζ .Since a second NS countable

space is necessary first NS countable. So corresponding to the point ζ , there exists an infinite soft sequence of NS open sets {h`, ∂in : i = i, 2, 3, } which is a soft sub-sequence

of the soft sequence. {hW, ∂ipn : n = 1, 2, 3, ...},and therefore also a soft sub-sequence

of the soft sequence {hW, ∂in : n = 1, 2, 3, ...}, such that ∩{h`, ∂ie i : i = i, 2, 3, } =

{(xe (a,b,c), `

parameter_{)}. Thus to each point (x}e (a,b,c), `

parameter_{) in x}crip_{,there corresponds a}

soft sub-sequence {h`, ∂ii : i = i, 2, 3, } of the soft sequence {hW, ∂in : n = 1, 2, 3, ...};

and two different points there correspond two such different soft sub-spaces {h`, ∂ii :

i = i, 2, 3, } . Hence the soft set of all points in xcrip ,i.e. the crisp set xcrip ,has the same cardinality as that of a certain soft sub-collection of the soft collection of all soft sub-space of the sequence {hW, ∂in : n = 1, 2, 3, ...}. Thus the cardinality of xcrip less

than or equal to C. In other words, the crisp set xcrip has the power of the continuum at most.

Theorem 5.9. Let hxcrip, =, ∂i be NSST such that it is NS second countable. Every soft uncountable soft subsets contains a point of condensation.

Proof. Since hxcrip_{, =, ∂i is NS second countable space and {hW, ∂i}

i : i = 1, 2, 3, ...} be a

soft countable NS α open base of hxcrip_{, =, ∂i. Let h℘, ∂i be a NS sub-set of x}crip_{such that}

h℘, ∂i does not contain any point of condensation. For each point (xe (a,b,c), `

parameter_{) ∝}

h℘, ∂i, (xe (a,b,c), `

parameter_{) is not point of condensation of h℘, ∂i. Hence there exists NS}

α open set hζ, ∂i containing (xe (a,b,c), `

parameter_{) such that hζ, ∂i}

e

∩h℘, ∂i is soft countable at most. ∃ a suffix n ((xe

(a,b,c), `

parameter_{)), such that (x}e (a,b,c), `

parameter_{) ∝ hW, ∂i} n(p) b

hζ, ∂i ;and then h℘, ∂i∩hW, ∂ie n(p)is also NS countable at most. But we can express h℘, ∂i in the form h℘, ∂i =∩{(xe

e (a,b,c), `

parameter_{) : (x}e (a,b,c), `

parameter_{) ∝ h℘, ∂ie∪{h℘, ∂i}

e

∩hW, ∂in((xe

(a,b,c),`parameter)):

(xe (a,b,c), `

parameter_{) ∝ h℘, ∂i} be at most a NS countable number of different suffices. So,}

h℘, ∂i is at most a NS union of NS uncountable soft sub-set; that is h℘, ∂i is at most a NS countable soft sub-set of eX. Consequently, if h℘, ∂i is soft uncountable soft subset of, then it must possess a point of condensation.

Theorem 5.10. hxcrip_{, =, ∂i be NSST such that it is NS second countable. If hΨ, ∂i is}

uncountable NS sub-set of hxcrip_{, =, ∂i , then the soft sub-set h$, ∂i , consisting of all}

those N points of hΨ, ∂i which are not points of condensation of hΨ, ∂i , is at most NS countable.

Proof. Since hxcrip, =, ∂i is NS second countable space, and {hW, ∂ii : i = 1, 2, 3, ...} be

a countable NS? open base of hxcrip, =, ∂i. Let {hV, ∂ii : i = 1, 2, 3, ...} be a soft

sub-sequence of the soft sub-sequence {hW, ∂ii : i = 1, 2, 3, ...}. consisting of all those soft sets

hW, ∂ij for which hW, ∂ij∩h$, ∂i is at most soft countable. Then {hV, ∂ie i}e∩h$, ∂i : i =

1, 2, 3, ...is at most NS countable .We shall establish that h$, ∂ = hΨ, ∂iie∩(hV, ∂i1∪hV, ∂ie 2∪, ....).e (xe_{(a,b,c), `}parameter_{) ∝ hΨ, ∂i ,there is not a point of condensation of hΨ, ∂i ;hence there}

exists a soft nbd hU, ∂i of (xe_{(a,b,c), `}parameter) , such that hU, ∂ie∩hΨ, ∂i is at most NS countable.Also,∃ a soft set hΨ, ∂ij, belonging to the soft sequence {hW, ∂ii : i =

1, 2, 3, ...},satisfying (xe_{(a,b,c), `}parameter_{) ∝ hW, ∂i}j b hU, ∂i. Then hW, ∂ij∩hΨ, ∂i is ate most NS countable, and so hW, ∂ij must be one of the soft sets hV, ∂ij and therefore

(xe<sub>(a,b,c), `</sub>parameter<sub>) ∝ hV, ∂i</sub>i . Again, since (xe(a,b,c), `

parameter

) ∝ hΨ, ∂i , it follows that (xe_{(a,b,c), `}parameter) = hΨ, ∂iie∩(hV, ∂i1∪hV, ∂ie 2∪, ....).. Next, lete

(y_(ae00_,b0_,c0₎, `parameter) = hΨ, ∂iie∩(hV, ∂i₁∪hV, ∂ie 2∪, ....). thene

(y_(ae00_,b0_,c0₎, `parameter) belomgs to some hV, ∂i_i. Now, as hV, ∂i_i is a soft nbd of

(y_(ae00_,b0_,c0₎, `parameter) ,and hV, ∂i_i∩hΨ, ∂i is at most NS countable.e

(y_(ae00_,b0_,c0₎, `parameter) can not be a point of condensation of hΨ, ∂i. Also, as

(y_(ae00_,b0_,c0₎, `parameter) ∝ hΨ, ∂i. It follows that (ye 0

(a0_,b0_,c0₎, `parameter) ∝ h$, ∂i.Thus

h$, ∂ = hΨ, ∂iie∩(hV, ∂i1∪hV, ∂ie 2∪, ....) = h$, ∂ = hΨ, ∂iie ∩(hV, ∂ie 1∪hV, ∂ie 2∪, ....).e and since each hΨ, ∂ie∪hV, ∂i1 is at most NS countable. It follows that h$, ∂i is also at

most NS countable.

6

Neutrosophic soft Monotonous Functions

Theorem 6.1. Let hχcrip_{, =, ∂i be NSST such that it is NS α}

2 space and hYcrip, =, ∂i

be anyNSST. Lethf, ∂i: hχcrip_{, =, ∂i → hχ}crip_{, =, ∂i be a soft fuction such that it is soft}

monotone and continuous. Then hχcrip_{, =, ∂i is also of characteristics of NS α} 2.

Proof. Suppose (xe

(a,b,c), `parameter)1, (xe(a,b,c), `parameter)2n χcrips.t. either (xe(a,b,c), `parameter)1

> (xe

(a,b,c), `parameter)2 or (xe(a,b,c), `parameter)1 < (xe(a,b,c), `parameter)2. Since hf, ∂i is soft

monotone. Let us suppose the monotonically increasing case. So, (xe

(a,b,c), `parameter)1 >

f(xe(a,b,c), `parameter)2or (xe(a,b,c), `parameter)1< f(xe(a,b,c), `parameter)2implies that f(xe(a,b,c), `parameter)1

> f (x2) or f(xe(a,b,c), `parameter)1< (xe(a,b,c), `parameter)2respectively. Suppose (ye/_(a/_,b/_,c/₎, `

parameter₎

1,(ye/_(a/_,b/_,c/₎, `

parameter₎ 2

n Ycripsuch that (ye/_(a/_,b/_,c/₎, `

parameter₎ 1(ye/_(a/_,b/_,c/₎, ` parameter<sub>)</sub> 2or (ye/<sub>(a</sub>/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, ` parameter₎ 1 < (ye/ (a/_,b/_,c/₎, ` parameter<sub>)</sub> 2So, (ye/<sub>(a</sub>/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, ` parameter₎ 1> (ye/_(a/_,b/_,c/₎, ` parameter<sub>)</sub> 2or (ye/<sub>(a</sub>/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, ` parameter₎ 1 < (ye/_(a/_,b/_,c/₎, ` parameter₎

2respectively such that (ye/_(a/_,b/_,c/₎, `

parameter_{) = f}

(xe_(a,b,c),`parameter₎₁,(ye/

(a/_,b/_,c/₎, `

parameter₎ 2

= f(xe_(a,b,c),`parameter₎₂.

hk1, ∂i and hk2, ∂i, n hχcrip, =, ∂i ⇒ f (hk1, ∂i) and hk2, ∂i, n hχcrip, =, ∂i. We claim

that f (hk1, ∂)ie∩f (hk2, ∂)i = 0(heχi,∂parameter). Otherwise ∃ (te//_(a//_,b//_,c//₎, `

parameter₎ 1 n f |(hk1, ∂i)∩f (hke 2, ∂i) ⇒ (t e// (a//_,b//_,c//₎, ` parameter₎

1f (hk1, ∂i) and (te//_(a//_,b//_,c//₎, `

parameter₎ 1

n f (hk2, ∂i), (te//_(a//_,b//_,c//₎, `

parameter₎

1n f (hk1, ∂i), f is soft one-one ∃ (te//_(a//_,b//_,c//₎, `

parameter₎ 2 n f (hk1, ∂i) s.t. (te//_(a//_,b//_,c//₎, ` parameter<sub>)</sub> 1= f ((te//<sub>(a</sub>//<sub>,b</sub>//<sub>,c</sub>//<sub>)</sub>, ` parameter₎ 2), (te//_(a//_,b//_,c//₎, ` parameter<sub>)</sub> 1 n f (hk2, ∂i) ⇒ ∃ (te//<sub>(a</sub>//<sub>,b</sub>//<sub>,c</sub>//<sub>)</sub>, ` parameter₎ 3 n f (hk2, ∂i) s.t. (te//_(a//_,b//_,c//₎, ` parameter<sub>)</sub> 1 = f ((te//<sub>(a</sub>//<sub>,b</sub>//<sub>,c</sub>//<sub>)</sub>, ` parameter₎ 3) ⇒ (te//_(a//_,b//_,c//₎, ` parameter<sub>)</sub> 2n f (hk1, ∂i), (te//<sub>(a</sub>//<sub>,b</sub>//<sub>,c</sub>//<sub>)</sub>, ` parameter₎ 2 n f (hk2, ∂i) ⇒ (te//_(a//_,b//_,c//₎, ` parameter₎

2 n f (hk1, ∂i) ∩f (hke 2, ∂i). This is contradiction because hk1, ∂i∩hke 2, ∂i = 0(hχi,∂parameter)e . Therefore f (hk1, ∂i)∩f (hke 2, ∂i) = 0(hχi,∂parameter)e .

Finally, (xe (a,b,c), `

parameter₎

1> (xe_(a,b,c), `parameter)2or (xe<sub>(a,b,c)</sub>, `parameter)1< (xe_(a,b,c), `parameter)2

⇒ (xe (a,b,c), `

parameter₎

16= (xe_(a,b,c), `parameter)2⇒ f ((xe<sub>(a,b,c)</sub>, `parameter)1) > f ((xe_(a,b,c), `parameter)2)

or f ((xe (a,b,c), `

parameter₎

1) < f ((xe_(a,b,c), `parameter)2) ⇒ f ((xe<sub>(a,b,c)</sub>, `parameter)1) 6= f ((xe_(a,b,c), `parameter)2).

Given a pair of points (ye/

(a/_,b/_,c/₎, ` parameter<sub>)</sub> 1, (ye/<sub>(a</sub>/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, ` parameter₎ 2n Ycrip 3 (ye/_(a/_,b/_,c/₎, ` parameter<sub>)</sub> 1 6= (ye/ (a/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, ` parameter₎

2 We are able to find out mutually exclusive NS α open sets

f (hk1, ∂i), f (hk2, ∂i) n hYcrip, =, ∂i s.t. (ye/_(a/_,b/_,c/₎, `

parameter₎

1n f (hk1, ∂i), (ye/_(a/_,b/_,c/₎, `

parameter₎ 2

n f (hk2, ∂i). This proves that hYcrip, =, ∂i is NS α(2) space.

Theorem 6.2. Let hχcrip_{, =, ∂i be NSST and hY}crip_{, =, ∂i be an-other NSST which}

satisfies one more condition of NS α2. Lethf, ∂i: hχcrip, =, ∂i → hYcrip, =, ∂i be a soft

function such that it is soft monotone and continuous. Then hχcrip_{, =, ∂i is also of}

characteristics of NS α2.

Proof. Suppose (xe (a,b,c), `

parameter₎

1, (xe_(a,b,c), `parameter)2n χcrips.t. either (xe<sub>(a,b,c)</sub>, `parameter)1

> (xe (a,b,c), `

parameter₎

2 or (xe_(a,b,c), `parameter)1 < (xe<sub>(a,b,c)</sub>, `parameter)2. Let us suppose

the monotonically increasing case. So, (xe (a,b,c), ` parameter<sub>)</sub> 1 > f(xe<sub>(a,b,c)</sub>, `parameter)2 or (xe (a,b,c), ` parameter₎

1< f(xe_(a,b,c), `parameter)2implies that f ((x<sub>(a,b,c)</sub>e , `parameter)1) > f ((xe_(a,b,c), `parameter)2)

or f ((xe

(a,b,c), `parameter)1) < f ((xe(a,b,c), `parameter)2) respectively. Suppose (ye/_(a/_,b/_,c/₎, `

parameter₎

1,(ye/_(a/_,b/_,c/₎, `

parameter₎ 2

n Ycripsuch that (ye/_(a/_,b/_,c/₎, `

parameter₎ 1> (ye/_(a/_,b/_,c/₎, ` parameter<sub>)</sub> 2or (ye/<sub>(a</sub>/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, ` parameter₎ 1 < (ye/ (a/_,b/_,c/₎, ` parameter<sub>)</sub> 2So, (ye/<sub>(a</sub>/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, ` parameter₎ 1> (ye/_(a/_,b/_,c/₎, ` parameter<sub>)</sub> 2or (ye/<sub>(a</sub>/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, ` parameter₎ 1 < (ye/_(a/_,b/_,c/₎, ` parameter₎

2 respectively such that

(ye/

(a/_,b/_,c/₎, `

parameter₎

1= f ((xe_(a,b,c), `parameter)1),(ye/<sub>(a</sub>/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, `

parameter₎

2= f ((xe_(a,b,c), `parameter)2)

s.t. (xe<sub>(a,b,c), `</sub>parameter)1= f−1(y1) and (xe/<sub>(a</sub>/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, `

parameter₎ 1= f−1((ye/_(a/_,b/_,c/₎, ` parameter<sub>)</sub> 2). Since (ye/ (a/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, ` parameter₎ 1, (ye/_(a/_,b/_,c/₎, ` parameter₎

2 nYcrip but hYcrip, =, ∂i is NS α2

space. So according to definition (ye/

(a/_,b/_,c/₎, ` parameter<sub>)</sub> 1 > (ye/<sub>(a</sub>/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, ` parameter₎ 2 or (ye/ (a/_,b/_,c/₎, ` parameter<sub>)</sub> 1 < (ye/<sub>(a</sub>/<sub>,b</sub>/<sub>,c</sub>/<sub>)</sub>, ` parameter₎

2. There definitely exists NS open sets

hk1, ∂i and hk2, ∂i n Ycrip, =, ∂i such that (ye/_(a/_,b/_,c/₎, `

parameter₎

1n hk1, ∂i and (ye/_(a/_,b/_,c/₎, `

parameter₎ 2