Ijset 2014 301 by Innovative Research Publications

International Journal of Scientific Engineering and Technology Volume No.3 Issue No.3, pp : 211 – 215

(ISSN : 2277-1581) 1 March 2014

Some Features of α-T2 Space in Intuitionistic Fuzzy Topology Rafiqul Islam Department of Mathematics, Pabna University of Science and Technology, Pabna-6600, Bangladesh Email: rafiqul.pust.12@gmail.com Abstract : The basic concepts of the theory of intuitionistic fuzzy topological spaces have been defined by D. Coker and co-workers. In this paper, we define new notions of Hausdorffness in the intuitionistic fuzzy sense, and obtain some new properties “good extension property” is one of them, in particular on convergence. MSC (2000): 54A40, 03E72. Key words: Fuzzy set, Intuitionistic fuzzy topology, Hausdorffness, IFS. Introduction The introduction of “intuitionistic fuzzy sets” is due to K.T Atanassov [1], and this theory has been developed by many authors [2-4]. In particular D. Coker has defined the intuitionistic fuzzy topological spaces, and several authors have studied this category [5-14]. Nevertheless, separation in intuitionistic fuzzy topological spaces is not studied. Only there exists a definition due to D. Coker. Definition: Let X be a non-empty set and I=[0,1]. A fuzzy set in X is a function u : X  I which assign to each element

x  X , a degree of membership , u( x)  I .

the collection of all mappings from X into I , i. e. the class of all fuzzy sets in X. A fuzzy topology on X is defined as a family t of members of IX, satisfying the following conditions: each i   ,then  i u i  t (iii) if u1 , u 2

for

 t then

u1  u 2  t .Then the pair ( X , t ) is called a fuzzy topological space (FTS) and the members of t are called t-open (or simply open) fuzzy sets. A fuzzy set v is called a t-closed (or simply closed) fuzzy set if 1  v  t . Example: Let X  {a, b, c, d} , t

 {0,1, u, v} ,where

1  {( a,1), (b,1), (c,1), (d ,1)} 0  {( a,0), (b,0), (c,0), (d ,0)} IJSET@2014

A is an object having the form A  { x,  A ( x), A ( x) : x  X } where the functions An intuitionistic fuzzy set (IFS)

 A : X  I and  A : X  I denote the degree of membership ( namely  A (x) ) and the degree of nonmembership (namely  A (x) ) of each element x  X to the set A , respectively and 0   A ( x)   A ( x)  1 for each x X . Definition: Let X be a nonempty set and  be a family of intuitionistic fuzzy sets in X. Then  is called an intuitionistic fuzzy

topology

on X

satisfy

the

following

conditions: (i )0,1  

(ii)G1  G2  for any G1 ,G2   , ( X , ) is called an intuitionistic fuzzy topological space (IFTS) and any IFS in  is known as an intuitionistic fuzzy open set (IFOS) in X .

Definition: Let I  [0,1] . X be a non-empty set and IX be

ui  t

Definition: (Atanassov [4]). Let X be a nonempty fixed set.

this case the pair

{(a,0.2),(b,0.4),(c,0.5)} is a fuzzy set in X.

(ii) if

fuzzy topological space.

(iii)  Gi  for any arbitrary family {Gi : i  J }   .In

 {a, b, c} and I  [0,1].If u(a)  0.2, u(b)  0.4, u(c)  0.5 then

Example:Let X

(i)1,0  t

u  {(a,0.2), (b,0.5), (c,0.7), (d ,0.9)} v  {(a,0.3), (b,0.5), (c,0.8), (d ,0.95)} Then ( X , t ) is a

( X , ) is called Hausdorff iff

Definition: An IFTS

x1 , x2  X and x1  x 2 exist G1

imply

that

there

 x, G1 ,  G1 , G2  x, G2 ,  G2  with

 G ( x1 )  1,  G ( x1 )  0  G ( x 2 )  1,  G ( x 2 )  0 and 1

G1  G2  0 . Definition: An IFTS

( X , ) is called (a) T2 (i) if for all

x1 , x2  X , x1  x 2 imply that there exist open sets

G1  x, G1 ,  G1 , G2  x, G2 ,  G2  that  G1 ( x1 )  1,  G1 ( x1 )  0 and

such

 G ( x 2 )  1,  G ( x 2 )  0 2

G1  G2  0 . Page 211

International Journal of Scientific Engineering and Technology Volume No.3 Issue No.3, pp : 211 – 215

  T2 (ii)

(b)

if for all x1 , x 2

there

 X with x1  x 2 imply that

exist

open

G1  x, G1 ,  G1 , G2  x, G2 ,  G2 

such

(ISSN : 2277-1581) 1 March 2014

by {u, v}  {cons tan ts} ,where

sets

u ( x1 )  1, u ( x2 )  0, v( x1 )  0, v( x2 )  1 .Again let  be

that

the indiscreat topology on X.Then for every

  I1 , the IFTS

 G ( x1 )  1,  G ( x1 )  0  G ( x 2 )  1,  G ( x 2 )  0 and

( X , ) is   T2 ( j ) . But the fuzzy topological space (X,T)

G1  G2   .

is not

(c)

  T2 (iii)

if for all

x1 , x2  X , x1  x 2

imply

there

such that exists G1  x, G1 ,  G1 , G2  x, G2 ,  G2 

 G ( x1 )   ,  G ( x 2 )   and G1  G2  0 . (d)

if for all

x1 , x2  X , x1  x 2

that

 be the intuitionistic fuzzy topology on X generated  G ( x)  1,  G ( x)  0 by {G1}  {cons tan ts} ,where

Theorem: If (X, T) be fuzzy topological space and

( X , ) be

  T2 ( j ) ,for

  I1 ,

J  i, ii, iii, iv.

  T2 (ii)

Theorem: Let

space.

some   I 1 , x1 , x 2

following implication:

that u ( x1 )  1  v( x2 ) and u v   .This implies that if

x1 , x2  X , x1  x 2 imply that there exist open sets

G1  x, G1 ,  G1 , G2  x, G2 ,  G2 

such

that

 G ( x1 )  1,  G ( x1 )  0  G ( x 2 )  1,  G ( x 2 )  0 and 1

Similarly, one can see that (X,T) is

 T2 (i ) .(X,T)

is 

 T2 (iii) .

(X,T) is

  T2 (i)  ( X , )

  T2 (iii) 

( X , )

IJSET@2014

X  {x1 , x2 } and (X,T) be the fuzzy topology X

Proof:

Let ( X , ) be T2 (i ) .We

  T2 (ii) . T2 (i) ,

prove

x1 , x2  X , x1  x 2 . Since ( X , ) is

Let

there

exist

open

G1  x, G1 ,  G1 , G2  x, G2 ,  G2 

 G ( x1 )  1,  G ( x1 )  0 1

and

that ( X , ) is

G1  G2  0 .

sets such

that

 G ( x 2 )  1,  G ( x 2 )  0 2

see

that

 G ( x1 )   ,  G ( x 2 )   , and G1  G2   for every 1

  I1 .Hence it is clear that ( X , )   T2 (iii) .

  T2 (ii)

and also

Further one can easily verify that

  T2 (iv)  ( X , ) is   T2 (iv) .

Example: Let

  T2 (iii)

G1  G2   .Hence we have ( X , ) is   T2 (ii) space. is 

  T2 (iv)

T2 (i)

 X with

, u, v  T such

for

( X , ) be an IFTS. Then we have the

x1  x 2 . Since (X,T) is fuzzy

x1  x 2 for all

  T2 ( j )

  T2 (ii)

Proof: First suppose that (X,T) is a fuzzy

  T2 (ii) space,for

( X , ) is   T2 ( j ) .

J  i, ii, iii, iv.

But the converse is not true.

with

the IFTS

But the fuzzy topological space (X,T) is not

corresponding intuitionistic fuzzy topological space(IFTS)

  T2 ( j ) ⇒ ( X , )

.Then for every

x1 , x2  X

Again

let

 G ( x1 )   ,  G ( x 2 )   and G1  G2   .

Let

for

J  i, ii, iii, iv. For this consider the following example.

by {u}  {cons tan ts} ,where u ( x)  1, u ( y)  0 .

G1  x, G1 ,  G1 , G2  x, G2 ,  G2  such

then (X,T) is

  T2 ( j )

not imply (X,T) is

generated

imply that there exist

  T2 ( j ) does

Example: Let X  {x, y} and T be the fuzzy topology on X

  T2 (iv)

for J  i, ii, iii, iv.

Remarks: Let (X,T) be the fuzzy topological space and ( X , ) be its corresponding IFTS. Then ( X , ) is

that

  T2 ( j )

generated

  T2 (ii)    T2 (iv)   T2 (iii)    T2 (iv) Page 212

International Journal of Scientific Engineering and Technology Volume No.3 Issue No.3, pp : 211 – 215

T2 (i)    T2 (iii)

X  {x1 , x2 } and G1 , G2  ( I  I ) X

Example(a): Let

 G ( x1 )  0.7,  G ( x1 )  0

where G1 ,G2 are defined by

 G ( x 2 )  0,  G ( x2 )  0.8 . Consider the intuitionistic fuzzy topology  on X generated by {G1 , G2 }  {cons tan ts} . For   0.4 , it is clear that and

( X , ) is   T2 (iii) but ( X , ) is neither   T2 (ii)

T2 (i) .

nor

X  {x1 , x2 } and G1 , G2  ( I  I ) X

G1 ,G2 are defined by

 G ( x1 )  1,  G ( x1 )  0 1

 G ( x 2 )  0,  G ( x 2 )  1 . Consider the intuitionistic  on fuzzy topology X generated by {G1 , G2 }  {cons tan ts} . For   0.5 , it is clear that and

( X , ) is   T2 (ii) but ( X , ) is neither   T2 (iii)

T2 (i) .

nor

X  {x1 , x2 } and G1 , G2  ( I  I ) X

G1 ,G2 are defined by  G ( x1 )  0.8,  G ( x1 )  0 1

 G ( x 2 )  0.5,  G ( x2 )  0.3 . Consider the intuitionistic fuzzy topology  on X generated by {G1 , G2 }  {cons tan ts} . For   0.4 , it is clear that and

( X , ) is   T2 (iv) but ( X , ) is neither   T2 (ii)

  T2 (iii) .

nor

Theorem: If

(c)

( X , ) is   T2 (ii) . Example: Let

X  {x1 , x2 } and G1 , G2  ( I  I ) X where

 G ( x1 )  1,  G ( x1 )  0 and

G1 ,G2 are defined by

 G ( x 2 )  1,  G ( x 2 )  0 . Consider the intuitionistic  on fuzzy topology X generated by {G1 , G2 }  {cons tan ts} . For   0.5,   0.8 , it is 2

( X , )

  T2 (ii)

but

( X , ) is not

  T2 (ii) . Further one can easily verify that   T2 (iii)    T2 (iii) and 0  T2 (iii)  0  T2 (iv) are true. This completes the proof. ̔Good extension ̓҆ property Now we discuss about the “good extension” property of

Since

x1 , x2  X , x1  x 2

( X , ) is   T2 (ii) , if for all imply that there exist open sets

 G ( x1 )  1,  G ( x1 )  0 IJSET@2014

f is

called lower semi continuous function. Definition: Let X be a non-empty set and t be a topology on X. Let    (t ) be the set of all lower semi continuous function (lsc) from (X, t) to ( I  I ) (with usual topology). Thus

 (t )

for each

  I1 .

It can be

is a intuitionistic fuzzy topology on X.

(X, t ).

G1  x, G1 ,  G1 , G2  x, G2 ,  G2  1

space. If {x : f ( x)   } is open for every real α, then

Let P be the property of a topological space (X , t) and FP be its intuitionistic fuzzy topological analogue. Then FP is called a “good extension” of P “ iff the statement (X, t) has P iff ( X ,  (t )) has FP” holds good for every topological space

( X , ) be   T2 (ii) . We prove that ( X , ) is

  T2 (ii) .

Definition: Let f be a real valued function on a topological

shown that

0  T2 (iii)  0  T2 (iv)

Proof: Let

G1  G2   as 0      1 . Hence it is clear that

and

G : X  I ,  G : X  I

( X , ) is IFTS and 0      1 then

  T2 (iii)    T2 (iii)

(b)

 G ( x 2 )  1,  G ( x 2 )  0

that

(t )  {G  (I  I ) X : [G 1 ( ,1], G 1[0, )]}where

  T2 (ii)    T2 (ii)

(a)

implies

T2 ( j ) for j = i , ii ,iii , iv.

Example(c): Let where

This

 G ( x1 )  1,  G ( x1 )  0

clear that

Example (b): Let where

G1  G2   .

and

Now we give some examples to show that none of the reverse implications are true in general.

(ISSN : 2277-1581) 1 March 2014

such

that

 G ( x 2 )  1,  G ( x 2 )  0 2

Theorem: Let (X, t) be a IFTS. Consider the following statements: (1). (X, t) be

T2 (i) space.

(2). ( X ,  (t )) be

T2 (i) space.

Page 213

International Journal of Scientific Engineering and Technology Volume No.3 Issue No.3, pp : 211 – 215 (3).

( X ,  (t )) be   T2 (ii) space.

(4). ( X ,  (t )) be

  T2 (iii)

space.

(5). ( X ,  (t )) be

  T2 (iv)

space.

(ISSN : 2277-1581) 1 March 2014

G1  x, G1 ,  G1 , G2  x, G2 ,  G2  t

exist open sets such

 G ( x1 )  1,  G ( x1 )  0

that

 G ( x 2 )  1,  G ( x 2 )  0 and G1  G2   .But for 2

every   I 1 , 1

1

{  G1 ( ,1],  G1 [0,  ) ,

Then the following implications are true

 G 1 ( ,1],  G 1[0,  ) : G1 , G2  t}  I  (t )

③

①

②

and

Proof: Let the intuitionistic fuzzy topological space (X, t) be

T2 (i) . Suppose x1 , x2  X with x1  x 2 . Since (X, t) is T2 (i) ,

there

exist

G1  x, G1 ,  G1 , G2  x, G2 ,  G2  t

such

that

1

G 1 ( ,1]  G 1 ( ,1]   , as G1  G2   1

is clear that ( X , I  (t )) is

(X, t) is

  T2 (iii)  ( X , I  (t ))

Conversely, suppose that

1G1 ,1G2   (t )

continuous function, there exist that 1

G1 ( x1 )

 1,1 G1 ( x1 )

 0 , 1

( x2 )

 1,1 G2 ( x2 )

such

 0 ,and

x1 , x2  X

( X ,  (t )) is T2 (i) space.

(3)  (5) and (4)  (5) .Hence proved.

T2 (i) .Let

( X , I  (t ))

T2 (i)

1

{  G1 ( ,1],  G1 [0,  ) , 2

such that 1

(X,t)

IFTS

and

1

x1  G1 ( ,1] , x2  G2 ( ,1] and

G 1 ( ,1]  G 1 ( ,1]   .

1

I  (t )  {  G1 ( ,1],  G1 [0,  ) ,

1

Again

since

1

{  G1 ( ,1],  G1 [0,  ) ,

 G 1 ( ,1],  G 1[0,  ) : G1 , G2  t} 2

( X , I  (t ))

 G 1 ( ,1],  G 1[0,  ) : G1 , G2  t}  I  (t )

Further it can be easily to show that (2)  (4) (2)  (3)

1

T2 (i) and (X, t) is

T2 (i) .

x1  x 2 .Since

Let

there,exist

1G1 G2  0 i, e 1G1  1G2  0 . Hence it is clear that the IFTS

Theorem:

T2 (i) . Further , one can easily

verify that

G1  G2  0 . But from the definition of the lower semi

and

.Hence it

  T2 (iv)  ( X , I  (t ))

1

x1  G1 (1] , x2  G2 ( ,1]

also

 G ( x 2 )  1,  G ( x 2 )  0

 G ( x1 )  1,  G ( x1 )  0 and

⑤ ④

 G 1 ( ,1],  G 1[0,  ) : G1 , G2  t}  I  (t ) 2

then (a) (X, t) is

  T2 (ii)  ( X , I  (t ))

(b) (X, t) is

  T2 (iii)  ( X , I  (t ))

T2 (i)

  T2 (iv)  ( X , I  (t ))

T2 (i)

G1  x, G1 ,  G1 , G2  x, G2 ,  G2  t

, so we get such

 G ( x1 )   ,  G ( x 2 )   and

that

1

G ( ,1]  G ( ,1]    1

(G1  G2 ) ( ,1]   i.e, G1  G2   . So we see 1

The reverse implications in (a) and (b) are not true in general.

that (X, t) is 

Proof: Let the intuitionistic fuzzy topological space (IFTS in

Now we have an example for non-implication.

short) (X, t) be a

  T2 (ii) .We

T2 (i) . Since (X, t) is

topological space ( X , I  (t )) is

  T2 (ii) , if for all x1 , x2  X IJSET@2014

shall prove that the

x1  x 2 imply that there

Example: Let

 T2 (iv) . X  {x1 , x2 } and G1 , G2  ( I  I ) X where

G1 ,G2 are defined by  G ( x1 )  0.8,  G ( x1 )  0.2 and 1

 G ( x2 )  0.1,  G ( x2 )  0.6 . Consider the intuitionistic 2

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International Journal of Scientific Engineering and Technology Volume No.3 Issue No.3, pp : 211 – 215

(ISSN : 2277-1581) 1 March 2014

References: fuzzy

topology

generated

{G1 , G2 }  {cons tan ts} . For   0.4 , it is clear that (X, t)

  T2 (ii)

is neither 1

  T2 (iii)

nor

.Now

 G 1 ( ,1],  G 1[0,  ) : G1 , G2  t} 2

Also, x1  G

1

K.T Atanassov: intuitionistic fuzzy sets, Fuzzy sets Syst. 20 (1986) ,

iii. K.T Atanassov : Review and new results on intuitionistic fuzzy sets, preprint IM-MFAIS-1-88, Sofia (1988). iv. K.T Atanassov: intuitionistic fuzzy sets. Theory and applications, Springer-Verlag (Heidelberg , New York , 1999)

1

( ,1] , x2  G2 ( ,1] and

1

G ( ,1]  G ( ,1]   1

ii. 87-96

1

I  (t )  {  G1 ( ,1],  G1 [0,  ) , 2

i. K.T Atanassov: intuitionistic fuzzy sets, in VII ITKR’s Session, Sofia (June 1983).

is clear that ( X , I  (t )) is This completes the proof.

, as

T2 (i) .

G1  G2   . Hence it

v. D. Coker: An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets Syst. 88 (1997) , 81-89 vi. D. Coker: An introduction to fuzzy subspaces in intuitionistic fuzzy topological spaces , J. Fuzzy Math. 4 (1996), 749-764. vii. D. Coker and M. Demirci: On intuitionistic fuzzy points, Notes IFS 1, 2 (1995), 79-84. viii. D. Coker and A.H. Es: On fuzzy compactness in intuitionistic fuzzy topological spaces, J. Fuzzy Math. 3 (1995), 899-909.

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