Vol 9, No 1 (2018)

(1)

Volume 9, No. 1, Jan. - 2018 (Special Issue)

International Journal of Mathematical Archive-9(1), 2018, 47-53

Available online throug

ISSN 2229 – 5046

International Journal of Mathematical Archive- 9(1), Jan. – 2018 47

ON INTUITIONISTIC

b

_{- OPEN SETS}

A. PRABHU

1

_{, A. VADIVEL}

2

_{AND J. SATHIYARAJ}

3

1

_{Research Scholar, Annamalai University,}

Annamalai Nagar, Chidambaram, Cuddalore 608001.

2,3

_{Assistant Professor, Department of Mathematics,}

Government Arts College (Autonomous), Karur – 639005,

E- mail: [email protected]1_{, [email protected]}2

ABSTRACT

In this paper, we introduce a new types of sets called intuitionistic

b

-open sets in intuitionistic topological space. Also intuitionistic

b

-interior and intuitionistic

b

-closure of an intuitionistic sets are defined and their properties are investigated with other existing forms of intuitionistic open sets.

Key words and phrases: Intuitionistic

b

-open (closed) sets, intuitionistic

b

-interior and intuitionistic

b

-closure.

AMS (2000) subject classification: 54A99.

1. INTRODUCTION

Ever since the introduction of fuzzy sets by Zadeh [8], the fuzzy concept has invaded almost all branches of mathematics, Atanassov [2], introduced the notion of intuitionistic fuzzy sets. Using this notion Coker [[3], [4], [5]] defined and studied intuitionistic topological spaces, intuitionistic open sets, intuitionistic closed sets and compactness on intuitionistic topological spaces. Also, he defined the closure and interior operators in intuitionistic topological spaces and established their properties. Many different forms of open sets have been introduced over the years in general topology. Andrijevic [1] introduced and studied about

b

-open sets in general topology. Palaniswamy et al., [7] introduced the concept of intuitionistic semi-open, intuitionistic

α

-open, intuitionistic pre-open and intuitionistic

β

-open sets in 2016. In this

paper, is to define and study the intuitionistic

b

-open sets in intuitionistic topological spaces. Also some properties of these are discussed.

2. PRELIMINARIES

The following definitions and results are essential to proceed further.

Definition 2.1: [3] Let

X

be a non empty fixed set. An intuitionistic set (briefly.

IS

)

A

is an object of the form

),

,

(

=

X

A

₁

A

₂

A

where

A

₁ and

A

₂ are subsets of

X

satisfying

A

₁

∩

A

₂

=

∅

.

The set

A

₁ is called the set of

members of

A

,

while

A

₂ is called the set of non-members of

A

.

The family of all

IS

’s in

X

will be denoted by

IS

(

X

).

Every crisp set

A

on a non-empty set

X

is obviously an intuitionistic set.

X

be a non-empty set,

A

=

(

X

,

A

₁

,

A

₂

)

and

B

=

(

X

,

B

₁

,

B

₂

)

be intuitionistic sets on

,

X

then

(2)

/ On Intuitionistic b - open Sets / IJMA- 9(1), Jan.-2018, (Special Issue)

2.

A

=

B

if and only if

A

⊆

B

and

B

⊆

A

.

3.

A

⊂

B

if and only if

A

₁

∪

A

₂

⊇

B

₁

∪

B

₂

.

4.

A

=

(

X

,

A

₂

,

A

₁

).

5.

A

∪

B

=

(

X

,

A

₁

∪

B

₁

,

A

₂

∩

B

₂

).

6.

A

∩

B

=

(

X

,

A

₁

∩

B

₁

,

A

₂

∪

B

₂

).

7.

A

−

B

=

A

∩

B

8.

φ

~

=

(

X

,

φ

,

X

)

and

X

~

=

(

X

,

X

,

φ

)

Corollary 2.1: [3] Let

A

,

B

,

C

and

A

_i be

IS

’s in

X

.

Then 1.

A

_i

⊆

B

for each

i

implies that



A

_i

⊆

B

.

2.

B

⊆

A

_i for each

i

implies that

B

⊆



A

_i

.

3.

 

A

_i

=

A

_i and

 

A

_i

=

A

_i

.

4.

A

⊆

B

⇐

B

⊆

A

.

5.

(

A

)

=

A

,

φ

~

=

X

~

and

X

~

=

φ

~

Definition 2.3: [5] An intuitionistic topology (briefly

IT

) on a non-empty set

X

is a family

τ

of

IS

’s in

X

satisfying the following axioms

1.

φ

~

,

X

~

∈

τ

.

2.

A

∩

B

∈

τ

for any

A

,

B

∈

τ

.

3.



A

_i

∈

τ

for an arbitrary family in

τ

.

In this case the pair

(

X

,

τ

)

is called intuitionistic topological space (briefly

ITS

) and the

IS

’s in

τ

are called the intuitionistic open set in

X

denoted by

I

(τ)

O

and the complement of an

I

(τ)

O

is called Intuitionistic closed set in

X

denoted by

I

(τ)

C

.

The family of all

I

(τ)

O

(resp.

I

(τ)

C

) sets in

X

will be denoted by

I

(τ)

O

(

X

)

(resp.

).

(

) (

X

C

I

τ )

(

X

,

τ

)

be an

ITS

and

A

∈

IS

(

X

).

Then the intuitionistic interior (resp. intuitionistic closure) of

A

are defined by int A( ) =



{K K: ∈I O X and K( )τ ( ) ⊆ A} (resp.

cl A

( ) =



{

K K

:

∈

I

( )τ

C X

( )

}

and A

⊆

K

)

In this study we use

I

(τ)

i

(

A

)

(resp.

I

(τ)

c

(

A

)

) instead of

int

(

A

)

(resp.

cl

(

A

)

).

Definition 2.5: Let

(

X

,

τ

)

be an

ITS

and an

IS

A

in

X

is said to be

1. intuitionistic regular-open [6] (briefly

I

(τ)

RO

) if

A

=

I

(τ)

i

(

I

(τ)

c

(

A

))

and intuitionistic

regular

-closed

(briefly

I

(τ)

RC

) if

I

(τ)

c

(

I

(τ)

i

(

A

))

=

A

.

2. intuitionistic pre-open [6] (briefly

I

(τ)

PO

) if

A

⊆

I

(τ)

i

(

I

(τ)

c

(

A

))

and intuitionistic pre-closed (briefly

PC

I

(τ) ) if

I

(τ)

c

(

I

(τ)

i

(

A

))

⊆

A

.

3. intuitionistic semi-open [6] (briefly

I

(τ)

SO

) if

A

⊆

I

(τ)

c

(

I

(τ)

i

(

A

))

and intuitionistic semi-closed (briefly

SC

I

(τ) ) if

I

(τ)

i

(

I

(τ)

c

(

A

))

⊆

A

.

4. intuitionistic

α

-open [7] (briefly

I

( )τ

α

O

) if

A

⊆

I

(τ)

i

(

I

(τ)

c

(

I

(τ)

i

(

A

)))

and intuitionistic

α

-closed

(3)

5. intuitionistic

β

-open [7] (briefly

I

( )τ

β

O

) if

A

⊆

I

( )τ

c I i I

(

( )τ

(

( )τ

c A

( )))

and intuitionistic

β

-closed (briefly

I

( )τ

β

C

) if

I i I

( )τ

(

( )τ

c I i A

(

( )τ

( )))

⊆

A

.

The family of all

I

(τ)

RO

(resp.

I

(τ)

RC

,

I

(τ)

PO

,

I

(τ)

PC

,

I

(τ)

SO

,

I

(τ)

SC

,

I

(τ)

α

O

,

I

(τ)

α

C

,

O

I

(τ)

β

and

I

(τ)

β

C

) sets in

X

will be denoted by

I

(τ)

RO

(

X

)

(resp.

I

(τ)

RC

(

X

),

I

(τ)

PO

(

X

),

(

) (

X

PC

I

τ

I

(τ)

SO

(

X

),

I

(τ)

SC

(

X

),

I

(τ)

α

O

(

X

),

I

(τ)

α

C

(

X

),

I

(τ)

β

O

(

X

)

and

I

(τ)

β

C

(

X

).

)

Definition 2.6: [6, 7] Let

(

X

,

τ

)

be an

ITS

and

A

be an

IS

(

X

),

then

1. intuitionistic regular-interior (resp. intuitionistic pre- interior, intuitionistic semi-interior, intuitionistic

α

-interior and

intuitionistic

β

-interior ) of

A

is the union of all

I

(τ)

RO

(

X

)

(resp.

I

(τ)

PO

(

X

),

I

(τ)

SO

(

X

),

I

(τ)

α

O

(

X

)

and

I

(τ)

β

O

(

X

)

) contained in

A

,

and is denoted by

I

(τ)

Ri

(

A

)

(resp.

I

(τ)

Pi

(

A

),

I

(τ)

Si

(

A

),

I

(τ)

α

i

(

A

)

and

).

(

) (

A

i

I

τ

β

)

i.e.

I

(τ)

Ri

(

A

)

=



{

G

:

G

∈

I

(τ)

RO

(

X

)

and

G

⊆

A

},

and

)

(

:

{

=

)

(

I

(τ)

Pi

A



G

∈

I

(τ)

PO

X

G

⊆

A

},

and

)

(

:

{

=

)

(

( ) ( )

A

G

X

SO

I

G

A

Si

I

τ



∈

τ

⊆

},

and

)

(

:

{

=

)

(

I

(τ)

α

i

A



G

∈

I

(τ)

α

O

X

G

⊆

A

}.

and

)

(

:

{

=

)

(

I

(τ)

β

i

A



G

∈

I

(τ)

β

O

X

G

⊆

A

2. intuitionistic regular-closure (resp. intuitionistic pre-closure, intuitionistic semi-closure, intuitionistic

α

-closure,

intuitionistic

β

-closure) of

A

is the intersection of all

I

(τ)

RC

(

X

)

(resp.

I

(τ)

PC

(

X

),

I

(τ)

SC

(

X

),

(

) (

X

C

I

τ

α

I

(τ)

β

C

(

X

)

) containing

A

,

and is denoted by

I

(τ)

Rc

(

A

)

(resp.

I

(τ)

Pc

(

A

),

I

(τ)

Sc

(

A

),

(

) (

A

c

I

τ

α

I

(τ)

β

c

(

A

).

)

i.e.

I

(τ)

Rc

(

A

)

=



{

G

:

G

∈

I

(τ)

RC

(

X

)

and

G

⊇

A

},

}

and

)

(

:

{

=

)

(

I

(τ)

Pc

A



G

∈

I

(τ)

PC

X

G

⊇

A

},

and

)

(

:

{

=

)

(

I

(τ)

Sc

A



G

∈

I

(τ)

SC

X

G

⊇

A

},

and

)

(

:

{

=

)

(

I

(τ)

α

c

A



G

∈

I

(τ)

α

C

X

G

⊇

A

}.

and

)

(

:

{

=

)

(

I

(τ)

β

c

A



G

∈

I

(τ)

β

C

X

G

⊇

A

Proposition 2.1: [7] Let

A

∈

IS

(

X

),

then 1.

I

( )τ

S iA

( ) =

A

∩

I

( )τ

c I i A

(

( )τ

( )).

2.

I

( )τ

Pi A

( ) =

A

∩

I i I

( )τ

(

( )τ

c A

( )).

3.

I

( )τ

S cA

( ) =

A

∪

I i I

( )τ

(

( )τ

c A

( )).

4.

I

( )τ

Pc A

( ) =

A

∪

I

( )τ

c I i A

(

( )τ

( )).

5.

I

( )τ

Pc I

(

( )τ

Pi A

( )) =

I

( )τ

Pi A

( )

∪

I

( )τ

c I i A

(

( )τ

( )).

3. ON INTUITIONISTIC

b

-OPEN SETS

Here intuitionistic

b

-open sets is defined and its relation with existing intuitionistic sets and basic properties are studied.

Definition 3.1: A subset

A

of an intuitionistic topological space

X

is said to be

1. intuitionistic (briefly

I

(τ)

bO

)

b

-open set if

A

⊆

I

(τ)

i

(

I

(τ)

c

(

A

))

∪

I

(τ)

c

(

I

(τ)

i

(

A

)).

(4)

The family of all intuitionistic

b

-open (resp.

b

-closed) in

X

will be denoted by

I

(τ)

bO

(

X

)

(resp.

I

(τ)

bC

(

X

)

.)

Remark 3.1 The complement of intuitionistic

b

-open sets are intuitionistic

b

-closed.

Theorem 3.1:

1. Every intuitionistic regular open (resp. open, semi-open, pre-open and

α

-open) set is intuitionistic

b

-open. 2. Every intuitionistic

b

-open set is intuitionistic

β

-open.

Proof:

1. Let

A

∈

I

( )τ

RO X

( ),

this implies that

A

=

I

( )τ

i I

(

( )τ

c A

( ))

which gives

)).

(

))

(

( ) ( ) ( ) )

(

A

i

I

c

I

A

c

I

i

I

A

⊆

τ τ

∪

τ τ

Let

A

∈

I O X

( )τ

( ),

this implies that

A

=

I i A

( )τ

( )

which gives

A

⊆

I

( )τ

c I i A

(

( )τ

( ))

and also

)).

(

))

(

( ) ( ) ( ) )

(

A

i

I

c

I

A

c

I

i

I

A

⊆

τ τ

∪

τ τ

Let

A

∈

I

( )τ

SO X

( ),

this implies that

A

⊆

I

( )τ

c I i A

(

( )τ

( ))

which gives

)).

(

))

(

( ) ( ) ( ) )

(

A

i

I

c

I

A

c

I

i

I

A

⊆

τ τ

∪

τ τ

Let

A

∈

I

( )τ

PO X

( ),

this implies that

A

⊆

I i I

( )τ

(

( )τ

c A

( ))

which gives

)).

(

))

(

( ) ( ) ( ) )

(

A

i

I

c

I

A

c

I

i

I

A

⊆

τ τ

∪

τ τ

Let

A

∈

I

( )τ

α

O X

( ),

this implies that

A

⊆

I i I

( )τ

(

( )τ

c I i A

(

τ

( )))

which gives

)).

(

))

(

( ) ( ) ( ) )

(

A

i

I

c

I

A

c

I

i

I

A

⊆

τ τ

∪

τ τ

2. Let

A

∈

I bO X

( )τ

( ),

this implies that

A

⊆

I i I

( )τ

(

( )τ

c A

( ))

∪

I

( )τ

c I i A

(

( )τ

( )),

that is either ( ) ( )

(

( ))

A

⊆

I i I

τ τ

c A

or

A

⊆

I

( )τ

c I i A

(

( )τ

( )),

in the first case, it is direct that

A

∈

i

( )τ

β

O X

( )

and in the other case using

A

⊆

I

( )τ

c A

( )

implies the remaining notion.

The converse of above theorem need not be true. It can be seen in the following examples.

Example 3.1: Let

X

=

{

a

,

b

,

c

}

and

τ

=

{(

X

,

φ

,

X

),

(

X

,

X

,

φ

),

(

X

,

φ

,

{

a

,

b

}),

( , ,{ }),

X

ϕ

b

( , ,{ , }),

X

ϕ

b c

( ,{ },{ , }), ( ,{ },{ }), ( ,{ , },{ })}.

X c

a b

X c

b

X a c

b

Then the

IS

1.

(

X

,

{

b

},

{

c

})

∈

I

(τ)

bO

(

X

)

but

(

X

,

{

b

},

{

c

})

∉

I

(τ)

RO

(

X

).

2.

(

X

,

{

b

},

{

c

})

∈

I

(τ)

bO

(

X

)

but

(

X

,

{

b

},

{

c

})

∉

I

(τ)

OS

(

X

).

3.

(

X

,

{

b

,

c

},

{

a

})

∈

I

(τ)

bO

(

X

)

but

(

X

,

{

b

},

{

c

})

∉

I

(τ)

PO

(

X

).

4.

(

X

,

{

b

,

c

},

φ

)

∈

I

(τ)

bO

(

X

)

but

(

X

,

{

b

},

{

c

})

∉

I

(τ)

α

O

(

X

).

X

=

{ }

a

,

b

and

τ

=

{

(

X

,

φ

,

X

),

(

X

,

X

,

φ

),

(

X

,

{

a

},

φ

)

}

then the

IS

(

X

,

{

a

},

{

b

})

∈

I

(τ)

bO

(

X

)

but

(

X

,

{

b

},

{

c

})

∉

I

(τ)

SO

(

X

).

The following diagram shows that the relationship between

I

(τ)

bO

and some other intuitionistic sets.

I

(τ)

PO

I

(τ)

RO

I

(τ)

O

I

(τ)

α

O

I

(τ)

bO

I

(τ)

β

O

I

(τ)

SO

B

(5)

(

X

,

τ

)

be an

ITS

and

{

A

_i

}

_i_∈_J be any arbitrary family of members from

I

(τ)

bO

(

X

)

then

).

(

) (

X

bO

I

A

i

τ

∈



Proof. Let

A

i

I

( )

bO

(

X

)

τ

∈

then

A

i

I

( )

c

(

I

( )

i

(

A

i

))

I

( )

i

(

I

( )

c

(

A

i

)).

τ τ τ

τ

_∪

⊆

Now we decompose the family into

two sub families such that

B

_i

=

{

K

:

K

is

a

member

in

A

_i

and

K

⊆

I

(τ)

c

(

I

(τ)

i

(

K

))}

and

C

i

= { :

G G is

( ) ( )

(

( ))}

i

a member in A and G

⊆

I

τ

i I

τ

c G

Clearly



A

i

=

(



B

i

)

∪

(



C

i

)

and we have

(

)

) (

X

SO

I

B

_i

∈

τ



and

C

i

I

( )

PO

(

X

).

τ

∈



Union of an

I

(τ)

SO

and

I

(τ)

PO

is in

I

(τ)

bO

(

X

).

Corollary 3.1: Let

(

X

,

τ

)

be an

ITS

and

{

A

_i

}

_i_∈_J be any arbitrary family of members from

I

(τ)

bC

(

X

)

then

).

(

) (

X

bC

I

A

i

τ

∈



Remark 3.2: The intersection of two

I

(τ)

bO

(

X

)

need not be

I

(τ)

bO

(

X

)

as shown by the following example.

X

=

{

a

,

b

,

c

}

and

τ

=

{(

X

,

φ

,

X

),

(

X

,

X

,

φ

),

(

X

,

{

φ

},

{ }), ( ,{ },{ , }),

b

X

ϕ

a b

( ,{ },

X

ϕ

{ , }), ( ,{ },{ , }), ( ,{ },{ }), ( ,{ , },

b c

X c

a b

X c

b

X a c

{

b

})}

then the

IS

(

X

,

{

φ

},

{

a

})

and

(

X

,

{

φ

},

{

c

})

are

)

(

) (

X

bO

I

τ but the intersection

(

X

,

{

φ

},

{

a

,

c

})

is not a

b

-open.

Theorem 3.3: Let

(

X

,

τ

)

be an

ITS

and

A

⊂

X

,

then 1.

(

I

(τ)

bc

(

A

))

C

=

I

(τ)

bi

(

A

C

).

2.

(

I

(τ)

bi

(

A

))

C

=

I

(τ)

bc

(

A

C

).

Proof.

(i) Let

A

∈

IS

(

X

)

and

(

X

,

τ

)

be an

ITS

.

Now

}

)

(

:

{

=

)

(

( )

) (

A

G

and

X

bC

I

G

A

bc

I

τ



∈

τ

⊇

[

]

C

A

G

and

X

bC

I

G

A

bc

I

(

))

=

{

:

(

)

}

(

(τ)



∈

(τ)

⊇

=

{

:

(

)

}

)

( C C

C C

A

G

and

X

bO

I

G

∈

τ

⊆



).

(

=

))

(

I

(τ)

bc

A

C

I

(τ)

bi

A

C

(ii) The similar proof can be given.

Theorem 3.4 In an

ITS

(

X

,

τ

),

for an

A

∈

IS

(

X

),

the following are equivalent: 1.

A

∈

I

(τ)

bO

(

X

).

2.

A

=

I

(τ)

Pi

(

A

)

∪

I

(τ)

Si

(

A

).

3.

A

⊆

I

(τ)

Pc

(

I

(τ)

Pi

(

A

)).

Proof. Let

A

∈

IS

(

X

)

in an

ITS

(

X

,

τ

)

(i)

⇒

(ii): Suppose that

A

∈

I

(τ)

bO

(

X

),

so that

A

⊆

I

(τ)

c

(

I

(τ)

i

(

A

))

∪

I

(τ)

i

(

I

(τ)

c

(

A

)).

Now,

)))

(

)))

(

=

)

(

)

(

( ) ( ) ( ) ( ) ( ) )

(

A

i

I

c

I

A

c

I

i

I

A

Si

I

A

Pi

I

τ

∪

τ

∩

τ τ

∪

∩

τ τ

=

(

))))

) ( ) ( ) (

A

i

I

c

I

i

I

A

∩

τ τ τ

(6)

⇒

(iii): Consider

)

(

)

(

=

I

( )

Pi

A

I

( )

Si

A

τ

∪

τ

)))

(

)

(

=

I

(τ)

Pi

A

∪

A

∩

I

(τ)

c

I

(τ)

i

A

))

(

)

(

( ) ( ) ) (

A

i

I

c

I

A

Pi

I

τ

∪

τ τ

⊆

))

(

=

I

(τ)

Pc

I

(τ)

Pi

A

.

.,

(

)).

) ( ) (

A

Pi

I

Pc

I

A

e

i

⊆

τ τ

(iii)

⇒

(i): Consider

))

(

( ) ) (

A

Pi

I

Pc

I

A

⊆

τ τ

))

(

)

(

=

I

(τ)

Pi

A

∪

I

(τ)

c

I

(τ)

i

A

))

(

)))

(

=

A

∩

I

(τ)

i

I

(τ)

c

A

∪

I

(τ)

c

I

(τ)

i

A

))

(

))

(

( ) ( ) ( ) ) (

A

i

I

c

I

A

c

I

i

I

τ τ

∪

τ τ

⊆

))

(

))

(

( ) ( ) ( ) ) (

A

i

I

c

I

A

c

I

i

I

A

⊆

τ τ

∪

τ τ

.

.,

(

).

) (

X

bO

I

A

e

i

∈

τ

This completes the proof.

Theorem 3.5: Let

A

be an

IS

in

X

,

then 1.

I

(τ)

bi

(

A

)

=

I

(τ)

Si

(

A

)

∪

I

(τ)

Pi

(

A

)

2.

I

(τ)

bc

(

A

)

=

I

(τ)

Sc

(

A

)

∩

I

(τ)

Pc

(

A

).

Proof:

(a) Let

A

∈

IS

(

X

),

(

X

,

τ

)

be an

ITS

.

Since

I

(τ)

bi

(

A

)

∈

I

(τ)

bO

(

X

),

we have,

)))

(

)))

(

)

(

( ) ( ) ( ) ( ) ( ) ( ) ) (

A

bi

I

c

I

i

I

A

bi

I

i

I

c

I

A

bi

I

τ

⊆

τ τ τ

∪

τ τ τ

(

))

(

))

) ( ) ( ) ( ) (

A

c

I

i

I

A

i

I

c

I

τ τ

∪

τ τ

⊆

))]

(

))

(

[

)

(

( ) ( ) ( ) ( ) ) (

A

c

I

i

I

A

i

I

c

I

A

bi

I

A

∩

τ

⊆

∩

τ τ

∪

τ τ

=

[

(

))]

[

(

))]

) ( ) ( ) ( ) (

A

c

I

i

I

A

i

I

c

I

A

∩

τ τ

∪

∩

τ τ

=

(

)

(

)

(

(2.1))

) ( ) (

n

Propositio

by

A

Pi

I

A

Si

I

τ

∪

τ

).

(

)

(

)

(

( ) ( ) ) (

A

Pi

I

A

Si

I

A

bi

I

τ

⊆

τ

∪

τ

But

I

(τ)

Si

(

A

)

⊆

I

(τ)

bi

(

A

)

and

I

(τ)

Pi

(

A

)

⊆

I

(τ)

bi

(

A

),

hence

I

(τ)

Si

(

A

)

∪

I

(τ)

Pi

(

A

)

⊆

I

(τ)

bi

(

A

).

Therefore

I

(τ)

bi

(

A

)

=

I

(τ)

Si

(

A

)

∪

I

(τ)

Pi

(

A

).

(b) Since

I

(τ)

bc

(

A

)

∈

I

(τ)

bC

(

X

),

we have,

)

(

)))

(

)))

(

( ) ( ) ( ) ( ) ( ) ( ) ) (

A

bc

I

A

bc

I

c

I

i

I

A

bc

I

i

I

c

I

τ τ τ

∩

τ τ τ

⊆

τ

(

))

(

))

(

)))

(

)))

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

A

bc

I

c

I

i

I

A

bc

I

i

I

c

I

A

c

I

i

I

A

i

I

c

I

τ τ

∩

τ τ

⊆

τ τ τ

∩

τ τ τ

⊆

I

(τ)

bc

(

A

)

(

))]

(

))

(

[

I

( )

c

I

( )

i

A

I

( )

i

I

( )

c

A

I

( )

bc

A

∪

τ τ

∩

τ τ

⊆

∪

τ

∩

[

A

∪

I

(τ)

i

(

I

(τ)

c

(

A

))]

⊆

I

(τ)

bc

(

A

)

(2.1))

(

).

(

)

(

)

(

( ) ( ) ) (

n

Propositio

by

A

bc

I

A

Pc

I

A

Sc

I

τ

∩

τ

⊆

τ

But

I

(τ)

bc

(

A

)

⊆

I

(τ)

Sc

(

A

)

and

I

(τ)

bc

(

A

)

⊆

I

(τ)

Pc

(

A

),

hence

I

(τ)

bc

(

A

)

⊆

I

(τ)

Sc

(

A

)

∩

I

(τ)

Pc

(

A

).

(7)

REFERENCES

1. Andrijevic D., On

b

-open sets, Mat. Vesnik, 48 (1996), 59-64.

2. Atanassov K., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1)(1986), 84-96.

3. Coker D., A note on intuitionistic sets and intuitionistic points, Turkish Journal of Mathematics, 20 (3) (1996), 343-351.

4. Coker D., An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88 (1997), 81-89. 5. Coker D., An introduction to intuitionistic topological spaces, BUSEFAL, 81 (2000), 51-56.

6. Girija S. and Ilango G, Some more results on intuitionistic semi open sets, Int. Journal of Engineering Research and Applications, 4(11) (2014), 70-74.

7. Palaniswamy B. and Varadharajan K.., On

I

(τ)

α

-open set and

I

(τ)

β

-open set in intuitionistic topological spaces, Maejo International Journal of Science and Technology, 10 (2) (2016), 187-196.

8. Zadeh L. A., Fuzzy sets, Information and Control, 8 (1965), 338-353.