Download 9. A VIEW ON INTUITIONISTIC…

Int Jr. of Mathematical Sciences & Applications
Vol. 4, No.1, January-June 2014
Copyright Mind Reader Publications
ISSN No: 2230-9888
www.journalshub.com
A VIEW ON INTUITIONISTIC
INTUITIONISTIC
*
*
OPEN SETS AND
CONTINUOUS FUNCTIONS
G. Vasuki
[email protected]
&
B. Amudhambigai
[email protected]
Department of Mathematics
Sri Sarada College for Women,
Salem-636 016, Tamil Nadu, India.
Abstract
In this paper the concepts of intuitionistic
*
open set, intuitionistic
open set, intuitionistic β *-open set, intuitionistic
*
*
continuous function, intuitionistic
introduced. Some interesting properties and characterization concerning intuitionistic
*
*
closed set, intuitionistic semi
*
*
compactness are
open sets intuitionistic
*
continuity, intuitionistic
T½ space and intuitionistic compactness are studied
Key words
Intuitionistic
continuous, Intuitionistic
*
-open set, intuitionistic semi
*
T½ space, Intuitionistic
-open, Intuitionistic β *-open, Intuitionistic
*
*
-
*
-compactness.
2010 AMS Subject Classification Primary: 54A05, 54A10, 54A20.
Introduction
Atanassov [2] introduced the concept of “Intuitionistic fuzzy sets” as a generalization of fuzzy sets, it
becomes a popular topic of investigation in the fuzzy set community. Later Coker [3] introduced the concept of
“intuitionistic sets” in 1996. This is a discrete form of intuitionistic fuzzy set where all the sets are entirely the
crisp sets. In 2000, he also introduced the concept of intuitionistic topological spaces with intuitionistic sets, and
investigated basic properties of continuous functions and compactness. He and his colleague also examined
separation axioms in intuitionistic topological spaces. The concept of
Erdal EKICI. In this paper, we obtain some characterizations of
*
open set was discussed by Rajesh and
*
open set,
*
continuous functions,
compactness in intuitionistic topological spaces. And examples are provided wherever necessary.
1 73
2. Preliminaries
Definition 2.1 [3] Let X be a nonempty set. An intuitionistic set (IS for short) A is an object having the form
1
2
1
2
1
2
A = 〈 X, A , A 〉 where A and A are subsets of X satisfying A I A = φ. The set A1 is called the set of
2
members of A, while A is called the set of non members of A. Every crisp set A on a non empty set X is
c
obviously an IS having the form 〈 X, A, A 〉.
1
2
1 2
Definition 2.2 [3] Let X be a non empty set. Let A = 〈 X, A , A 〉 and B = 〈 X, B ,B 〉 be an IS’s on X. And let
1
2
{ Ai= 〈 X, Ai , Ai 〉 : i ∈ J } be an arbitrary family of IS’s in X. Then
1
1
2
(i) A ⊆ B iff A ⊆ B and B ⊆ A
(iv) I Ai = 〈 X, I
, U
2 c
〉
2
2
(ii) A = B iff A ⊆ B and B ⊆ A
(iii)
(v) U Ai = 〈 X, U
(vi)
, I
〉 (viii)φ∼ = 〈 X, φ, X 〉 and X∼ = 〈 X, X, φ 〉
A = 〈 X, A2, A1 〉
1
c
A = 〈 X, I A , (A1) 〉
(vii) ◊A = 〈 X, ( A ) , A 〉 1
2
Definition 2.3 [3] Let f be a function from a set X to a set Y. Let A = 〈 X, A , A 〉 be an IS in X and B = 〈 Y,
1
2
-1
-1
-1
1
-1
2
B , B 〉 an IS in Y. Then the pre image f (B) is an IS in X defined by f ( B ) = 〈 X, f ( B ), f ( B ) 〉 and
1
2
2
2 c
c
the image f ( A ) is an IS in Y defined by f ( A) = 〈 Y, f ( A ), f−( A ) 〉 where f−(A ) =〈 f ( ( A ) ) 〉 .
Definition 2.4 [4] An intuitionistic topology ( IT for short ) on a non empty set X is a family τ of IS’s in X
satisfying the following axioms.
(i) φ∼, X∼ ∈ T
(ii) G1 I G2 ∈ T for any G1, G2 ∈ T.
(iii) U Gi ∈ T for any arbitrary family { Gi / i∈J } ⊆ T where ( X, T ) is called an intuitionistic topological
space ( ITS for short ) and any intuitionistic set in T is called an intuitionistic open set ( IOS for short ) in X. The
complement A of an IOS A is called an intuitionistic closed set ( ICS for short ) in X.
1
2
Definition 2.5 [4] Let ( X, T ) be an ITS and A = 〈 X, A , A 〉 be an IS in X. Then the interior and closure of A
are defined by
cl( A ) = I { K / K is an ICS in X and A ⊆ K }
int( A ) = U { G/G is an IOS in X and G ⊆ A }
Theorem 2.1 [4] For any IS A in 〈 X,T 〉 the following properties hold:
cl (
A ) = int( A )
int ( A ) = cl( A )
Definition 2.6 [3] A family
is a cover of a set B iff B is a subset of the union
U { A : A ∈ }; that is, iff each member of B belongs to some member of . The family is an open cover of
B iff each member of is an open set.
Definition 2.7 [3] Let ( X, T ) and ( Y, T′ ) be any two ITS’s and let f : X → Y be a function. Then f is said to
be intuitionistic continuous iff the pre image of each IS in T′ is an IS in T.
Definition 2.8 [2] Let ( X, T ) and ( Y, T′ ) be any two ITS’s and let f : X → Y be a function. Then f is said to
be intuitionistic open iff the image of IS in T is IS in T′.
Definition 2.9 [4] A subset A of a space ( X, T ) is called
(i)
∧
∧
g -closed if cl( A ) ⊆ U whenever A ⊆U and U is semi-open in ( X, T ). The complement of g - closed
∧
set is said to be g -open.
∧
(ii) *g-closed set if cl ( A ) ⊆ U whenever A ⊆ U and U is g -open in ( X, T ). The complement of *gclosed set is said to be *g-open.
(iii) #g semi-closed (briefly #gs-closed) set if scl ( A ) ⊆ U whenever A ⊆ U and U is *g-open in ( X, T ).
The complement of #gs-closed set is said to be #gs-open.
(iv) -closed set if cl ( A ) ⊆ U whenever A ⊆ U and U is an #gs-open in ( X, T ). The complement of closed set is said to be -open.
Definition 2.10 [1] A smooth fuzzy topological space ( X, T ) is called
(i) fuzzy s-closed if each fuzzy regular closed cover of ( X, T ) has a finite
sub-cover.
(ii) fuzzy s-Lindelof if each fuzzy regular closed cover of ( X, T ) has a countable sub-cover.
(iii) fuzzy countable s-closed if each countable fuzzy regular closed cover of
( X, T ) has a finite sub-cover.
2 74
3. INTUITIONISTIC *-OPEN SETS
1
2
1
2
Definition 3.1.1 Let ( X, T ) be an intuitionistic topological space and A = 〈 X, A , A 〉, B = 〈 Y, B , B 〉 be an
ISs in X. Then A is said to be
∧
∧
(i) I g -closed if cl( A ) ⊆ B whenever A ⊆ B and B is an intuitionistic semi open. The complement of I g ∧
closed set is said to be I g -open.
∧
(ii) I*g-closed if cl ( A ) ⊆ B whenever A ⊆ B and B is an I g -open. The complement of I*g- closed set is
said to be I*g-open.
(iii) I#g semi-closed if scl ( A ) ⊆ B whenever A ⊆ B and B is an I*g-open. The complement of I#g semiclosed set is said to be I#g semi-open.
(iv) I -closed if cl ( A ) ⊆ B whenever A ⊆ B and B is an I #gs-open. The complement of I -closed set is
said to be I -open.
1
2
Definition 3.1.2 Let ( X, T ) be an ITS and A = 〈 X, A , A 〉 be an intuitionistic set. Then A is said to be an
intuitionistic semi open ( resp. intuitionistic semi closed ) set ( ISOS for short ) ( resp. ISCS ) if A ⊆cl( int( A ) )
( resp. A ⊇ int( cl ( A ) ).
1
2
Definition 3.1.3 Let (X, T) be an ITS and A = 〈 X, A , A 〉 be an intuitionistic set. Then A is said to be an
intuitionistic β open ( resp. intuitionistic β closed ) set ( IβOS for short ) ( resp. IβCS ) if A ⊆ cl ( int ( cl( A ) )
( resp. A ⊇ int ( cl int ( A ) ) ).
1
2
1
2
Definition 3.1.4 Let ( X, T ) be an intuitionistic topological space and A = 〈 X, A , A 〉, B = 〈 Y, B , B 〉 be an
ISs in X. Then A is said to be
∧
(i) Isemi g -closed if cl ( int ( A ) ) ⊆ B whenever A ⊆B and B is an intuitionistic semi open. The
∧
∧
complement of intuitionistic semi g - closed set is said to be Isemi g -open.
∧
(ii) Isemi*g-closed if cl ( int ( A ) ) ⊆ B whenever A ⊆ B and B is an Isemi g -open. The complement of
Isemi* g-closed set is said to be Isemi* g-open
(iii) Isemi# g-closed if scl ( sint ( A ) ) ⊆ B whenever A ⊆ B and B is an Isemi* g-open. The complement of
Isemi# g-closed set is said to be Isemi# g-open.
(iv) Isemi -closed if cl ( int ( A ) ) ⊆ B whenever A ⊆ B and B is an Isemi #g-open. The complement of
Isemi -closed set is said to be Isemi -open set.
1
2
Definition 3.1.5 Let (X, T) be an intuitionistic topological space and A=〈 X, A ,A 〉, B = 〈 Y, B1, B2 〉 be an ISs
in X. Then A is said to be
∧
(i) Iβ g -closed if cl( int ( cl ( A ) ) ) ⊆ B whenever A ⊆B and B is intuitionistic semi open set. The
∧
complement of intuitionistic β g - closed set is said to be
∧
Iβ g -open.
∧
(ii) Iβ*g-closed if cl ( int ( cl ( A ) ) ) ⊆ B whenever A ⊆ B and B is an Iβ g -open. The complement of
Iβ*g-closed set is said to be Iβ*g-open.
(iii) Iβ#g-closed if scl ( sint ( scl( A ) ) ) ⊆ B whenever A ⊆ B and B is an Iβ*g-open. The complement of
Iβ#g-closed set is said to be Iβ#g-open.
(iv) Iβ -closed if cl ( int ( cl ( A ) ) ) ⊆ B whenever A ⊆ B and B is an Iβ#g-open. The complement of
Iβ -closed set is said to be Iβ -open set.
Proposition 3.1.1 Every I -closed set is an Isemi -closed set.
Proof : Proof follows from the definition.
Remark 3.1.1 The reverse inclusion of proposition 3.1.1 does not hold as shown in the Example 3.1.1
Example 3.1.1 Let X = { a, b, c } and consider the intuitionistic topology T = { φ∼, X∼, A, B } where
A = 〈 X, { a }, { b, c } 〉, B = 〈 X, {a, b} , { c } 〉. Now consider any intuitionistic set D = 〈 { X, { b }, { c }.
Clearly D is an Isemi -closed set but not I -closed set.
Proposition 3.1.2 Every I -closed set is an Iβ -closed set.
Proof : Follows directly from the fact that definition.
Remark 3.1.2 The reverse inclusion of proposition 3.1.2 does not hold as shown in the Example 3.1.2.
3 75
Example 3.1.2 Let X = { a, b, c , d } and consider the intuitionistic topology T = { φ∼, X∼, A, B } where
A = 〈 X, { a, b, c }, { d } 〉, B = 〈 X, { b } , { a, c, d} 〉. Now consider any intuitionistic set D = 〈 { X, { c },
{ a, b, d }. Clearly D is an Iβ -closed set but not I -closed set.
Proposition 3.1.3 Every Iβ -closed set is an Isemi -closed set.
Proof : Follows from the definition.
Remark 3.1.3 The reverse inclusion of proposition 3.1.3. does not hold as shown in the Example 3.1.3.
Example 3.1.3 In example 3.1.1 the intuitionistic set D is Isemi -closed but not Iβ -closed set.
Remark 3.1.4 The complement of the propositions 3.1.1, 3.1.2 and 3.1.3 can be given as below:
(i) Every I -open set is an Isemi -open
(ii) Every I -open set is an Iβ -open
(iii) Every Iβ -open set is an Isemi -open
Remarks 3.1.5 Clearly, the above discussions give the following implications.
1
2
Definition 3.1.6 Let (X,T) be an intuitionistic topological space and let A=〈X,A ,A 〉,
1
2
B = 〈 Y, B1, B2 〉, C = 〈 X, C , C 〉 be an ISs in X. Then A is said to be an I *-open set ( briefly I *OS ) if
A = B I C where B is I -open and C is an intuitionistic open set.
1
2
Definition 3.1.7 Let (X,T) be an intuitionistic topological space and let A=〈 X,A ,A 〉,
1
2
B = 〈 Y, B1, B2 〉, C = 〈 X, C , C 〉 be an ISs in X. Then A is said to be an Iβ *-open set ( briefly Iβ *OS ) if
A = B I C where B is Iβ -open and C is an intuitionistic open set.
1
2
Definition 3.1.8 Let (X,T) be an intuitionistic topological space and let A=〈X,A , A 〉,
1
2
B = 〈 Y, B1, B2 〉, C = 〈 X, C , C 〉 be an ISs in X. Then A is said to be an Isemi *-open set ( briefly IS *OS ) if
A = B I C where B is IS -open and C is an intuitionistic open set.
Proposition 3.1.4 Every I *OS is IS *OS.
Proof : The proof holds directly from the fact that every IOS is ISOS.
Remark 3.1.5 The reverse inclusion of proposition 3.1.4 does not hold as shown in the Example 3.1.4.
Example 3.1.4 In example 3.1.1 the intuitionistic set D = X∼ I D is an IS *OS but not I *OS .
Proposition 3.1.5 Every I *OS is Iβ *OS.
Proof : The proof holds directly from the fact that every IOS is IβOS.
Remark 3.1.6 The reverse inclusion of proposition 3.1.5 does not hold as shown in the Example 3.1.5.
Example 3.1.5 In example 3.1.2 the intuitionistic set D = X∼ I D is Iβ *OS but not I *OS .
Proposition 3.1.6 Every Iβ *OS is IS *OS.
Proof : The proof holds directly from the fact that every IβOS is ISOS.
Remark 3.1.7 The reverse inclusion of proposition 3.1.6 does not hold as shown in the Example 3.1.6.
Example 3.1.6 In example 3.1.1 the intuitionistic set D = X I D is IS *OS but not Iβ *OS .
Remarks 3.1.8 Clearly, the above discussions give the following implications.
4. I *-CONTINUOUS FUNCTIONS
Definition 4.1.1 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces and let f : (X, T) → (Y, S)
-1
be a function. Then f is said to be I *-continuous function ( briefly I *CF) if f ( A ) is I *OS in ( X, T ) for
every IOS A in ( Y, S ).
4 76
Definition 4.1.2 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces and let f : (X, T) → (Y, S)
-1
be a function. Then f is said to be Isemi *-continuous function ( briefly IS *CF) if f ( A ) is IS *OS in ( X, T )
for every IOS A in ( Y, S ).
Definition 4.1.3 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces and let f : (X, T) → (Y, S)
-1
be a function. Then f is said to be Iβ *-continuous function ( briefly Iβ *CF) if f ( A ) is Iβ *OS in ( X, T ) for
every IOS A in ( Y, S ).
Definition 4.1.4 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces.
-1
A function f : ( X, T ) → ( Y, S ) is said to be an I *-irresolute function if f ( A ) is an I *OS in ( X, T ) for
*
every I OS A in ( Y, S ).
Definition 4.1.5 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces. A function f : ( X, T ) →
-1
(Y, S) is said to be an Isemi *irresolute function if f (A) is an IS *OS in (X, T) for every IS *OS A in (Y, S).
Definition 4.1.6 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces. A function f : ( X, T ) →
-1
( Y, S ) is said to be an Iβ * irresolute function if f (A) is an Iβ *OS in (X, T) for every Iβ *OS A in (Y, S).
Definition 4.1.7 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces. A function f : ( X, T ) →
-1
( Y, S ) is said to be an intuitionistic semi irresolute function if f (A) is an ISOS in (X, T) for every ISOS A in
(Y, S).
Definition 4.1.8 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces. A function f : ( X, T ) →
-1
(Y, S) is said to be an intuitionistic β-irresolute function if f (A) is an IβOS in (X, T) for every IβOS A
in (Y, S).
1
2
Definition 4.1.9 Let ( X, T ) be an ITS and A = 〈 X, A , A 〉 be an IS in 〈 X, T 〉. Then the intuitionistic *
*
interior and intuitionistic closure of A are defined to be
I *int ( A )
=
U { G : G is an I *OS in X and G ⊆ A }
I *cl ( A )
=
I { K : K is an I *CS in X and A ⊆ K }
Proposition 4.1.1 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces. Let f : (X, T) → (Y, S)
be a function. Then the following statements are equivalent:
(a) f is an I *-irresolute functions.
(b) f ( I *cl ( A ) ) ⊆ I *cl ( f( A ) ) for every A in ( X, T )
-1
-1
(c) I *clf ( B ) ⊆ f ( I *cl( B ) ) for every B in ( Y, S )
Proposition 4.1.2 Every I *continuous function is Isemi * continuous.
Remark 4.1.1 The converse of proposition 4.1.2. need not be true as shown in the example 4.1.1.
Example 4.1.1 Let X = { a, b, c } and Y = { b, c }. Consider an intuitionistic topologies T and S of X and Y
respectively as follows
T = { φ∼, X∼, A, B } and S = { φ∼, Y∼, D } where A = 〈 X, { a }, { b, c } 〉, B = 〈 X, { b }, { c } 〉 and
D = 〈 Y, { c }, { b } 〉. Let f : ( X, T ) → ( Y, S ) be an identity function. Then the inverse image of every ICS in
-1
( Y, S ) is IS *CS in (X, T ). But f ( D ) = D = D I X∼ is not I *CS. Therefore, every IS *CF need not be
I *CF.
Proposition 4.1.3 Every I *CF is Iβ *CF.
Proof : The proof is simple.
Remark 4.1.2 The reverse inclusion of proposition 4.1.3. need not be true as shown in the example 4.1.2.
Example 4.1.2 Let X = Y = { a, b, c, d }. Consider an intuitionistic topologies T and S respectively as follows
T = { φ∼, X∼, A, B } and S = { φ∼, Y∼, D } where A = 〈 X, { a }, { b, c } 〉, B = 〈 X, { a, b }, { c } 〉 and
D = 〈 Y, { a, b, d } , { c } 〉. Let f : ( X, T )→( Y, S ) be an identity function. Then the inverse image of every
-1
ICS in (Y, S) is Iβ *CS in (X, T). But f ( D ) = D = D I X∼ is Iβ *CS but not I *CS. Therefore, every Iβ *CF
need not be I *CF.
Proposition 4.1.4 Every Iβ *CF is IS *CF
Proof : The proof is simple.
Remark 4.1.3 The reverse inclusion of proposition 4.1.4. need not be true as shown in the example 4.1.3.
-1
Example 4.1.3 In example 4.1.2, f ( D ) = D = D I X∼ is IS *CS but not Iβ *CS. Therefore, every IS *CF
need not be Iβ *CF.
Remark 4.1.4 Clearly, the above discussions give the following implications.
5 77
Proposition 4.1.5 Let ( X, T ) and ( Y, S ) and ( Z, R ) be any three ITS’s. Let f : ( X, T ) → ( Y, S ) be an
I *-irresolute function and g : ( Y, S ) → ( Z, R ) be an I *CF then gof is I *CF.
Proposition 4.1.6 Let ( X, T ) and ( Y, S ) and ( Z, R ) be any three ITS’s. Let f : ( X, T ) → ( Y, S ) be an Isemi
*
-irresolute function and g : ( Y, S ) → ( Z, R ) be an IS *CF then gof is Isemi *continuous function.
Proof : The proof is similar to the proof of proposition 4.1.5.
Proposition 4.1.7 Let ( X, T ) and ( Y, S ) and ( Z, R ) be any three ITS’s. Let f : ( X, T ) → ( Y, S ) be an
Iβ *-irresolute function and g : ( Y, S ) → ( Z, R ) be an Iβ *CF then gof is Iβ * continuous.
Proof : The proof is similar to the proof of proposition 4.1.5.
Definition 4.1.10 An ITS ( X, T ) is called an intuitionistic *T½ space (briefly I *T½S ) if every I * OS in
( X, T ) is IOS ( resp. I * CS in ( X, T ) is ICS ).
Definition 4.1.11 An ITS ( X, T ) is called an intuitionistic *Tβ space (briefly I * TβS ) if every I * OS in
( X, T ) is IβOS.
Definition 4.1.12 An ITS ( X, T ) is called an intuitionistic- *Ts space (briefly I * TsS ) if every I *OS in
( X, T ) is ISOS.
Proposition 4.1.8 Let ( X, T ) and ( Y, S ) and ( Z, R ) be any three intuitionistic topological spaces. Let
f : ( X, T ) → ( Y, S ) be an intuitionistic semi irresolute function and g : ( Y, S ) → ( Z, R ) be an I *continuous
function. If ( Y, S ) is an I *TsS then gof is an intuitionistic semi continuous.
Proposition 4.1.9 Let ( X, T ) and ( Y, S ) and ( Z, R ) be any three intuitionistic topological spaces. Let
f : ( X, T ) → ( Y, S ) be an intuitionistic β irresolute function and g : ( Y, S ) → ( Z, R ) be an I *continuous
function. If ( Y, S ) is an I *-TβS then gof is an intuitionistic β continuous.
Proof : The proof is similar to the proof of proposition 4.1.8.
Proposition 4.1.10 Let ( X, T ) and ( Y, S ) and ( Z, R ) be any three intuitionistic topological spaces. Let
f : ( X, T ) → ( Y, S ) be an I *continuous function and
g : ( Y, S ) → ( Z, R ) be an I *irresolute function. If ( Y, S ) is an I *T½ S then gof is an I * irresolute function.
5. I *-COMPACT SPACES
Definition 5.1.1 A class { Ai, i∈ J } of an intuitionistic open sets in ( X, T ) is said to be intuitionistic open
cover ( briefly, IOC ) of ( X, T ) if i J Ai X .
Definition 5.1.2 An ITS ( X, T ) is said to be an intuitionistic compact space ( briefly IComS ) if every IOC has
a finite subcover.
Definition 5.1.3 A class { Ai, i∈ J } of an I *OS’s in ( X, T ) is said to be an intuitionistic * open cover
( briefly, I *OC ) of ( X, T ) if i J Ai X .
Definition 5.1.4 An ITS ( X, T ) is said to be an intuitionistic *compact space ( briefly, I *ComS ) if every
I *OC has a finite subcover.
Proposition 5.1.1 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces. Let f : ( X, T ) → ( Y, S
) be an I *continuous and bijective function. If ( X, T ) is an IComS and I *T1/2S then ( Y, S ) also an IComS.
Proof : Let { Ai, , i∈ J } be a class of IOS’s in Y such that
-1
i J
Ai Y . Then by hypothesis, f ( Ai ) are I *OS
-1
in ( X, T ). Since ( X, T ) is an I *T1/2 space, f ( Ai ) are IOSs in ( X, T ). Since ( X, T ) is an IComS, there
exists a finite subset J0 of J such that
−
1
U f (A i )
⎛
⎞
−1
f (A i ) ⎟⎟ = U A i . Hence
⎜ iU
⎝ ∈J0
⎠ i∈J0
X . Then Y = f(X ) = f ⎜
i∈J 0
( Y, S ) is an IComS.
Proposition 5.1.2 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces. Let f : ( X, T ) → (Y, S)
be an I *irresolute and bijective function. If ( X, T ) is an I *ComS then ( Y, S ) is also and I *ComS.
6 78
Proof : Let { Ai, , i∈ J } be a class of I *OS’s in Y such that
-1
i J
Ai Y . Then by hypothesis, f ( Ai ) is an
I *OS in X. Since ( X, T ) is an I *ComS, there exists a finite subset J0 of J such that
−
1
U f (A i )
X . Then
i∈J 0
⎛
⎞
−1
*
⎟
f
(A
)
U
i ⎟ = U A i . Hence ( Y, S ) is an I ComS.
⎜ i∈J
⎝ 0
⎠ i∈J0
Y =f(X )= f ⎜
Definition 5.1.5 Let ( X, T ) be an ITS. An intuitionistic set A of X is called an intuitionistic regular open set
( briefly, IROS ) if A = int ( cl( A ) ). The complement of IROS is called an intuitionistic regular closed set
( briefly IRCS ).
Definition 5.1.6 Let ( X, T ) be an ITS. An intuitionistic *closed cover ( briefly, I *CC ) of ( X, T ) is the
*
collection { Ai, , i∈ J } where each Ai is an intuitionistic
closed such that
−
1
U f (A i )
X .
i∈J
Definition 5.1.7 An ITS ( X, T ) is called an
(i) intuitionistic S-Closed (briefly, ISC) if each IRCC of (X, T) has a finite subcover.
(ii) intuitionistic S-Lindelof if each IRCC of ( X, T ) has a countable subcover.
(iii) intuitionistic countable S-closed if each countable IRCC of ( X, T ) has a finite subcover.
Definition 5.1.8 An ITS ( X, T ) is called
(i) I *S-closed if each I *CC of ( X, T ) has a finite subcover.
(ii) I *S-Lindelof if each I *CC of ( X, T ) has a countable sub cover.
(iii) I *countable S-closed if each countable I *CC of ( X, T ) has a finite subcover.
Definition 5.1.9 An ITS ( X, T ) is called an
(i) intuitionistic strongly S-closed if for each collection {Ai, i∈ J } where each Ai is an ICS with
U Ai
i∈J
X , there exists a finite subset Jo of
U Ai
X .
i∈J 0
(ii) intutitionistic strongly S-Lindelof if for each collection {Ai, i∈ J } where each Ai is an ICS with
U A X , there exists a countable subset J0 of J such that U A i X .
i∈J
i
i∈J 0
(iii) intuitionistic strongly S-closed if for each cocutable collection {Ai, i∈ J } where Ai is an ICS with
U A X , there exists a finite subset J0 of J such that U A i X .
i∈J
i
i∈J 0
Proposition 5.1.3 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces. Let f : ( X, T ) → ( Y, S )
be an I *-continuous and bijective. If ( X, T ) is an I *S-closed ( resp. I *S-Lindelof, and I * countable Sclosed ) space then ( Y, S ) is an intuitionistic strongly S-closed ( resp. intuitionistic strongly S-Lindelof ,
intuitionistic countable strongly S-closed ) space.
Definition 5.1.10 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces. Any function f : ( X, T )
→( Y, S ) is called an I *closed function (briefly I *CF ) if for each ICS A in ( X, T ), f( A ) is I *CS in (Y, S).
Proposition 5.1.4 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces. Let f : ( X, T ) → ( Y, S )
be an I *C and bijective function. If ( Y, S ) is an I *S-closed (resp. I *S-Lindelof, and I * countable S-closed)
then ( X, T ) is an intuitionistic strongly S-closed ( resp. intuitionistic strongly S-Lindelof and intuitionistic
countable strongly S-closed )
Proposition 5.1.5 Let ( X, T ) and ( Y, S ) be any two intuitionistic opological spaces. Let f : ( X, T ) → ( Y, S )
be an I *-continuous and bijective. If ( X, T ) is an I *T½ space and an intuitionistic strongly S-closed ( resp. an
intuitionistic strongly S-Lindelof an intuitionistic countable strongly S-closed ) then ( Y, S ) is also an
intuitionistic strongly S-closed ( resp. intuitionistic strongly S-Lindelof and intuitionistic countable strongly
S-closed ) space.
Proposition 5.1.6 Let ( X, T ) and ( Y, S ) be any two intuitionistic topological spaces. Let f:( X, T ) → ( Y, S )
be an I *CF and bijective. If ( Y, S ) is an I *T½ space and an intuitionistic strongly S-closed ( resp.
intuitionistic strongly S-Lindelof and intuitionistic countable strongly S-closed ) then ( X, T ) is also an
intuitionistic strongly S-closed (intuitionistic strongly S-Lindelof and intuitionistic countable strongly S-closed)
space.
7 79
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
% -locally closed sets and fuzzy Gδ- g% -locally continuous functions,
B. Amudhambigai, r-fuzzy Gδ- g
International Journal of Mathematical Sciences and Applications, Vol. I(3) (2011) 1339-1348.
K. Atanassov, “Intuitionistic fuzzy sets”, VII ITKR session, Sofia (September, 1983) (in Bulgarian).
Dogan Coker, A note on intuitionistic sets and intuitionistic points, Turkish J. Math. 20(1996), 343 351.
Dogan Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems
88(1997), 81 - 89.
S. Jafari, T,Noiri, N.Rajesh and M.L.Thivagar, Another generalization of closed sets, Kochi Journal of
Mathematics, Vol.III(2008).
L.A. Zadeh, Fuzzy sets, Information and Control, 8(1965), 338-353.
ACKNOWLEDGEMENT
The authors would like to thank the UGC for the publication of his paper and the refree for their
valuable suggestions towards the improvement of this paper.
8 80