Neutrosophic Generalized Regular Star Compact, Connected, Regular and

Normal Topological Spaces

RAJA MOHAMMAD LATIF

Prince Mohammad bin Fad University, College of Sciences and Human Studies, Department of

Mathematics and Natural Sciences, Al Khobar, SAUDI ARABIA

Abstract: Real-life structures always include indeterminacy. The Mathematical tool which is well-known in dealing with

indeterminacy is Neutrosophic. Smarandache proposed the approach of Neutrosophic sets. Neutrosophic sets deal with uncertain

data. The notion of Neutrosophic set is generally referred to as the generalization of intuitionistic fuzzy set. In 2022, S. Pious

Missier, A. Anusuya and J. Martina Jency introduced and studied the concepts of Neutrosophic generalized regular star closed

 

eu closed*-Ngr

sets and Neutrosic generalized regular star

 

eu open*-Ngr

sets in Neutrosophic topological spaces and

studied some of their properties and characterizations as well as analyzed the relationships between these newly introduced sets

and the already existing neutrosophic sets. In this paper, we introduce the notions of

eu compact spaces,*-Ngr

eu Lindelof*-Ngr

space, countably

eu compact*-Ngr

spaces,

eu connected*-Ngr

spaces,

eu separated*-Ngr

sets,

eu connected*- - -N Super gr

spaces,

eu disconnected*- - -N Extrem ely gr

spaces, and

eu connected*- - -N Strongly gr

spaces,

eu egular*-NRgr

spaces, strongly

egular

eu *-

NR

gr

spaces,

eu *-N gr N orm al

spaces, and strongly

eu *-N gr N orm al

spaces by using

open

eu *-Ngr

sets and

closed

eu *-Ngr

sets

in Neutrosophic topological spaces. We study the basic properties and fundamental characteristics of these spaces in

Neutrosophic topological spaces.

Keywords:

closed

eu *-Ngr

set,

open

eu *-Ngr

set,

compact

eu *-Ngr

space,

Lindelof space,

eu *Ngr -

Countably compact space,

eu *Ngr -

connected

eu *-Ngr

space,

separated

eu *-Ngr

set,

eu connected*N Super gr- - -

space,

eu disconnected*N Extrem ely gr- - -

space,

eu connected*- - -N Strongly gr

space,

egular

eu *-NRgr

space, Strongly

egular

eu *-NRgr

space,

Normal

eu *-Ngr

space, Strongly

ormal

eu *-Ngr N

space

Received: September 29, 2022. Revised: July 24, 2023. Accepted: August 19, 2023. Published: September 19, 2023.

1. Introduction

Many real-life problems in Business, Finance, Medical

Sciences, Engineering, and Social Sciences deal with

uncertainties. There are difficulties in solving the uncertainties

in data by traditional mathematical models. There are

approaches such as fuzzy sets, intuitionistic fuzzy sets, vague

sets, and rough sets, Nano sets, micro sets which can be

treated as mathematical tools to avert obstacles in dealing with

ambiguous data. But all these approaches have their implicit

crisis in solving the problems involving indeterminant and

inconsistent data due to the inadequacy of parameterization

tools. Molodtsov introduced the soft set theory. Smarandache

studies neutrosophic sets as an approach to solving issues that

cover unreliable, indeterminacy, and persistent data.

Applications of neutrosophic topology depend upon the

properties of neutrosophic closed sets, neutrosophic interior

and closure operators, and neutrosophic open sets. In 2022, S.

Pious Missier, A. Anusuya and J. Martina Jency introduced

and studied the concepts of Neutrosophic generalized regular

star closed

 

eu closed*-Ngr

sets and Neutrosic generalized

regular star

 

eu open*-Ngr

sets in Neutrosophic topological

spaces and studied some of their properties and

characterizations as well as analyzed the relationships between

these newly introduced sets and the already existing

neutrosophic sets. In this paper, we introduce the notions of

eu compact spaces,*-Ngr

eu Lindelof*-Ngr

space,

countably

eu compact*-Ngr

spaces,

eu connected*-Ngr

spaces,

eu separated*-Ngr

sets,

eu connected*- - -N Super gr

spaces,

eu disconnected*- - -N E xtrem ely gr

spaces, and

eu connected*- - -N Strongly gr

spaces,

eu egular*-NRgr

spaces, strongly

egular

eu *-

NR

gr

spaces,

eu *-N gr N orm al

spaces, and strongly

eu *-N gr N orm al

spaces by using

open

eu *-Ngr

sets and

closed

eu *-Ngr

sets in Neutrosophic topological spaces.

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

32

Volume 3, 2023

We study the basic properties and fundaments characteristics

of these spaces in Neutrosophic topological spaces.

2. Preliminaries

Definition 2.1.

Let

X

be a non-empty fixed set. A

neutrosophic set

 

eu

briefly set-N

P

is an object having a

form

     

 

, , , : ,

P P P

P x x x x x X

  

where

 

-



Px

represents the degree of membership,

 

-

Px



represents the degree of indeterminacy, and

 

-

Px



represents the degree of non-membership.

Definition 2.2.

A neutrosophic topology on a non-empty

set

X

is a family

N

of neutrosophic subsets of

X

satisfying

 

i

0 , 1 .

Neu Neu N



 

ii

N

GH

for every

,,N

GH

 

iii





jN

jJ

G

for

every

 

:.

jN

G j J T

Then the pair

 

,N

X

is called a

neutrosophic topological space

 

eu

briefly Top Space-,N

. The elements of

N

are

called neutrosophic open

 

eu

briefly open-N

sets in

.X

A

eu setN-

A

in

X

is called a neutrosophic closed

 

eu

briefly closed-N

set if and only if its complement

C

A

is a

eu openN-

set.

Definition 2.3.

Let

 

,N

X

be a

eu Top Space-N

and

A

be a

eu set-.N

Then

 

i

The neutrosophic interior of

,A

denoted by

 

eu Int AN

is

the union of all

eu open-N

subsets of

.A

 

ii

The neutrosophic closure of

A

denoted by

 

euCl AN

is

the intersection of all

eu closed-N

sets containing A.

Definition 2.4.

Let

A

be a

eu subset-N

of a

eu Top Space-N

 

,.

N

X

Then

A

is said to be a

 

i

neutrosophic regular

 

eu

briefly N-R egular

open set

if

 

.

eu eu

A Int Cl A



NN

 

ii

neutrosophic regular

 

eu

briefly N-R egular

closed set if

A

 

.

eu eu

Cl Int A





NN

Clearly

eu

N-R egular

open sets and

eu

N-R egular

closed sets in

 

,N

X

are complements of

each other.

Definition 2.5.

Let

 

,N

X

be a

eu Top Space-N

and

A

be a

eu set-N

of

.X

Then

A

is said to be a neutrosophic

generalized closed

 

eu

briefly closed-Ng

set if

 

euCl A GN

whenever

AG

and

G

is

eu open-N

set

in

 

,.

N

X

The complement of a

eu closed-Ng

set is called

a

eu open-Ng

set in

 

,.

N

X

Definition 2.6.

Let

 

,N

X

be a

eu Top Space-N

and

A

be a

eu set-N

of

.X

Then

A

is said to be a neutrosophic

generalized regular star closed

 

resp. closed

eu *-Ngr

set if

 

Cl A G

eu -N reg

whenever

AG

and

G

is

open

eu -N g

set in

 

,.

N

X

The complement of a

closed

eu *-Ngr

set is called a

neutrosophic generalized regular star open

 

briefly open

eu *-Ngr

set in

 

,.

N

X

The family of all

open

eu *-Ngr

 

resp. closed

eu *-Ngr

in

a

eu Top Space-N

 

,N

X

is denoted by

 

N

O X,T

eu *-Ngr

 

N

resp. C X,T .

eu *-Ngr

Definition 2.7.

Let

 

,N

X

be a

eu Top Space-N

and

A

be a

eu set-N

of

.X

Then

 

i

the neutrosophic

generalized regular star closure of

A

is denoted and defined

   

 

N

by : Cl A F C X,T & A F .

eu eu

  **

--NNgr gr

 

ii

the neutrosophic generalized regular star interior of

A

is denoted and defined by:

   

 

N

Int A G O X,T & G A .

eu eu

  **

--NNgr gr

Theorem 2.8.

Every

eu closedN-

 

resp. open

eu -N

set in a

eu Top Space-N

 

,N

X

is a

closed*

eugrN-

 

resp. open

eu *-Ngr

set in

 

,.N

X

Theorem 2.9.

In a

eu Top Space-N

 

,N

X

we have

the following conditions.

 

i

eu

0N

and

eu

1N

are

open

eu *-Ngr

sets in

 

,N

X

.

 

ii

The intersection of any two

closed

eu *-Ngr

sets is

closed

eu *-Ngr

set in

 

,N

X

.

 

iii

The union of any two

open

eu *-Ngr

sets is

open

eu *-Ngr

set in

 

,N

X

.

Theorem 2.10.

Let

 

,N

X

be a

eu Top Space-.N

Then for any

eu subsets-N

A

and

B

of

,X

we have

 

i

   

eu eu

Int A A Cl A**--NNgr gr

 

ii

A

is

eu open*-Ngr

set in

X

if and only if

 

eu Int A A.*-Ngr

 

iii

A

is

eu closed*-Ngr

set in

X

if and only if

 

eu Cl A A.*-Ngr

 

iv

   

eu eu eu

Int Int A Int A .  



* * *- - -N N Ngr gr gr

 

v

   

eu eu eu

Cl Cl A Cl A .  



* * *- - -N N Ngr gr gr

 

vi

If

,AB

then

   

eu eu

Int A Int B**--NNgr gr

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

33

Volume 3, 2023

 

vii

If

,AB

then

   

eu eu

Cl A Cl B**--NNgr gr

 

viii

 

cc

eu eu

Cl A Int A**--N = Ngr gr

 

iX

 

cc

eu eu

Int A Cl A**--N = Ngr gr

 

X

 

00

eu eu

eu Int ,*-NN

Ngr

 

11*eu eu

eu Intgr NN

N-

 

Xi

 

00

eu eu

eu Cl ,*-NN

Ngr

 

11

eu eu

eu Cl *-NN

Ngr

 

Xii

 

eu Int A B *-Ngr

   

eu eu

Int A Int B**

--NNgr gr

 

Xiii

   

eu eu

Cl A Cl B **

--NNgr gr

 

eu Cl A B*-Ngr

 

Xiv

   

eu eu

Int A Int B **

--NNgr gr

 

eu Int A B*-Ngr

 

Xiii

 

eu Cl A B *-Ngr

   

eu eu

Cl A Cl B**

--NNgr gr

Definition 2.11.

A function

   

: , ,

NN

f X Y



is

called a

continuous*

eugrN-

function if

 

1

fB

is a

open*

eugrN-

 

resp. closed*

eugrN-

set in

,X

for every

eu openN-

 

eu

resp. closedN-

set

B

in

.Y

Definition 2.12.

A function

   

: , ,

NN

f X Y



is

called a

irresolute*

eugrN-

function if

 

1

fB

is a

open*

eugrN-

 

resp. closed*

eugrN-

set in

,X

for every

open*

eugrN-

 

resp. closed*

eugrN-

set

B

in

.Y

3 Neutrosophic Generalized

Regular Star Compact Spaces

In this section, we introduce

compact*

eugrN-

space,

Lindelof*

eugrN-

space, and countably

compact

eu *Ngr -

space and investigate their basic

properties and characterizations.

Definition 3.1.

A collection

 

:

i

A i I

of

eu open-N

 

resp. open*

eugrN-

sets in a

eu Top Space-N-

 

,N

X

is called a

eu open-N

 

resp. open*

eugrN-

cover of a subset

B

of

X

if

 

:

i

B A i I

holds.

Definition 3.2.

A subset

B

of a

eu Top Space-N-

 

,N

X

is said to be

eu compact-N

 

resp. compact*

eugrN-

relative to

 

,,N

X

if for every

collection

 

:

i

A i I

of

eu open-N

 

resp. open*

eugrN-

subsets of

 

,N

X

such that

 

:

i

B A i I

there exists a finite subset

0

I

of

I

such

that

 

0

:.



i

B A i I

Definition 3.3.

A subset

B

of a

eu Top Space-N-

 

,N

X

is called

eu compact-N

 

resp compact*

eugrN-

if

B

is

eu compact-N

 

resp. compact*

eugrN-

as a subspace of

.X

Theorem 3.4.

A

closed*

eugrN-

subset of a

compact*

eugrN-

 

,N

X

is

compact*

eugrN-

relative to

 

,.N

X

Proof .

Let

A

be a

closed*

eugrN-

subset of a

compact*

eugrN-

space

 

,.N

X

Then

C

A

is

open*

eugrN-

in

 

,.N

X

Let

 



i

A :i IS=

be a

open*

eugrN-

cover of

A

by

open*

eugrN-

subsets of

 

,.N

X

Then

 

C

AS*= S

is a

open*

eugrN-

cover of

 

,.N

X

That is

 

.



C

i

iI

X A A

By hypothesis

 

,N

X

is

compact*

eugrN-

and hence

S*

is reducible

to a finite subcover of

 

,N

X

say

12

....... ,

n

C

i i i

X A A A A

.

k

i

ASS*

Then

12

....... .

n

i i i

A A A A

Thus a

open*

eugrN-

cover S of

A

contains a finite subcover. Hence

A

is

compact*

eugrN-

relative to

 

,.N

X

Theorem 3.5.

A

eu Top Space-N-

 

,N

X

is

compact*

eugrN-

if and only if every family of

closed*

eugrN-

sets of

 

,N

X

having finite intersection

property has a non-empty intersection.

Proof .

Suppose

 

,N

X

is

compact.*

eugrN-

Let

 

:

i

A i I

be a family of

closed*

eugrN-

sets with finite

intersection property. Suppose

.





i

iI

A

Then

.–

i

iI

X A X

This implies

 

.–



i

iI X A X

Thus

the cover

 

:–

i

X A i I

is a

open*

eugrN-

cover of

 

,.N

X

Then the

open*

eugrN-

cover

 

:–

i

X A i I

has a finite subcover say

 

0

:–

i

X A i I

for some finite subset

0

I

of

.I

This implies

 

0

,–



i

iI

X X A

which implies

 

0

––





i

iI

X X A

which implies

0

.





i

iI A

This

contradicts the assumption. Hence

.





i

iI

A

Conversely,

suppose

 

,N

X

is not

compact.*

eugrN-

Then there

exists a

open*

eugrN-

cover of

 

,N

X

say

 

:

i

G i I

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

34

Volume 3, 2023

having no finite subcover. This implies for any finite

subfamily

 

: 1,2,...,

i

G i n

of

 

:,

i

G i I

we have

1,



n

i

iGX

which implies

1,––



n

i

X G X X

which

implies

 

1.–





n

i

iXG

Then the family

 

:–

i

X G i I

of

closed*

eugrN-

sets has a finite

intersection property. Also, by assumption

 

i

iI XG



–

which implies

–





i

iI

XG

so that

.



i

iI

GX

This implies

 

:

i

G i I

is not a cover for

 

,.N

X

This contradicts the fact

 

:

i

G i I

is a cover for

 

,.N

X

Therefore a

open*

eugrN-

cover

 

:

i

G i I

of

 

,N

X

has a finite subcover

 

: 1,2,..., .

i

G i n

Hence

 

,N

X

is a

compact.*

eugrN-

Theorem 3.6.

The image of a

compact*

eugrN-

space

under a

irresolute*

eugrN-

mapping is

compact.*

eugrN-

Proof .

Let

   

: , ,





NN

f X Y

be a

irresolute*

eugrN-

mapping from a

compact*

eugrN-

space

 

,N

X

onto a

Top Space*

eugrN--

 

,.



N

Y

Let

 

:

i

A i I

be a

open*

eugrN-

cover of

 

,.



N

Y

Then

 

 

1:



i

f A i I

is a

open*

eugrN-

cover of

 

,N

X

,

since

f

is

irresolute.*

eugrN-

As

 

,N

X

is

compact,*

eugrN-

the

open*

eugrN-

cover

 

 

1:



i

f A i I

of

 

,N

X

has a finite subcover

 

 

1: 1,2,..., .



i

f A i n

Therefore

 

1

1.





n

i

X f A

Then

 

1,



n

i

f X A

that is

1.



n

i

YA

Then

 

12

, ,..., n

A A A

is a finite subcover of

 

:

i

A i I

for

 

,.



N

Y

Hence

Y

is a

compact*

eugrN-

space.

Definition 3.7.

A

eu Top Space-N-

 

,N

X

is

countably

compact*

eugrN-

if every countable

open*

eugrN-

cover of

 

,N

X

has a finite subcover.

Definition 3.8.

A

eu Top Space-N-

 

,N

X

is said to

be

Hausdorff*

eugrN-

if whenever

 

,,

x

  

and

 

,,r s t

y

are distinct points of

 

,,N

X

there exist disjoint

open*

eugrN-

sets

A

and

B

of

X

such that

 

,,

xA

  



and

 

,, .

r s t

yB

Theorem 3.9.

Let

 

,N

X

be a

eu Top Space-N-

and

 

,



N

Y

be a

Hausdorff*

eugrN-

space. If

   

: , ,





NN

f X Y

is

irresolute*

eugrN-

injective

mapping, then

 

,N

X

is

Hausdorff.*

eugrN-

Proof .

Let

 

,,

  

x

and

 

,,r s t

y

be any two distinct

eu pointsN-

of

 

,.N

X

Then

 

,,

  

fx

and

 

,,r s t

fy

are distinct

eu pointsN-

of

 

,,



N

Y

because

f

is injective. Since

 

,



N

Y

is

Hausdorff ,*

eugrN-

there are disjoint

open*

eugrN-

sets

G

and

H

in

 

,



N

Y

containing

 

,,

  

fx

and

 

,,r s t

fy

respectively.

Since

f

is

irresolute*

eugrN-

and

,



GH

we have

 

1

fG

and

 

1

fH



are disjoint

open*

eugrN-

sets in

 

,N

X

such that

 

1

,,

  



x f G

and

 

1

,, .

r s t

y f H





Hence

 

,N

X

is

Hausdorff.*

eugrN-

Theorem 3.10.

If

   

: , ,





NN

f X Y

is

irresolute*

eugrN-

and bijective and if

X

is

compact*

eugrN-

and

Y

is

Hausdorff ,*

eugrN-

then

f

is a

homeomorphism.*

eugrN-

Proof .

We have to show that the inverse function

g

of

f

is

irresolute.*

eugrN-

For this we show that if

A

is

open*

eugrN-

in

 

,N

X

then the pre-image

 

1

gA

is

open*

eugrN-

in

 

,.



N

Y

Since the

open*

eugrN-

 

or closed*

eugrN-

sets are just the complements of

closed*

eugrN-

 

resp. open*

eugrN-

subsets, and

   

11

.––



g X A Y g A

We see that the

irresolute*

eugrN-

mapping of

g

is equivalent to: if

B

is

closed*

eugrN-

in

 

,N

X

then the pre-image

 

1

gB

is

closed*

eugrN-

in

.Y

To prove this, let

B

be a

closed*

eugrN-

subset of

.X

Since

g

is the inverse of

,f

we have

   

1,

g B f B

hence we have to show that

 

fB

is a

closed*

eugrN-

set in

.Y

By theorem 3.4,

B

is

compact.*

eugrN-

By Theorem 3.6, implies that

 

fB

is

copmpact.

eu *Ngr -

Since

Y

is

Hausdorff

eu *Ngr -

space implies that

 

fB

is

closed*

eugrN-

in

 

,.



N

Y

Definition 3.11.

A

eu Top Space-N-

 

,N

X

is said to

be

Lindelof*

eugrN-

space if every

open*

eugrN-

cover

of

 

,N

X

has a countable subcover.

Theorem 3.12.

Every

compact*

eugrN-

space is a

Lindelof*

eugrN-

space.

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

35

Volume 3, 2023

Proof .

Let

 

,N

X

be

compact.*

eugrN-

Let

 

:

i

A i I

be a

open*

eugrN-

cover of

 

,.N

X

Then

 

:

i

A i I

has a finite subcover

 

: 1,2,..., ,

i

A i n

since

 

,N

X

is

copmpact.*

eugrN-

Since every finite

subcover is always a countable subcover and therefore,

 

: 1.2,...,

i

A i n

is countable subcover of

 

:

i

A i I

for

 

,.N

X

Hence

 

,N

X

is

Lindelof*

eugrN-

space.

Theorem 3.13.

The image of a

Lindelof*

eugrN-

space

under a

irresolute*

eugrN-

mapping is

Lindelof.*

eugrN-

Proof .

Let

   

: , ,





NN

f X Y

be a

irresolute*

eugrN-

mapping from a

Lindelof*

eugrN-

space

 

,N

X

onto a

eu Top Space-N-

 

,.



N

Y

Let

 

:

i

A i I

be a

open*

eugrN-

cover of

 

,.



N

Y

Then

 

 

1:



i

f A i I

is a

open*

eugrN-

cover of

 

,,N

X

since

f

is

irresolute.*

eugrN-

As

 

,N

X

is

Lindelof ,*

eugrN-

the

open*

eugrN-

cover

 

 

1:



i

f A i I

of

 

,N

X

has a countable subcover

 

 

1

0

:

i

f A i I



for some countable subset

0

I

of

.I

Therefore

 

0

1i

iI

X f A







which implies

 

0

,

i

iI

f X Y A





that is

 

0

:

i

A i I

a countable

subcover of

 

:

i

A i I

for

 

,.



N

Y

Hence

 

,



N

Y

is

Lindelof*

eugrN-

space.

Theorem 3.14.

Let

 

,N

X

be

Lindelof*

eugrN-

and

countably

copmpact*

eugrN-

space. Then

 

,N

X

is

copmpact.*

eugrN-

Proof .

Let

 

:

i

A i I

be a

open*

eugrN-

cover of

 

,.N

X

Since

 

,N

X

is

Lindelof*

eugrN-

space.

Hence

 

:

i

A i I

has a countable subcover

 

:.

n

i

An

Therefore,

 

:

n

i

An

is a countable subcover of

 

,N

X

and

 

:

n

i

An

is a subfamily of

 

:

i

A i I

and

so

 

:

n

i

An

is a countable

open*

eugrN-

cover of

 

,.N

X

Again since

 

,N

X

is countably

copmpact,*

eugrN-

 

:

n

i

An

has a finite subcover

 

: 1,2,..., .

k

i

A k n

Therefore

 

: 1,2,...,

k

i

A k n

is a

finite subcover of

 

:

i

A i I

for

 

,.N

X

Hence

 

,N

X

is

copmpact*

eugrN-

space.

Theorem 3.15.

A

eu Top Space-N-

 

,N

X

is

copmpact*

eugrN-

if and only if every basic

open*

eugrN-

cover of

 

,N

X

has a finite subcover.

Proof .

Let

 

,N

X

be

copmpact.*

eugrN-

Then every

open*

eugrN-

cover of

 

,N

X

has a finite subcover.

Conversely, suppose that every basic

open*

eugrN-

cover

of

 

,N

X

has a finite subcover and let

 

:



GC=

be

any

open*

eugrN-

cover of

 

,.N

X

If

 

:



DB=

is any

open*

eugrN-

base for

 

,,N

X

then each



G

is

union of some members of

B

and the totality of all such

members of

B

evidently a basic

open*

eugrN-

cover of

 

,.N

X

By hypothesis this collection of members of

B

has

a finite subcover,

 

: 1, 2,..., .





i

D i n

For each



i

D

in this

finite subcover, we can select a

i

G



from

C

such that

.





ii

DG

It follows that the finite subcollection

 

: 1, 2,..., ,

i

G i n





which arises in this way is a subcover of

C.

Hence

 

,N

X

is

copmpact.*

eugrN-

4Neutrosophic Generalized

Regular Star Connected Spaces

In this section, we introduce and study the notions of

connected*

eugrN-

spaces,

separated*

eugrN-

sets,

eu connected*Super grN - - -

spaces,

eu disconnected*E xtrem ely grN - - -

spaces, and

eu connected*Strongly grN - - -

spaces in

eu Top Spaces.N--

Definition 4.1.

A

eu Top SpaceN--

 

,N

X

is

disconnected*

eugrN-

if there exist

open*

eugrN-

sets

,A

B

in

,X

0,

Neu

A

0Neu

B

such that

1Neu

AB

and

0.

Neu

AB

If

 

,N

X

is not

disconnected*

eugrN-

then it is said to be

connected.*

eugrN-

Theorem 4.2.

A

eu Top SpaceN--

 

,N

X

is

connected*

eugrN-

space if and only if there exists no

nonempty

open*

eugrN-

sets

U

and

V

in

 

,N

X

such

that

.

C

UV

Proof .

Necessity: Let

U

and

V

be two

open*

eugrN-

sets in

 

,N

X

such that

0,

Neu

U

0Neu

V

and

.

C

UV

Therefore

C

V

is a

closed*

eugrN-

set. Since

0,

Neu

U

1.

Neu

V

This implies

V

is a proper

eu subset-N

which is both

open*

eugrN-

set and

closed*

eugrN-

set in

.X

Hence

X

is not a

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

36

Volume 3, 2023

connected*

eugrN-

space. But this is a contradiction to our

hypothesis. Thus, there exist no nonempty

open*

eugrN-

sets

U

and

V

in

,X

such that

.

C

UV

Sufficiency: Let

U

be both

open*

eugrN-

and

closed*

eugrN-

set of

X

such that

0,

Neu

U

1.

Neu

U

Now let

.

C

VU

Then

V

is a

open*

eugrN-

set and

1.

Neu

V

This implies

0,

CNeu

UV

which is a

contradiction to our hypothesis. Therefore

X

is

connected*

eugrN-

space.

Theorem 4.3.

A

eu Top Space-N-

 

,N

X

is

connected*

eugrN-

space if and only if there do not exist

nonempty

eu subsets-N

U

and

V

in

X

such that

,

C

UV

 

  



*C

eu

V Cl UgrN-

and

 

.  



*C

eu

U Cl VgrN-

Proof .

Necessity: Let

U

and

V

be two

eu subsets-N

of

 

,N

X

such that

0,

Neu

U

0Neu

V

and

,

C

UV

 

  



*C

eu

V Cl UgrN-

and

 

.  



*C

eu

U Cl VgrN-

Since

 





*C

eu Cl UgrN-

and

 





*C

eu Cl VgrN-

are

open*

eugrN-

sets in

,X

so

U

and

V

are

open*

eugrN-

sets in

.X

This implies

X

is not a

connected*

eugrN-

space, which is a contradiction.

Therefore, there exist no nonempty

open*

eugrN-

sets

U

and

V

in

,X

such that

,

C

UV

 

  



*C

eu

V Cl UgrN-

and

 

.  



*C

eu

U Cl VgrN-

Sufficiency: Let

U

be both

open*

eugrN-

and

closed*

eugrN-

set in

X

such that

0,

Neu

U

1.

Neu

U

Now by taking

C

VU

we obtain a contradiction to our

hypothesis. Hence

X

is

connected*

eugrN-

space.

Theorem 4.4.

Let

   

: , ,

NN

f X Y



be a

irresolure*

eugrN-

surjection and

X

be

connected.*

eugrN-

Then

Y

is

connected.*

eugrN-

Proof .

Assume that

Y

is not

connected,*

eugrN-

then

there exist nonempty

open*

eugrN-

sets

U

and

V

in

Y

such that

1Neu

UV

and

0.

Neu

UV

Since

f

is

irresolure*

eugrN-

mapping,

 

10,

N

A f U





 

10,

Neu

B f V





which are

open*

eugrN-

sets in

X

and

     

1 1 1 1 1 ,

  



Neu Neu

f U f V f

which implies

1.

Neu

AB

Also

     

1 1 1 0 0 ,

Neu Neu

f U f V f

  



which implies

0.

Neu

AB

Thus, 𝑋 is

disconnected,*

eugrN-

which is a contradiction to our

hypothesis. Hence

Y

is

connected.*

eugrN-

Definition 4.5.

Let

A

and

B

be nonempty

eu subsets-N

in a

eu Top SpaceN--

 

,.

N

X

Then

A

and

B

are said

to be

separated*

eugrN-

if

   

0.

eu eu Neu

Cl A B A Cl B     

   

**

NNgr gr--

Remark 4.6.

Any two disjoint non-empty

closed

eu *Ngr -

sets are

separated.*

eugrN-

Proof .

Suppose

A

and

B

are disjoint non-empty

closed*

eugrN-

sets. Then

 

  



*

eu Cl A BgrN-

 

0.

eu Neu

A Cl B A B   



*-Ngr

This shows that

A

and

B

are

separated.*

eugrN-

Theorem 4.7.

 

i

Let

A

and

B

be two

separated*

eugrN-

subsets of a

eu Top Space-N-

 

,N

X

and

,CA

.DB

Then

C

and

D

are also

separated.*

eugrN-

 

ii

Let

A

and

B

be both

separated*

eugrN-

subsets of a

eu Top SpaceN--

 

,N

X

and let

C

H A B

and

.

C

G B A

Then

H

and

G

are also

separated.*

eugrN-

Proof .

 

i

Let

A

and

B

be two

separated

eu *-Ngr

sets

in

eu Top SpaceN--

 

,.

N

X

Then

   

0.

eu Neu eu

Cl A B A Cl B     

   

**

NNgr gr--

Since

CA

and

,DB

then

 

*

eu Cl CgrN-

 

*

eu Cl AgrN-

and

   

.**

eu eu

Cl D Cl Bgr grNN--

This implies that,

 

  



*

eu Cl C DgrN-

 

0

eu Neu

Cl A B  



*-Ngr

and hence

 

0.

eu Neu

Cl C D  



*-Ngr

Similarly

 

  



*

eu Cl D CgrN-

 

0

eu Neu

Cl B A  



*-Ngr

and hence

 

0.

eu Neu

Cl D C  



*

Ngr -

Therefore

C

and

D

are

separated.*

eugrN-

 

ii

Let

A

and

B

be both

open*

eugrN-

subsets of

.X

Then

C

A

and

C

B

are

closed*

eugrN-

sets. Since

,

C

HB

then

 

*

eu Cl HgrN-

 

*CC

eu Cl B BgrN-

and so

 

0.

eu Neu

Cl H B *-Ngr

Since

,GB

then

   

0.

eu eu Neu

Cl H G Cl H B     

   

**

--NNgr gr

Thus, we have

 

0.

eu Neu

Cl H G  



*-Ngr

Similarly,

 

0.

eu Neu

Cl G H  



*-Ngr

Hence

H

and

G

are

separated.*

eugrN-

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

37

Volume 3, 2023

Theorem 4.8.

Two

eu subsetsN-

A

and

B

of a

eu Top SpaceN--

 

,N

X

are

separated*

eugrN-

if

and only if there exist

open*

eugrN-

sets

U

and

V

in

X

such that

,AU

BV

and

0Neu

AV

and

0.

Neu

BU

Proof .

Let

A

and

B

be

separated.*

eugrN-

Then

   

0.

eu Neu eu

A Cl B B Cl A     

   

**

NNgr gr--

Let

 

*C

eu

V Cl AgrN-

and

 

.*C

eu

U Cl BgrN-

Then

U

and

V

are

open*

eugrN-

sets in

X

such that

,AU

BV

and

0Neu

AV

and

0.

Neu

BU

Conversely, let

U

and

V

be

open*

eugrN-

sets such that

,AU

BV

and

0,

Neu

AV

0.

Neu

BU

Then

C

AV

and

C

BU

and

C

V

and

C

U

are

closed.

eu *Ngr -

This implies

 

*

eu Cl AgrN-

 

CC

eu Cl V V*-Ngr

C

B

and

 

.  **

C C C

eu eu

Cl B Cl U U Agr grNN--

That is,

 

*C

eu Cl A BgrN-

and

 

.*C

eu Cl B AgrN-

So

 

0

eu Neu

A Cl B 





*

Ngr -

 

.

eu Cl A B





*

Ngr -

Hence

A

and

B

are

separated.*

eugrN-

Theorem 4.9.

Every two

separated*

eugrN-

sets are

always disjoint.

Proof .

Let

A

and

B

be

separated.*

eugrN-

Then

   

0.

eu Neu eu

A Cl B Cl A B     

   

**

NNgr gr--

Now,

 

0.

eu Neu

A B A Cl B   



*

Ngr -

Therefore

0N

AB

and hence

A

and

B

are disjoint.

Theorem 4.10.

A

eu Top SpaceN--

 

,N

X

is

connected*

eugrN-

if and only if

1,

Neu

AB

where

A

and

B

are

separated

eu *-Ngr

sets.

Proof .

Assume that

 

,N

X

is

connected*

eugrN-

space. Suppose

1,

Neu

AB

where

A

and

B

are

separated*

eugrN-

sets. Then

   

0.

eu eu Neu

Cl A B A Cl B     

   

**

--NNgr gr

Since

 

,*

eu

A Cl AgrN-

we have

 

0.

eu Neu

A B Cl A B   



*-Ngr

Therefore

 

*C

eu Cl A B AgrN-

and

 

*

eu Cl BgrN-

.

C

AB

Hence

 

*

eu

A Cl AgrN-

and

 

.*

eu

B Cl BgrN-

Therefore

A

and

B

are

closed*

eugrN-

sets and hence

C

AB

and

C

BA

are

disjoint

open*

eugrN-

sets. Thus

0,

Neu

A

0Neu

B

such

that

1Neu

AB

and

0,

Neu

AB

A

and

B

are

open*

eugrN-

sets. That is

X

is not

connected,*

eugrN-

which is a contradiction to

X

is a

connected*

eugrN-

space. Hence

1Neu

is not the union of any two

separated*

eugrN-

sets.

Conversely, assume that

1Neu

is not the union of any two

separated*

eugrN-

sets. Suppose

X

is not

connected.*

eugrN-

Then

1,

Neu

AB

where

0,

Neu

A

0Neu

B

such that

1,

Neu

AB

A

and

B

are

open*

eugrN-

sets in

.X

Since

C

AB

and

,

C

BA

 

0

C

eu Neu

Cl A B B B   



*-Ngr

and

 

  



*

eu

A Cl BgrN-

0.

CNeu

AA

That is

A

and

B

are

separated*

eugrN-

sets. This is a contradiction.

Therefore

X

is

connected.

eu *Ngr -

Definition 4.11.

A

eu Top SpaceN--

 

,N

X

is

eu disconnected*Super grN - - -

if there exists a

eu open*R egular grN - - -

set

A

in

X

such that

0N

A

and

1.

Neu

A

A

eu Top SpacesN--

 

,N

X

is called

eu connected*Super grN - - -

if

X

is not

eu disconnected.*Super grN - - -

Theorem 4.12.

Let

 

,N

X

be a

eu Top Space.N--

Then following assertions are equivalent:

 

i

X

is

eu connected.*Super grN - - -

 

ii

For each

open*

eugrN-

set

0Neu

U

in

,X

we have

 

1.

eu Neu

Cl U *-Ngr

 

iii

For each

closed*

eugrN-

set

1Neu

U

in

,X

we have

 

0.

eu Neu

Int U *-Ngr

 

iv

There do not exist

open*

eugrN-

subsets

U

and

V

in

 

,,

N

X

such that

0,

Neu

U

0Neu

V

and

.

C

UV

 

v

There do not exist

open*

eugrN-

subsets

U

and

V

in

 

,,

N

X

such that

0,

Neu

U

0,

Neu

V

 

C

eu

V Cl U*-Ngr

and

 

.

C

eu

U Cl V*-Ngr

 

vi

There do not exist

closed

eu *-Ngr

subsets

U

and

V

in

 

,,

N

X

such that

1,

Neu

U

1,

Neu

V

 

C

eu

V Int U*-Ngr

and

 

.*C

eu

U Int VgrN-

Proof .

   

i ii :

Assume that there exists a

open

eu *-Ngr

set

0Neu

A

such that

 

1.

eu Neu

Cl A *-Ngr

Now take

 

.

eu eu

B Int Cl A  



**--NNgr gr

Then

B

is a proper

eu open*- - -N R egular gr

set in

X

which contradicts

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

38

Volume 3, 2023

that

X

is

eu connected.*- - -N Super gr

Therefore

 

1.

eu Neu

Cl A *-Ngr

   

ii iii :

Let

1Neu

A

be a

closed

eu *-Ngr

set in

.X

Then

C

A

is

open

eu *-Ngr

set in

X

and

0.

CNeu

A

Hence

by hypothesis,

 

1,

C

eu Neu

Cl A *-Ngr

and so

 

1.

C

eu eu Neu

Cl A Int A**--NNgr gr

This

implies that

 

0.

eu Neu

Int A *-Ngr

   

iii iv :

Let

A

and

B

be

open

eu *-Ngr

sets in

X

such that

0Neu

AB

and

.C

AB

Since

C

B

is

closed

eu *-Ngr

set in

X

and

0Neu

B

implies

1,

CNeu

B

we obtain

 

0.

C

eu N

Int B *-Ngr

But, from

,

C

AB

 

0  *

Neu eu

A Int AgrN-

 

0,*C

eu Neu

Int BgrN-

which is a contradiction.

   

iv i :

Let

01

Neu Neu

A

be

eu open*- - -N R egular gr

set in

.X

Let

 

.

C

eu

B Cl A*-Ngr

Since

 

eu eu

Int Cl B 





**--NNgr gr

 

C

eu eu eu

Int Cl Cl A





* * *- - -N N Ngr gr gr

 

C

eu eu eu

Int Int Cl A







* * *- - -N N Ngr gr gr

 

.

C

eu eu

Int A Cl A B   



**--NNgr gr

Also we get

0,

Neu

B

since otherwise, we have

0Neu

B

and this implies

 

0.*C

eu Neu

Cl AgrN-

That implies

 

1.*

eu Neu

Cl AgrN-

That shows that

 

eu eu

A Int Cl A   



**--NNgr gr

 

1 1 .

eu eu

eu N N

Int *-Ngr

That is

1,

Neu

A

which is a

contradiction. Therefore

0N

B

and

.

C

AB

But this is a

contradiction to

 

iv .

Therefore

 

,N

X

is

eu connected space.*- - -N Super gr

   

i v :

Let

A

and

B

be

eu open*grN-

sets in

 

,N

X

such that

0,

Neu

AB

 

,

C

eu

B sCl A  



N g

 

.  



*C

eu

A Cl BgrN-

Now

 

,

   



  



* * *

*

C

eu eu eu

C

eu

Int Cl A Int B

Cl B A

gr gr gr

gr

N N N

N

- - -

-

0Neu

A

and

1,

Neu

A

since if

1,

Neu

A

then

 

1.  



*C

N eu Cl BgrN-

This implies

 

0.

eu Neu

Cl B *-Ngr

But

0.

Neu

B

Therefore

1Neu

A

implies that

A

is proper

eu open*R egular grN - - -

set in

 

,,

N

X

which is a contradiction to

 

i.

Hence

 

v

is true.

   

v i :

Let

A

be

eu open*R egular grN - - -

set in

 

,N

X

such that

 

  



**

eu eu

A Int Cl Agr grNN--

and

0 1 .

Neu Neu

A

Now take

 

.  



*C

eu

B Cl AgrN-

In

this case we get

0Neu

B

and

B

is

eu open*N R egular gr- - -

set in

 

,.

N

X

 

C

eu

B Cl A  



*Ngr -

and

 

C

eu Cl B  



*-Ngr

 

C

eu eu

Cl Cl A







**--NNgr gr

 

C

eu eu

Int Cl A







**--NNgr gr

 

eu eu

Int Cl A





**--NNgr gr

.A

But this is a

contradiction. Therefore

 

,N

X

is

eu connected space.*- - -N Super gr

   

v vi :

Let

A

and

B

be two

eu closed*- - -N R egular gr

sets in

 

,N

X

such that

1,

Neu

AB

 

,

C

eu

B Int A  



*-Ngr

 

.

C

eu

A Int B  



*-Ngr

Take

C

CA

and

,

C

DB

C

and

D

become

eu open*- - -N R egular gr

sets in

 

,N

X

with

0,

Neu

CD

 

,

C

eu

D Int C  



*-Ngr

 

,

C

eu

C Int D  



*-Ngr

which is a contradiction to

 

v.

Hence

 

vi

is true.

   

vi v :

It can be easily proved by the similar way as in

   

v vi .

Definition 4.13.

A

eu Top SpaceN--

 

,N

X

is said

to be

eu disconnected*- - -N Extrem ely gr

if the

closure

eu *-Ngr

of every

open

eu *Ngr -

set in

 

,N

X

is

open

eu *-Ngr

set in

.X

Theorem 4.14.

Let

 

,N

X

be a

eu Top Space.N--

Then the following statements are equivalent.

 

i

X

is

eu disconnected space.*E xtrem ely grN - - -

 

ii

For each

closed*

eugrN-

set

,A

 

eu Int A*-Ngr

is

closed

eu *-Ngr

set.

 

iii

For each

open

eu *-Ngr

set

,A

 

*

eu Cl AgrN-

 

.

C

eu eu

Cl Cl A





**--NNgr gr

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

39

Volume 3, 2023

 

iv

For each

open*

eugrN-

sets

A

and

B

with

 

,*C

eu Cl A BgrN-

 

*

eu Cl AgrN-

 

.



*C

eu Cl BgrN-

Proof .

   

i ii :

Let

A

be any

closed*

eugrN-

set in

 

,N

X

. Then

C

A

is

open*

eugrN-

set. So

 

i

implies

that

 

  



**

C

eu eu

Cl A Int Agr grNN--

is

open*

eugrN-

set. Thus

 

*

eu Int AgrN-

is

closed*

eugrN-

set in

 

,.

N

X

   

ii iii :

Let

A

be

open*

eugrN-

set. Then we have

 







**

C

eu eu

Cl Cl Agr grNN--

 

.





**

C

eu eu

Cl Int Agr grNN--

Since

A

is

open*

eugrN-

set. Then

C

A

is

closed*

eugrN-

set. So,

by

 

ii

 

*C

eu Int AgrN-

is

closed*

eugrN-

set. That is

 







**

C

eu eu

Cl Int Agr grNN--

 







**

C

eu eu

Cl Cl Agr grNN--

 







**

C

eu eu

Cl Int Agr grNN--

 







*C

C

eu Int AgrN-

 

.*C

eu Int AgrN-

Hence we

obtain

 

*

eu Cl AgrN-

which implies that

 

eu Cl A *Ngr -

 

.





**

C

eu eu

Cl Cl Agr grNN--

   

iii iv :

Let

A

and

B

be any two

open*

eugrN-

sets

in

 

,N

X

such that

 

.*C

eu Cl A BgrN-

Then

     

 

C

eu eu eu

iii Cl A Cl Cl A







* * *- - -N N Ngr gr gr

 

.

C

CC

C

eu eu

Cl B Cl B













**--NNgr gr

   

iv i :

Let

A

be any

open

eu *-Ngr

set in

 

,.

N

X

Let

 

.



*C

eu

B Cl AgrN-=

Then

 

.*C

eu Cl A BgrN-

Then (iv) implies

   

.  



**

C

eu eu

Cl A Cl Bgr grNN--

Since

 

*

eu Cl BgrN-

is

closed*

eugrN-

set, this implies

that

 

*

eu Cl AgrN-

is

open*

eugrN-

set. This implies

that

 

,N

X

is

eu disconnected space.*E xtrem ely grN - - -

Definition 4.15.

A

eu Top Space.N--

 

,N

X

is

eu connected,*Strongly grN - - -

if there does not exist

any nonempty

closed*

eugrN-

sets

A

and

B

in

X

such

that

0.

Neu

AB

Theorem 4.16.

Let

   

: , ,

NN

f X Y



be a

irresolute*

eugrN-

surjection and

X

be a

eu connected space.*Strongly grN - - -

Then

Y

is

eu connected.*

eu

Strongly grNN- - -

Proof .

Assume that

Y

is not

eu connected,*

eu

Strongly grNN- - -

then there exist

nonempty

closed*

eugrN-

sets

U

and

V

in

Y

such that

0,

Neu

U

0,

Neu

V

and

0.

Neu

UV

Since

f

is

irresolute*

eugrN-

mapping,

 

10,

Neu

A f U





 

10,

Neu

B f V





which are

closed*

eugrN-

sets in

X

and

     

1 1 1 0 0 ,

Neu Neu

f U f V f

  



which implies

0.

Neu

AB

Thus, 𝑋 is not

eu connected,*Strongly grN - - -

which is a

eu connected.*Strongly grN - - -

Hence it follows that

this is contradiction to our hypothesis. Consequently

Y

is

eu connected.*Strongly grN - - -

5. Neutrosophic Generalized

Regular Star Topological Spaces

In this section, we define

egular*

eugrNR-

spaces and

egular*

eu

Strongly grNR--

spaces by using

open*

eugrN-

sets and

closed*

eugrN-

sets in

eu Top Spaces.N--

We study their basic properties and

characterizations.

Definition 5.1.

A

eu Top SpaceN--

 

,N

X



is said to

be

egular*

eugrNR-

if for each

closed*

eugrN-

set

A

and a

eu point-N

 

,, ,xA

  



there exist disjoint

open*

eugrN-

sets

U

and

V

such that

,AU

 

,, .xV

  



Theorem 5.2.

Let

 

,N

X



be a

eu Top Space.N--

Then

the following statements are equivalent:

 

i

X

is

egular.*

eugrNR-

 

ii

For every

 

,,

xX

  



and every

open*

eugrN-

set

G

containing

 

,, ,x

  

there exists a

open*

eugrN-

set

U

such that

 

,, .  *

eu

x U Cl U G

  

grN-

 

iii

For every

closed*

eugrN-

set

,F

the intersection of

all

closed*

eugrN-

neighbourhoods*

eugrN-

of

F

is

exactly

.F

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

40

Volume 3, 2023

 

iv

For any

eu set-N

A

and a

open*

eugrN-

set

B

such that

0,

Neu

AB

there exists a

eu open*grN-

set

U

such that

0Neu

AU

and

 

.*

eu Cl U BgrN-

 

v

For every non-empty

eu set-N

A

and

closed*

eugrN-

set

B

such that

0,

Neu

AB

there exist

disjoint

open*

eugrN-

sets

U

and

V

such that

0Neu

AU

and

.BV

Proof .

   

i ii :

Suppose

X

is

egular.*

eugrNR-

Let

 

,,

xX

  



and let

G

be a

open*

eugrN-

set containing

 

,, .x

  

Then

 

,,

C

xG

  



and

C

G

is

closed.*

eugrN-

Since

X

is

egular,*

eugrNR-

there exist

open*

eugrN-

sets

U

and

V

such that

0Neu

UV

and

 

,, ,xU

  



.

C

GV

It follows that

C

U V G

and hence

 

.  **

CC

eu eu

Cl U Cl V V Ggr grNN--

That is

 

,, .  *

eu

x U Cl U G

  

grN-

   

ii iii :

Let

F

be any

closed*

eugrN-

set and

 

,, .xF

  



Then

C

F

is

open*

eugrN-

and

 

,, .

C

xF

  



By assumption, there exists a

open

eu *Ngr -

set

U

such

that

 

,, .  *C

eu

x U Cl U F

  

grN-

Thus

 

.*CC

eu

F Cl U UgrN-

Now

C

U

is

closed*

eugrN-

and

neighbourhood*

eugrN-

of

F

which does not contain

 

,, .x

  

So, we get the intersection of

all

closed*

eugrN-

neighbourhoods*

eugrN-

of

F

to be

exactly equal to

.F

   

iii iv :

Suppose

0Neu

AB

and

B

is

open*

eugrN-

set. Let

 

,, .x A B

  



Since

B

is

open,*

eugrN-

C

B

is

closed*

eugrN-

and

 

,, .

C

xB

  



By using

 

iii ,

there exists a

closed,*

eugrN-

neighbourhood*

eugrN-

V

of

C

B

such that

 

,, .xV

  



Now for the

neighbourhood*

eugrN-

V

of

,

C

B

there

exists a

open*

eugrN-

set

G

such that

.

C

B G V

Take

.

C

UV

Thus

U

is a

open*

eugrN-

set containing

 

,, .x

  

Also

0Neu

AU

and

 

.*C

eu U G BgrN-

   

iv v :

Suppose

A

is a non-empty set and

B

is a

closed*

eugrN-

set such that

0.

Neu

AB

Then

C

B

is

open*

eugrN-

set and

0.

CNeu

AB

By our assumption,

there exists a

open*

eugrN-

U

such that

0Neu

AU

and

 

.*C

eu Cl U BgrN-

Take

 

.*C

eu

V Cl UgrN-=

Since

 

*

eu Cl UgrN-

is

closed,*

eugrN-

V

is

open.*

eugrN-

Also

BV

and

UV

   

 

0.

C

eu eu Neu

Cl U Cl U **

--NNgr gr

   

v i :

Let

S

be

closed*

eugrN-

set and

 

,, .xS

  



Then

 

 

,, 0.

Neu

Sx

  



By

 

v,

there exist disjoint

open*

eugrN-

sets

U

and

V

such that

 

 

,, 0Neu

Ux

  



and

.SV

That is

U

and

V

are

disjoint

open

eu *Ngr -

sets containing

 

,,

x

  

and

S

respectively. This proves that

 

,N

X



is

egular.*

eugrNR-

Corollary 5.3.

Let

 

,N

X



be a

eu Top Space.N--

Then the following statements are equivalent:

 

i

X

is

egular.*

eugrNR-

 

ii

For every

 

,,

xX

  



and every

open

eu *-Ngr

set

G

containing

 

,, ,x

  

there exists a

open

eu *-Ngr

set

U

such that

 

,, .  *

eu

x U Cl U G

  

grN-

 

iii

For every

closed

eu *-Ngr

,F

the intersection of all

eu closed,-N

eu neighbourhoods-N

of

F

is exactly

.F

 

iv

For any

eu set-N

A

and a

eu open-N

set

B

such that

0,

Neu

AB

there exists a

open

eu *Ngr -

set

U

such that

0Neu

AU

and

 

.

eu Cl U B*-Ngr

 

v

For every non-empty

eu set-N

A

and a

eu closed-N

set

B

such that

0,

Neu

AB

there exist disjoint

open

eu *-Ngr

sets

U

and

V

such that

0N

AU

and

.BV

Proof .

Since every

eu open-N

set is

open

eu *-Ngr

and

follows from Theorem 5.2.

Theorem 5.4.

A

eu Top SpaceN--

 

,N

X



is

egular

eu *-NRgr

if and only if every

 

,,

xX

  



and

every

neighbourhood

eu *-Ngr

N

containing

 

,, ,x

  

there exists a

open

eu *Ngr -

set

V

such that

 

,, .

eu

x V Cl V N

  

  *Ngr -

Proof .

Let

X

be a

egular

eu *NRgr -

space. Let

N

be

any

neighbourhood

eu *Ngr -

of

 

,, .x

  

Then there exists

a

open

eu *Ngr -

set

G

such that

 

,, .x G N

  



Since

C

G

is

closed

eu *Ngr -

set and

 

,, ,

C

xG

  



by

definition there exist

open

eu *Ngr -

sets

U

and

V

such

C

GU

and

 

,,

xV

  



and

0Neu

UV

so that

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

41

Volume 3, 2023

.

C

VU

It follows that

 

.

CC

eu eu

Cl V Cl U U**NNgr gr--

Also

C

GU

implies

.

C

U G N

Hence

 

,, .

eu

x V V N

  

  *Ngr -

Conversely, suppose for

every

 

,,

xX

  



and every

neighbourhood

eu *Ngr -

N

containing

 

,, ,x

  

there exists a

open

eu *Ngr -

set

V

such that

 

,, .

eu

x V Cl V N

  

  *Ngr -

Let

F

be

any

closed

eu *Ngr -

set and

 

,, .xF

  



Then

 

,, .

C

xF

  



Since

C

F

is

open

eu *Ngr -

set,

C

F

is

neighbourhood

eu *Ngr -

containing

 

,, .x

  

By

hypothesis there exists a

open

eu *Ngr -

set

V

such that

 

,,

xV

  



and

 

.

C

eu Cl V F*Ngr -

This implies that

 

.

C

eu

F Cl V*Ngr -

Then

 

C

eu Cl V*Ngr -

is a

open

eu *Ngr -

set containing

.F

Also

 

0.

C

eu Neu

V Cl V *

Ngr -

Hence

 

,Neu

X



is

egular.

eu *NRgr -

Theorem 5.5.

A

eu Top SpaceN--

 

,N

X



is

egular

eu *NRgr -

if and only if for each

closed

eu *Ngr -

set

F

of

X

and each

 

,, ,

C

xF

  



there exist

open

eu *Ngr -

sets

U

and

V

of

X

such that

 

,,

xU

  



and

FV

and

   

0.

eu eu Neu

Cl U Cl V    

   

**

NNgr gr--

Proof .

Suppose

 

,N

X



is

egular.*

eugrNR-

Let

F

be a

closed*

eugrN-

set in

X

and

 

,, .xF

  



Then there

exist

open*

eugrN-

sets

 

,,

x

U

  

and

V

such that

 

,,

,, ,

x

xU

  



FV

and

 

,, 0.

x Neu

UV

  



This

implies that

 

,,

0.

eu Neu

x

U Cl V

  

  



*Ngr -

Also

 

*

eu Cl VgrN-

is a

closed*

eugrN-

set and

 

,, .

eu

x Cl V

  

*Ngr -

Since

 

,N

X



is

egular,*

eugrNR-

there exist

open*

eugrN-

sets

G

and

H

of

X

such that

 

,, ,xG

  



 

*

eu Cl V HgrN-

and

0.

N

GV

This implies

 

  



*

eu Cl G HgrN-

 







*C

eu Cl H HgrN-

0.

CNeu

HH

Take

.UG

Now

U

and

V

are

open*

eugrN-

sets in

X

such that

 

,,

xU

  



and

.FV

Also

   

 

0.

eu eu

eu Neu

Cl U Cl V

Cl G H



   

   







**

*

NN

N

gr gr

gr

--

-

Conversely, suppose for each

closed

eu *Ngr -

set

F

of

X

and each

 

,, ,

C

xF

  



there exist

open

eu *Ngr -

sets

U

and

V

of

X

such that

 

,,

xU

  



and

FV

and

   

0.

eu eu Neu

Cl U Cl V    

   

**

NNgr gr--

Now

   

0.

eu eu Neu

U V Cl U Cl V     

   

**

NNgr gr--

Therefore

0.

Neu

UV

This proves that

 

,N

X



is

egular.*

eugrNR-

Theorem 5.6.

Let

   

: , ,

NN

f X Y





be a bijective

function. If

f

is

irresolute,*

eugrN-

open*

eugrN-

and

X

is

egular,*

eugrNR-

then

Y

is

egular.*

eugrNR-

Proof .

Suppose

 

,N

X



is

egular.*

eugrNR-

Let

S

be

any

closed*

eugrN-

set in

Y

such that

 

,, .

r t s

yS

Since

f

is

irresolute,*

eugrN-

 

1

fS



is

closed*

eugrN-

set

in

.X

Since

f

is onto, there exists

 

,,

xX

  



such that

   

 

, , , , .

r t s

y f x

  



Now

 

, , , ,r t s

f x y S

  



implies

that

 

1

,, .x f S

  





Since

X

is

egular,*

eugrNR-

there exist

open*

eugrN-

sets

U

and

V

in

X

such that

 

,, ,xU

  



 

1

f S V



and

0.

Neu

UV

Now

 

,,

xU

  



implies that

 

,,

f x f U

  



and

 

1

f S V



implies that

 

.S f V

Also

0N

UV

implies that

 

0Neu

f U V 

which implies that

   

0.

Neu

f U f V 

Since

f

is a

open*

eugrN-

mapping,

 

fU

and

 

fV

are disjoint

open*

eugrN-

sets in

Y

containing

 

,,r t s

y

and

S

respectively. Thus

Y

is

egular.*

eugrNR-

Theorem 5.7.

Let

 

,N

X



be a

egular*

eugrNR-

space. Then

 

i

Every

open*

eugrN-

set in

X

is a union of

closed*

eugrN-

sets.

 

ii

Every

closed*

eugrN-

set in

X

is an intersection of

open*

eugrN-

sets.

Proof .

 

i

Suppose

X

is

egular.*

eugrNR-

Let

G

be a

open*

eugrN-

set and

 

,, .xG

  



Then

C

FG

is

closed*

eugrN-

set and

 

,, .xF

  



Since

X

is

PROOF

DOI: 10.37394/232020.2023.3.6

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42

Volume 3, 2023

egular,*

eugrNR-

there exist disjoint

open*

eugrN-

sets

 

,,

x

U

  

and

V

in

X

such that

 

,,

,, x

xU

  



and

.FV

Since

   

, , , , 0,

x x Neu

U F U V

     



we have

 

,, .

C

x

U F G

  



Take

 

,,

.*

eu x

x

V Cl U

  

grN-

Then

 

,,

x

V

  

is

closed*

eugrN-

set and

 

,, 0.

xN

VV

  



Now

FV

implies that

   

, , , , 0.

x x Neu

V F V V

     



It

follows that

 

,,

,, .

C

x

x V F G

  

  

This proves that

 

 

,, ,,

:.

x

G V x G

     



Thus

G

is a union of

closed*

eugrN-

sets.

 

ii

Follows from

 

i

and set

theoretic properties.

Theorem 5.8.

Let

   

: , ,

NN

f X Y





be a

continuous*

eugrN-

and

eu closed-N

injection from a

eu Top SpaceN--

 

,N

X



into a

eu egular-NR

space

 

,.

N

Y



If every

closed*

eugrN-

set in

X

is

eu closed,-N

then

X

is

egular.*

eugrNR-

Proof .

Let

 

,,

xX

  



and

A

be a

closed*

eugrN-

set

in

X

such that

 

,, .xA

  



Then by assumption,

A

is

eu closed-N

in

.X

Since

f

is

eu closed,-N

 

fA

is a

eu closed-N

set in

Y

such that

 

,, .f x f A

  



Since

Y

is

eu egular,-NR

there exist disjoint

eu open-N

sets

G

and

H

in

Y

such that

 

,,

f x G

  



and

 

.f A H

Since

f

is

continuous,*

eugrN-

 

1

fG



and

 

1

fH



are disjoint

open*

eugrN-

sets in

X

containing

 

,,

x

  

and

A

respectively. Hence

X

is

egular.*

eugrNR-

Theorem 5.9.

Let

   

: , ,

NN

f X Y





be a

eu continuous,-N

open*

eugrN-

bijection of a

eu egular-NR

space

X

into a

eu space-N

Y

and if every

closed*

eugrN-

set in

Y

is

eu closed,-N

then

Y

is

egular.*

eugrNR-

Proof .

Let

 

,,r t s

yY

and

B

be a

closed*

eugrN-

set in

Y

such that

 

,, .

r t s

yB

Since

f

is a bijection. So there

exists a unique point

 

,,

xX

  



such that

 

, , , , .

r t s

f x y

  



Then by assumption,

B

is

eu closed-N

in

.Y

Since

f

is a

eu continuous-N

bijection,

 

1

fB



is a

eu closed-N

set in

X

such that

 

1

,, .x f B

  





Since

X

is

eu egular,-NR

there exist

disjoint

eu open-N

sets

G

and

H

in

X

such that

 

,,

xG

  



and

 

1.f B H



Since

f

is

open,

eu *Ngr -

 

fG

and

 

fH

are disjoint

open

eu *Ngr -

sets in

Y

such that

 

, , , ,r t s

f x y f G

  



and

 

.B f H

Hence

Y

is

egular.

eu *NRgr -

Definition 5.10.

A

eu Top SpaceN--

 

,N

X



is said to

be

strongly

egular

eu *NRgr -

if for each

closed

eu *Ngr -

set

A

and a point

 

,, ,xA

  



there exist

disjoint

eu open-N

sets

U

and

V

such that

AU

and

 

,, .xV

  



Proposition 5.11.

 

i

Every

strongly

egular*

eugrNR-

space is

egular.*

eugrNR-

 

ii

Every

strongly

egular*

eugrNR-

space is strongly

eu egular.-NR

Proof .

 

i

Suppose

 

,N

X



is

strongly

egular.*

eugrNR-

Let

F

be a

closed*

eugrN-

set and

 

,, .xF

  



Since

X

is

strongly

egular,*

eugrNR-

there

exist disjoint

eu open-N

sets

U

and

V

such that

 

,,

xU

  



and

.FV

Since every

eu open-N

set is

open,

eu *Ngr -

so

U

and

V

are

open*

eugrN-

sets. This implies that

X

is

egular.*

eugrNR-

 

ii

This can be proved similarly as

 

i.

Definition 5.12.

A

eu Top SpaceN--

 

,N

X



is said to

be strongly*

egular.*

eugrNR-

if for each

eu closed-N

set

A

and a point

 

,, ,xA

  



there exist disjoint

open*

eugrN-

sets

U

and

V

such that

,AU

 

,, .xV

  



Proposition 5.13.

Every

egular*

eugrNR-

eu Top SpaceN--

 

,N

X



is strongly*

egular.*

eugrNR-

Proof .

Suppose

 

,N

X



is

egular.*

eugrNR-

Let

F

be

a

eu closed-N

set and

 

,, .xF

  



Then

F

is

closed.*

eugrN-

Since

X

is

egular,*

eugrNR-

there

exist disjoint

open*

eugrN-

sets

U

and

V

such that

 

,,

xU

  



and

.FV

This implies that

X

is strongly*

egular.*

eugrNR-

Theorem 5.14.

Let

 

,N

X



be a

eu Top Space.N--

Then

the following statements are equivalent:

 

i

X

is strongly

egular.*

eugrNR-

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

43

Volume 3, 2023

 

ii

For every

 

,,

xX

  



and every

open*

eugrN-

set

G

containing

 

,, ,x

  

there exists a

eu open-N

set

U

such that

 

,, .  

eu

x U Cl U G

  

N

 

iii

For every

closed*

eugrN-

set

,F

the intersection of

all

eu closed,-N

eu neighbourhoods-N

F

is exactly

.F

 

iv

For any

eu set-N

A

and a

open*

eugrN-

set

B

such

that

0,

Neu

AB

there exists a

eu open-N

set

U

such that

0Neu

AU

and

 

.

euCl U BN

 

v

For every non-empty

eu set-N

A

and

closed

eu *Ngr -

set

B

such that

0,

N

AB

there exist disjoint

eu open-N

sets

U

and

V

such that

0Neu

AU

and

.BV

Proof .

   

i ii :

Suppose

X

is strongly

egular.*

eugrNR-

Let

 

,,

xX

  



and let

G

be a

open*

eugrN-

set containing

 

,, .x

  

Then

 

,,

C

xG

  



and

C

G

is

closed.*

eugrN-

Since

X

is

egular,*

eugrNR-

there exist

eu open-N

sets

U

and

V

such that

0Neu

UV

and

 

,, ,xU

  



.

C

GV

It

follows that

C

U V G

and hence

 

.  

CC

eu eu

Cl U Cl V V GNN

That is

 

,, .  

eu

x U Cl U G

  

N

   

ii iii :

Let

F

be a

closed*

eugrN-

set and

 

,, .xF

  



Then

C

F

is

open

eu *Ngr -

set and

 

,, .

C

xF

  



By assumption, there exists a

eu open-N

set

U

such that

 

,, .   C

eu

x U Cl U F

  

N

Thus

 

.

CC

eu

F Cl U UN

Now

C

U

is

eu closed,-N

eu neighbourhoodN-

of

F

which does not contain

 

,, .x

  

So, the intersection of all

eu closed,-N

eu neighbourhoodsN-

of

F

is exactly

.F

   

iii iv :

Suppose

0Neu

AB

and

B

is

open*

eugrN-

set. Let

 

,, .x A B

  



Since

B

is

open,*

eugrN-

C

B

is

closed

eu *Ngr -

and

 

,, .

C

xB

  



By using

 

iii ,

there exists a

eu closed,-N

eu neighbourhoodN-

V

of

C

B

such that

 

,, .xV

  



Now

for the

eu neighbourhoodN-

V

of

,

C

B

there exists a

eu open-N

set

G

such that

.

C

B G V

Take

.

C

UV

Thus

U

is a

eu open-N

set containing

 

,, .x

  

Also

0Neu

AU

and

 

.

C

euCl U G BN

   

iv v :

Suppose

A

is a non-empty set and

B

is

closed*

eugrN-

set such that

0.

Neu

AB

Then

C

B

is

open

eu *Ngr -

set and

0.

CNeu

AB

By our assumption,

there exists a

eu open-N

set

U

such that

0Neu

AU

and

 

.C

euCl U BN

Take

 

.

C

eu

V Cl UN=

Since

 

euCl UN

is

eu closed,-N

V

is

eu open.-N

Also

BV

and

   

 

0.

C

eu eu N

U V Cl U Cl U

NN

   

v i :

Let

S

be

closed*

eugrN-

set and

 

,, .xS

  



Then

 

 

,, 0.

Neu

Sx

  



By

 

v,

there exist disjoint

eu open-N

sets

U

and

V

such that

 

 

,, 0Neu

Ux

  



and

.SV

That is

U

and

V

are disjoint

eu open-N

sets

containing

 

,,

x

  

and

S

respectively. This proves that

 

,N

X



is strongly

egular.*

eugrNR-

Theorem 5.15.

A

eu Top Space.N--

 

,N

X



is

egular*

eugrNR-

if and only if for each

closed*

eugrN-

set

F

of

X

and each

 

,, ,

C

xF

  



there exist

eu open-N

sets

U

and

V

of

X

such that

 

,,

xU

  



and

FV

and

   

0.

eu eu Neu

Cl U Cl V NN

Proof .

Suppose

 

,N

X



is strongly

eu egular.*grNR-

Let

F

be a

closed*

eugrN-

set in

X

and

 

,, .xF

  



Then there exist

eu open-N

sets

 

,,

x

U

  

and

V

such that

 

,,

,, ,

x

xU

  



FV

and

 

,, 0.

x Neu

UV

  



This

implies that

 

,,

0.

eu N

x

U Cl V

  

N

Also

 

euCl VN

is

a

eu closed-N

set and

 

,, .eu

x Cl V

  

N

Since

 

,N

X



is strongly

egular,*

eugrNR-

there exist

eu open-N

sets

G

and

H

of

X

such that

 

,, ,xG

  



 



euCl V HN

and

0.

Neu

GV

This implies

 

0.

CC

eu eu Neu

Cl G H Cl H H H H  

NN

Take

.UG

Now

U

and

V

are

eu open-N

sets in

X

such that

 

,,

xU

  



and

.FV

Also

     

0.

eu eu eu Neu

Cl U Cl V Cl G HN N N

Thus

   

0.

eu eu Neu

Cl U Cl V NN

Conversely, suppose for each

closed*

eugrN-

set

F

of

X

and each

 

,, ,

C

xF

  



there exist

eu open-N

sets

U

and

V

of

X

such that

 

,,

xU

  



and

FV

and

   

0.

eu eu Neu

Cl U Cl V NN

Now

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

44

Volume 3, 2023

   

0.

eu eu Neu

U V Cl U Cl VNN

Therefore

0.

Neu

UV

This proves that

 

,N

X



is strongly

egular.*

eugrNR-

Theorem 5.16.

A

eu Top Space.N--

 

,N

X



is strongly

egular*

eugrNR-

if and only if every pair consisting of a

eu compact-N

set and a disjoint

closed*

eugrN-

set can

be separated by

eu open-N

sets.

Proof .

Let

 

,N

X



be strongly

egular*

eugrNR-

and let

A

be a

eu compact-N

set, and B be a

closed*

eugrN-

set

such that

0.

Neu

AB

Since

X

is strongly

egular,*

eugrNR-

for each

 

,, ,xA

  



there exist disjoint

eu open-N

sets

 

,,

x

U

  

and

 

,,

x

V

  

such that

 

,,

,, ,

x

xU

  



 

,, .

x

BV

  



Obviously,

 

 

,, ,,

:

x

U x A

     



is a

eu open-N

covering of

.A

Since

A

is

eu compact,-N

there exists a finite set

FA

such

that

 

 

,, ,,

:

x

A U x F

     



and

 

 

,, ,,

:.

x

B V x F

     



Put

 

 

,, ,,

:

x

U U x F

     



and

 

 

,, ,,

:.

x

V V x F

     



Then

U

and

V

are

eu open-N

sets in

.X

Also

0.

N

UV

Otherwise, if

 

,, ,x U V

  



then

   

, , , ,r s t

xU

  



for some

 

,,r s t

xF

and

   

, , , , .

r s t

x V V

  



This implies that

 

   

, , , ,

,, ,

r s t r s t

xx

x U V

  



which is a contradiction to

   

, , , , .

r s t r s t

xx

UV





Thus

U

and

V

are disjoint

eu open-N

sets containing

A

and

B

respectively.

Conversely, suppose every pair consisting of a

eu compact-N

set and a disjoint

closed*

eugrN-

set can

be separated by

eu open-N

sets. Let

F

be a

closed*

eugrN-

set and

 

,, .xF

  



Then

 

 

,,

x

  

is

eu compact-N

set of

X

and

 

 

,, 0.

Neu

xF

  



By our

assumption, there exist disjoint

eu open-N

sets

U

and

V

such that

 

,,

xU

  



and

.FV

This proves that

X

is

strongly

egular.*

eugrNR-

Corollary 5.17.

If

X

is a strongly

egular*

eugrNR-

space,

A

is a

eu compact-N

subset of

X

and

B

is a

open*

eugrN-

set containing

,A

then there exists a

eu egular-NR

open set

V

such that

 

.  

eu

A V Cl V BN

Proof .

Let

X

be strongly

egular*

eugrNR-

and let

A

be

a

eu compact-N

set, and

B

be

open*

eugrN-

set with

.AB

Then

C

B

is

closed*

eugrN-

set such that

0.

CNeu

BA

Since

X

is a strongly

egular*

eugrNR-

space, then there exist disjoint

eu open-N

sets

G

and

H

such that

AG

and

.

C

BH

Take

 

.  



eu eu

V Int Cl GNN

Then

   

 

eu eu eu eu

Cl V Cl Int Cl G







N N N N

 

eu eu

Cl Cl G  



NN

 

.

euCl GN

Since

G

is a

eu open-N

set and

 

,eu

G Cl GN

we have

 



eu

G Int GN

 

.  



eu eu

Int Cl G VNN

This implies

that

   

.

eu eu

Cl G Cl VNN

It follows that

   



eu eu

Cl V Cl GNN

and

   

   

   

eu eu eu eu

Int Cl V Int Cl GN N = N N

.V

Thus

V

is

eu egular-NR

open. Now

 



eu

A G Int G=N

 

.







eu eu

Int Cl G VNN

This

implies that

AV

and

   

  

C

eu eu

Cl V Cl G H BNN

implies that

 

.  

eu

A V Cl V BN

Theorem 5.18.

Let

   

: , ,

NN

f X Y





be a bijective

function. If

f

is

irresolute,*

eugrN-

eu open-N

and

X

is strongly

egular,*

eugrNR-

then

Y

is strongly

egular.*

eugrNR-

Proof .

Suppose

 

,N

X



is strongly

egular.*

eugrNR-

Let

S

be a

closed*

eugrN-

set in

Y

such that

 

,, .

r t s

yS

Since

f

is

irresolute,*

eugrN-

 

1

fS



is

closed*

eugrN-

set in

.X

Since

f

is onto, there exists

 

,,

xX

  



such that

   

 

, , , , .

r t s

y f x

  



Now

 

, , , ,r t s

f x y S

  



implies that

 

1

,, .x f S

  





Since

X

is strongly

egular.*

eugrNR-

there exist

eu open-N

sets

U

and

V

in

X

such that

 

,, ,xU

  



 

1

f S V



and

0.

Neu

UV

Now

 

,,

xU

  



implies

that

 

,,

f x f U

  



and

 

1

f S V



implies that

 

.S f V

Also

0Neu

UV

implies that

 

0N

f U V 

which implies that

   

0.

Neu

f U f V 

Since

f

is a

eu open-N

mapping,

 

fU

and

 

fV

are disjoint

eu open-N

sets in

Y

containing

 

,,r t s

y

and

S

respectively.

Thus

Y

is strongly

egular.*

eugrNR-

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

45

Volume 3, 2023

6. Neutrosophic Generalized

Regular Star Normal Spaces

In this section, we introduce

ormal

eu *N gr N-

and strongly

ormal

eu *N gr N-

spaces and study their properties and

characteristics.

Definition 6.1.

A

eu Top SpaceN--

 

,N

X

is said to

be

ormal*

eugr NN -

if for any two disjoint

closed*

eugrN-

sets

A

and

,B

there exist disjoint

open*

eugrN-

sets

U

and

V

such that

AU

and

.BV

Theorem 6.2.

Let

 

,N

X

be a

eu Top Space.N--

Then

the following statements are equivalent:

 

a

X

is

ormal.*

eugr NN -

 

b

For every

closed*

eugrN-

set

A

in

X

and every

open*

eugrN-

set

U

containing

,A

there exists a

open*

eugrN-

set

V

containing

A

such that

 

.*

eu Cl V UgrN-

 

c

For each pair of disjoint

closed*

eugrN-

sets

A

and

B

in

,X

there exists a

open*

eugrN-

set

U

containing

A

such that

 

0.

eu Neu

Cl U B  



*

Ngr -

 

d

For each pair of disjoint

closed*

eugrN-

sets

A

and

B

in

,X

there exist

open*

eugrN-

sets

U

and

V

containing

A

and

B

respectively such that

   

0.

eu eu Neu

Cl U Cl V    

   

**

NNgr gr--

Proof .

   

a b :

Let

U

be a

open

eu *Ngr -

set

containing the

closed*

eugrN-

set

.A

Then

C

BU

is a

closed*

eugrN-

set disjoint from

.A

Since

X

is

ormal,*

eugr NN -

there exist disjoint

open*

eugrN-

sets

V

and

W

containing

A

and

B

respectively. Then

 

*

eu Cl VgrN-

is disjoint from

.B

Since if

 

,, ,

r t s

yB

the

set

W

is a

open*

eugrN-

set containing

 

,,r t s

yB

disjoint

from

.V

Hence

 

.*

eu Cl V UgrN-

   

b c :

Let

A

and

B

be disjoint

closed*

eugrN-

sets

in

.X

Then

C

B

is a

open*

eugrN-

set containing

.A

By

 

b,

there exists a

open*

eugrN-

set

U

containing

A

such that

 

.*C

eu Cl U BgrN-

Hence

 

0.

eu Neu

Cl U B  



*

Ngr -

This proves

 

c.

   

c d :

Let

A

and

B

be disjoint

closed*

eugrN-

sets

in

.X

Then by

 

c,

there exists a

open*

eugrN-

set

U

containing

A

such that

 

0.  



*

eu N

Cl U BgrN-

Since

 

*

eu Cl UgrN-

is

closed,*

eugrN-

B

and

 

*

eu Cl UgrN-

are disjoint

closed*

eugrN-

sets in

.X

Again by

 

c,

there exists a

open*

eugrN-

set

V

containing

B

such that

 

eu Cl U





*

Ngr -

 

0

eu Neu

Cl V 





*Ngr -

This proves

 

d.

   

d a :

Let

A

and

B

be disjoint

closed*

eugrN-

sets

in

.X

By

 

d,

there exist

open*

eugrN-

sets

U

and

V

containing

A

and

B

respectively such that

   

0.

eu eu Neu

Cl U Cl V    

   

**

NNgr gr--

Since

   

,    

   

**

eu eu

U V Cl U Cl Vgr grNN--

U

and

V

are disjoint

open*

eugrN-

sets containing

A

and

B

respectively. Hence the result of

 

a

follows.

Theorem 6.3.

A

eu Top SpaceN--

 

,N

X

is

ormal*

eugr NN -

if and only if for every

closed*

eugrN-

set

F

and

open*

eugrN-

set

W

containing

,F

there

exists a

open*

eugrN-

set

U

such that

 

.  *

eu

F U Cl U WgrN-

Proof .

Let

 

,N

X

be

ormal.*

eugrN-N

Let

F

be a

closed*

eugrN-

set and let

W

be a

open*

eugrN-

set

containing

.F

Then

F

and

C

W

are disjoint

closed*

eugrN-

sets. Since

X

is

ormal,

eu *N gr N-

there

exist disjoint

open

eu *Ngr -

sets

U

and

V

such that

FU

and

.

C

WV

Thus

.

C

F U V W  

Since

C

V

is

closed,*

eugrN-

so

 

.  **

CC

eu eu

Cl U Cl V V Wgr grNN--

Thus

 

.  *

eu

F U Cl U WgrN-

Conversely, suppose the condition holds. Let

G

and

H

be

two disjoint

closed*

eugrN-

sets in

.X

Then

C

H

is a

open

eu *Ngr -

set containing

.G

By assumption, there

exists a

eu open*grN-

set

U

such that

 

.  *C

eu

G U Cl U HgrN-

Since

U

is

open*

eugrN-

and

 

Cl U

is

closed.*

eugrN-

Then

 

*C

eu Cl UgrN-

is

open.*

eugrN-

Now

 

*C

eu Cl U HgrN-

implies

 

.*C

eu

H Cl UgrN-

Also we have

 



*C

eu

U Cl UgrN-

   

 

0.

C

eu eu Neu

Cl U Cl U **

NNgr gr--

That is

U

and

 

*C

eu Cl UgrN-

are disjoint

open*

eugrN-

sets

containing

G

and

H

respectively. This shows that

 

,N

X

is

ormal.*

eugrN-N

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

46

Volume 3, 2023

Theorem 6.4.

Let

 

,N

X

be a

eu Top Space.N--

Then

the following statements are equivalent:

 

a

X

is

.*

eugr NN -orm al

 

b

For any two

open*

eugrN-

sets

U

and

V

whose

union is

1,

Neu

there exist

closed*

eugrN-

subsets

A

of

U

and

B

of

V

such that

1.

Neu

AB

Proof .

   

a b :

Let

U

and

V

be two

open*

eugrN-

sets in a

*

eugr NN -orm al

space

X

such that

1.

N

UV

Then

C

U

and

C

V

are disjoint

closed*

eugrN-

sets. Since

X

is

,*

eugr NN -orm al

then there exist disjoint

open*

eugrN-

sets

G

and

H

such that

C

UG

and

.

C

VH

Let

C

AG

and

.

C

BH

Then

A

and

B

are

closed*

eugrN-

subsets of

U

and

V

respectively such that

1.

Neu

AB

This proves

 

b.

   

b a :

Let

A

and

B

be disjoint

closed*

eugrN-

sets

in

.X

Then

C

A

and

C

B

are

open*

eugrN-

sets whose

union is

1.

Neu

By

 

b,

there exist

closed*

eugrN-

sets

E

and

F

such that

,

C

EA

C

FB

and

1.

Neu

EF

Then

C

E

and

C

F

are disjoint

open*

eugrN-

sets containing

A

and

B

respectively. Therefore

X

is

.*

eugr NN -orm al

Definition 6.5.

A

eu Top SpaceN--

 

,N

X

is said to

be strongly

*

eugr NN -orm al

if for every pair of disjoint

eu closed-N

sets

A

and

B

in

,X

there are disjoint

open*

eugrN-

sets

U

and

V

in

X

containing

A

and

B

respectively.

Theorem 6.6.

Every

*

eugr NN -orm al

space is strongly

.

eu *N gr N-orm al

Proof .

Suppose

X

is

.*

eugr NN -orm al

Let

A

and

B

be disjoint

eu closed-N

sets in

.X

Then

A

and

B

are

closed*

eugrN-

in

.X

Since

X

is

ormal,*

eugrN-N

there exist disjoint

eu open-N

sets

U

and

V

containing

A

and

B

respectively. Since every

eu open-N

set is

open*

eugrN-

set. Therefore

U

and

V

are

open*

eugrN-

sets in

.X

This implies that

X

is strongly

.*

eugr NN -orm al

Theorem 6.7.

Let

 

,N

X

be a

eu Top Space.N--

Then

the following statements are equivalent:

 

a

X

is strongly

.*

eugr NN -orm al

 

b

For every

eu closed-N

set

F

in

X

and every

eu open-N

set

U

containing

,F

there exists a

open*

eugrN-

set

V

containing

F

such that

 

.*

eu Cl V UgrN-

 

c

For each pair of disjoint

eu closed-N

sets

A

and

B

in

,X

there exists a

open*

eugrN-

set

U

containing

A

such that

 

0.

eu Neu

Cl U B *Ngr -

Proof .

   

a b :

Let

U

be a

eu open-N

set containing

eu closed-N

set

.F

Then

C

HU

is a

eu closed-N

set

disjoint from

.F

Since

X

is strongly

ormal,*

eugr NN -

there exist disjoint

open*

eugrN-

sets

V

and

W

containing

F

and

H

respectively. Then

 

*

eu Cl VgrN-

is

disjoint from

,H

since if

 

,, ,

r t s

yH

the set

W

is a

open*

eugrN-

set containing

 

,,r t s

y

disjoint from

.V

Hence

 

.*

eu Cl V UgrN-

   

b c :

Let

A

and

B

be disjoint

eu closed-N

sets

in

.X

Then

C

B

is a

eu open-N

set containing

.A

By

 

b,

there exists a

open*

eugrN-

set

U

containing

A

such that

 

.*C

eu Cl U BgrN-

Hence

 

0.

eu Neu

Cl U B *Ngr -

This proves

 

c.

   

c a :

Let

A

and

B

be disjoint

closed*

eugrN-

sets

in

.X

By (c), there exists a

open*

eugrN-

set

U

containing

A

such that

 

0.

eu Neu

Cl U B  



*

Ngr -

Take

 

.*C

eu

V Cl UgrN-

Then

U

and

V

are disjoint

open*

eugrN-

sets containing

A

and

B

respectively. Thus

X

is strongly

ormal.*

eugr NN -

Theorem 6.8.

Let

 

,N

X

be a

eu Top Space.N--

Then

the following statements are equivalent:

 

a

X

is strongly

ormal.*

eugrN-N

 

b

For any two

eu open-N

sets

U

and

V

whose union is

1,

N

there exist

closed*

eugrN-

subsets

A

of

U

and

B

of

V

such that

1.

Neu

AB

Proof .

   

a b :

Let

U

and

V

be two

eu open-N

sets in a strongly

ormal*

eugr NN -

space

X

such that

1.

Neu

UV

Then

C

U

and

C

V

are disjoint

eu closed-N

sets. Since

X

is strongly

ormal,*

eugrN-N

then there exist

disjoint

open*

eugrN-

sets

G

and

H

such that

C

UG

and

.

C

VH

Let

C

AG

and

.

C

BH

Then

A

and

B

are

closed*

eugrN-

subsets of

U

and

V

respectively such

that

1.

Neu

AB

   

b a :

Let

A

and

B

be disjoint

eu closed-N

sets in

.X

Then

C

A

and

C

B

are

eu open-N

sets such that

1.

CC

Neu

AB

By

 

b,

there exist

closed*

eugrN-

sets

PROOF

DOI: 10.37394/232020.2023.3.6

Raja Mohammad Latif

E-ISSN: 2732-9941

47

Volume 3, 2023

G

and

H

such that

,

C

GA

C

HB

and

1.

N

GH

Then

C

G

and

C

H

are disjoint

open*

eugrN-

sets

containing

A

and

B

respectively. Therefore,

X

is strongly

ormal.*

eugr NN -

7. Conclusion

Topology is an important and major area of mathematics, and

it can give many relationships between other scientific areas

and mathematical models. Recently, many scientists have

studied the

eu setN-

theory, which is initiated by Molodtsov

and easily applied to many problems having uncertainties

from social life. In the present work, we have continued to

study the properties of

eu Top Spaces.N--

We introduced the

idea of new types of

eu compactness,N-

eu cconnectedness,N-

eu RegularN-

spaces, and

eu NormalN-

spaces defined in terms of

open

eu *Ngr -

and

closed

eu *Ngr -

sets in a

eu Top Space--N

 

,N

X

namely,

compact

eu *Ngr -

spaces,

Lindelof

eu *Ngr -

spaces, countably

compact

eu *Ngr -

spaces,

connected

eu *Ngr -

spaces,

cseparated

eu *Ngr -

sets,

eu connected*N Super gr- - -

spaces,

eu disconnected*N Extrem ely gr- - -

spaces, and

eu connected*N Strongly gr- - -

spaces,

egular

eu *NRgr -

spaces, strongly

egular

eu *NRgr -

spaces,

Normal

eu *Ngr -

spaces, and strongly

ormal

eu *Ngr -N

spaces. Also, several

of their topological properties are investigated. Finally, some

effects of various kinds of

functions

eu

N-

on them are

studied. and have established several interesting properties.

Because there exist compact connections between

sets

eu

N-

and information systems, we can use the results deducted from

the studies on

eu Top SpaceN--

to improve these kinds of

connections. We see that this research work will help

researchers enhance and promote further study on the

eu TopologyN-

to carry out a general framework for their

applications in practical life.

Acknowledgment

The author is highly and gratefully indebted to Prince

Mohammad Bin Fahd University Al Khobar Saudi Arabia, for

providing excellent research facilities during the preparation

of this research paper.

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[2] Babu V Amarendra and J. Aswini, Separation Axioms in

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and Applications in Mathematical Sciences, Volume 20,

Issue 6, (2021), pp. 1115 – 1128.

[3] K. Bageerathi and P. Jeya Puvaneswari, Neutrosophic

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The author is highly and gratefully indebted to Prince

Mohammad Bin Fahd University Al Khobar Saudi Arabia, for

providing excellent research facilities during the preparation

of this research paper.

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DOI: 10.37394/232020.2023.3.6

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