An approach to pre-separation axioms in neutrosophic soft topological spaces

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C om mun.Fac.Sci.U niv.A nk.Ser. A 1 M ath. Stat. Volum e 69, N umb er 2, Pages 1389–1404 (2020) D O I: 10.31801/cfsuasm as.749946

ISSN 1303–5991 E-ISSN 2618–6470

Received by the editors: Ju n e 09, 2020; Accepted: S eptem ber 20, 20 20

AN APPROACH TO PRE-SEPARATION AXIOMS IN NEUTROSOPHIC SOFT TOPOLOGICAL SPACES

Ahu AÇIKGÖZ and Ferhat ESENBEL

Department of Mathematics, Balikesir University, 10145 Balikesir, TURKEY

Abstract. In this study, we introduce the concept of neutrosophic soft pre-open (neutrosophic soft pre-closed) sets and pre-separation axioms in neutro-sophic soft topological spaces. In particular, the relationship between these separation axioms are investigated. Also, we give a new de…nition for neu-trosophic soft topological subspace and de…ne neuneu-trosophic soft pre irresolute soft and neutrosophic pre irresolute open soft functions.

1. Introduction

In 2005, Smarandache introduced the concept of a neutrosophic set [20] as a generalization of classical sets, fuzzy set theory [20] (see also [10]) , intuitionistic fuzzy set theory [4] (see also [14]) etc. By using this theory of neutrosophic set, some scientists made researches in many areas of mathematics [7, 18]. Many in-herent di¢ culties exist in classical methods for the inadequacy of the theories of parametrization tools. So, classical methods are insu¢ cient in dealing with sev-eral practical problems in some other disciplines such as economics, engineering, environment, social science, medical science, etc. In 1999, Molodtsov [16] pointed out the inherent di¢ culties of these theories. A di¤erent approach was initiated by Molodtsov for modeling uncertainties. This approach was applied in some other directions such as smoothness of functions, game theory, operations research, Rie-mann integration, Perron integration and so on. The theory of soft topological spaces was introduced by Shabir and Naz [19] for the …rst time in 2011. Soft topo-logical spaces were de…ned over an initial universe with a …xed set of parameters and showed that a soft topological space gives a parameterized family of topological

2020 Mathematics Subject Classi…cation. 54A05, 54D10, 54D15.

Keywords and phrases. Neutrosophic pre open soft set, neutrosophic soft pre interior point, neutrosophic soft pre cluster point, neutrosophic soft pre-separation axioms, neutrosophic soft subspace.

[email protected] author; [email protected] 0000-0003-1468-8240; 0000-0001-2345-6789.

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spaces. In [1, 2, 5, 6, 9, 11, 13], some scientists made researches and did theoretical studies in soft topological spaces. In 2013, Maji [15] de…ned the concept of neutro-sophic soft sets for the …rst time. Then, Deli and Broumi [12] modi…ed this concept. In 2017, Bera presented neutrosophic soft topological spaces in [8].

In this study, our pupose is to adapt the concepts of neutrosophic pre open soft set, neutrosophic pre closed soft set to neutrosophic soft topological spaces. Then, we de…ne neutrosophic soft pre interior point, neutrosophic soft pre cluster point, neutrosophic soft pre interior operator and neutrosophic soft pre closure operator. By using these de…nitions and concepts, the concept of pre-separation axioms of neutrosophic soft topological spaces is introduced. Furthermore, we analyze proper-ties of neutrosophic soft pre Ti-spaces (i = 0; 1; 2; 3; 4) and focus on some relations

between them. Characterization theorems of them are also proved. We hope that, the …ndings in this document will help scientists to enhance and promote the fur-ther studies on neutrosophic soft topology to carry out a general framework for their applications in practical life.

2. Preliminaries

In this section, we present the basic de…nitions and theorems related to neutro-sophic soft set theory.

De…nition 1. [20] A neutrosophic set A on the universe set X is de…ned as: A = fhx; TA(x) ; IA(x) ; FAxi : x 2 Xg,

where

T ,I, F : X ! ] 0; 1+_{[ and} ₀ _T

A(x) + IA(x) + FA(x) 3+.

De…nition 2. [16] Let X be an initial universe, E be a set of all parameters, and P (X) denote the power set of X. A pair (F; E) is called a soft set over X, where F is a mapping given by F : E ! P (X). In other words, the soft set is a parameterized family of subsets of the set X. For e 2 E, F (e) may be considered as the set of e-elements of the soft set (F; E), or as the set of e-approximate elements of the soft set, i.e.

(F; E) = f(e; F (e)) : e 2 E; F : E ! P (X)g.

After the neutrosophic soft set was de…ned by Maji [15], this concept was modi…ed by Deli and Broumi [12] as given below:

De…nition 3. [12] Let X be an initial universe set and E be a set of parameters. Let P (X) denote the set of all neutrosophic sets of X. Then a neutrosophic soft set

e

F ; E over X is a set de…ned by a set valued function eF representing a mapping e

F : E ! P (X), where eF is called the approximate function of the neutrosophic soft set F ; E . In other words, the neutrosophic soft set is a parametrized family ofe some elements of the set P (X) and therefore it can be written as a set of ordered pairs:

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e F ; E =

n e;

D

x; T_{F (e)}_e (x) ; I_{F (e)}_e (x) ; F_{F (e)}_e (x) E

: x 2 X : e 2 E o

where T_{F (e)}_e (x), I_{F (e)}_e (x), F_{F (e)}_e _{(x) 2 [0; 1] are respectively called the} truth-membership,

indeterminacy-membership and falsity-membership function of eF (e). Since the supremum of each T , I, F is 1, the inequality

0 TA(x) + IA(x) + FA(x) 3

is obvious.

De…nition 4. [8] Let F ; Ee be a neutrosophic soft set over the universe set X. The complement of F ; Ee is denoted by F ; Ee

c

and is de…ned by: e F ; E c = n e; D

x; F_{F (e)}_e (x) ; 1 I_{F (e)}_e (x) ; T_{F (e)}_e (x) E : x 2 X : e 2 E o . It is obvious that h e F ; E c ic = F ; E .e

De…nition 5. [15] Let F ; Ee and G; Ee be two neutrosophic soft sets over the universe set X. F ; Ee is said to be a neutrosophic soft subset of G; Ee if

T_{F (e)}_e (x) T_G(e)_e (x), I_{F (e)}_e (x) I (x), F_{F (e)}_e (x) F_G(e)_e _{(x), 8e 2 E, 8x 2 X.} It is denoted by F ; Ee G; E .e F ; Ee is said to be neutrosophic soft equal to G; Ee if F ; Ee G; Ee and G; Ee F ; E . It is denoted bye F ; Ee =

e G; E .

De…nition 6. [3] Let Fe1; E and Fe2; E be two neutrosophic soft sets over the

universe set X. Then their union is denoted by Fe1; E [ Fe2; E = Fe3; E and

is de…ned by: e F3; E = n e; D x; T_F_e 3(e)(x) ; I (x) ; FFe3(e)(x) E : x 2 X : e 2 E o ; where T_F_e 3(e)(x) = max n T_F_e 1(e)(x) , TFe2(e)(x) o , IFe3(e)(x) = max n IFe1(e)(x) , IFe2(e)(x) o , F_F_e 3(e)(x) = min n F_F_e 1(e)(x) , FFe2(e)(x) o .

De…nition 7. [3] Let Fe1; E and Fe2; E be two neutrosophic soft sets over the

universe set X. Then their intersection is denoted by Fe1; E \ eF2; E = Fe3; E

and is de…ned by: e F3; E = n e; Dx; T_F_e 3(e)(x) ; I (x) ; FFe3(e)(x) E : x 2 X : e 2 Eo; where

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T_F_e 3(e)(x) = min n T_F_e 1(e)(x) , TFe2(e)(x) o , I_F_e 3(e)(x) = min n I_F_e 1(e)(x) , IFe2(e)(x) o , F_F_e 3(e)(x) = max n F_F_e 1(e)(x) , FFe2(e)(x) o .

De…nition 8. [3] A neutrosophic soft set F ; Ee over the universe set X is said to be a null neutrosophic soft set if T_{F (e)}_e (x) = 0, I_{F (e)}_e (x) = 0, F_{F (e)}_e (x) = 1; 8e 2 E, 8x 2 X. It is denoted by 0(X;E).

De…nition 9. [3] A neutrosophic soft set F ; Ee over the universe set X is said to be an absolute neutrosophic soft set if T_{F (e)}_e (x) = 1, I_{F (e)}_e (x) = 1, F_{F (e)}_e (x) = 0; 8e 2 E, 8x 2 X It is denoted by 1(X;E).

Clearly 0c

(X;E)= 1(X;E) and 1c(X;E)= 0(X;E):

De…nition 10. [3] Let N SS (X; E) be the family of all neutrosophic soft sets over the universe set X and N SS (X; E). Then is said to be a neutrosophic soft topology on X if:

1. 0(X;E) and 1(X;E) belong to ,

2. the union of any number of neutrosophic soft sets in belongs to ,

3. the intersection of a …nite number of neutrosophic soft sets in belongs to . Then (X; ; E) is said to be a neutrosophic soft topological space over X. Each member of is said tobe a neutrosophic soft open set [3].

De…nition 11. [3] Let (X; ; E) be a neutrosophic soft topological space over X and e

F ; E be a neutrosophic soft set over X. Then F ; Ee is said to be a neutrosophic soft closed set i¤ its complement is a neutrosophic soft open set.

De…nition 12. [3] Let N SS (X; E) be the family of all neutrosophic soft sets over the universe set X. Then neutrosophic soft set xe_{( ; ; )} is called a neutrosophic soft point for every x 2 X, 0 < , , 1, e 2 E and is de…ned as follows:

xe_{( ; ; )}(e0) (y) = (

( ; ; ) ; if e0_{= e and y = x}

(0; 0; 1) ; if e0_{6= e or y 6= x}

It is clear that every neutrosophic soft set is the union of its neutrosophic soft points.

De…nition 13. [3] Let F ; Ee be a neutrosophic soft set over the universe set X. We say that xe

( ; ; )2 eF ; E read as belonging to the neutrosophic soft set F ; Ee

whenever

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De…nition 14. [3] Let xe

( ; ; ) and ye

0

( 0; 0_; 0)be two neutrosophic soft points. For

the neutrosophic soft points xe

( ; ; ) and ye

0

( 0; 0_; 0) over a common universe X, we

say that the neutrosophic soft points are distinct points if xe

( ; ; )\ ye

0

( 0; 0_; 0) =

0(X;E). It is clear that xe( ; ; ) and ye

0

( 0; 0_; 0) are distinct neutrosophic soft points

if and only if x 6= y or e 6= e0.

De…nition 15. [7] Let F ; Ee 1 , eG; E2 be two neutrosophic sets over the

uni-versal set X. Then their cartesian product is another neutrosophic set K; Ee 3 =

e

F ; E1 G; Ee 2 , where E3 = E1 E2 and eK (e1; e2) = eF (e1) G (ee 2). The

truth, indeterminacy and falsity membership of K; Ee 3 are given by 8e1 2 E1,

8e22 E2, 8x 2 X, T_K(e_e 1;e2)(x) = min n T_{F (e}_e 1)(x) , TG(ee 2)(x) o , IK(ee 1;e2)(x) = min n IF (ee 1)(x) , IG(ee 2)(x) o , F_K(e_e 1;e2)(x) = max n F_{F (e}_e 1)(x) , FG(ee 2)(x) o .

This de…nition can be extended for more than two neutrosophic soft sets.

De…nition 16. [7] A neutrosophic soft relation eR between two neutrosophic soft sets F ; Ee 1 and G; Ee 2 over the common universe X is the neutrosophic soft

subset of F ; Ee 1 G; Ee 2 . Clearly, it is another neutrosophic soft set R; Ee 3

where E3 E1 E2 and

e

R (e1; e2) = eF (e1) G (ee 2) for (e1; e2) 2 E3.

De…nition 17. [7] Let F ; Ee 1 , G; Ee 2 be two neutrosophic sets over the

univer-sal set X and f be a neutrosophic soft relation de…ned on F ; Ee 1 G; Ee 2 . Then

f is called neutrosophic soft function if f associates each element of F ; Ee 1 with

the unique element of G; Ee 2 . We write f : F ; Ee 1 ! eG; E2 as a neutrosophic

soft function or a mapping. For xe

( ; ; )2 eF ; E1 and ye 0 ( 0_; 0_; 0₎2 eG; E2 when xe ( ; ; ) ye 0 ( 0; 0_; 0) 2 f, we denote it by f xe( ; ; ) = ye 0 ( 0; 0_; 0). Here F ; Ee 1

and G; Ee 2 are called domain and codomain respectively and ye

0

( 0_; 0_; 0₎is the image

of xe

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De…nition 18. [8] Let (X; ; E) be a neutrosophic soft topological space and e

F ; E _{2 NSS (X; E) be arbitrary. Then the interior of} F ; Ee is denoted by e

F ; E and is de…ned as: e

F ; E =S n eG;E : eG;E F ; E ;e G; Ee ₂ o

i.e., it is the union of all open neutrosophic soft subsets of F ; E .e De…nition 19. [8] Let (X; ; E) be a neutrosophic soft topological space and

e

F ; E _{2 NSS (X; E) be arbitrary. Then the closure of} F ; Ee is denoted by e

F ; E and is de…ned as: e

F ; E =T n eG;E : eG;E F ; E ;e G; Ee c₂ o

i.e., it is the intersection of all closed neutrosophic soft super sets of F ; E .e 3. Some Properties

De…nition 20. A subset F ; Ee of a neutrosophic soft topological space (X; ; E) is said to be neutrosophic pre open soft, if F ; Ee F ; Ee . The family of all neutrosophic pre open soft sets of (X; ; E) is denoted by N SP O (X). The family of all neutrosophic pre open soft sets of (X; ; E) containing a neutrosophic soft point xe

( ; ; ) is denoted by N SP O X; xe( ; ; ) .

De…nition 21. A neutrosophic soft point xe

( ; ; ) of a neutrosophic soft topological

space

(X; ; E) is said to be neutrosophic soft pre interior point of a neutrosophic soft set e

F ; E , if there exists G; Ee _{2 NSP O X; x}e

( ; ; ) such that xe( ; ; )* eG; E c

and G; Ee F ; E .e

De…nition 22. The set of all neutrosophic soft pre interior points of F ; Ee is said to be neutrosophic soft pre interior of F ; Ee and denoted by N SP int F ; E .e De…nition 23. The complement of a neutrosophic pre open soft set is called neutro-sophic pre closed soft. The intersection of all neutroneutro-sophic pre closed soft sets con-taining a neutrosophic soft set F ; Ee is called neutrosophic pre closure of F ; Ee and is denoted by N SP cl F ; E .e

De…nition 24. A neutrosophic soft point xe

( ; ; ) of a neutrosophic soft topological

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(X; ; E) is said to be neutrosophic soft pre cluster point of a neutrosophic soft set e

F ; E , if G; Ee * eF ; E

c

for any G; Ee _{2 NSP O X; x}e_{( ; ; )} .

De…nition 25. A neutrosophic soft topological space (X; ; E) is said to be a neu-trosophic soft pre T0-space if for every pair of distinct neutrosophic soft points

xe

( ; ; ), ye

0

( 0_; 0_; 0₎there exist neutrosophic pre-open soft sets F ; E ,e G; Ee such

that xe ( ; ; )2 eF ; E , y₍e00_; 0_; 0₎2 F ; Ee c or xe_{( ; ; )} _{2 e}G; E c , y₍e00_; 0_; 0₎2 eG; E .

De…nition 26. Let (X; ; E) be a neutrosophic soft topological space and Y X. Let H; Ee be a neutrosophic soft set over Y such that

T_H(e)_e (x) = ( 1; _{if x 2 Y} 0; if x =_{2 Y} I_H(e)_e (x) = ( 1; _{if x 2 Y} 0; if x =_{2 Y} F_H(e)_e (x) = ( 1; _{if x 2 Y} 0; if x =_{2 Y} for any e 2 E. Let Y = n e H; E \ eF ; E : F ; Ee ₂ o

, then (Y; Y; E) is called

neutro-sophic soft subspace of (X; ; E). If H; Ee ₂ (resp. H; Ee c ₂ ), then (Y; Y; E) is called neutrosophic open (resp.closed) soft subspace of (X; ; E).

Theorem 27. A neutrosophic soft subspace (Y; Y; E) of a neutrosophic soft pre

T0-space

(X; ; E) is neutrosophic soft pre T0.

Proof. Let xe

( ; ; ), ye

0

( 0_; 0_; 0₎be two distinct neutrosophic soft points in (Y; Y; E).

Then, these neutrosophic soft points are also in (X; ; E). Hence, there exist neu-trosophic pre-open soft sets F ; E ,e G; Ee in such that xe

( ; ; ) 2 F ; E ,e ye0 ( 0_; 0_; 0₎ 2 F ; Ee c or xe ( ; ; ) 2 G; Ee c , ye0 ( 0_; 0_; 0₎2 G; E . Lete H; Ee be a

neutrosophic soft set over Y as described in De…nition 26. Thus, _{H; E \ e}e F ; E and _{H; E \ e}e G; E are neutrosophic pre-open soft sets in (Y; Y; E) such that

xe ( ; ; )2 H; E \ ee F ; E , ye 0 ( 0_; 0_; 0₎2 h e H; E \ eF ; E ic or

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xe_{( ; ; )}₂ h e H; E \ eG; E ic , y₍e00_; 0_; 0₎2 H; E \ ee G; E . Therefore, (Y; Y; E)

is neutrosophic soft pre T0.

xe_{( ; ; )}, y₍e00_; 0_; 0₎ there exists neutrosophic pre-open soft sets F ; Ee and G; Ee

such that xe ( ; ; )2 eF ; E , ye0 ( 0_; 0_; 0₎2 F ; Ee c and xe ( ; ; ) 2 eG; E c , ye0 ( 0_; 0_; 0₎2 eG; E .

T1-space

Proof. It is similar to the proof of Theorem 27.

Theorem 30. Every neutrosophic soft point with the truth-membership value 1, the

indeterminacy-membership value 1 and falsity-membership value 0, is neutrosophic pre-closed soft in a neutrosophic soft topological space (X; ; E) if and only if (X; ; E) is neutrosophic soft pre T1.

Proof. ()) Suppose that xe

( ; ; ) and ye

0

( 0_; 0_; 0₎ be two distinct neutrosophic soft

points of (X; ; E). Then, xe

( ; ; ) xe(1;1;0) and ye 0 ( 0_; 0_; 0₎ ye 0 (1;1;0). By hypoth-esis, ye0 (1;1;0) and ye 0

(1;1;0) are neutrosophic pre-closed soft sets. Then,

h xe (1;1;0) ic and hye0 (1;1;0) ic

are neutrosophic pre-open soft sets such that xe ( ; ; ) 2 h ye0 (1;1;0) ic , ye0 ( 0_; 0_; 0₎ 2 hh ye0 (1;1;0) icic and xe ( ; ; ) 2 hh xe (1;1;0) icic , ye0 ( 0_; 0_; 0₎ 2 h xe (1;1;0) ic . Therefore, (X; ; E) is neutrosophic soft pre T1.

(() Suppose that (X; ; E) is neutrosophic soft pre T1. Let xe_(1;1;0) be a

neutro-sophic soft point with the truth-membership value 1, the indeterminacy-membership value 1 and falsity- membership value 0. Take any neutrosophic soft point ye0

( 0; 0_; 0)2

h xe

(1;1;0)

ic

. It is easily seen that xe

(1;1;0) and ye

0

( 0_; 0_; 0₎ are distinct. From our

as-sumption, there exist neutrosophic pre-open soft sets F ; Ee and G; Ee such that xe (1;1;0)2 eF ; E , ye 0 ( 0_; 0_; 0₎2 eF ; E c and xe (1;1;0) 2 G; Ee c , ye0 ( 0_; 0_; 0₎ 2 G; E . Then, ye e 0 ( 0_; 0_; 0₎ 2 G; Ee h xe (1;1;0) ic .

This means that hxe (1;1;0)

ic

is neutrosophic pre-open soft. Therefore, xe (1;1;0) is

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xe

( ; ; ), ye

0

( 0_; 0_; 0₎ there exists neutrosophic pre-open soft sets F ; Ee and G; Ee

such that xe ( ; ; )2 eF ; E , y₍e00_; 0_; 0₎ 2 F ; Ee c , y₍e00_; 0_; 0₎ 2 G; E , xe e_{( ; ; )} 2 G; Ee c and F ; Ee e G; E c .

For a neutrosophic soft topological space (X; ; E) we have the following diagram:

neutrosophic soft pre T2 space

#

Converse statements may not be true as shown in the examples below;

Example 32. _{Let X = fx; yg be a universe, E = fa:bg be a parameteric set and} e

Fa; E be a neutrosophic soft set de…ned as eFa(a) = fhx; a; a; 1 ai ; hy; a; a; 1 aig

and e

Fa(b) = fhx; 0; 0; 1i ; hy; a; a; 1 aig for any 2 (0; 1]. Then, the family

= 0(X;E); 1(X;E) [

n e

Fa; E : a 2 (0; 1]

o

is a neutrosophic soft topology over X. So, (X; ; E) is a neutrosophic soft topo-logical space. (X; ; E) is a neutrosophic soft pre T0 -space but not a neutrosophic

soft pre T1 –space. Because, xb_{(0:9, 0:6, 0:2)} and ya_{(0:8, 0:7, 0:4)} are distinct

neutro-sophic soft points in (X; ; E) and there doesn’t exist any neutroneutro-sophic pre-open soft set that contains xb

(0:9, 0:6, 0:2) but doesn’t contain ya(0:8, 0:7, 0:4).

Example 33. _{Let X = fx; yg be a universe, E = fa:bg be a parameteric set and} e

F ; E be a neutrosophic soft set de…ned as e_{F (a) = fhx; 0; 0; 1i ; hy; 0; 0; 1ig and} e

F (b) = fhx; 0; 0; 1i ; hy; 0; 0; 0:9ig. Then, the family = n

0(X;E); 1(X;E); F ; Ee

o is a neutrosophic soft topology over X. So, (X; ; E) is a neutrosophic soft topo-logical space. (X; ; E) is a neutrosophic soft pre T1 –space. But, it is not a

neu-trosophic soft pre T2 –space for the existence of distinct neutrosophic soft points

xa

(0:5; 0:5; 0:1)and yb(0:4, 0:4, 0:6).

Theorem 34. Let (X; ; E) be a neutrosophic soft topological space. (X; ; E) is neutrosophic soft pre T2-space if and only if for any pair of distinct neutrosophic

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soft points xe_{( ; ; )}, ye₍00_; 0_; 0₎, there exists a neutrosophic pre-open soft set F ; Ee such that xe_{( ; ; )} _{2 e}F ; E , ye₍00_; 0_; 0₎2 eF ; E c and y₍e00_; 0_; 0₎2 h N SP cl F ; Ee ic . Proof. ()) Let xe ( ; ; ) and ye 0

( 0; 0_; 0) be two distinct neutrosophic soft points in

(X; ; E) . Since (X; ; E) is a neutrosophic soft pre T2-space, there exist two

neutrosophic pre-open soft sets F ; Ee and G; Ee such that xe

( ; ; ) 2 F ; E ,e

ye0

( 0_; 0_; 0₎ 2 G; Ee and F ; Ee G; Ee

c

. So, it is implied that ye0

( 0_; 0_; 0₎ 2

e

F ; E c. Since G; Ee c is a neutrosophic pre-closed soft set, N SP cl F ; Ee e

G; E

c

. This means that, G; Ee h N SP cl F ; Ee ic . So, ye0 ( 0_; 0_; 0₎2 h N SP cl F ; Ee ic .

(()Take any pair of distinct neutrosophic soft points xe

( ; ; ), ye

0

( 0_; 0_; 0₎ in

(X; ; E). From our assumption, there exists a neutrosophic pre-open soft set e F ; E such that xe ( ; ; ) 2 F ; E ,e ye 0 ( 0; 0_; 0) 2 F ; Ee c and ye0 ( 0; 0_; 0) 2 h N SP cl F ; Ee ic . Since h N SP cl F ; Ee ic

is a neutrosophic pre-open soft set and e

F ; E hh

N SP cl F ; Ee icic

, (X; ; E) is neutrosophic soft pre T2-space.

Theorem 35. A neutrosophic soft subspace (Y; Y; E) of neutrosophic soft pre

T2 space

Proof. Let (X; ; E) be a neutrosophic soft pre T2-space, Y X and (Y; Y; E) be

a neutrosophic soft subspace. Take any distinct neutrosophic soft points xe_{( ; ; )} and y₍e00_; 0_; 0₎ in (Y; Y; E).

So, these neutrosophic soft points are also contained in (X; ; E). Hence, there exist neutrosophic pre-open soft sets F ; Ee and G; Ee in such that xe

( ; ; )2

e F ; E ,

y₍e00_; 0_; 0₎2 eG; E and F ; Ee G; Ee

c

. Let H; Ee be a neutrosophic soft set over Y as described in De…nition 26. Then, _{H; E \ e}e F ; E and _{H; E \ e}e G; E are neutrosophic pre-open soft sets in (Y; Y; E) such that xe_{( ; ; )} 2 H; E \e

e F ; E , ye₍00_; 0_; 0₎2 H; E \ ee G; E and H; E \ ee F ; E h e H; E \ eG; E ic . This means that (Y; Y; E) is neutrosophic soft pre T2.

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De…nition 36. Let (X; ; E) be a neutrosophic soft topological space, G; Ee be a neutrosophic pre-closed soft set and xe

( ; ; ) be a neutrosophic soft point such

that xe

( ; ; ) 2 F ; Ee c

. If there exist neutrosophic pre-open soft sets G; Ee and e

K; E such that xe

( ; ; ) 2 G; E ,e F ; Ee K; Ee and K; Ee G; Ee c

, then (X; ; E) is said to be a neutrosophic soft pre regular space.

De…nition 37. A neutrosophic soft topological space (X; ; E) is said to be a strong neutrosophic soft pre T1-space if every neutrosophic soft point is a neutrosophic

pre-closed soft set in (X; ; E).

De…nition 38. A neutrosophic soft pre regular space (X; ; E) is said to be a neutrosophic soft pre T3 space if it is also a strong neutrosophic soft pre T1 space.

Theorem 39. Every neutrosophic soft pre T3 space is a neutrosophic soft pre

T2 space.

Proof. Let xe_{( ; ; )} and y₍e00_; 0_; 0₎ be two distinct neutrosophic soft points of a

neutrosophic soft pre T3-space (X; ; E). Then, ye

0

( 0_; 0_; 0₎ is neutrosophic

pre-closed soft set and xe ( ; ; ) 2 h ye0 ( 0_; 0_; 0₎ ic

. From the neutrosophic soft pre-regularity, there exist

disjoint neutrosophic pre-open soft sets G; Ee and K; Ee such that xe_{( ; ; )} ₂ e G; E and ye0 ( 0; 0_; 0) K; E . Thus, xe e( ; ; ) 2 eG; E and ye 0 ( 0; 0_; 0)2 K; E . Therefore,e

(X; ; E) is neutrosophic soft pre T2-space

T3 space (X; ; E) is neutrosophic soft pre T3.

Proof. Let (X; ; E) be a neutrosophic soft pre T3-space, Y X and (Y; Y; E)

be a neutrosophic soft subspace. Let xe

( ; ; ) be any neutrosophic soft point in

(Y; Y; E). It is obvious that xe( ; ; )is also a neutrosophic soft point in (X; ; E).

Since (X; ; E) is a strong neutrosophic soft pre T1-space, xe_{( ; ; )} is a

neutro-sophic pre-closed soft set in (X; ; E). Consider the neutrosphic soft set H; Ee over Y de…ned in De…nition 26. It is easily seen that H; Ee _{\ x}e

( ; ; ) is

neu-trosophic pre-closed soft in (Y; Y; E). This means that (Y; Y; E) is a strong

neutrosophic soft pre-T1-space. Now, we must show that (Y; Y; E) is also a

neu-trosophic soft pre-regular space. Let G; Ee be a neutrosophic pre-closed soft set in (Y; Y; E) and xe( ; ; ) be a neutrosophic soft point in (Y; Y; E) such that

xe

( ; ; ) 2 G; Ee c

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pre-closed soft set F ; Ee in (X; ; E). Hence, xe_{( ; ; )} ₂ h e H; E \ eF ; E ic . So, xe_{( ; ; )} ₂ H; Ee c [ F ; Ee c

. Because of the description of the

neutro-sophic soft set H; Ee in De…nition 5.2, it is clear that xe

( ; ; ) 2= H; Ee c . This means that xe ( ; ; ) 2 F ; Ee c

. From the neutrosophic soft pre-regularity of

(X; ; E), there exists neutrosophic pre-open soft sets K; Ee and L; Ee such that xe_{( ; ; )} ₂ K; E ,e F ; Ee L; Ee and K; Ee L; Ee

c

. This implies that _{H; E \ e}e K; E and _{H; E \ e}e L; E are neutrosophic pre-open soft sets in (Y; Y; E) such that xe_{( ; ; )}2 H; E \ ee K; E , F ; Ee H; E \ ee L; E and

e

H; E \ eK; E H; Ee _{\ e}L; E

c

. Therefore, (Y; Y; E) is neutrosophic soft

pre T3.

De…nition 41. Let (X; ; E) be a neutrosophic soft topological space, Fe1; E and

e

F2; E be neutrosophic pre-closed soft sets such that Fe1; E Fe2; E c

. If

there exist neutrosophic pre-open soft sets G; Ee and K; Ee such that Fe1; E

e

G; E , Fe2; E K; Ee and G; Ee K; Ee c

, then (X; ; E) is said to be a neutrosophic soft pre normal space.

De…nition 42. A neutrosophic soft pre normal space (X; ; E) is said to be a neutrosophic soft pre T4-space, if it is also a strong neutrosophic soft pre T1 space.

Theorem 43. Let (X; ; E) be a fuzzy soft topological space. Then, the following statements are equivalent:

(1) (X; ; E) is a neutrosophic soft pre normal space.

(2) For every neutrosophic pre closed soft set K; Ee and neutrosophic preopen soft set L; Ee such that K; Ee L; E , there exists a neutrosophic pre opene soft set F ; Ee such that K; Ee F ; E , N SP cle F ; Ee L; E .e

Proof. (1) ) (2) Let K; Ee be a pre closed soft set and L; Ee be a fuzzy pre open soft set such that K; Ee L; E . Then,e K; E ,e L; Ee care neutrosophic pre closed soft sets such that L; Ee

c

e K; E

c

. It follows from (1), there exist neutrosophic pre open soft sets F ; Ee and G; Ee such that K; Ee F ; E ,e

e

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closed soft, N SP cl F ; Ee G; Ee

c

. So,

N SP cl F ; Ee L; E . Therefore, the neutrosophic pre open soft sete F ; Ee satis…es the conditions.

(2) ) (1) Let Fe1; E and Fe2; E be neutrosophic pre-closed soft sets such

that e F1; E Fe2; E c , where Fe2; E c

is neutrosophic pre open soft. From our

hy-pothesis, there exists a neutrosophic pre open soft set F ; Ee such that Fe1; E

e F ; E and N SP cl F ; Ee Fe2; E c . So, Fe2; E h N SP cl F ; Ee ic, Fe1; E F ; Ee and h

N SP cl F ; Ee ic F ; Ee c, wherehN SP cl F ; Ee icand F ; E are are neutro-e sophic pre open soft sets. Thus, (X; ; E) is neutrosophic soft pre normal space.

Theorem 44. A neutrosophic pre closed neutrosophic soft subspace (Y; Y; E) of

a neutrosophic soft pre normal space (X; ; E) is neutrosophic soft pre normal.

Proof. Let F ; Ee and G; Ee be neutrosophic pre closed soft sets in (Y; Y; E)

such that F ; Ee G; Ee c. Consider the neutrosphic soft set H; Ee over Y de…ned in De…nition 26. Then, H; Ee is neutrosophic pre closed soft in (X; ; E),

e

F ; E = _{H; E \ e}e K; E and G; Ee = _{H; E \ e}e L; E for some neutrosophic pre closed soft sets K; Ee and L; Ee in (X; ; E). Hence, _{H; E \ e}e K; E and

e

H; E \ eL; E are neutrosophic pre closed soft sets in (X; ; E) and _{H; E \}e e K; E h e H; E \ eL; E ic

. Since (X; ; E) is neutrosophic soft pre normal, there exist neutrosophic pre open sets M ; Ef and N ; Ee such that

e H; E \ eK; E M ; E ,f _{H; E \ e}e L; E N ; Ee and M ; Ef N ; Ee c . So, e

H; E \ fM ; E and _{H; E \ e}e N ; E are neutrosophic pre open sets in (Y; Y; E)

such that F ; Ee _{H; E \ f}e M ; E , G; Ee _{H; E \ e}e N ; E and e H; E _{\ f}M ; E h e H; E \ eN ; E ic

. Therefore, (Y; Y; E) is neutrosophic

soft pre normal.

De…nition 45. Let (X; 1; E), (Y; 2; K) be neutrosophic soft topological spaces

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f : (X; 1; E) ! (Y; 2; K) be a neutrosophic soft function. The function f is said

to be neutrosophic pre irresolute soft, if f 1 _{G; E}e ₂

1 for any G; Ee 2 2.

De…nition 46. Let (X; 1; E), (Y; 2; K) be neutrosophic soft topological spaces

and

f : (X; 1; E) ! (Y; 2; K) be a neutrosophic soft function. The function f is said to

be neutrosophic pre irresolute open soft, if f F ; Ee ₂ 2 for any F ; Ee 2 1.

Theorem 47. Let (X; 1; E) and (Y; 2; K) be neutrosophic soft topological spaces

and

f : (X; 1; E) ! (Y; 2; K) be a neutrosophic soft function which is bijective,

neu-trosophic pre irresolute soft and neuneu-trosophic pre irresolute open soft. If (X; 1; E)

is a neutrosophic soft pre normal space, then (Y; 2; K) is also a neutrosophic soft

pre normal space.

Proof. Let F ; Ee and G; Ee be neutrosophic pre closed soft sets in (Y; 2; K)

such that e

F ; E G; Ee c. Since f is neutrosophic pre irresolute soft, then f 1 _{F ; E}_e

and

f 1 G; Ee are neutrosophic pre closed soft sets in (X; 1; E) such that

f 1 _{F ; E}e h_f 1 _{G; E}e ic_{. Since (X;}

1; E) is a neutrosophic soft pre

nor-mal space, there exist neutrosophic pre open soft sets K; Ee and L; Ee such that f 1 _{F ; E}_e _{K; E , f}_e 1 _{G; E}_e _{L; E}_e _and _{K; E}_e _{L; E}_e c_{. It} follows that e F ; E = f h f 1 F ; Ee i f K; Ee , G; Ee = f h f 1 G; Ee i f L; Ee and f K; Ee f L; Ee c = h f L; Ee ic

. From the fact that f is neutrosophic pre irresolute open soft, f K; Ee and f L; Ee are neutro-sophic pre open soft sets such that F ; Ee f K; Ee , G; Ee f L; Ee and f K; Ee

h

f L; Ee ic

. This means that (Y; 2; K) is a neutrosophic

soft pre normal space.

4. Conclusion

The notions of neutrosophic pre open soft sets, neutrosophic pre closed soft sets, neutrosophic pre soft interior, neutrosophic pre soft closure, neutrosophic soft pre-interior point, neutrosophic soft pre-cluster point and neutrosophic soft pre separation axioms are introduced, and some properties of the notions are studied. Also, several interesting properties have been established. Additionally, a new

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approach is made to the concept of neutrosophic soft topological subspace. Since topological structures on neutrosophic soft sets have been introduced by many scientists, we generalize the pre topological properties to the neutrosophic soft sets which may be useful in some other disciplines. For the existence of compact connections between soft sets and information systems [17, 21] the results obtained from the studies on neutrosophic soft topological space can be used to develop these connections. We hope that many researchers will bene…t from the …ndings in this document to further their studies on neutrosophic soft topology to carry out a general framework for their applications in practical life.

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