Download Intuitionistic Fuzzy Contra Strong Precontinuity

Faculty of Sciences and Mathematics, University of Niš, Serbia
Available at: http://www.pmf.ni.ac.yu/filomat
Filomat 21:2 (2007), 273–284 _____________________________________
INTUITIONISTIC FUZZY CONTRA STRONG PRECONTINUITY
Biljana Krsteska and Erdal Ekici
Abstract. The concept of intuitionistic fuzzy contra strongly precontinuous mapping are
introduced and studied. Properties and relationship of intuitionistic fuzzy contra strongly
precontinuous mapping are established. Some applications to fuzzy compact spaces are given.
Keywords and phrases: intuitionistic fuzzy topology, intuitionistic fuzzy contra strongly
precontinuous mapping.
2000 Mathematics Subject Classification: 54 A 40.
Received: April 30, 2007.
1. Introduction
The concept of intuitionistic fuzzy set is introduced by Atanassov in his classic
paper [1]. Coker [2] defined the intuitionistic fuzzy topological spaces. As a
continuation of this work, Joen et al [6] obtained some interesting results about the
intuitionistic -continuity and the intuitionistic fuzzy precontinuity.
Dontctev [3] introduced the notion of contra continuous mapping. Ekici and
Kerre [4] introduced the concept of fuzzy contra continuous mapping. The authors
[10,11] introduced the notions of intuitionistic fuzzy contra continuous mapping and
fuzzy contra strongly preconuinuous mapping. In the Section 3 as, a generalization of
this classes, we introduce new weaker form of intuitionistic fuzzy contra continuity
called intuitionistic fuzzy contra strongly precontinuity. Also, we produce some
characterization theorems for that mapping. In the Section 4 we consider some
properties concerning fuzzy compactness.
2. Preliminaries
We introduce some basic notions and results that are used in the sequel.
Definition 2.1. [1] Let X be a nonempty fixed set and I the closed interval [0,1].
An intuitionistic fuzzy set (IFS) A is an object of the following form
A  { x,  A ( x ),  A ( x ) | x  X}
where the mapping  A : X  I and  A : X  I denote the degree of membership
(namely  A ( x ) ) and the degree of nonmembership (namely  A ( x ) ) for each element
x  X to the set A, respectively, and 0   A ( x )   A ( x )  1 for each x  X.
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Obviously, every fuzzy set A on a nonempty set X is an IFS of the following
form
A  { x,  A ( x ),1   A ( x ) | x  X}.
Definition 2.2. [1] Let A and B be IFSs of the form
A  { x,  A ( x ),  A ( x ) | x  X} and B  { x,  B ( x ),  B ( x ) | x  X}. Then
(i) A  B if and only if  A ( x )   B ( x ) and  A ( x )   B ( x );
(ii) A  { x,  A ( x ),  A ( x ) | x  X};
(iii) A  B  { x,  A ( x )   B ( x ),  A ( x )   B ( x ) | x  X};
(iv) A  B  { x,  A ( x )   B ( x ),  A ( x )   B ( x ) | x  X}.
A  x,  A ,  A 
We
will
use
the
notation
instead
of
A  { x,  A ( x ),  A ( x ) | x  X}. A constant fuzzy set  taking value   [0,1] will
be denote by . The IFSs 0 ~ and 1~ are defined by 0 ~  { x, 0,1 | x  X} and
1~  { x,1, 0 | x  X}.
Let , [0,1] such that     1. An intuitionistic fuzzy point (IFP) p(,) is
an IFS defined by
(, ) x  p
p(,) (x)  
.
(0,1) otherwise
Let f be a mapping from an ordinary set X into an ordinary set Y.
If B  { y,  B ( y),  B ( y) | y  Y}
is an IFS in Y, then the inverse image of B under f is IFS defined by
f 1 ( B)  { x, f 1 ( B )(x ), f 1 (  B )(x ) | x  X}.
The image of IFS A  { x,  A ( x ),  A ( x ) | x  X} under f is IFS defined by
f ( A)  { y, f ( A )( y), f (  A )( y) | y  Y}
where
 sup  A ( x )
f 1 ( y)  0
xf 1 ( y )
f ( A )( y)  
 0
otherwise
and
 inf  A ( x )
f 1 ( y)  0
xf 1 ( y )
f ( A )( y)  

otherwise,
 1
for each y  Y.
Definition 2.3. [2] An intuitionistic fuzzy topology (IFT) in Coker’s sense on a
nonempty set X is a family  of IFSs in X satisfying the following axioms:
(T1 ) 0 ~ ,1~  ;
( T2 ) G1  G 2   for any G1 , G 2  ;
( T2 )  G i   for arbitrary family {G i | i  I}  .
274
In this paper by ( X, ) or simply by X we will denote the Coker’s intuitionistic
fuzzy topological space (IFTS). Each IFS in  is called intuitionistic fuzzy open set
(IFOS) in X. The complement A of an IFOS A in X is called an intuitionistic fuzzy
closed set (IFCS) in X.
Definition 2.4.[2] Let A be an IFS in IFTS X. Then
int A  {G | G is an IFOS in X and G  A} is called an intuitionistic fuzzy
interior of A;
clA  {G | G is an IFCS in X and G  A} is called an intuitionistic fuzzy
closure of A.
Definition 2.5. [5] Let A be an IFS of an IFTS X. Then A is called
(1) an intuitionistic fuzzy regular open set (IFROS) if A  int( clA ) ;
(2) an intuitionistic fuzzy -open set (IFOS) if A  int(cl(int A)) ;
(3) an intuitionistic fuzzy semiopen set (IFSOS) if A  cl(int A) ;
(4) an intuitionistic fuzzy preopen set (IFPOS) if A  int( clA ) .
Definition 2.6. [5] Let A be an IFS of an IFTS X. Then A is called
(1) an intuitionistic fuzzy regular closed set (IFRCS) if A is an IFROS;
(2) an intuitionistic fuzzy -closed set (IFCS) if A is an IFOS;
(3) an intuitionistic fuzzy semiclosed set (IFSCS) if A is an IFSOS;
(4) an intuitionistic fuzzy preclosed set (IFPCS) if A is an IFPOS.
Definition 2.7. [7] Let A be an IFS in IFTS X. Then
p int A  {G | G is an IFPOS in X and G  A} is called an intuitionistic fuzzy
preinterior of A;
pclA  {G | G is an IFPCS in X and G  A} is called an intuitionistic fuzzy
preclosure of A.
Definition 2.8. [7] An IFS A in an IFTS X is called an intuitionistic fuzzy
strongly preopen set (IFSPOS) if A  int( pclA ).
The complement A of an IFSPOS A in X is called an intuitionistic fuzzy
strongly preclosed set (IFSPCS) in X.
Theorem 2.1. [6] An IFS of an IFTS X is -open if and only if it is both IFSOS
and IFSPOS.
Definition 2.9.[7] Let A be an IFS in IFTS X. Then
sp int A  {G | G is an IFSPOS in X and G  A} is called an intuitionistic
fuzzy strong preinterior of A;
spclA  {G | G is an IFSPCS in X and G  A} is called an intuitionistic fuzzy
strong preclosure of A.
Definition 2.10. [2,7,8] A mapping f : X  Y from an IFTS X into an IFTS Y
is called
(1) intuitionistic fuzzy continuous if f 1 ( B) is an IFOS in X, for each IFOS B in
Y;
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(2) intuitionistic fuzzy strongly precontinuous if f 1 ( B) is an IFSPOS in X, for
each IFOS B in Y;
(3) intuitionistic fuzzy irresolute strongly precontinuous if f 1 ( B) is an IFSPOS
in X, for each IFSPOS B in Y.
Definition 2.11. [2,8] A mapping f : X  Y from an IFTS X into an IFTS Y is
called
(1) intuitionistic fuzzy open (intuitionistic fuzzy closed) if f ( A) is an IFOS
(IFCS) in Y, for each fuzzy open (fuzzy closed) set A of X;
(2) intuitionistic fuzzy irresolute strongly preopen (intuitionistic fuzzy irresolute
strongly preclosed) if f ( A) is an IFSPOS in Y, for each IFSPOS A in X.
Definition 2.12. [4,11] A mapping f : X  Y from an IFTS X into an IFTS Y is
called
(1) intuitionistic fuzzy contra continuous if f 1 ( B) is an IFOS in X, for each
IFCS B in Y;
(2) intuitionistic fuzzy contra -continuous if f 1 ( B) is an IFOS in X, for each
IFCS B in Y;
(3) intuitionistic fuzzy contra precontinuous if f 1 ( B) is an IFPOS in X, for
each IFCS B in Y.
Definition 2.13. [2,9] Let X be an IFTS. A family { x,  Gi ,  Gi | i  I} of
IFOSs (IFPOSs, IFSPOSs) in X satisfies the condition 1~  { x,  Gi ,  Gi | i  I} is
called a fuzzy open (fuzzy preopen, fuzzy strongly preopen) cover of X.
A finite subfamily of a fuzzy open (fuzzy preopen, fuzzy strongly preopen) cover
{ x,  Gi ,  Gi | i  I} of X which is also a fuzzy open (fuzzy preopen, fuzzy strongly
preopen) cover of X is called a finite subcover of { x,  Gi ,  Gi | i  I}.
An IFTS X is called fuzzy compact (fuzzy precompact, fuzzy strongly
precompact) if every fuzzy open (fuzzy preopen, fuzzy strongly preopen) cover of X
has a finite subcover.
Definition 2.14. [9] An IFTS X is called
(1) fuzzy strongly precompact if each fuzzy strongly preopen cover of X has a
finite subcover for X;
(2) fuzzy strongly pre-Lindelof if each fuzzy strongly preopen cover of X has a
countable subcover for X;
(3) fuzzy countable strongly precompact if each countable fuzzy strongly
preopen cover of X has a finite subcover for X.
Definition 2.15. [9] An IFTS X is called
(1) fuzzy precompact if each fuzzy preopen cover of X has a finite subcover for
X;
(2) fuzzy pre-Lindelof if each fuzzy preopen cover of X has a countable subcover
for X;
276
(3) fuzzy countable precompact if each countable fuzzy preopen cover of X has a
finite subcover for X.
3. Intuitionistic fuzzy contra strong precontinuity
Definition 3.1. A mapping f : X  Y from an IFTS X into an IFTS Y is called
intuitionistic fuzzy contra strongly precontinuous if f 1 ( B) is an IFSPOS in X, for
each IFCS B in Y.
Remark 3.1. From the definition above it is not difficult to conclude that the
following diagram of implications is true.
intuitionistic fuzzy contra continuity

intuitionistic fuzzy contra -continuity

intuitionistic fuzzy contra strong precontinuity

intuitionistic fuzzy contra precontinuity
The following example shows that the converse statement may not be true.
Example 3.1. Let X  {a, b, c} and A, B, C be fuzzy sets of X defined as
follows:
 a b c  a b c 
A  x, 
,
, ,  , , 
 0,3 0,2 07   0,1 0,1 0,1 
 a b c  a b c 
B  x, 
, , ,  , , 
 0,8 0,8 04   0,1 0,1 0,1 
 a b c  a b c 
C  x, 
, ,
,  , , 
 0,2 0,3 0,4   0,1 0,1 0,1 
We put   {0 ~ , A, B, A  B, A  B,1~ }, 1  {0 ~ , B,1~ },  2  {0 ~ , A c ,1~ } and
 3  {0 ~ , C,1~ }. Then, the mapping f  id : ( X, 1 )  ( X,  2 ) is intuitionistic fuzzy
contra precontinuous but it is not intuitionistic fuzzy contra strongly precontinuous.
Similarly, the mapping f  id : ( X, )  ( X,  3 ) is intuitionistic fuzzy contra strongly
precontinuous but f is neither intuitionistic fuzzy contra continuous nor intuitionistic
fuzzy contra -continuous. 
Theorem 3.1. Let f : X  Y be a mapping from an IFTS X into an IFTS Y.
Then, the following statements are equivalent:
(i) f is an intuitionistic fuzzy contra strongly precontinuous mapping;
(ii) f 1 ( B) is an IFSPCS in X, for each IFOS B in Y.
Proof. (i)(ii) Let f be any intuitionistic fuzzy contra strongly precontinuous
mapping and let B be any IFOS in Y. Then, B is an IFCS in Y. According to the
277
assumption f 1 ( B) is an IFSPOS in X. Since f 1 ( B)  f 1 ( B) we obtain that f 1 ( B)
is an IFSPCS in X.
The converse can be prove by the same token. ■
Theorem 3.2. Let f : X  Y be a mapping from an IFTS X into an IFTS Y.
Suppose that one of the following properties holds:
(i) f 1 (clB)  int( pclf 1 ( B)), for each IFS B in Y;
(ii) cl( p int f 1 ( B))  f 1 (int B), for each IFS B in Y;
(iii) f (cl( p int A))  int f ( A), for each IFS A in X;
(iv) f (clA )  int f ( A), for each IFPOS A in X.
Then, f is intuitionistic fuzzy contra strongly precontinuous.
Proof. (i)  (ii) It can be prove by using the complement.
(ii)  (iii) Let A be any IFS in X. We put B  f (A). Then A  f 1 (B). From
the assumption follows that cl(pint A)  cl(pint f 1 (B))  f 1 (int B). Therefore
f (cl(pint A))  int B  int f (A).
(iii)  (iv) Let A be any IFPOS in X. Then, pinA  A. From the assumption
we
obtain that f (clA)  f (cl(pint A))  int f (A).
Suppose that (iv) holds. Let B be any IFOS in Y. Then, p int f 1 (B)
IFPOS in X. According to the assumption we have
is an
f (cl(pint f 1 (B))  int f (pint f 1 (B))  int f (f 1 (B))  int B  B.
Hence cl( p int f 1 ( B))  f 1 ( B), so f 1 ( B) is an IFSPCS in X. From the Theorem 3.1.
(ii) follows that f is an intuitionistic fuzzy contra strongly precontinuous mapping. ■
Theorem 3.3. Let f : X  Y be a mapping from an IFTS X into an IFTS Y.
Suppose that one of the following properties holds:
(i) f (spclA)  int f (A), for each IFS A in X;
(ii) spclf -1 (B)  f -1 (intB), for each IFS B in Y;
(iii) f -1 (clB)  spintf -1 (B), for each IFS B in Y.
Then, f is intuitionistic fuzzy contra strongly precontinuous.
Proof. (i)  (ii) Let B be any IFS in Y. From the assumption we have
f(spclf -1 (B))  intf (f -1 (B))  intB.
Therefore spclf -1 (B)  f 1 (f (spclf -1 (B))  f 1 (intB).
(ii)  (iii) It can be prove by using the complement.
Suppose that (iii) holds. Let B be any IFCS in X. Then B=clB. From the
assumption
we
have
f -1 (B) = f -1 (clB)  spintf -1 (B). It
follows
that
f -1 (B)  spintf -1 (B). Hence f 1 (B) is an IFSPOS in X, so f is an intuitionistic fuzzy
contra strongly precontinuous mapping. ■
278
Theorem 3.4. Let f : X  Y be a bijective mapping from an IFTS X into an
IFTS Y. The mapping f is intuitionistic fuzzy contra strongly precontinuous if
clf ( A)  f (sp int A), for each IFS A in X.
Proof. Let B be any IFCS in X. Then, clB  B. Since f is surjective, from the
assumtion
follows
that
Therefore
f (sp int f 1 ( B))  clf (f 1 ( B))  clB  B.
f 1 (f (sp int f 1 ( B)))  f 1 ( B).
Since
f
is
a
injective
mapping
we
have
1
1
1
1
1
1
1
sp int f ( B)  f (f (sp int f ( B))  f ( B). Hence sp int f ( B)  f ( B), so f ( B)
is an IFSPOS in X. Thus f is an intuitionistic fuzzy contra strongly precontinuous
mapping. ■
Theorem 3.5. Let f : X  Y be an intuitionistic fuzzy contra strong
precontinuous mapping from an IFTS X into an IFTS Y. Then the following statements
holds:
(i) spclf 1 ( B)  f 1 (int( pclB), for each IFSPOS B in Y.
(ii) f 1 (cl( p int B))  sp int f 1 ( B), for each IFSPCS B in Y.
Proof. (i) Let B be any IFSPOS in Y. Then f 1 ( B)  f 1 (int( pclB)). Since
f 1 (int( pclB )) is an IFSPCS in X, we have spclf 1 ( B)  f 1 (int( pclB)).
(ii) It can be prove in a similar manner as (i). ■
The following theorem gives some local characterizations of the intuitionistic
fuzzy contra strong precontinuous mappings.
Theorem 3.6. Let f : X  Y be a mapping from an IFTS X into an IFTS Y.
Then the following statements are equivalent:
(i) f is an intuitionistic fuzzy contra strongly precontinuous mapping.
(ii) for each IFP p ( ,) in X and IFCS B containing f ( p ( , ) ), there exists
IFSPOS A in X containing p ( ,) such that A  f 1 ( B).
(iii) for each IFP p ( ,) in X and IFCS B containing f ( p ( , ) ), there exists
IFSPOS A in X containing p ( ,) such that f ( A)  B.
Proof. (i)(ii) Let f be an intuitionistic fuzzy contra strongly precontinuous
mapping, let B be any IFCS in Y and let p ( ,) be an IFP in X and such that
f ( p ( ,) )  B. Then p( ,)  f 1 ( B)  sp int f 1 ( B). Let A  sp int f 1 ( B). Then A is an
IFSPOS and A  sp int f 1 ( B)  f 1 ( B).
(ii)(iii) The results follows from the evident relations f ( A)  f (f 1 ( B))  B.
(iii)(i) Let B be any IFCS in Y and let p ( ,) be an IFP in X such that
p( ,)  f 1 ( B). Then f ( p ( ,) )  B. According to the assumption there exists an
IFSPOS
A
in
1
X
such
that
p ( ,)  A
1
and
f ( A)  B.
p( ,)  A  f f (A)  f ( B). Therefore p( ,)  A  sp int A  sp int f
279
1
Hence
( B). Since
p ( ,) is an arbitrary IFP and f 1 ( B) is the union of all IFPs in f 1 ( B), we obtain that
f 1 ( B)  sp int f 1 ( B). Thus f is an intuitionistic fuzzy contra strongly precontinuous
mapping. ■
Theorem 3. 7. Let f : X  Y and g : Y  Z are mappings, where X, Y and Z
are IFTSs. Then, the following statements hold:
(i) if f is an intuitionistic fuzzy contra strongly precontinuous mapping and g is
an intuitionistic fuzzy continuous mapping, then gf is an intuitionistic fuzzy contra
strongly precontinuous mapping;
(ii) if f is an intuitionistic fuzzy contra strongly precontinuous mapping and g is
an intuitionistic fuzzy contra continuous mapping, then gf is an intuitionistic fuzzy
strongly precontinuous mapping;
(iii) if f is an intuitionistic fuzzy irresolute strongly precontinuous mapping and g
is an intuitionistic fuzzy contra strongly precontinuous mapping, then gf is an
intuitionistic fuzzy contra strongly precontinuous mapping;
(iv) if f is an intuitionistic fuzzy irresolute strongly preopen (intuitionistic fuzzy
irresolute strongly preclosed) surjective mapping and gf is an intuitionistic fuzzy contra
strongly precontinuous mapping, then g is an intuitionistic fuzzy contra strongly
precontinuous mapping.
(v) if gf is an intuitionistic fuzzy contra strongly precontinuous mapping and g is
an intuitionistic fuzzy open (fuzzy closed) injective mapping, then f is a fuzzy strongly
precontinuous mapping.
Proof. The proof of the statements (i), (ii) and (iii) follows from the relation
1
(gf ) (C)  f 1 (g 1 (C)), for each IFS C in Z.
The proof of the condition (iv) follows from the fact that for any surjective
mapping g holds g 1 (C)  f (gf ) 1 (C), for each IFS C in Z.
The proof of the condition (v) follows from the fact that for any injective
mapping g holds f 1 ( B)  (gf ) 1 (g( B)), for each IFS B in Y. ■
Corollary 3.8. Let X, X 1 and X 2 be IFTSs and pi : X  X1  X 2 (i  1,2) are
the projections of X1  X 2 onto X i . If f : X  X1  X 2 is an intuitionistic fuzzy
contra strongly precontinuous, then p i f are also intuitionistic fuzzy contra strongly
precontinuous mappings.
Proof. It follows from the facts that projections are intuitionistic fuzzy
continuous mappings. ■
Theorem 3.9. Let f : X  Y be a mapping from an IFTS X into an IFTS Y. If
the graph g : X  X  Y of f is intuitionistic fuzzy contra strongly precontinuous, then
f is intuitionistic fuzzy contra strongly precontinuous.
Proof. For each IFOS B in Y holds f 1 ( B)  1  f 1 ( B)  g 1 (1  B). Since g is
an intuitionistic fuzzy contra strongly precontinuous mapping and 1 B is an IFOS in
X  Y, f 1 ( B) is an IFSPCS in X, so f is an intuitionistic fuzzy contra strongly
precontinuous mapping. ■
280
Theorem 3. 10. Let f : X  Y be a mapping from an IFTS X into an IFTS Y.
The mapping f is intuitionistic fuzzy contra -continuous if and only if it is both
intuitionistic fuzzy contra semicontinuous and intuitionistic fuzzy contra strongly
precontinuous.
Proof. It follows from the Theorem 2.1. ■
4. Applications to fuzzy compact spaces
Definition 4.1. An IFTS X is called
(1) fuzzy S-closed if each fuzzy regular closed cover of X has a finite subcover
for X;
(2) fuzzy S-Lindelof if each fuzzy regular closed cover of X has a countable
subcover for X;
(3) fuzzy countable S-closed if each countable fuzzy regular closed cover of X
has a finite subcover for X.
Remark 4.1. From the definitions above we may conclude that
1) every fuzzy S-closed IFTS is fuzzy S-Lindelof.
2) every fuzzy S-closed IFTS is fuzzy countably S-closed.
Definition 4.2. An IFTS X is called
(1) fuzzy strongly S-closed if each fuzzy closed cover of X has a finite subcover
for X;
(2) fuzzy strongly S-Lindelof if each fuzzy closed cover of X has a countable
subcover for X;
(3) fuzzy countable strongly S-closed if each countable fuzzy closed cover of X
has a finite subcover for X.
Remark 4.2. It is not difficult to conclude that
1) every fuzzy strongly S-closed IFTS is fuzzy strongly S-Lindelof.
2) every fuzzy strongly S-closed IFTS is fuzzy countably strongly Sclosed.
Definition 4.3. An IFTS X is called
(1) fuzzy almost compact if each fuzzy open cover of X has a finite subcover the
closure of whose members cover X;
(2) fuzzy almost Lindelof if each fuzzy open cover of X has a countable subcover
the closure of whose members cover X;
(3) fuzzy countable almost compact if each countable fuzzy open cover of X has
a finite subcover the closure of whose members cover X.
Remark 4.3. From the definitions above we may conclude that
1) every fuzzy almost compact IFTS is fuzzy almost Lindelof.
2) every fuzzy almost compact IFTS is fuzzy countably almost compact.
Theorem 4.1. Let f : X  Y be an intuitionistic fuzzy contra strongly
precontinuous mapping from an IFTS X onto an IFTS Y. If X is fuzzy strongly
precompact (fuzzy strongly pre-Lindelof, fuzzy countable strongly precompact), then Y
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is fuzzy strongly S-closed (fuzzy strongly S-Lindelof, fuzzy countable strongly Sclosed).
Proof. Let {G i | i  I} be any fuzzy closed cover of Y. Then 1~  G i . From

iI


the relation 1~  f 1  G i  follows that 1~  f 1 (G i ), so {f 1 (G i ) | i  I} is a


iI
 iI 
fuzzy strongly preopen cover of X. Since X is fuzzy strongly precompact, there exists a


finite subcover {f
1
(G i ) | i  1,2,..., n}. Therefore 1~ 
n
 f 1 (G i ). Hence
i 1
n
 n
 n
1~  f  (f 1 (G i )   f (f 1 (G i )  G i ,


i 1
 i 1
 i 1
so Y is fuzzy strongly S-closed. ■



Corollary 4.2. Let f : X  Y be an intuitionistic fuzzy contra strongly
precontinuous mapping from an IFTS X onto an IFTS Y. If X is fuzzy precompact
(fuzzy pre-Lindelof, fuzzy countable precompact), then Y is fuzzy strongly S-closed
(fuzzy strongly S-Lindelof, fuzzy countable strongly S-closed).
Theorem 4.3. Let f : X  Y be an intuitionistic fuzzy contra strongly
precontinuous mapping from an IFTS X onto an IFTS Y. If X is fuzzy strongly
precompact (fuzzy strongly pre-Lindelof, fuzzy countable strongly precompact), then Y
is fuzzy S-closed (fuzzy S-Lindelof, fuzzy countable S-closed).
Proof. It follows from the statement that each IFRCS is an IFCS. ■
Corollary 4.4. Let f : X  Y be an intuitionistic fuzzy contra strongly
precontinuous mapping from an IFTS X onto an IFTS Y. If X is fuzzy precompact
(fuzzy pre-Lindelof, fuzzy countable precompact), then Y is fuzzy S-closed (fuzzy SLindelof, fuzzy countable S-closed).
Theorem 4.5. Let f : X  Y be an intuitionistic fuzzy contra strongly
precontinuous mapping from an IFTS X onto an IFTS Y. If X is fuzzy strongly
precompact (fuzzy strongly pre-Lindelof, fuzzy countable strongly precompact), then Y
is fuzzy almost compact (fuzzy almost Lindelof, fuzzy countable almost compact).
Proof. Let {G i | i  I} be any fuzzy open cover of Y. Then 1~  G i . It

iI


From the relation 1~  f 1  clG i  follows that


iI
 iI

1
1
f (clG i ), so {f (clG i ) | i  I} is a fuzzy strongly preopen cover of X. Since
follows that 1~ 
1~ 

 clG i .

iI
X is fuzzy strongly precompact, there exists a finite subcover {f 1 (clG i ) | i  1,2,..., n}.
Therefore 1~ 
n
 f 1 (clG i ). Hence
i 1
282
n
 n
 n
1~  f  (f 1 (clG i )   f (f 1 (clG i )  clG i ,


i 1
 i 1
 i 1
so Y is fuzzy almost compact. ■



Corollary 4.6. Let f : X  Y be an intuitionistic fuzzy contra strongly
precontinuous mapping from an IFTS X onto an IFTS Y. If X is fuzzy precompact
(fuzzy pre-Lindelof, fuzzy countable precompact), then Y is fuzzy almost compact
(fuzzy almost Lindelof, fuzzy countable almost compact).
We urge the interested reader to find example/s for the notions introduced
in this section and to show how they are related to each other and the other
related types of intuitionistic fuzzy compact notions.
References
[1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986),
87-96.
[2] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy
Sets and Systems 88 (1997), 81-89.
[3] J. Dontchev, Contra-continuous functions and strongly S-closed spaces, Inter.
J. Math. Math. Sci., 19 (1996), 303-310.
[4] E. Ekici and E. Kerre, On fuzzy contra-continuities, Advances in Fuzzy
Mathematics, 1 (2006), 35-44.
[5] H. Gurcay and D. Coker, On fuzzy continuity in intuitionistic fuzzy
topological spaces, J. Fuzzy Math. 5 (1997), 365-378.
[6] J.K. Joen, Y.B. Jun and J.H. Park, Intuitionistic fuzzy alpha-continuity and
intuitionistic fuzzy precontinuity, Int. J. Math. Sci., 19 (2005), 3091–3101.
[7] B. Krsteska and S. E. Abbas, Intuitionistic fuzzy strongly preopen sets and
intuitionistic strong precontinuity, submitted to the Hacet. J. Math. Stat.
[8] B. Krsteska and S. Abbas, Intuitionistic fuzzy strongly irresolute
precontinuous mappings in Coker’s spaces, to apear in Kragujevac J. Math., 2007.
[9] B. Krsteska and S. Abbas, Fuzzy strong preccompactness in Coker’s sense,
submitted in Math. Moravica.
[10] B. Krsteska and E. Ekici, Fuzzy contra strong precontinuity, submitted in
Ind. J. Math.
[11] B. Krsteska and E. Ekici, Intuitionistic fuzzy contra continuity, submitted in
Tamkang J. Math.
Biljana Krsteska
Faculty of Mathematics and Natural Science, Gazi Baba b.b. P.O. 162
1000 Skopje, Macedonia
283
e-mail address: [email protected]
Erdal Ekici
Department of Mathematics, Canakkale Onsekiz Mart University,
Terzioglu Campus, 17020 Canakkale, Turkey
e-mail address: [email protected]
284