G J E S R X{x,x,x}   (x)0.8,(x)0.5,(x)0.2

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G ^LOBAL J ^{OURNAL OF} E ^NGINEERING S ^{CIENCE AND} R ^ESEARCHES

A NOTE ON FG- INTERIOR

J.K. Maitra¹, Sujeet Chaturvedi¹& Lakshmi Narayan Mishra^*2

1

Department of Mathematics and Computer Science, R.D. University, Jabalpur 482001

*2

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore 632 014, Tamil Nadu, India

Corresponding Author^*

- Lakshmi Narayan Mishra

ABSTRACT

In this paper we have obtained significant properties of Fuzzy g-Interior of a set in fuzzy generalized topological space.

Keywords: Fuzzy sets, Fuzzy topology, Generalized fuzzy g-Interior.

I. INTRODUCTION

The concept of generalized topological spaces was introduced and investigated by A. Csaszar. We introduce a new class of Fuzzy g-Interior of a set in fuzzy generalized topological space Also we investigate some of their basic properties and produced many interesting theorems.

II. PRELIMINARIES

Definition 2.1 Let X be a (non-empty) universal crisp set. A fuzzy topology on X is a non empty collection



of fuzzy sets on X satisfies the following conditions

(i) Fuzzy sets0 and 1 belong to



(ii) Any arbitrary union of members of



is in



(iii) A finite intersection of members of



is in



Here0 and 1 represent the Zero Fuzzy Set and the Whole Fuzzy set on X, defined as,

0(x) 0, x X   

and

1(x) 1, x X   

. The pair (X,



) is called Fuzzy Topological Space on X, For Convenience, we shall denote the fuzzy topological space simply as X.

Example 2.1 Let

X {x ,x ,x }  1 2 3

be the universal crisp set and



be a fuzzy set defined on X as

1 2 3

(x ) 0.8, (x ) 0.5, (x ) 0.2

     

. Then we can see that the collection

{0,A,1}

satisfies all the three conditions of fuzzy topology on X. Hence

  {0, ,1} 

is a fuzzy topology on X and

{X, } 

is a fuzzy topological space.

Proposition 2.1 Let

(X, ) 

be a fuzzy topological space and



let be a fuzzy set in X. Then (i)

 , x

are fuzzy closed set in X.

(ii) Arbitrary intersection of fuzzy closed sets is a fuzzy closed set.

(iii) Finite union of fuzzy closed sets is a fuzzy closed set.

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31

proof : (i) let



and X are fuzzy g-closed set it follow that their complement X and



are fuzzy g-closed set in X.

Proof: (ii). Let (X, ) be a fuzzy topological space and let

  ^

_{j j J}_ be the collection fuzzy closed sets in X.

Where J is any index set then

  ^

^c_{j j}__J is a collection of fuzzy open sets in X. This implies



j J j_



^c

 

^c



j J j_



^c



is a fuzzy closed set in X.

(iii). Let

 

1

,

2 be two fuzzy closed sets in X this means



1c _and

_

₂^c are fuzzy open sets in X. Therefore

 

2

1c c 1 2 c

    

is a fuzzy open set in X. Hence



1



2 is a closed set in X.

Definition 2.2 Let (X, ) be a fuzzy topological space and let



be a fuzzy set in X. Then closure of fuzzy set



is denoted by

cl ( ) 

and is defined to the intersection of all fuzzy closed sets in X containing



.

Remark 2.2 : We note that

cl ( )   inf K { :   K K ,

^c

  }

Thus closure of a fuzzy set



is the smallest fuzzy closed set containing.

Proposition 2.2 Let



be a fuzzy set in a fuzzy topological space (X, ), then is



a fuzzy closed if Cl (



_{) =}



Proof:- Suppose that X is a fuzzy closed set in X. Since Cl(



) is the intersection of all fuzzy closed sets in X containing



And

  

follows that Cl(



_)≤



. As we know that



_≤Cl(



). Thus, we find that Cl(



₎₌



_{Thus we}

find that Cl(



₎₌



Conversely, suppose that Cl(



₎₌



Then by the definition of closure of fuzzy sets it follows that Cl(

  

) is a fuzzy closed set. Thus



is a fuzzy closed set in X.

Proposition 2.3 Let (X,



) be a fuzzy topological space and

  1 2 ,

be a two fuzzy sets in X. Then ; i

Cl( )   

ii

Cl(x) x 

iii If

  1, 2 then cl( ) cl( )    ₁ ₂

iv

cl(  1    2 ) cl( ) cl( ) 1  2

v

cl(  1    2 ) cl( ) cl( ) 1  2

vi

cl(cl( )) cl( )  1   1

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Proposition 2.4: Let X be a topological space and

  ^ _{j j J} _

be a family of subsets of X. Then (i)

j J _ Cl( ) Cl(   j j J j _  )

(ii)

Cl( j J j _ A )  j J _ Cl( )  j

Definition 2.3: Let (X, ) be a topological space and let



be a fuzzy set in X. Then interior of fuzzy set



is denoted by

int( ) 

. and is defined to be the union of all fuzzy open sets in X which are contained in



. Remark 2.2 : We note that Int. (



) = Sup {O:O≤



, ). Thus interior of a fuzzy set



is the largest fuzzy open set contained in





) be a fuzzy topological space and

  1 2 ,

be two fuzzy sets in X. Then ; i

Int( )   

ii

Int(x) x 

iii If

  1, 2 then Int( ) Int( )   ₁  ₂

iv

Int(  1   2 ) int( ) Int( )  1  2

v

Int( ) int( ) Int(  1   1  2  2 )

vi

Int(Int( )) Int( )   1  1

proposition 2.6 Let (X, ) be a fuzzy topological space and let {



} _∈ be the collection fuzzy closed sets in X.

Where J is any index set then

(i) ∪ int

( ) 

_j



int ( ∪



j) (ii)

Int( j ) Int( ) j

j J j J

 

  

(X, ) 

be a fuzzy topological space and



let be a fuzzy set in X. Then ( i)

Int 1       1 Cl( ) 

(ii)

Cl 1       1 Int( ) 

Proof:- We have

Int ( ) U   j j 

where



is a fuzzy open set in X and

 j   ,   j J.

This implies

that

c

j j j

1 int 1  (  )   U    ,

_where

{ }  ^c

is the family of fuzzy closed sets containing

(1   )

Further, we have This implies

Int 1 (   )

=

1  Cl(  )

Hence, by the definition of closure of fuzzy set we get

Cl(1   )

=

1 Int( ).  

G-interior in generalized fuzzy topological spaces

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33

Definition 3.1: Let X be a (non-empty) universal set. A fuzzy topology on X is a non empty collection



Fg _of fuzzy sets on X satisfying the conditions

(i) Fuzzy sets0 and 1 belong to



(ii) if

{ }  j

for

j J 

is any familyof fuzzy sets on X and

  j Fg , j J  

then

j J j     Fg

the pair (x,

 Fg

)is called fuzzy generalized topological . the element of family

 Fg

are called fuzzy g-open sets and their complements are called fuzzy g-closed sets.

Definitions 3.2

Let

( X , 

_Fg

)

be a topological space and let



be a generalized fuzzy set in X. Then the g- interior of fuzzy set



is denoted by .

I ( ) Fg 

and is defined to be the union of all fuzzy g-open sets in X which are contained in



.

Remark 3.2 We note that

I ( ) Fg 

= Sup {O:O≤



, ). Thus g-interior of a fuzzy set



is the largest fuzzy g-open set contained in



Proposition 3.1 Let



be a fuzzy set in fuzzy generalized topological space

(x,  Fg )

_then



is fuzzy g - open if and only if

I Fg (    )

Proof: Suppose



is a fuzzy g-open set in X. Since

I ( ) Fg 

is the union of all fuzzy g-open sets in X

contained in



and it

  

follows that

^I _Fg   ^{  }

As we know that

^I _Fg   ^{  }

. Thus we find that

I Fg (    )

Conversely, suppose that

I ( ) Fg 

then by definition of g -interior of fuzzy set, it follows that

I ( ) Fg 

is a fuzzy g- open set. Thus



is a fuzzy g-open set in X.

 Fg

) be a generalized fuzzy topological space and

  1 2 ,

be two fuzzy sets in X.

Then ;

i

I ( ) Fg   

ii

I (x) x Fg 

iii If

  1, 2 then I ( ) I ( ) _Fg   _Fg 

iv

I ( Fg 1    2 ) I ( ) I ( ) Fg 1  Fg  2

v

I ( ) I ( ) I ( Fg 1  Fg 1   Fg  2  2 )

vi

I (I ( )) I ( ) Fg Fg 1   Fg 1 

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proof : (i) since



and X are fuzzy g-open sets from let (x,

 Fg

) be a generalized fuzzy topological space and let



be a fuzzy set in X. then



is fuzzy g-open set if and only if

I ( ) Fg 

=



. we have

I ( )

_Fg

  

_and

Fg

(X)

I  X

_.

(ii) Suppose



₁

 

₂ in x since



₂

 IFg ( ) 

₂ and



₁

 

₂ we have



₁

 I

_Fg

( ) 

₂ _{, now}

_{I ( )} ₂

Fg 

is a fuzzy g-open set and is

IFg ( )  1

the largest fuzzy g-open set containing



we find that

Fg 1 Fg 2

I ( ) I ( )   

(iii) Since



¹

 

¹



² _,



₂

 

₁



₂ _{we have}

I Fg ( )  1  I Fg (  1  2 )

and

2 1 2

( ) ( )

Fg Fg

I   I  

this implies

I ( ) I ( ) Fg 1  Fg  2  I

_Fg

₍  

₁ ₂

₎

(iv) Since



¹



²

 

¹ _and



₁



₂

 

₂ _we _have

Fg 1 2 Fg 1 Fg 2

I (    ) I ( ) I ( )  

(v) since

I

_Fg

( ) 

1 is a g- fuzzy open set in X. if follow that

Fg Fg 1 Fg 1

I (I ( )) I ( )   

Example 3.1 let X = {x1,x2} let

   1 2 3 , , and be  4

fuzzy sets for X defined as

1 {(x ,0.4),(x ,0.6) 1 2

    ₁ ^c {(x 0.6),(x 0.4)} _1, _2,

 2

= {(x1, 0.5), (x2, 0.3)}



₂^c = {(x1, 0.5), (x2, 0.7)}

 3

={(x1, 0.3), (x2, 0.4)}

c

 3

, = {(x1, 0.6), (x2, 0.7)

 4

= {(x1, 0.5), (x2, 0.6)}

c

 4

= {(x1, 0.5), (x2, 0.4)}

Then

 Fg

_{= {0,}

_{   } _{1 2 3 4} _{, , ,}

,1} is a fuzzy topology on X. It is easy to see that

c

 g

the collection of closed sets in X is given by

Fg 1 c 1 2 2

I ( ) (x ,0.5),(x ,0.3)    

Fg 2 c 1 2 3

I ( ) (x ,0.5),(x ,0.6)    

   

 

Fg 1 1 2 1

I ( ) x , 0.4 , x , 0.6    

and

I( ) x , 0.5 , x , 0.3   2   1   2     2

Fg 1 2 3

I (     )

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35

Fg 1 Fg 2 4

I ( ) I ( )    

Fg 1 Fg 2 4

I ( ) I ( )    

Fg 1 Fg 2 Fg 1 2

I ( ) I ( ) I (      )

proposition 3.3 Let

( , X Fg  )

be a generalized fuzzy topological space and let {



} _∈ be the collection fuzzy g-open sets in X. Where J is any index set then

(i) _{j J Fg}_

I ( ) 

_j

 I

_Fg

(

_{j J}_

 )

(ii)

I Fg ( j J j  ( )   j J Fg  I ( ) 

(X, )  Fg

be a generalized fuzzy topological space and



let be a fuzzy set in X. Then (i)

^{I 1} _Fg  ^{   }  ^{1 C ( )} _Fg ^

(ii)

^{C 1} _Fg  ^{   }  ^{1 I ( )} _Fg ^

Proof:- We have

C 1 _Fg       1 I ( ) _Fg 

where



is a fuzzy open set in X and

 j   ,   j J.

This implies that

c

Fg j j j

1 I  (  ) 1   U    ,

_where

{ }  ^c

is the family of fuzzy g-open sets containing

(1   )

Further, we have This implies

F Fg ( ) C (1 g

I 1      )

. Hence, by the definition of g-open of fuzzy set we get

Fg Fg

C (1     ) 1 I ( ) 

. REFERENCES

1. ´A. Csaszar, Generalized topology, generalized continuity, Acta Math. Hungar. 96 (2002),no. 4, 351–357.

2. Andrejevic D. “On the topology generated by pre-open sets”, Mat Vasnik 38 (1987), 367-376.

3. Azad K.K “Fuzzy Continuity, Fuzzy Almost Continuity and Fuzzy Weaker Continuity”, (1981).

4. Bin Shahana“On Fuzzy Strongly Continuity and Pre-Continuity” Fuzzy Sets and Systems.

(1991),303308

5. Shrivastava M., Maitra J.K. and Shukla M. “On Fuzzy Strongly Pre-continuous Maps”, (2007).

6. Singal M.K. and Prakash N. “Fuzzy Semi Open Sets”, J. India Math. Soc. 63 (1997) No. 1-4, 7. Thakur S.S and S. Singh “On Fuzzy Semi-Pre-open Sets and Fuzzy Semi-pre-continuity, Fuzzy

Sets and System” 98, 383-391.

8. Yalvac T.H. “Semi-Interior and Semi-Closure Fuzzy Sets”, J. Math. Anal. Appl. 132 (1988), 356-364.

9. Zadeh, L.A., Fuzzy Sets, Inform And Control, 8 (1965), 338-353.

10. Palaniappan N. “Fuzzy Topology”, Narosa Publishing House, New Delhi (2002)

G J E S R X{x,x,x}   (x)0.8,(x)0.5,(x)0.2

G LOBAL J OURNAL OF E NGINEERING S CIENCE AND R ESEARCHES

A NOTE ON FG- INTERIOR

Department of Mathematics and Computer Science, R.D. University, Jabalpur 482001

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore 632 014, Tamil Nadu, India

- Lakshmi Narayan Mishra













0(x) 0, x X   

1(x) 1, x X   



X {x ,x ,x }  1 2 3



1 2 3

(x ) 0.8, (x ) 0.5, (x ) 0.2

     

{0,A,1}

  {0, ,1} 

{X, } 

(X, ) 



 , x





  

  





 







 

,





 

    









cl ( ) 



cl ( )   inf K { :   K K ,

  }















  





















  





  1 2 ,

Cl( )   

Cl(x) x 

  1, 2 then cl( ) cl( )    1 2

cl(  1    2 ) cl( ) cl( ) 1  2

cl(  1    2 ) cl( ) cl( ) 1  2

cl(cl( )) cl( )  1   1

   j j J 

j J  Cl( ) Cl(   j j J j   )

G ^LOBAL J ^{OURNAL OF} E ^NGINEERING S ^{CIENCE AND} R ^ESEARCHES

  ^

  ^

_

  1, 2 then cl( ) cl( )    ₁ ₂

  ^ _{j j J} _

j J _ Cl( ) Cl(   j j J j _  )

Cl( j J j _ A )  j J _ Cl( )  j

  1, 2 then Int( ) Int( )   ₁  ₂

{ }  ^c

^I _Fg   ^{  }

^I _Fg   ^{  }