FUZZY PRE SEMI LAMBDA- CLOSED SETS IN FUZZY SCALABLE STRUCTURE SPACE

(1)

STRUCTURE SPACE

Dr.D.Amsaveni1, S. Meenakshi2 and J. Tamilmani3

1

Department of Mathematics (Assistant Professor), Sri Sarada College for Women, Salem - 636 016, Tamilnadu, India

2,3

Department of Mathematics (Research Scholars),Sri Sarada College for Women, Salem - 636 016, Tamilnadu, India

Abstract

In this paper, a new class of sets called **

X

FQ pre semi-

λ

closed sets in fuzzy scalable structure space is introduced and its possible interrelations with other type of **

X

FQ closed sets are studied with necessary counter examples. Similar types of interrelations for continuous functions are also been discussed.

Keywords: Fuzzy scalable structure space, **

X

FQ pre semi-

λ

closed set, **

X

FQ continuous function. AMS Mathematics Subject Classification (2010): 54A40, 03E72

I. Introduction

Fuzzy topological spaces are enriched by additional structures in order to give a more realistic representation of real life phenomena and computational process and at the same time, to provide for utilization of the powerful technique need a scale because all measurements and the majority of computations are performed in a definite space and problems of scalability are important both for physical theories and computational algorithms. The fuzzy concepts have penetrated almost all branches of Mathematics since the introduction of the concept of fuzzy set by Zadeh [11]. The theory of fuzzy topological spaces was introduced and developed by Chang [6]. M.Burgin [4 , 5] introduced the concept of discontinuity structures in topological spaces and also investigated the concept of scale

Q

in a topological space. This structure serves as a scale for the initial topology on X. In this paper, the notion of **

X

FQ pre semi-

λ

closed sets and FQ*X* continuous functions in fuzzy

scalable structure spaces are introduced and studied. Some of their properties are investigated.

1. Preliminaries

Definition 1.1 [11]

Let X be a non-empty set and

I

be the unit interval. A fuzzy set in X is an element of the set

x

I

of all functions from X to

I

.

A fuzzy topology is a family T of fuzzy sets in

(

X,T

)

which satisfies the following conditions:

(a)

0 ,

1 ∈

T

;

(b)

if

A

,

B

∈

T

,

then

A

I

B

∈

T

;

(2)

Then, T is called a fuzzy topology for X and the pair

(

X,T

)

is a fuzzy topological space (abbreviated as fts). Every member of T is called a T-open fuzzy set. A fuzzy set is T-closed if and only if its complement is T-open.

Definition 1.3

A fuzzy set

µ

of a fts

(

X,T

)

is called:

1. a fuzzy pre-open set [3] if

µ

≤

int(

cl

(

µ

))

and a fuzzy pre-closed set if

cl

(int(

µ

))

≤

µ

.

2. a fuzzy semi-open set [2] if

µ

≤

cl

(int(

µ

))

and a fuzzy semi-closed set if

int(

cl

(

µ

))

≤

µ

.

3. a fuzzy semi-pre open set [9] if

µ

≤

cl

(int(

cl

(

µ

)))

and a fuzzy semi-pre closed if

.

)))

(int

int(

cl

µ

≤

µ

4. a fuzzy regular open set [2] if

µ

=

int(

cl

(

µ

))

and a fuzzy regular closed set if

µ

=

cl

(int(

µ

)).

5. a fuzzy generalized open (fg-open) set [1] if

int(

µ

)

≤

ω

whenever µ≤ω and ω is fuzzy open set

in (X,T). The complement of fuzzy g-open set is called fuzzy g-closed set in (X,T). 6. a fuzzy b-open (fb-open) set [2] iff

µ

≤

(int

cl

µ

)

∨

(

cl

int

µ

)

and a fuzzy b-closed 7. (fb-closed) set iff

µ

≥

(

cl

int

µ

)

∧

(int

cl

µ

).

8. a fuzzy t-set [10] if

int(

µ

)

=

int

cl

(

µ

)

.

9. a fuzzy B-set [10] if

µ

=

µ

₁∧

µ

₂where

µ

₁ is fuzzy open and

µ

₂ is a fuzzy t-set.

a. The class of all fuzzy pre-open (resp. fuzzy semi-open, fuzzy semi-pre open, fuzzy regular open, fuzzy generalized open, fuzzy b-open, fuzzy t-set and fuzzy B set) sets in a fuzzy topological space

(

X,T

)

is denoted by FPO(X) (resp. FSO(X), FSPO(X), FRO(X), FGO(X), FbO(X), Ft(X) and FB(X) ).

A subset B of a space X is a Λ-set ifB=BΛ, where BΛ=

∧

{

U U >BU∈T

}

,

, .

A subset A of a space X is called

λ

-closed if A = B∧C, where B is a Λ-set and C is a closed set.

A subset A of a space X is called

λ

-open if Ac = X\A is

λ

-closed.

Definition 1.7[9]

Let

µ

be a fuzzy set in a fts X. Then fuzzy semi pre - interior of

µ

is denoted and defined by spint(

µ

) =∨

{

ω

:

ω

≤

µ

,

µ

∈FSPO(X)

}

.

Definition 1.8[8]

A fuzzy set

µ

of a fts X is called a fuzzy pre semi open set if spint(

µ

)≥ ω, whenever

ω

µ≥ and ωis a fuzzy g-closed set in ( X,T ). The class of all fuzzy pre semi open sets is denoted by FPSO(X).

The complement of a fuzzy pre semi open set is called a fuzzy pre semi closed set in ( X, T ).

Let X be a topological space with a topology TX and TQ be a subset of TX, i.e., TQ consists of open sets from X.

A scale or discontinuity structure QX = Q on X is a mapping X →

TQ

2 that satisfies the following conditions:

(SC1) For all points x from X, if A is an element of Q(x), then x∈A.

(3)

2. Fuzzy Pre semi -

λ

closed sets in fuzzy scalable structure space

Definition 2.1

Let X be a non empty set, τi:i∈J be the fuzzy topologies associated with X and let *

X

τ be

the collection of all fuzzy open sets of(X,τi)and a function QX:

*

X

X→τ be the fuzzy scale. A fuzzy scalable structure is denoted by **

X

Q and defined by **

{

₀_,₁

}

*

X

X Q

Q = U where

{

Q x x X

}

QX = X( ): ∈

* _.

Definition 2.2

Let X be a non empty set, *

X

τ be the collection of all fuzzy open sets of X and **

X

Q be a fuzzy scalable structure on X. Then, the triad

(

X

,

τ

X*,Q*X*)is called a fuzzy scalable structure space.

Every member of **

X

Q is called a fuzzy **

X

Q open set (briefly, **

X

FQ open). Its complement is said to be a fuzzy **

X

Q closed set (briefly, **

X

FQ closed).

Definition 2.3

Let

₍

_,

*, **)

X X Q

X

τ

be a fuzzy scalable structure space. The fuzzy **

X

Q interior and fuzzy **

X

Q closure of a fuzzy set

µ

of X are defined and denoted as follows:

(i) **int(_µ)

{

_ω:_ω ** _ω _µ

}

.

≤ =

∨

isaFQ opensetand

FQX X

(ii) FQ*X*cl(µ)=

∧

{

ω:ωisaFQ*X*closed setandω≥µ

}

.

Definition 2.4

A fuzzy subset

µ

of a fuzzy scalable structure space

(

X

,

τ

*X,Q*X*) is called: 1. a fuzzy **

X

Q pre-open (in short, **

X

FQ pre-open) set, if **int( ** ( ))

µ

µ≤FQX FQXcl and a fuzzy * *

X

Q

pre-closed (in short, **

X

FQ pre-closed) set if FQX**cl(FQ*X*int(µ))≤µ. 2. a fuzzy **

X

Q semi-open (in short, **

X

FQ semi-open) set if µ ≤FQ*X*cl(FQX**int(µ))and a fuzzy * *

X

Q semi-closed (in short, **

X

FQ semi-closed) set if FQ*X*int(FQ*X*cl(µ))≤µ.

3. a fuzzy **

X

Q semi-pre open (in short, **

X

FQ sp-open) set if ** ( **int( ** ( )))

µ

µ ≤FQ_Xcl FQ_X FQ_Xcl and a fuzzy **

X

Q semi-pre closed (in short, **

X

FQ sp-closed) if FQ_X**int(FQ_X**cl(FQ*_X*intµ)))≤µ. 4. a fuzzy **

X

Q regular open (in short, **

X

FQ regular-open) set if **int( ** ( ))

µ

µ =FQ_X FQ_Xcl and a fuzzy *

*

X

Q regular closed (in short, **

X

FQ regular-closed) set if ** ( **int( )).

µ µ = FQXcl FQX

5. a fuzzyQ*X* generalized open (in short,

* *

X

FQ _{g-open) set if} **int(µ)≤ω

X

FQ _wheneverµ≤ω_and

ω is fuzzy Q*X*_{open set in}₍ _, **₎ X

Q

X . The complement of fuzzyQ*X*_{g-open set is called fuzzy}

* *

X

Q

g-closed (in short,FQ*X*_{g-closed) set in}

(

,

*_, **₎ X X Q

X

τ

.

6. a fuzzyQ*X*_{b-open (in short,}

* *

X

FQ _{b-open) set if}µ ≤(FQX**intFQX**clµ)∨(FQ*X*clFQ*X*intµ)_{and a}

fuzzyQX**_{b-closed (in short}

* *

X

FQ _{b-closed) set if}µ ≥(FQ*X*clFQX**intµ)∧(FQ*X*intFQX**clµ).

7. a fuzzyQ*X*_{t-set (in short,}

* *

X

FQ _{t-set) if} **int(_µ) **int ** (_µ) cl FQ FQ

FQX = X X _.

8. a fuzzy **

X

Q B-set (in short, **

X

FQ B-set) if

µ

=

µ

₁∧

µ

₂where

µ

₁ is a **

X

FQ t - set and

µ

₂is a **

X

FQ open set.

The class of all **

X

FQ pre-open (resp. **

X

FQ semi-open, **

X

FQ semi-pre open, **

X

FQ regular open, *

*

X

FQ generalized open, **

X

FQ b open, **

X

FQ t-set and **

X

FQ B set) sets in a fuzz scalable structure space

(

X

,

τ

*X,Q*X*) is denoted by

* *

X

FQ PO(X) (respectively **

X

FQ SO(X), **

X

FQ SPO(X), **

X

FQ

RO(X), **

X

FQ GO(X), **

X

FQ bO(X), **

X

FQ t(X) and **

X

(4)

Definition 2.5

A fuzzy subset µof a fuzzy scalable structure space

(

X

,

τ

*X,Q*X*) is called: 1. a fuzzyQ*X*_-Λ_{set (in short,}

* *

X

FQ _-_Λ_{set) if}µ= **µΛ

X

FQ _{, where}FQ*_X*_µΛ=

∧

{

_ω,_ω>_µ,_ω∈Q_X**

}

.

2. a fuzzyQ*X*_{pre semi -}Λ_{set (in short,}

* *

X

FQ _{pre semi -}_Λ_{set) if}µ=FQXPSµΛ

* *

,

where

{

, , (

)

}

. *

* *

*

X PSO FQ PS

FQX =

∧

> ∈ X

Λ _ω _ω _µ _ω µ

3. a fuzzy Q*X* _{regular -} Λ _{set (in short,}

* *

X

FQ _{regular -} _Λ _{set) if} µ=FQXRµΛ

* *

, where

{

, , ** (

)

}

. *

*

X RO FQ R

FQ_X Λ =

∧

> ∈ _X ω µ ω ω µ

4. a fuzzy Q*X* b - Λ set (in short,

* *

X

FQ _{b -} _Λ _{set) if} µ =FQXbµΛ

* *

, where

{

, , ** (

)

}

. *

*

X bO FQ b

Q

F X =

∧

> ∈ X

Λ _ω _ω _µ _ω µ

Definition 2.6

A fuzzy subset µof a fuzzy scalable structure space *, **)

,

(

X

τ

X QX is called:

1. a fuzzy **

X

Q

λ

-closed (in short, **

X

FQ

λ

- closed) if

µ

=

µ

₁∧

µ

₂, where

µ

₁ is a **

X

FQ Λ-set and

2

µ

is a **

X

FQ closed set. 2. a fuzzy **

X

Q pre semi

λ

X

FQ pre semi

λ

-closed) if

µ

=

µ

₁∧

µ

₂, where

µ

₁is a *

*

X

FQ pre semiΛ-set and

µ

₂is a **

X

Q regular

λ

X

FQ regular

λ

-closed) if

µ

=

µ

₁∧

µ

₂, where

µ

₁is a *

*

X

FQ regularΛ-set and

µ

2is a * *

X

Q b -

λ

closed (in short, **

X

FQ b -

λ

closed) if

µ

=

µ

₁∧

µ

₂, where

µ

₁is a **

X

FQ b -Λset and

µ

₂is a **

X

FQ closed set. The complement of a **

X

FQ

λ

-closed (resp. **

X

FQ pre semi

λ

-closed, **

X

FQ regular

λ

-closed and *

*

X

FQ b -

λ

closed) set is called a **

X

FQ

λ

-open(resp. **

X

FQ pre semi

λ

- open , **

X

FQ regular

λ

-open and **

X

FQ b -λopen).

Proposition 2.1

Every **

X

FQ -closed set in fuzzy scalable structure space

₍

_,

*, **)

X X Q

X

τ

is a **

X

FQ semi pre closed set.

Remark 2.1

The converse part of the above Proposition 2.1 need not be true as shown in the following example.

Example 2.1

Let X=

{

a,b

}

be a non empty set and *

X

τ

be the collection of all fuzzy open sets in X. Let }

, , 1 , 0

{ 1 2

1

µ

τ

= and τ₂ ={0,1,µ₃}be the fuzzy topologies on X where

µ

1,

µ

2,

µ

3∈IX defined as

; 6 . 0 ) ( , 7 . 0 )

( ₁

1 a =

µ

b =

µ

₂(a)=0.3,

µ

₂(b)=0.2;

µ

₃(a)=0.5,

µ

₃(b)=0.4. Now, define *

: X

X X

Q →

τ

by QX(a)=

µ

1; QX(b)=µ3. Then, QX is a fuzzy scale and { 1, 3}. *

µ µ =

X

Q Let

} , , 1 , 0

{ 1 3 *

*

µ µ =

X

Q be a fuzzy scalable structure on X. Clearly, the triad

₍

_,

*, **)

X X Q

X

τ

is fuzzy scalable structure space. Here the fuzzy set

µ

2 is a

* *

X

FQ semi pre closed but not **

X

FQ closed.

Proposition 2.2

Every **

X

FQ semi pre closed set in fuzzy scalable structure space *, **)

,

(

X

τ

X QX is a

* *

X

FQ pre semi closed set.

Remark 2.2

(5)

Example 2.2

Let X=

{

a,b

}

X

τ

, , 1 , 0

{ 1 2

1

µ

τ

= and τ₂ ={0,1,µ₃}be the fuzzy topologies on X where

µ

1,

µ

2,

µ

3∈IX defined as

; 3 . 0 ) ( , 2 . 0 )

( 1

1 a =

µ

b =

µ

2(a)=0.7,

µ

2(b)=0.7;

µ

3(a)=0.3,

µ

3(b)=0.4. Now, define *

: X

X X

Q →

τ

by Q_X(a)=

µ

₁; QX(b)=µ₃. Then, QX is a fuzzy scale and { 1, 3}. *

µ µ =

X

Q Let

} , , 1 , 0

{ 1 3 *

*

µ µ =

X

₍

_,

*, **)

X X Q

X

τ

µ

₂ is a **

X

FQ pre semi closed but not semi pre **

X

FQ closed.

Proposition 2.3

Every **

X

FQ closed set in fuzzy scalable structure space

₍

_,

*, **)

X X Q

X

τ

is a **

X

Remark 2.3

Example 2.3

In example 2.1, the fuzzy set

µ

₂ is a **

X

FQ pre semi closed but not **

X

FQ closed.

Proposition 2.4

Every **

X

FQ regular closed set in fuzzy scalable structure space

(

X

,

τ

*X,Q*X*)is a * *

X

Remark 2.4

Example 2.4

µ

₂ is a **

X

FQ regular closed.

Proposition 2.5

Every **

X

FQ regular closed set in fuzzy scalable structure space

(

X

,

τ

*X,Q*X*)is a * *

X

Remark 2.5

Example 2.5

µ

₂ is **

X

FQ regular closed.

Proposition 2.6

Every **

X

FQ b - closed set in fuzzy scalable structure space *, **)

,

(

X

τ

X QX is a

* *

X

Remark 2.6

Example 2.6

µ

₂ is a **

X

FQ b-closed.

Proposition 2.7

Every **

X

FQ b - closed set in fuzzy scalable structure space

₍

_,

*, **)

X X Q

X

τ

is a **

X

Remark 2.7

(6)

Example 2.7

µ

₂ is a **

X

FQ b-closed.

Proposition 2.8

Every **

X

FQ closed set in fuzzy scalable structure space

₍

_,

*, **)

X X Q

X

τ

is: 1. a **

X

FQ pre semi-λclosed set 2. a **

X

FQ t-set 3. a **

X

FQ B-set 4. a **

X

FQ b-λclosed set 5. a **

X

FQ

λ

- closed set 6. a **

X

FQ regular- λ closed set

Remark 2.8

The converse part of the above Proposition 2.8 need not be true as shown in the following examples.

Example 2.8

Let X=

{

a,b

}

X

τ

, , , , 1 , 0

{ ₁ ₂ ₄ ₅

1 µ µ µ µ

τ = and τ₂ ={0,1,µ₃}be the fuzzy topologies on X where

µ

1,

µ

2,

µ

3,

µ

4∈IX defined as

µ

1(a)=0.4,

µ

1(b)=0.3;

µ

2(a)=0.4,

µ

2(b)=0.1;

µ

3(a)=0.8,

µ

3(b)=0.1;

µ

4(a)=0.3,

; 3 . 0 ) ( 4 b =

µ

5(a)=0.3,

µ

5(b)=0.1.Now, define

*

: X

X X

Q →

τ

by Q_X(a)=

µ

₁;QX(b)=µ3.Then,

X

Q is a fuzzy scale and { 1, 3}. *

µ µ =

X

Q Let {0,1, 1, 3} *

*

µ µ =

X

Q be a fuzzy scalable structure on X.

Clearly, the triad

₍

_,

*, **)

X X Q

X

τ

ω

defined by

4 .

0 )

(

,

4 .

0 )

(

a

=

ω

b

=

ω

is a **

X

FQ pre semi-λclosed, **

X

FQ t-set, **

X

FQ B –set and **

X

FQ b -λset but not *

*

X

FQ closed set. The fuzzy set

µ

₂ is **

X

FQ -λclosed but not **

X

FQ closed.

Example 2.9

Let X=

{

a,b

}

X

τ

, , , 1 , 0

{ 1 2 4

1

µ

τ

= and τ2 ={0,1,µ3}be the fuzzy topologies on X where

X

I ∈

4 3 2 1,

µ

,

µ

,

µ

defined

as

µ

₁(a)=0.6,

µ

₁(b)=0.4;

µ

₂(a)=0.1,

µ

₂(b)=0.3;

µ

₃(a)=0.9,

µ

₃(b)=0.4;and

µ

₄(a)=0.3, 4

. 0 ) ( 4 b =

µ

. Now, define *

: X

X X

Q →

τ

by QX(a)=

µ

2;QX(b)=µ3.Then,QX is a fuzzy scale and

}. , { 2 3 *

µ µ =

X

Q Let {0,1, 2, 3} *

*

µ µ =

X

)

, **

*

,

(

X

τ

X QX is fuzzy scalable structure space. Here the fuzzy set

ω

defined by

4 .

0 )

(

,

1 .

0 )

(

a

=

ω

b

=

ω

is a **

X

FQ regular-λclosed but not **

X

FQ closed.

Proposition 2.9

Every **

X

FQ

λ

- closed set is a **

X

FQ pre semi-

λ

closed set in fuzzy scalable structure space ).

, **

*

,

(

X

τ

_X Q_X

Remark 2.9

Example 2.10

In example 2.8 the fuzzy set

µ

₄is a **

X

FQ pre semi-λclosed but not **

X

FQ

λ

- closed set in

)

, **

*

,

(

X

τ

X QX .

Proposition 2.10

Every **

X

FQ regular -

λ

closed set is a **

X

FQ pre semi-

λ

closed set in fuzzy scalable structure

(7)

Remark 2.10

The converse part of the above Proposition 2.10 need not be true as shown in the example.

Example 2.11

In example 2.8 the fuzzy set

µ

₁is a **

X

FQ pre semi-λ closed but not **

X

FQ regular -

λ

closed

set in

(

X

,

τ

*X,Q*X*). Proposition 2.11

Every **

X

FQ b -

λ

closed set is a **

X

FQ pre semi-

λ

closed set in fuzzy scalable structure space

)

, **

*

,

(

X

τ

X QX .

Remark 2.11

Example 2.12

In example 2.8 the fuzzy set ω defined by

ω

(

a

)

=

0 ,

ω

(

b

)

=

0 .

6

is a **

X

FQ pre semi-λ closed but not **

X

FQ b -

λ

closed set in

₍

_,

*, **) X

X Q

X

τ

.

Every **

X

FQ t-set is a **

X

FQ B-set in the fuzzy scalable structure space

(

X

,

τ

*_X,Q*_X*).

Remark 2.12

Example 2.13

Let X=

{

a,b

}

X

τ

, , 1 , 0

{ ₁ ₂

1

µ

τ

= and τ₂ ={0,1,µ₂,µ₃}be the fuzzy topologies on X where

µ

1,

µ

2,

µ

3∈IX defined as

; 3 . 0 ) ( , 2 . 0 )

( 1

1 a =

µ

b =

µ

2(a)=0.5,

µ

2(b)=0.6and

µ

3(a)=0.1,

µ

3(b)=0.2

.

Now, define *

: X

X X

Q →

τ

by QX(a)=

µ

2; QX(b)=

µ

2. Then, QX is a fuzzy scale and { 2}.

* _µ

=

X

Q Let

} , 1 , 0 { ₂ *

* ₌ _µ

X

Q be a fuzzy scalable structure on X. Clearly, the triad *, **)

,

(

X

τ

X QX is a fuzzy scalable

structure space. Here the fuzzy set

µ

₂is a **

X

FQ B-set but not a **

X

FQ t - set.

Every **

X

FQ regular -λ closed set is a **

X

FQ t -set in the fuzzy scalable structure space )

, **

*

,

(

X

τ

X QX .

Remark 2.13

Example 2.14

Let X=

{

a,b

}

be a non empty set and τ*_Xbe the collection of all fuzzy open sets in X. Let

} , , 1 , 0

{ 1 2

1

µ

τ

= and τ₂ ={0,1,µ₃}be the fuzzy topologies on X where 1

3 2 1, ,

X

I ∈

µ

are defined as ;

2 . 0 ) ( , 6 . 0 )

( ₁

1 a =

µ

b =

µ

₂(a)=0.7,

µ

₂(b)=0.2;

µ

3(a)=0.1,

µ

3(b)=0.8 . Now define : *_X

X X

Q →τ by ( ) ₂;

1 a =µ

Q_X QX1(b)=µ3.Then, QX is a fuzzy scale and { 2, 3}. *

µ µ =

X

Q Let

} , , 1 , 0

{ ₂ ₃

* *

µ µ

= X

(

X

,

τ

*X,Q*X*)is a fuzzy scalable structure space. Here the fuzzy set ω defined by

ω

(

a

)

=

0 .

8 ,

ω

(

b

)

=

0 .

2

is a **

X

FQ t -set but not **

X

(8)

Every **

X

FQ regular -λclosed set is a **

X

FQ b -λclosed set in the fuzzy scalable structure space )

, **

*

,

(

X

τ

X QX .

Remark 2.14

Example 2.15

In example 2.14, the fuzzy set ω defined by

ω

(

a

)

=

0 .

3 ,

ω

(

b

)

=

0 .

2

is a **

X

FQ b -λclosed set but not a **

X

FQ regular -λclosed set.

Every **

X

FQ λ - closed set is a **

X

FQ b -λ closed set in the fuzzy scalable structure space )

, **

*

,

(

X

τ

_X Q_X .

Remark 2.15

Example 2.16

ω

defined by

ω

(

a

)

=

0 .

8 ,

ω

(

b

)

=

0 .

2

is a **

X

FQ b -λclosed set but not a **

X

FQ regular -λclosed set.

Remark 2.16

From the above discussion we have the following implications are true:

Figure: 1 Relationship between **

X

FQ - pre semi

λ

closed set and other existing **

X

FQ - closed sets

3.

**

X

FQ

Pre semi -

λ

continuity in fuzzy scalable structure spaces

Definition 3.1

Let

(

X

1

,

τ

*X1,Q*X*1) and , ) * * * 2

,

2 2

(

X

τ

X QX be any two fuzzy scalable structure spaces. Then the

function f : * , **) 1

,

1 1

(

X

τ

_X Q_X → * , **)

2

,

2 2

(

X

τ

_X Q_X is said to be **

X

FQ continuous if for every ** 2

X

FQ - closed set in

(

X

2

,

τ

X*2,QX**2) there exists a

* *

1

X

FQ - closed set in

(

X

1

,

τ

*X1,Q*X*1). Definition 3.2

Let

(

X

1

,

τ

*X1,Q*X*1) and , ) * * * 2

,

2 2

(

X

τ

X QX be any two fuzzy scalable structure spaces. Then the

function f : * , **) 1

,

1 1

(

X

τ

X QX → , )

* * * 2

,

2 2

(

X

τ

X QX is said to be:

1. **

X

FQ pre semi-λ continuous if for every **

2

X

FQ - closed set in

(

X

2

,

τ

*X2,Q*X*2) there exists a

* *

1

X

FQ

pre semi-λ- closed set in * , **)

1

,

1 1

(

X

τ

_X Q_X .

* *

X

FQ pre semi-λ

closed set *

*

X

FQ t-set

* *

X

FQ B-set

* *

X

FQ b-λclosed set

* *

X

FQ –closed set

* *

X

FQ

λ

closed set

* *

X

FQ regular- λ

(9)

2. **

X

FQ t -continuous if for every **

2

X

FQ - closed set in ₂ * , **)

2 2

,

(

X

τ

X QX there exists a

* *

1

X

FQ t -closed

set in * , **)

1

,

1 1

(

X

τ

X QX .

3. **

X

FQ B -continuous if for every **

2

X

FQ - closed set in

(

X

2

,

τ

*X2,Q*X*2) there exists a

* *

1

X

FQ B -closed

set in * , **)

1

,

1 1

(

X

τ

_X Q_X .

4. **

X

FQ b -λcontinuous if for every **

2

X

FQ - closed set in

(

X

2

,

τ

X*2,Q*X*2) there exists a

* *

1

X

FQ b -λ

closed set in

(

X

1

,

τ

X*1,QX**1).

5. **

X

FQ λ- continuous if for every **

2

X

FQ - closed set in * , **)

2

,

2 2

(

X

τ

_X Q_X there exists a **

1

X

FQ λ- closed

set in ₁ * , **)

1 1

,

(

X

τ

X QX .

6. **

X

FQ regular -λcontinuous if for every **

2

X

FQ - closed set in * , **)

2

,

2 2

(

X

τ

X QX there exists a fuzzy

* *

1

X

FQ regular -λclosed set in

(

X

1

,

τ

X*1,Q*X*1).

Proposition 3.1

Let

(

X

1

,

τ

*X1,Q*X*1) and , )

* * * 2

,

2 2

(

X

τ

X QX be any two fuzzy scalable structure spaces. If a

function f : * , **)

1

,

1 1

(

X

τ

_X Q_X → * , **)

2

,

2 2

(

X

τ

_X Q_X is **

X

FQ continuous function then it is:

1. a **

X

FQ pre semi-λcontinuous function.

2. a **

X

FQ t-continuous function.

3. a **

X

FQ B-continuous function.

4. a **

X

FQ b-λcontinuous function.

5. a **

X

FQ λ- continuous function.

6. a **

X

FQ regular- λcontinuous function.

Remark 3.1

The converse part of the above Proposition 3.1 need not be true as shown in the following examples.

Example 3.1

Let X1=

{

a,b

}

be a non empty set and

*

1 X

τ

be the collection of all fuzzy open sets in X1. Let

} , , , , 1 , 0

{ ₁ ₂ ₄ ₅

1

µ

τ

= and τ2 ={0,1,µ3}be the fuzzy topologies on X1 where

1

5 4 3 2

1, , , ,

X

I ∈

µ

are defined as

µ

₁(a)=0.4,

µ

₁(b)=0.3;

µ

₂(a)=0.4,

µ

₂(b)=0.1;

µ

₃(a)=0.8,

, 3 . 0 ) ( ; 1 . 0 )

( ₄

3 b =

µ

a =

µ

₄(b)=0.3;

µ

5(a)=0.3,

µ

5(b)=0.1.Now, define :

*

1 1

1 X

X X

Q →τ by

; )

( 1

1 a =

µ

Q_X ( ) ₃.

1 b =µ

QX Then,QX1is a fuzzy scale and { 1, 3}.

*

1 = µ µ

X

Q Let {0,1, 1, 3}

* *

1 = µ µ

X

Q be a

fuzzy scalable structure on X1. Clearly, the triad

(

X

1

,

τ

X*1,Q*X*1) is a fuzzy scalable structure space.

Let X2=

{

u,v

}

*

2 X

τ

} , , 1 , 0

{ ₁ ₂

3

ω

τ

= and τ₄ ={0,1,ω₃}be the fuzzy topologies on X2 where 1, 2, 3 2

X

I ∈

ω

are defined

as

ω

1(u)=0.2,

ω

1(v)=0.2;

ω

2(u)=0.4,

ω

2(v)=0.3; ω3(u)=0.6,

ω

3(v)=0.7

.

Now, define

: *

2 2

2 X

X X

Q →τ by ( ) ₁;

2 u =

ω

Q_X ( ) ₃.

2 v =ω

Q_X Then,

2

X

Q is a fuzzy scale and * { ₁, ₃}.

2 = ω ω

X

Q Let

} , , 1 , 0

{ ₁ ₃

* *

2 = ω ω

X

Q be a fuzzy scalable structure on X2. Clearly, the triad

(

X

2

,

τ

X*2,Q*X*2)is a fuzzy

scalable structure space.

Let f : * , **)

1

,

1 1

(

X

τ

_X Q_X → * , **)

2

,

2 2

(

X

τ

_X Q_X be a **

X

FQ continuous function defined by

f(a) = u and f(b) = v. Then the function ‘f’ is **

X

FQ pre semi-λcontinuous and **

X

FQ b -λcontinuous

but not **

X

(10)

Example 3.2

Let X₁=

{

a,b

}

1 X

τ

} , , 1 , 0

{ 1 2

1

µ

τ

= and τ₂ ={0,1,µ₃}be the fuzzy topologies on X1 where 1, 2, 3 1

X

I ∈

µ

are defined as

; 2 . 0 ) ( , 6 . 0 ) ( 1

1 a =

µ

b =

µ

2(a)=0.7,

µ

2(b)=0.2;

µ

3(a)=0.1,

µ

3(b)=0.8 . Now define

: ₁ * 1

1 X

X X

Q →τ by ( ) ₂;

1 a =µ

Q_X QX1(b)=µ3.Then,QX1 is a fuzzy scale and { 2, 3}.

*

1 = µ µ

X Q Let } , , 1 , 0

{ ₂ ₃

* *

1 = µ µ

X

(

X

1

,

τ

Let X₂=

{

u,v

}

2 X

τ

} , , 1 , 0

{ ₁ ₂

3

ω

τ

X

I ∈

ω

are defined

as

ω

1(u)=0.3,

ω

1(v)=0.8;

ω

2(u)=0.2,

ω

2(v)=0.3; ω3(u)=0.6,

ω

3(v)=0.7

.

Now, define

: ₂ * 2

2 X

X X

Q →τ by QX₂(u)=

ω

1; QX2(v)=ω1. Then,QX2 is a fuzzy scale and { 1}.

*

2 = ω

X Q Let } , 1 , 0 { ₁ * *

2 = ω

X

(

X

2

,

τ

Let f :

(

X

1

,

τ

*X1,Q*X*1)→ , )

* * * 2

,

2 2

(

X

τ

X QX be a

* *

X

FQ regular -λcontinuous but not **

X

FQ continuous.

Example 3.3

Let X₁=

{

a,b

}

1 X

τ

} , , 1 , 0

{ 1 2

1

µ

τ

X

I ∈

µ

defined as

; 6 . 0 ) ( , 5 . 0 ) ( 1

1 a =

µ

b =

µ

2(a)=0.2,

µ

2(b)=0.1;

µ

3(a)=0.3,

µ

3(b)=0.2

.

Now, define

: 1 *1

1 X

X X

Q →τ by QX1(a)=µ1; QX1(b)=µ3.Then, QX1 is a fuzzy scale and { 1, 3}.

*

1 = µ µ

X Q Let } , , 1 , 0

{ ₁ ₃

* *

1 = µ µ

X

(

X

1

,

τ

X*1,Q*X*1) is a fuzzy

Let X₂=

{

u,v

}

2 X

τ

} , , 1 , 0

{ ₁ ₂

3

ω

τ

= and τ₄={0,1,ω₃}be the fuzzy topologies on X2 where 1, 2, 3 2

X

I ∈

ω

defined as

; 3 . 0 ) ( , 4 . 0 ) ( 1

1 u =

ω

v =

ω

2(u)=0.3,

ω

2(v)=0.2; ω3(u)=0.7,

ω

3(v)=0.6

.

Now, define

: ₂ * 2

2 X

X X

Q →τ by ( ) ₁;

2 u =ω

Q_X QX2(v)=ω3.Then,QX2 is a fuzzy scale and { 1, 3}.

*

2 = ω ω

X Q Let } , , 1 , 0

{ ₁ ₃

* *

2 = ω ω

X

(

X

2

,

τ

Let f : ₁ * , **)

1 1

,

(

X

τ

X QX → , )

* * * 2

,

2 2

(

X

τ

X QX be a

* *

X

FQ t-continuous and **

X

FQ B-continuous but not **

X

FQ continuous.

Example 3.4

Let X1=

{

a,b

}

*

1 X

τ

} , , , , 1 , 0

{ ₁ ₂ ₄ ₅

1

µ

τ

= and τ₂ ={0,1,µ₃}be the fuzzy topologies on X1 where 1, 2, 3, 4, 5 1

X I ∈

µ

defined as

µ

1(a)=0.4,

µ

1(b)=0.3;

µ

2(a)=0.4,

µ

2(b)=0.1;

µ

3(a)=0.8,

µ

3(b)=0.1;

µ

4(a)=0.3, ; 3 . 0 ) ( 4 b =

µ

5(a)=0.3,

µ

5(b)=0.1.Now, define :

*

1 1

1 X

X X

Q →τ by ( ) ₁;

1 a =µ

Q_X QX1(b)=µ3. Then,

1

X

1 = µ µ

X

Q Let ** {0,1, ₁, ₃}

1 = µ µ

X

Q be a fuzzy scalable structure

(11)

Let X2=

{

u,v

}

2 X

τ

} , , 1 , 0

{ ₁ ₂

3

ω

τ

= and τ4={0,1,ω3}be the fuzzy topologies on X2 where

2

3 2 1, ,

X

I ∈

ω

defined as

; 9 . 0 ) ( , 6 . 0 )

( ₁

1 u =

ω

v =

ω

₂(u)=0.4,

ω

₂(v)=0.3; ω₃(u)=0.8, ω₃(v)=0.9

.

Now, define

: *

2 2

2 X

X X

Q →τ by ( ) ₁;

2 u =ω

Q_X QX2(v)=ω3.Then, QX2 is a fuzzy scale and { 1, 3}.

*

2 = ω ω

X

Q Let

} , , 1 , 0

{ 1 3

* *

2 = ω ω

X

(

X

2

,

τ

Let f :

(

X

1

,

τ

*X1,Q*X*1)→ , )

* * * 2

,

2 2

(

X

τ

X QX be a

* *

X

FQ λ- continuous but not **

X

FQ continuous.

Proposition 3.2

Let * , **)

1

,

1 1

(

X

τ

_X Q_X and * , **)

2

,

2 2

(

X

τ

_X Q_X be any two fuzzy scalable structure spaces. If a

function f :

(

X

1

,

τ

*X1,Q*X*1) → , )

* * * 2

,

2 2

(

X

τ

X QX is

* *

X

FQ λ - continuous then it is **

X

FQ pre semi-λ

continuous. Remark 3.2

Example 3.5

In example 3.1, the function f is **

X

FQ pre semi λ- continuous but not **

X

FQ λ- continuous.

Proposition 3.3

Let

(

X

1

,

τ

*X1,Q*X*1) and , )

* * * 2

,

2 2

(

X

τ

function f : ₁ * , **)

1 1

,

(

X

τ

X QX → , )

* * * 2

,

2 2

(

X

τ

X QX is

* *

X

FQ regular λ- continuous then it is **

X

FQ pre

semi-λcontinuous.

Remark 3.3

Example 3.6

X

FQ pre semi λ - continuous but not **

X

FQ regular -λ

continuous. Proposition 3.4

Let

(

X

1

,

τ

*X1,Q*X*1) and , )

* * * 2

,

2 2

(

X

τ

1

,

1 1

(

X

τ

_X Q_X → * , **)

2

,

2 2

(

X

τ

_X Q_X is **

X

FQ b -λ continuous then it is **

X

FQ pre semi-λ

Example 3.7

Let X1=

{

a,b

}

*

1 X

τ

} , , , , 1 , 0

{ ₁ ₂ ₄ ₅

1

µ

τ

= and τ₂ ={0,1,µ₃}be the fuzzy topologies on X1 where 1, 2, 3, 4, 5 1

X

I ∈

µ

are defined as

µ

1(a)=0.4,

µ

1(b)=0.3;

µ

2(a)=0.4,

µ

2(b)=0.1;

µ

3(a)=0.8,

, 3 . 0 ) ( ; 1 . 0 )

( ₄

3 b =

µ

a =

µ

4(b)=0.3;

µ

5(a)=0.3,

µ

5(b)=0.1. Now, define :

*

1 1

1 X

X X

Q →τ by

; )

( ₁

1 a =µ

Q_X QX1(b)=µ3.Then, QX1 is a fuzzy scale and { 1, 3}.

*

1 = µ µ

X

Q Let ** {0,1, ₁, ₃}

1 = µ µ

X

Q be a

(12)

Let X2=

{

u,v

}

2 X

τ

} , , 1 , 0

{ ₁ ₂

3

ω

τ

= and τ4={0,1,ω3}be the fuzzy topologies on X2 where

2

3 2 1, ,

X

I ∈

ω

defined as

; 4 . 0 ) ( , 1 )

( ₁

1 u =

ω

v =

ω

₂(u)=0.4,

ω

₂(v)=0.3; ω₃(u)=0.3, ω₃(v)=0.3

.

Now, define

: *

2 2

2 X

X X

Q →τ by ( ) ₁;

2 u =ω

Q_X QX2(v)=ω3.Then, QX2 is a fuzzy scale and { 1, 3}.

*

2 = ω ω

X

Q Let

} , , 1 , 0

{ 1 3

* *

2 = ω ω

X

(

X

2

,

τ

Let f :

(

X

1

,

τ

*X1,Q*X*1)→ , )

* * * 2

,

2 2

(

X

τ

X QX be a fuzzy

* *

X

FQ function defined by f(a) = u and f(b)

= v. Then the function ‘f’ is **

X

FQ pre semi-λcontinuous but not **

X

FQ b-λcontinuous.

Proposition 3.5

Let * , **)

1

,

1 1

(

X

τ

_X Q_X and * , **)

2

,

2 2

(

X

τ

_X Q_X be any two fuzzy scalable structure spaces. If a

function f :

(

X

1

,

τ

*X1,Q*X*1)→ , )

* * * 2

,

2 2

(

X

τ

X QX is

* *

X

FQ t- continuous then it is **

X

FQ B -continuous.

Remark 3.5

Example 3.8

In example 2.13, define the mapping f : * , **)

1

,

1 1

(

X

τ

_X Q_X → * , **)

1

,

1 1

(

X

τ

_X Q_X by f(a) = a and f(b)

= b. Then f is **

X

FQ B - continuous but not **

X

FQ t -continuous.

Proposition 3.6

Let

(

X

1

,

τ

*X1,Q*X*1) and , )

* * * 2

,

2 2

(

X

τ

1

,

1 1

(

X

τ

_X Q_X → * , **)

2

,

2 2

(

X

τ

_X Q_X is **

X

FQ regular -λ continuous then it is **

X

FQ t

-continuous.

Remark 3.6

Example 3.9

Let X₁=

{

a,b

}

1 X

τ

} , , 1 , 0

{ 1 2

1

µ

τ

X

I ∈

µ

are defined as

; 2 . 0 ) ( , 6 . 0 )

( 1

1 a =

µ

b =

µ

2(a)=0.7,

µ

2(b)=0.2;

µ

3(a)=0.1,

µ

3(b)=0.8 . Now define

: ₁ * 1

1 X

X X

Q →τ by QX1(a)=µ2; QX1(b)=µ3.Then, QX1 is a fuzzy scale and { 2, 3}.

*

1 = µ µ

X

Q Let

} , , 1 , 0

{ ₂ ₃

* *

1 = µ µ

X

(

X

1

,

τ

Let X₂=

{

u,v

}

2 X

τ

} , , 1 , 0

{ ₁ ₂

3

ω

τ

X

I ∈

ω

are defined

as

ω

1(u)=0.2,

ω

1(v)=0.8;

ω

2(u)=0.2,

ω

2(v)=0.3; ω3(u)=0.9,

ω

3(v)=0.8

.

Now, define

: ₂ * 2

2 X

X X

Q →τ by QX₂(u)=

ω

1;QX2(v)=ω3.Then, QX2 is a fuzzy scale and { 1, 3}.

*

2 = ω ω

X

Q Let

} , , 1 , 0

{ ₁ ₃

* *

2 = ω ω

X

(

X

2

,

τ

Let f :

(

X

1

,

τ

*X1,Q*X*1)→ , )

* * * 2

,

2 2

(

X

τ

X QX be a

* *

X

FQ t - continuous but not **

X

FQ regular -λ

(13)

Proposition 3.7

Let * , **)

1

,

1 1

(

X

τ

X QX and , )

* * * 2

,

2 2

(

X

τ

function f : ₁ * , **)

1 1

,

(

X

τ

X QX → , )

* * * 2

,

2 2

(

X

τ

X QX is

* *

X

FQ regular -λ continuous then it is **

X

FQ b -λ

Example 3.10

Let X1=

{

a,b

}

*

1 X

τ

} , , 1 , 0

{ ₁ ₂

1

µ

τ

X

I ∈

µ

are defined as

; 2 . 0 ) ( , 6 . 0 )

( ₁

1 a =

µ

b =

µ

₂(a)=0.7,

µ

₂(b)=0.2;

µ

₃(a)=0.1,

µ

₃(b)=0.8 . Now define

: ₁ * 1

1 X

X X

Q →τ by ( ) ₂;

1 a =µ

Q_X QX1(b)=µ3.Then, QX1 is a fuzzy scale and { 2, 3}.

*

1 = µ µ

X

Q Let

} , , 1 , 0

{ 2 3

* *

1 = µ µ

X

(

X

1

,

τ

Let X2=

{

u,v

}

*

2 X

τ

} , , 1 , 0

{ ₁ ₂

3

ω

τ

X

I ∈

ω

are defined

as

ω

₁(u)=0.3,

ω

₁(v)=0.8;

ω

₂(u)=0.2,

ω

₂(v)=0.3; ω3(u)=0.9,

ω

3(v)=0.7

.

Now, define

: *

2 2

2 X

X X

Q →τ by ( ) ₁;

2 u =

ω

Q_X ( ) ₃.

2 v =ω

Q_X Then,

2

X

2 = ω ω

X

Q Let

} , , 1 , 0

{ 1 3

* *

2 = ω ω

X

(

X

2

,

τ

Let f : * , **)

1

,

1 1

(

X

τ

_X Q_X → * , **)

2

,

2 2

(

X

τ

_X Q_X be a **

X

FQ b -λ continuous but not **

X

FQ regular -λ

continuous. Proposition 3.8

Let ₁ * , **)

1 1

,

(

X

τ

X QX and , )

* * * 2

,

2 2

(

X

τ

1

,

1 1

(

X

τ

_X Q_X → * , **)

2

,

2 2

(

X

τ

_X Q_X is **

X

FQ λ- continuous then it is **

X

FQ b -λcontinuous.

Remark 3.8

Example 3.11

X

FQ b -λcontinuous but not **

X

FQ λcontinuous.

Remark 3.9

From the above discussion we have the following implications are true:

Figure: 2 Relationship between **

X

FQ - pre semi

λ

continuity and other existing **

X

FQ continuities *

*

X

FQ pre semi-λ

continuous *

*

X

FQ t-continuous

* *

X

FQ B -continuous

* *

X

FQ b-λcontinuous

* *

X

FQ – continuous

* *

X

FQ

λ

continuous

* *

X

FQ regular- λ

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[8] S. Murugesan and P. Thangavelu , Fuzzy Pre-semi-closed Sets, Bull. Malays. Math. Sci. Soc. 31 (2008), 223–232. [9] S. S. Thakur and S. Singh, On fuzzy semi-preopen sets and fuzzy semipre-continuity, Fuzzy Sets and Systems,

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