Download GENERALISED FUZZY CONTINUOUS MAPS IN FUZZY TOPOLOGICAL SPACES Author: Ravi Pandurangan

International Journal of Mathematical Sciences and Applications
Vol. 1 No. 2 (May, 2011)
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GENERALISED FUZZY CONTINUOUS MAPS IN
FUZZY TOPOLOGICAL SPACES
Dr. Ravi Pandurangan
Chaitanya Bharathi Institute of Technology, Proddatur, Kadapa District,
Andhra Pradesh, India – 516362, [email protected]
Abstract
The study of continuity and its weaker forms constitutes an established branch of
investigation in general topological spaces. Recently some researchers have tried to
extend these studies to the broader framework of fuzzy topological spaces. N. Ajmal [7]
proposed a theory of localization in the study of continuity and its weaker forms in fuzzy
setting. Using two notions of membership of a fuzzy point to a fuzzy set, neighborhood
structure of fuzzy point [8] and quasi-neighborhood structure of a fuzzy point [8], an
investigation of fuzzy continuity, fuzzy  - continuity, fuzzy  - continuity functions,
and fuzzy almost quasi continuous functions have been carried out in the present study.
The purpose of this chapter is to introduce and study the concepts of generalized fuzzy
continuous maps, which includes the class of fuzzy continuous maps.
INTRODUCTION
Let f : X  Y , A  I x , B  I x then f(A) is a fuzzy set in Y, defined by

sup A( x) / x  f
f ( A)( y )  
0, iff 1 ( y )  
And f
1
1

( y) , f
1
( y)  
( B) is a fuzzy set in X, defined by f
1
( B( x))  B( f ( x)) , x  X .
Let f :  X ,   Y ,   . Then
(1) f
1
(2) f ( f
B    f B , for any fuzzy set B in Y ,   .
1
c
1
(3) A  f
B )  B , for any fuzzy set B in Y ,   .
1
( f ( A)) , for any fuzzy set A in  X ,  .
(4) Let p x be a fuzzy point of X, A be a fuzzy set in  X ,  and B be a fuzzy set
in Y ,   . Then we have
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562
(i) If f ( p x )qB , then p x qf
1
( B) .
(ii) If p x qA , then f ( p x )qf ( A)
(5) Let A and B be fuzzy sets in  X ,  and Y ,   respectively. Let p x be a fuzzy
point in  X ,  . Then we have
(i) p x  f
1
( B ) if f ( p x )  B
(ii) f ( p x )  f ( A) if p x  A
Definition 1.1
Let f be a mapping from a fts  X ,  into a fts Y ,   . f is said to be fuzzy
continuous if and only if the inverse image of any fuzzy set is a matter of  .
Theorem 1.2
A mapping f from a fts  X ,  into a fts Y ,   is fuzzy continuous if and only if for
each fuzzy point p x in  X ,  and each fuzzy neighborhood  of f  p x  in Y ,   , there
is a fuzzy neighborhood  of p x in  X ,  such that f     .
Proof:
Let f be a fuzzy continuous. Let p x be a fuzzy point in X and  be a fuzzy
neighborhood of f  p x  in Y. There is a member v of  such that f  p x   v   and so,
 
f p x  f
1
(v )  f
1
( ) . f is fuzzy continuous and v   implies f
1
v   
and so
  f 1   is fuzzy neighborhood of p x in X and f    f  f 1     .
Conversely, let the given condition hold for f. Let    and p x be any fuzzy
point in f 1   . Then f  p x   f ( f
1
( ))   . By hypothesis there is a fuzzy
neighborhood  of p x such that f     , i.e.,   f
So, p x    f
1
  . As
1
 f   
1
f 1   =   px : px is a fuzzy point of f 1  
 
  .
  . Taking union of all such relations as
p x runs over f 1   , we obtain
1
1
 is a fuzzy neighborhood of p x there is a member v
of  such that p x  v   . So, p x  v  f
 v  f
f

GENERALISED FUZZY…
So, f
1
563
   v  . Hence f is fuzzy continuous.
Definition 1.3
A function f from a fts  X ,  into a fts Y ,   is said to be fuzzy continuous at a
fuzzy point p x in X if corresponding to any fuzzy neighborhood  of f( p x ) in Y ,  
there is a fuzzy neighborhood  in  X ,  such that f     .
Proposition 1.4
A mapping f from a fts  X ,  into a fts Y ,   is fuzzy continuous if and only if f
is fuzzy continuous at every fuzzy point p x in X.
Proof:
Since a fuzzy set is the union of all its fuzzy points the Proposition follows
directly from the Theorem 1.2.
Proposition 1.5
If a fuzzy point p x in a fts  X ,  belongs to the closure of a fuzzy set  in X,
then every fuzzy neighborhood of p x intersects  .
Proof:
Let p x  Cl .
If
then
every
fuzzy
neighborhood
of
p x intersects  .
Let p x  Cl . Let  be an open fuzzy neighborhood of p x . If possible, let     0 . Let
the support of  be F. Then F  support of  =  . Let us define a fuzzy set v in X as
follows:
0

v x   cl  x 
mincl ( x),1   ( x)

for those values of F for which   x   1
for x  F
for those values of F for which   x   1
Thus obviously,   v . But v  cl    is a closed set and v  cl ( ) , which implies a
contradiction. So     0 must be hold. The converse of the above Proposition is not
true.
Definition 1.6
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A fuzzy point p of a fuzzy set  in a fuzzy topological space  X ,X  is said to be
a cluster point of  if every fuzzy neighborhood containing a fuzzy point having same
support as p has non-null intersection with  .
Proposition 1.7
Every fuzzy point of the closure of a fuzzy set in a fts is a cluster point of the
fuzzy set.
PRODUCT FUZZY TOPOLOGY
Let  X 1 ,X 1  and  X 2 ,X 2  be two fuzzy topological spaces. Let X  X 1  X 2 be
the Cartesian product of X 1 and X 2 and pi be the projection of X onto X i . The product
fuzzy space of X 1 and X 2 is a set X ( X  X 1  X 2 ) equipped with the fuzzy topology
generated by the family
p
1
1

1
(  ), p2 ( ) :   X 1 and   X 2 .
The family B       :   X 1 and    X 2 from a base for the product
topology X on X.
Result 1.8
If  is a fuzzy set in a fts X and  is a fuzzy set in a fts Y, then
Cl   Cl   Cl (   ) and Int   Int   Int (   ) .
Definition 1.9
A fuzzy space X is product related to another fuzzy space Y if for any fuzzy set v
of
X
and
 of Y whenever
c  v and  c   implies c  1  1   c  v   ,
where   X and   Y , there exists 1  X and 1  Y such that
 c  v or 
c

and 1  1  1  1  c  1  1   c
c
c
Result 1.10
If X and Y are fuzzy topological spaces such that x is product related to Y, then
for a fuzzy set  of X and a fuzzy set  of Y,
(a)
Cl      Cl   Cl 
(b)
Int      Int   Int 
GENERALISED FUZZY…
GENERALISED FUZZY CONTINUOUS MAPS IN FUZZY TOPOLOGICAL
SPACES
Levine [9] introduced the concept of generalized closed sets of a topological
space and a class of topological spaces called T1 / 2 - spaces. Dunhamm and Levine further
studied some properties of generalized closed sets and T1 / 4 - spaces. Maki and Umehara
studied the characterizations of T1 / 2 - spaces and T  - spaces by using generalized new
sets. Balachandran, Sundran and Maki studies generalized continuous maps in
topological spaces. Zadeh [1] introduced fuzzy sets. Chang [10] and Lowen [4]
introduced the concept of fuzzy topological spaces and studied many properties of fuzzy
topological spaces.
The purpose of this chapter is to introduce and study the concepts of generalized
fuzzy continuous maps, which includes the class of fuzzy continuous maps.
Based on the notion of generalized closed set introduced and studied by Norman
Levine [9], we give the following definition:
Definition 1.11
Let ( X , ) be a fuzzy topological space. A fuzzy set  in X is called generalized
fuzzy closed (in short gfc)  Cl     whenever    and  is fuzzy open.
Theorem 1.12
If 1 and 2 are gfc then 1  2 is a gfc set.
Proof:
Follows from the definition of gfc and the fact that Cl (1  2 )  1  2 .
However, the intersection of two gfc sets is not generalized fuzzy closed sets.
Theorem 1.13
If  is gfc set and     Cl   , then  is a gfc set.
Proof:
Let  be a fuzzy open set such that    . Since    ,    and  is
gfc,   Cl   . But Cl ( )  Cl (  ) since Cl ( )   and   Cl   . Hence,  is a gfc set.
Theorem 1.14
In a fuzzy topological space  X , ,  F (the family of all fuzzy closed sets) 
565
Ravi Pandurangan
566
Every fuzzy subset of X is a gfc set.
Proof:
Suppose that every fuzzy set of X is gfc. Let X  F. Then since    and  is gfc
we have Cl ( )   , but   Cl ( ) and therefore Cl ( )   , i.e.,    ,   F.
Also if    , then     F and hence 1     , i.e., F   . Thus, we have seen
that F   . The converse is easy.
Theorem 1.15
Let (X, F) be a fuzzy compact (Lindale, countably compact) space and suppose
that  is a gfc set of X. Then  is fuzzy compact (Lindelof, countably compact).
We prove the case for fuzzy compactness as the proof is similar for other cases.
Let
 I be a family of fuzzy open sets in X such that   VI  . Since
 is a gfc
set and VI   is fuzzy open, it follows that   V I   . But  is fuzzy compact and
therefore it follows that Cl ( )     1    2  ...........  n
for some natural
number n  N .
Theorem 1.16
If  X ,  is fuzzy regular and  is fuzzy compact, then  is a gfc set.
Proof:
Suppose    ,    . As X is fuzzy regular, we can write   V I  , I being an
index
set, VI Cl (   )   .
Hence,
for
some
finite
subset
0   we
will
have   V 0   Cl (V 0  )   . This shows that Cl ( )   , i.e.,  is a gfc set.
Definition 1.17
A fuzzy set  is called generalized fuzzy open (in short gfo) 1   is gfc. We
shall now prove some properties of generalized fuzzy open sets.
Theorem 1.18
A fuzzy set  is gfo    Int  whenever  is fuzzy closed and    .
Proof:
Let  be a gfo set and  be a fuzzy control set such that    . Now
    1    1   and
1   is
gfc
 1   1  .
1  (1   )  1  (1   )   . But 1  (1   )  Int ( ) . Thus we find   Int  .
That
is,
GENERALISED FUZZY…
567
Conversely, suppose that  is a fuzzy set such that   Int  whenever  is
fuzzy closed and    . We claim 1   is gfc – set. So let 1     where  is fuzzy
open.
Now 1      1     . Hence, by assumption we must have 1    Int  ,
i.e., 1  Int    . But 1  Int   Cl (1   ) ; hence, Cl (1   )   . This shows that 1   is
a gfc set.
Remark 1.19
i)
The union of two sets is not generally gfo.
ii)
The intersection of any two-gfo sets is gfo.
Theorem 1.20
If Int      and if  is gf-open, then  is gf-open.
Proof:
Given Int      , we have 1    1    1  Int   Cl (1   ) . As  is gf-open,
1-  is gf-closed and so it follows by Theorem 1.18 that 1   is gf-closed i.e.,  is gfopen.
Theorem 1.21
If  is a gf-closed set in X and if f : X  Y is f-continuous and f-closed the f(  )
is gf-closed in Y.
Proof:
If f ( )   where  is f-open in Y, then   f
f
1
(  ) is f-open, Cl ( )  f
1
1
(  ) . Since  is gf-closed and
(  ) , i.e., f ( )   . Now by assumption, f (Cl ( )) is f-
closed and Cl ( f ( ))  Cl ( f (Cl ( )))  f (Cl ( ))   . This means f(  ) is gf-closed.
Definition 1.22
A map f : X  Y is called generalized fuzzy continuous (in short gf-continuous)
if the inverse image of every fuzzy closed set in Y is gf – closed in X.
The following are the properties of gf – continuous functions.
Theorem 1.23
If f : X  Y is fuzzy continuous then f is gf – continuous. The converse of this
Theorem is not true.
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568
Theorem 1.24
Let f :  X , 1   Y , 2  be a map. Then the following are equivalent:
a)
f is gf – continuous.
b)
The inverse image of each fuzzy set in Y is gf – open in X.
Based on Dunhaam’s introduction of generalized closure operator, we define the
generalized fuzzy closure operator Cl* for any fuzzy set  in  X ,  as follows:
Cl * ( )      and  is gf  closed .
Theorem 1.25
Let f : X  Y be gf – continuous. Then f [Cl * ( X )]  Cl[ f ( )] where  is any
fuzzy set in X.
Proof:
Now Cl f ( ) is a fuzzy closed set in Y. As f is gf – continuous, f
–
closed
in
X.
  f 1 (Cl f ( ))
1
(Cl ( )) is gf
and Cl * ( )  f 1 [Cl f ( )] .
Hence,
f [Cl * ( X )]  Cl[ f ( )] .
The converse of this Theorem is not true.
Definition 1.26
A fuzzy topological space  X ,  is said to be fuzzy T1 / 2 if every generalized
fuzzy closed set in x is a fuzzy closed in X.
Theorem 1.27
Let f : X  Y and Let g : Y  Z be mappings and Y be fuzzy T1 / 2 . If f and g
are gf – continuous, then the composition g  f is gf – continuous. This Theorem is not
valid if Y is not fuzzy T1 / 2 .
FUZZY  - CONTINUOUS MAPPINGS
Definition 1.28
A fuzzy point p x is a fuzzy  - cluster point of a fuzzy set  in a fts X if and
only if every fuzzy regularly open set containing a fuzzy point q x (having same support
as p) has non-null intersection with  .
Definition 1.29
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Let  be a fuzzy subset of a fts X. Let  be fuzzy subset of X satisfying the
following conditions:
(a) every fuzzy point p x in  is a fuzzy  - cluster point of  ,
(b) if v is a fuzzy set of X such that   v , then there is a fuzzy point p x in v which
is not a  - cluster point of  .
Any fuzzy subset of X having same support as  is defined to be a  - closure of  .
We denote a  - cluster point of a fuzzy set  in a fts by   . Thus a property related
with the notation   will always imply that the property holds for all  - closures of  .
Definition 1.30
A fuzzy set  of a fts X is said to be fuzzy  - closed if  is equal to one of its
 - closures. Complement of a fuzzy  - closed set is said to be fuzzy  - open.
Remark 1.31
 - Closure of a fuzzy set in a fts is not unique. However, it is unique up to its
support. Any fuzzy regularly open set of a fts X is a fuzzy open set of X. Therefore, the
set of all fuzzy regularly open sets a subset of the set of all fuzzy open sets of a fts. So, if
a fuzzy point p x of a fuzzy set  in a fts X is a fuzzy cluster point  , then p x is also a
fuzzy  - cluster point of  .
A fuzzy regularly closed set is not always a fuzzy  - closed set.
Definition 1.32
A mapping f from a fts  X ,  into a fts Y ,   is said to be a fuzzy  - continuous


at a fuzzy point Px in X, if for each fuzzy open neighborhood  of f( Px ), there exists
an open fuzzy neighborhood  of Px such that f Int Cl   Int Cl   .

Theorem 1.33
Let f be a mapping form a fts  X ,  into a fts Y ,   . Then the following two
conditions are equivalent:
(a) f is fuzzy  - continuous.
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570

(b) For each fuzzy point Px in X and each fuzzy regular open set  containing

f( Px ), there exists a fuzzy regular open set  containing Px

such that
f     .
Proof:
a   b  :
Let  be a fuzzy point in X and  be a fuzzy regularly open set

containing f( Px ). Since every fuzzy regularly open set is a fuzzy set, it follows from (a)
that
there
exists
a
fuzzy
open
neighborhood
1 of
Px

such
that,
f Int Cl 1   Int Cl     . Taking Int Cl 1  =  .
b   a  : Let
Px

be a fuzzy point of X and  be a fuzzy open neighborhood
containing f( Px ). Then Int Cl   is a fuzzy regular open set containing f( Px ) since


  Int Cl   . By (b), there is a fuzzy regularly open set  containing Px  such that
f    Int Cl   .
Theorem 1.34
Let f be a surjection from a fts
 X , 
into a fts Y ,   . Then the following
implications hold.
(a) f is fuzzy  - continuous.
(b) f ([ ] )  [ f ( )] for every fuzzy set  in X.
(c) [ f
1
(  )]  f 1 [  ] for every fuzzy set  in Y.
(  ) is fuzzy  - closed in X.
(d) For every fuzzy  - closed set  in Y, f
1
(e) For every fuzzy  - open set  in Y, f
(  ) is fuzzy  - open in X.
1
Proof:
a   b  :
Let p be a fuzzy point in Y such that p   f [ ]  then p=f(q),
where q  [ ] . Let f : X  Y be map and A be a fuzzy subset of X. Let q be a fuzzy
point of f(A) defined by
q(y)=a
for y  y q
a  0,1 and
q(y)=0
for y  y q
y Y 
GENERALISED FUZZY…
571
Then a  q ( y q )  f ( A)( y q ) =
Choose  0 so small that a<
Sup A( z )
z  f 1 ( y q )
Sup A( z ) 
z  f 1 ( y q )
But there is a z  f 1 ( yq ) such that
a<
Sup A( z )  A( z1 )
z  f 1 ( y q )
Let us define a fuzzy point p of X such that p(x)=0 for x  z1 and p(x)=0 for
x  z1 ( x  X ) . Then p  A and
Sup p( z )

1
z  f ( yq )
f ( p)( y )  
0
1

if f
1
( y ) is non  void
otherwise
So that f(p)(y)=a for y  y q and f(p)(y)=0 for y  y q
 y  Y  . Therefore f(p)=q.
Let v be a fuzzy regularly open set containing p. Then there is a fuzzy regular
open set  containing q such that f (  )  v . q  [ ]      0 , i.e., f (    )  0 .
But f (    )  f (  )  f ( )  0 and f (  )  v  v  f ( )  0  p  [ f ( )] .
a   c  :
f ([ f
1
f
1
(  )] )  [ f ( f
  is a fuzzy set in X for every fuzzy set
1
c   d  : Let
(c)  [ f
1
(  )]  f
1
d   e : Let
in Y. (d)  f
1
(  ))] , so that f ([ f
1
 in Y. By (b),
(  )] )  [  ] . Thus [ f
1
(  )]  f
1
([  ] ) .
 be a fuzzy  - closed set in Y. Then [  ]   .
([  ] )  f
1
(  ) . So, f 1 (  ) is fuzzy  - closed set in X.
 be a fuzzy  - open set in Y. Then  c is a fuzzy  - closed set
(  c ) is fuzzy  - closed set in X. But f
1
( c )  ( f
1
(  )) c so f 1 (  ) is
fuzzy  - open in X.
Theorem 1.35
If f is fuzzy  - continuous mapping from a fts
 X ,  into a fts Y ,   and g is a
fuzzy  - continuous mapping from a fts Y ,   into a fts Z ,   then so is g  f .
Proof:
Ravi Pandurangan
572
It is a direct consequence of Theorem 1.33 and Theorem 1.34.
Theorem 1.36
Let X 1 , X 2 and Y1 ,Y2 be fuzzy topological spaces such that Y1 is product related
to Y2 and X 1 is product related to X 2 . Then the product
f1  f 2 : X 1  X 2  Y1  Y2
of fuzzy  - continuous mappings
f 1 : X 1  Y1 and
f 2 : X 2  Y2 is fuzzy  -
continuous.
Proof:
p1 , p 2 be a fuzzy point in X 1  X 2 . Let  be a fuzzy open set in
Let
Y1  Y2 containing ( f 1 ( p1 ), f 2 ( p 2 )) . Then  is the union      , where A and B are
indexing sets and  ’s and   ’s fuzzy open sets of Y1 and Y2 respectively. To obtain the
result we have to show that there exists a fuzzy open set  in X 1  X 2 containing
( p1 , p 2 ) such that  f 1  f 2 Int Cl    Int Cl  
i.e., Int Cl     f 1  f 2 
1
Int Cl  
……………….(1)
Now Cl   Cl      
 Cl (    )
So
Int Cl    Int  Cl     
  Int Cl     
  Int Cl    ( Int Cl   )
Now  f 1  f 2 
1
Int Cl  
  f 1  f 2  [ Int Cl   Int Cl   ]
1
 [ f 1  f 2 
 [ f 1
 f1
1
1
1
Int Cl 

 Int Cl   ]
Int Cl    f 2 1 ( Int Cl  )]
Int Cl   f
1
1
2
( Int Cl  1 )]
[Where  1  A and 1  B are such that ( f ( p1 ), f ( p 2 ))  1   1 ]
GENERALISED FUZZY…
573
 Int Cl 1  Int Cl  1  Int cl ( 1   1 ) .
Considering 1   1 =  , we get (1).
Theorem 1.37
Let X 1  X 2 be the product space of two fuzzy topological spaces  X 1 , X 1  and
 X 2 , X 2 
and let  X , X  be a fts. If f : X  X 1  X 2 is fuzzy  - continuous, then
pi  f (i  1, 2) is also fuzzy  - continuous, where pi : X 1  X 2  X i (i=1, 2) is the
projection of X 1  X 2 onto X i .
Proof:
Let p be a fuzzy point in X and  be an fuzzy open set in X i containing p i of p.
Since
pi
is
fuzzy
certainly f ( p)  pi
1
continuous,
1
pi ( )
is
fuzzy
open
in
X1  X 2
  . Since f is fuzzy  - continuous, there is a fuzzy open set in
and
 in
X containing p such that
1
f ( Int (Cl  ))  Int pi (Cl  )
So
1
pi [ f ( Int (Cl  ))]  pi ( Int ( pi (Cl  )))
1
 pi ( pi ( Int Cl  ))
 Int (Cl  )
So, pi  f ( Int (Cl  ))  Int (Cl  ) .
Definition 1.38
A mapping f from fts  X ,  into a fts Y ,   is called a fuzzy almost continuous
mapping if f
1
(  )   X for each fuzzy open set  of Y.
The definition illustrates fuzzy continuity implies fuzzy almost continuity.
Theorem 1.39
For a function f form fts  X ,  into a fts Y ,   , the following are true:
(1) If Y is fuzzy semi-regular space and f is fuzzy  - continuous, then f is fuzzy
continuous.
Ravi Pandurangan
574
(2) If X is fuzzy semi-regular space and f is fuzzy almost continuous then f is fuzzy
 - continuous.
Proof:
(1) Let p be a fuzzy point in X and let  be an open fuzzy set in Y containing
f(p). Y is fuzzy semi-regular implies    , where  ’s fuzzy regular open sets and
f ( p )   for some  and f is fuzzy  - continuous implies there is a fuzzy regular open
set  containing p such that f (  )     . Then f is fuzzy continuous.
(2) Let p  X and  be a fuzzy regular open set in Y containing f(p). Then
f
1
( ) is fuzzy open in X, and f
X, p  f
1
1
( )    , where  ’s are fuzzy regular open sets in
( )  p   ( for some  )  f
1
( ) . Therefore f ( )  f ( f
1
( ))   . So, f
is  - continuous.
REFERENCES
1. ZADEH.LA, Fuzzy Sets, Information and Control, 8, (1965), 338-353.
2. BELLMAN.R and GIERTY.M, On the analytic formation of the theory of fuzzy sets.
Information Sciences 5 (1973), 149-156.
3. DIB.K.A, the Fuzzy topological spaces on a fuzzy space, Fuzzy Sets and Systems
108(1999), 103-110.
4. LOWEN.R. Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl.
56(1976), 621-633.
5. PALANIAPPAN.N, Fuzzy Topology, Narosa Publishing House, New Delhi, 2002.
6. DIB.K.A, On Fuzzy spaces and fuzzy group theory, Inform. Sci. 80(3-4), (1994),
253-282.
7. AJMAL.N.TYAGI.K, On fuzzy almost continuous functions, Fuzzy Sets and Systems
41(1991), 221-232.
8. HU CHING-MING, Fuzzy topological spaces, Journal of Mathematical Analysis and
Applications 110, (1985) 141-178.
9. LEVINE.N, Semi-open sets and semi continuity in topological spaces Amer. Math. J
22(1982), 55-60.
10. CHANG.C.L, Fuzzy topological spaces, J. Math. Anal. Appl. 24(1968), 182-190.
SYMBOLS
Cl
-
Closure
Int
-
Interior
fts
-
fuzzy topological space
gfc
-
generalized fuzzy closed
gfo
-
generalized fuzzy open
GENERALISED FUZZY…
FSO
-
fuzzy semi open
f.a.o.S
-
fuzzy almost open (semi)
f.a.o.H
-
fuzzy almost open (Pre semi)
fsCl(A)
-
fuzzy semi closure of A
fsInt(A)
-
fuzzy semi Interior of A
f.w.q.c
-
fuzzy weakly quasi continuous
f.s.w.c
-
fuzzy sub weakly continuous
f.a.w.c
-
fuzzy almost weakly continuous
FPO
-
fuzzy pro open
f.w.c
-
fuzzy weakly continuous
f.w.a.c
-
fuzzy weakly alpha/almost continuous
f.w.c.c
-
fuzzy weakly completely continuous
f.f.c
-
fuzzy faintly continuous
f.s.w.c
-
fuzzy sub weakly continuous
575