Download ON Compactly fuzzy

Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014
ON Compactly fuzzy b-k-closed sets
Hanan Ali Hussein
Dept.of Math. / College of Education for Girls / Kufa University
[email protected]
Abstract
In this paper ,we have dealt with the concepts of ( fuzzy b-irresolute ), (fuzzy st-b-compact ) and
(fuzzy b-continuous ) functions , and find that every fuzzy b-irresolute is fuzzy b-continuous and the
converse is not true in general .We have also remember the concept of fuzzy T2 -space and we have
proved that every fuzzy b-compact subset of an fuzzy T2 -space is fuzzy b-closed , and if X isn’t T2 space , then need not that every fuzzy b-compact set is fuzzy b-closed . Finally we defined compactly
fuzzy b-closed set and proved that every fuzzy b-closed subset of a space X is compactly fuzzy bclosed .
Keywords: fuzzy b-open ,compactly fuzzy b-closed set, compactly fuzzy b-k-closed set .
‫الخالصة‬
‫الاوبا اا‬, b- ‫ ومو الللوق نمنوا تف اود الواواب الاوبا اا المتو اا‬b-‫في بحثنا هذا تناولنا مفهوو الممموةوا الاوبا اا المفتوحوا‬
‫ وا الف ووغ ر و صووحا‬b- ‫ تكووو اووبا اا ممووتم‬b- ‫ووا اووبا اا مت و اا‬
‫ وومووانا‬, b- ‫ والاووبا اا الممووتم‬b- ‫الم صوصووا بةووو‬
‫ تكوو اوبا اا مقةةوا‬b- ‫وا ممموةوا اوبا اا م صوصوا‬
T2 ‫ وب هنا انق في الفااء‬T2 ‫ اااً تناولنا مفهو الفااء الابا ي‬. ً‫اائما‬
‫ وة فنوا‬b- ‫ اوبا اا مقةةوا‬b- ‫ فةاغ ش ط تكو ا ممموةا اوبا اا م صوصوا‬T2 ‫ وامتنتمنا انق إذا ل ا الفااء م نوع‬, b‫ بشو و ا م ص ووون ف ووي الفا وواء الت ول ووومي‬b-K- ‫ووا ا ووبا اا مقةة ووا‬
‫ بشو و ا م ص ووون وب هن ووا ا‬b-K- ‫الممموة ووا الا ووبا اا المقةة ووا‬
. b- ‫الابا ي تكو ابا اا م صوصا مقةةا‬
‫ الممموةوا الاوبا اا المقةةووا‬, ‫ بشو ا م صوون‬b- ‫ الممموةوا الاوبا اا المقةةوا‬, b– ‫الاووبا اا المفتوحوا‬
‫الكلمات االاالاةا الممموةوا‬
. ‫ بش ا م صون‬b-K-
Introduction
Zadeh in [ L.A.Zadeh, 1965 ] introduced the fundamental concept of fuzzy
sets. The study of fuzzy topology was introduced by Chang in
[ C.L.Chang, 1968].The theory of fuzzy topological spaces was subsequently
developed by several authors. In this paper the concepts of compactly fuzzy b-closed
set and compactly fuzzy b-k-closed set in fuzzy topological spaces are introduced and
studied . Throughout this paper X and Y means fuzzy topological spaces. The
definition of fuzzy sets, fuzzy topological spaces and other concepts by Zadeh and
Chang in [ L.A.Zadeh, 1965 ] and [ C.L.Chang, 1968] . This paper includes three
sections . In the first section we recall the concepts of fuzzy interior and fuzzy closure
of fuzzy open set, in the second section we have dealt with the concepts of fuzzy bopen ,fuzzy b-closed set and some their propositions, in section three we dealt with "
fuzzy b-irresolute function " , " fuzzy Strongly b-compact function " , and " fuzzy bcontinuous function " , " compactly fuzzy b-closed " , " compactly fuzzy b-k-closed "
and some theorems related to theme .
1.Prilimeries
1.1 Definition [S.Dang,A.Behra and S.Nanda, 1994]
A fuzzy point x in X is a fuzzy set defined as follows :
ify  x

x ( y)  
yx
0
Where 0    1;  is called its value and x is support of x .
1.2 Definition[G.J.Klir,U.S.clair,B.Yuan,1997]
Let A be a fuzzy set of the fuzzy topological space X , the support of A is
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the elements x whose membership value great than 0,
i.e supp(A)  {x  X : A( x)  0} .
1.3 Definition[M.H.Rashid and D.M.Ali,2008]
A fuzzy point x belong to a fuzzy set A in X ( denoted by: x  A ) if and only if
  A(x) ,for some x  X .
1.4 Definition[S.Dang,A.Behra and S.Nanda, 1994]
Let A and B are fuzzy sets in X . Then A  B if and only if x  B for all x  A .
1.5 Definition[D.H.Foster,1979]
Let A and B are fuzzy sets in X ,then :
1) A  B if and only if A( x)  B( x) x  X .
2) A  B if and only if A( x)  B( x) x  X .
3) Z  A  B if and only if Z ( x)  min{ A( x), B( x)} x  X .
4) D  A  B if and only if D( x)  max{ A( x), B( x)} x  X .
5) E  Ac ( The complement of A )if and only if E ( x)  1  A( x) x  X .
1.6 Definition[S.Carlson,2005]
Let f be a function from X to Y ,and let B be a fuzzy set in Y ,then the inverse
image of B under f is the fuzzy set f 1 ( B) in X with membership function defined
by the rule : f 1 ( B)( x)  B( f ( x)) for x  X .(i.e; f 1 ( B)  B  f )
For a fuzzy set A in X ,the image of A under f is the fuzzy set f ( A) in Y with
membership function f ( A)( y ) , y  Y defined by :

f ( A)( y )  sup A( x)
 xf -1 ( y )
0
if f 1 ( y ) is not empty
otherewise
where f 1 ( y)  {x : f ( x)  y} .
1.7 Definition[D.H.Foster,1979]
The union (respectively intersection ) of the fuzzy sets Ai is defined by :
iI
( Ai )( x)  sup{ Ai ( x) : i  I }
iI
( Ai )( x)  inf{ Ai ( x) : i  I }
iI
1.8 Theorem[A.B.Saeid,2006]
Let X and Y be two fuzzy topological spaces and let f : X  Y be a
function ,let { Ai }iI , {B j } jJ be families of fuzzy sets in X and Y respectively ,then :
1) f (  Ai )   f ( Ai ) .
iI
iI
2) f (  Ai )   f ( Ai ) .
iI
iI
1
3) f (  B j )   f 1 ( B j ) .
jJ
4) f
1
jJ
(  Bj )   f
jJ
jJ
1
(B j ) .
1.9 Definition[C.L.Change,1968]
Let A be a fuzzy set in X ,then :
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1) The union of all fuzzy open sets contained in A is called the fuzzy interior of A and
denoted by A0 .
i.e; A0  {B : B  A, B  T } .
2) The intersection of all fuzzy closed sets containing A is called the fuzzy closure of
A ,and denoted by A .i.e; A  {B : A  B, B c  T } .
1.10 Remarks[C.L.Change,1968]
1) The interior of a fuzzy set A is the largest fuzzy open set contained in A , and
trivially a fuzzy set is fuzzy open set if and only if A  A0 .
2)The closure of a fuzzy set A is the smallest fuzzy closed set containing A , and
trivially a fuzzy set A is fuzzy closed if and only if A  A .
1.11 Theorem[X.Tang,2004]
Let A and B are two fuzzy sets in X ,then :
1) 0 X  0 X ,1X  1X .
2) A0  A  A .
3) if A  B, then A  B .
4) if A  B, then A0  B 0 .
1.12 Definition[A.A.Nouh,2005]
A fuzzy set A in X is called quasi-coincident with a fuzzy set B in X , denoted
by AqB if and only if A( x)  B( x)  1 , for some x  X . If A is not quasi-coincident
with B ,then A( x)  B( x)  1 for every x  X and denoted by A q~ B .
1.13 Lemma[B.Sikn,1998]
Let A and B be fuzzy sets in X ,then:
1) if A  B  0 ,then A q~ B .
X
2) A q~ B if and only if A  B c .
1.14 Proposition[B.Sikn,1998]
If A is a fuzzy set in X , then x  A if and only if x q~ A c .
2.fuzzy b-open set and fuzzy net
2.1 Definition[B.Sikn,1998]
A fuzzy set A of a fuzzy topological space X is said to be fuzzy b-open if and only if
0
___
0
A  A  A .The complement of fuzzy b-open set is defined to be fuzzy b-closed set .
2.2 Proposition
Every fuzzy open set is fuzzy b-open .
Proof:Let A be a fuzzy open set .
0
Since A  A ,then A 0  A ………..(1)
___
Since A0  A0 ……….(2)
___
0
Then from (1) and (2) ,we have A0  A0  A .
0
Since A be a fuzzy open set , then A  A .
___
0
Then A  A0  A , then A is fuzzy b-open set □.
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The converse of above proposition is not true in general as the following
example .
2.3 Example
Let X  {a, b} and let T  {0 X ,1X , A1} be a fuzzy topology on X , and let
A1 : X  I be a fuzzy open set defined by A1 (a)  0.3 , A1 (b)  0.4 .
Let A2 : X  I be a fuzzy set defined by A2 (a)  0.6 , A2 (b)  0.4 .
Then A2 be a fuzzy b-open set, but it isn't fuzzy open set .
2.4 Remark
From proposition 2.2 and examples we can get that every fuzzy closed set is
fuzzy b-closed, but the converse is not true in general .
2.5 Definition[S.S,Benchalli and Jenifer J. Karnel,2010]
X
Let X be a fuzzy topological space and let A I , the union of all fuzzy b-open sets
b
contained in
A is called the fuzzy b-interior of A as denoted by A 0 ,
b
i.e A  {B : B  A, B is a fuzzy b - open set } .
2.6 Definition[S.S,Benchalli and Jenifer J. Karnel,2010]
X
Let X be a fuzzy topological space and let A I , the intersection of all fuzzy
0
b
b-closed sets containing A is called the fuzzy b-closure of A as denoted by A ,i.e
b
A  {B : A  B, B is a fuzzy b - closed set } .
2.7 Remark
b
0
0
1) A  A
b
2) A  A
2.8 Definition[S.S,Benchalli and Jenifer J. Karnel,2010]
A fuzzy set A in a fuzzy topological space X is called b-quasi neighborhood of
a fuzzy point x in X if and only if there exists fuzzy b-open set B such that x qB
and B  A .
2.9 Definition
Let X be a fuzzy topological space and x be a fuzzy point in X , then the family
N xbq consisting of all b-quasi neighborhoods of x is called the system of b-quasi
neighborhoods of x .
2.10 Definition[H.W.Mohammed,2011]
A set D is called a directed set if there is a relation  on D satisfying :
1) n  n for each n  D
2) if n1  n2 and n2  n3 ,then n1  n3
3) if n1 , n2  D ,then there is some n3  D with n1  n3 and n2  n3 .
2.11 Definition[H.W.Mohammed,2011]
A mapping S : D  FP( X ) is called a fuzzy net in X and is denoted by
{S (n) : n  D} where D is a directed set , if S ( n)  xnn for each n  D , x  X ,and
 n  (0,1] ,then the fuzzy net S is denoted as {xnn : n  D} or simply {xnn } .
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2.12 Definition[A.A.Nouh,2005]
A fuzzy net   { ymm : m  E} in X is said to be fuzzy subnet of a fuzzy net
S  {xnn : n  D} if and only if there is a function f : E  D such that
1)   S  f ,that is yi i  xff((ii)) for each i  E ,where E is a directed set .
2) for each n  D ,there exist some m  E such that f (m)  n .
We shall denoted a fuzzy subnet of a fuzzy net {xnn : n  D} by {xff((mm)) : m  E} .
2.13 Definition[A.A.Nouh,2005]
Let ( X , T ) be a fuzzy topological space and let S  {xnn : n  D} be a fuzzy net in
X and A I X .Then S is said to be :
1) Eventually with A if and only if m D such that xnn qA , n  m .
2) Frequently with A if and only if n  D , m D , m  n , and xmm qA .
2.14 Definition[A.A.Nouh,2005]
Let ( X , T ) be a fuzzy topological space and let S  {xnn : n  D} be a fuzzy net in
X and x  FP(X ) .Then S is said to be :
1) convergent to x and denoted by S  x ,if S eventually with A , A  N xq , x is
called a limit point of S .
2) has a cluster point x and denoted by S  x , if S is frequently with A ,
A  N xq .
2.15 Remark[H.W.Mohammed,2011]
If xnn  x , then xnn  x .
2.16 Theorem[H .W.Mohammed,2011]
Let ( X , T ) be a fuzzy topological space and let x  FP(X ) and A I X .Then
x  A if and only if there exists a fuzzy net in A convergent to x .
2.17 Theorem[H .W.Mohammed,2011]
Let ( X , T ) be a fuzzy topological space and let x  FP(X ) and A I X .Then
x  A if and only if there exists a fuzzy net xnn in A and xnn  x .
2.18 Definition
Let ( X , T ) be a fuzzy topological space and let S  {xnn : n  D} be a fuzzy net in
X and x  FP(X ) .Then S is said to be :b
1) b-convergent to x and denoted by S  x ,if A  N xbq , m D such that
xnn qA ,  n  m , x is called b-limit point of S .
b
2) has a b-cluster point
x and denoted by
S  x , if
A  N xbq
n  D , m D , m  n such that xmm qA .
2.19 Remark
Let ( X , T ) be a fuzzy topological space, then :b
b
1) if xnn  x ,then xnn  x .
b
2) if xnn is a fuzzy net in X , x  FP(X ) such that xnn  x ,then xnn  x .
b
3) if xnn is a fuzzy net in X , x  FP(X ) such that xnn  x ,then xnn  x .
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2.20 Proposition
Let ( X , T ) be a fuzzy topological space, and A I X ,if there exists a fuzzy net xnn
b
b
in A such that xnn  x ,then x  A .
Proof:b
Let xnn be a fuzzy net in A such that xnn  x ,then for every B  N xbq ,there exists
m D such that xnn qB for all n  m .
Since x n  A , then x n q~A c see proposition (1.14).
n
n
b
Then AqB , therefore x  A □.
3.The main Results
3.1 Definition
A family  of fuzzy sets is called a cover of a fuzzy set A if and only if
A  {Ci : i  } , and it is called a fuzzy b-open cover if each member C i is a fuzzy
b-open set .A sub cover of A is a sub family of  which is also a cover of A .
3.2 Definition
Let B be a fuzzy set in a fuzzy topological space X .Then B is said to be a fuzzy bcompact set if for every b-open cover of B has a finite sub cover .Let B  X ,
then X is called a fuzzy b-compact space, i.e X is a fuzzy b-compact space if for
every i   and  Bi  1 X , then there are finite many indices i1 , i2 ,..., in   such that
i
n
 Bij  1 X .
j 1
3.3 Proposition
Every fuzzy b-compact set in a fuzzy topological space X is fuzzy compact .
Proof:Let A be a fuzzy b-compact set, and let {Ci : i  } be a fuzzy open cover of A .
Then A   C i .
i
Then {Ci : i  } is a fuzzy b-open cover of A , see proposition (2.2).
Since A is fuzzy b-compact set, then there are finite many indices i1 , i2 ,..., in  
n
such that A   Cij
j 1
Then A is fuzzy compact set □.
3.4 Definition[A.S.Mashour,E.E.Kerre and M.H.Ghanim,1984]
A fuzzy topological space X is said to be fuzzy T2 -space, if for every pair fuzzy
points x  , y with sup p ( x  )  sup p ( y ) , there exist fuzzy b-open sets U and V
such that x U  ( y ) , y  V  ( x ) and Uq~V .
c
c
3.5 Theorem [H.W.Mohammed,2011]
Every fuzzy compact set of a fuzzy T2 - topological space is fuzzy closed.
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3.6 Theorem
Every fuzzy b-compact set of a fuzzy T2 - topological space is fuzzy b-closed.
Proof:Let A be a fuzzy b-compact set in a fuzzy T2 -space X .
Then A is a fuzzy compact set. see proposition (3.3) .
Then by above theorem, we have A is a fuzzy closed set .
Since every fuzzy closed set is fuzzy b-closed, see Remark (2.4) .
Then A is fuzzy b-closed set □.
If X isn't fuzzy T2 -space, then need not that every fuzzy b-compact set is fuzzy
b-closed set as the following example :3.7 Example
Let X  {a, b} and let T  {0 X ,1X , A} be a fuzzy topology on X where
A(b)  0.2
A : X  I defined by A(a)  0.1,
Then A is fuzzy b-compact set, but not fuzzy b-closed set .
3.8 Proposition[H.W.Mohammed,2011]
A fuzzy topological space ( X , T ) is fuzzy compact if and only if every fuzzy net
in X has a convergent fuzzy subnet .
3.9 Proposition
A fuzzy topological space ( X , T ) is fuzzy b-compact if and only if every fuzzy
net in X has a b-convergent fuzzy subnet .
ProofBy proposition (3.3), proposition (3.8) and remark (2.19 (2) ) □.
3.10 Theorem
In any space, the intersection of a fuzzy b-compact set with a fuzzy b-closed set is
fuzzy b-compact.
Proof:Let A be a fuzzy b-compact set and B be a fuzzy b-closed set in X .( To prove that
A  B is fuzzy b-compact set )
Let xnn be a fuzzy net in A  B , then xnn is a fuzzy net in A and xnn is a fuzzy net in
B.
Since xnn is a fuzzy net in A and A is a fuzzy b-compact, and since every b-compact
set is compact, then xnn is a fuzzy net in A and A is a fuzzy compact.
Then by proposition (3.9) we have xnn  x and x  A .
b
Then x  A and xnn  x by remark (2.19 (2) ).
Since xnn is a fuzzy net in B , and since xnn  x , then by remark (2.19 (2) ) we
b
b
have xnn  x , and by proposition (2.20) we have x  B .
b
Since B is fuzzy b-closed, then x  B and xnn  x .
Then A  B is fuzzy b-compact set, see proposition (3.9) □.
3.11 Proposition
Let Y be a fuzzy subspace of a fuzzy topological space X , and let A be a fuzzy
set in Y , then A is a fuzzy b-compact relative to X if and only if A is a fuzzy bcompact relative to Y .
Proof:1930
Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014
 Let A is a fuzzy b-compact relative to X , and let {V :   } be a collection of
fuzzy b-open sets relative to Y which covers A .
Then A   V .
 
Then there exist G fuzzy open relative to X , such that V  Y  G for any    .
Then A   G , and {G :   } is a fuzzy open cover of A relative to X .
 
Then {G :   } is a fuzzy b-open cover of A relative to X ,see proposition (2.2) .
Since A is a fuzzy b-compact relative to X , then there exists a finitely many indices
n
1 ,  2 ,...,  n   such that A   Gi .
i 1
Since A  Y , then A  Y  A  Y  (G1  G 2  ...  Gn )
 (Y  G 1 )  (Y  G 2 )  ...  (Y  Gn )
n
Since Y  Gi  Vi (i  1,2,...n) , we obtain A   Vi .
i 1
Then A is a fuzzy b-compact relative to Y .
 Let A is a fuzzy b-compact relative to Y , and let {G :   } be a collection of
fuzzy b-open sets relative to X which covers A .
Then A   G .
 
Since A  Y , we have A  Y  A  Y  (  G )   (Y  G ) .
 
 
Since Y  G is a fuzzy b-open set relative to Y .
Then the collection {Y  G :   } is a fuzzy b-open cover of A relative to Y .
Since A is a fuzzy b-compact relative to Y , we must have
A  (Y  G1 )  (Y  G 2 )  ...  (Y  Gn ) for some choice of finitely many indices
1 ,  2 ,...,  n   .
n
Then A   Gi ,So A is fuzzy b-compact set relative to X □.
i 1
3.12 Definition
Let X and Y are two fuzzy topological spaces, the function f : X  Y is called
1
fuzzy b-irresolute function if f ( A) is fuzzy b-open set in X for every fuzzy b-open
set A in X .
3.13 Definition
Let X and Y are two fuzzy topological spaces, the function f : X  Y is called a
1
strongly fuzzy b-compact (st-b-compact) function if and only if f ( A) is fuzzy bcompact set in X for every fuzzy b-compact set A in X .
3.14 Example
The identity function is strongly fuzzy b-compact .
3.15 Definition[S.S,Benchalli and Jenifer J. Karnel,2010]
Let X and Y are two fuzzy topological spaces, the function f : X  Y is called
1
fuzzy b-continuous if and only if f ( A) is fuzzy b-open ( b-closed ) set in X for
each fuzzy open ( fuzzy closed ) set A in Y .
3.16 Theorem
Every fuzzy b-irresolute function is fuzzy b-continuous .
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Proof :Let f : X  Y be a fuzzy b-irresolute function and let A be a fuzzy open set in Y ,
then A is fuzzy b-open set in Y .
1
Since f is fuzzy b-irresolute function, then f ( A) is a fuzzy b-open set in X □ .
The converse of the above theorem is not true in general as the following
example.
3.17 Example
Let X  {a, b, c} , Y  {x, y, z} be fuzzy sets, and let T1  {0 X ,1X , A, B} ,
T2  {0Y ,1Y } be topologies on X and Y respectively, A and B are fuzzy sets defined
as follows :-
A(a)  0.7
B( x)  0.1
A(b)  0.2
A(c)  0.1
B( y )  0.7
B( z )  0.2
Then the function f : X  Y which defined by f (a)  x
f (b)  y
f (c)  z is
fuzzy b-continuous .
Proof:-
0Y ,1Y are all fuzzy open sets in Y ,and f 1 (0Y )  0 X , f 1 (1Y )
X ,and since every fuzzy open set is fuzzy b-open set .
Then f is fuzzy b-continuous function, but f is not fuzzy
B
Y
because
a
fuzzy
set
in
which
B( x)  0.2
B( y )  0.6
B( z )  0.2 is a fuzzy
1
f ( B)  {a0.2 , b0.6 , c0.2 } is not fuzzy b-open set.
are fuzzy open set in
b-irresolute function
is
defined
by
b-open set, but
3.18 Proposition
Let f : X  Y be a fuzzy b-irresolute function, if A is a fuzzy b-compact set in
X , then f ( A) is fuzzy b-compact set .
Proof:Let {Ci : i  } be a fuzzy b-open cover of f ( A) .
Then f ( A)   C i .
i
Since f is fuzzy b-irresolute, then f
collection { f
1
1
(C i ) is a fuzzy b-open set in X ,hence the
(C i ) : i  } is a fuzzy b-open cover of
A
, in
X , i.e
A  f 1 ( f ( A))  f 1 (  )   f 1 (Ci ) .
i
i
Since A is fuzzy b-compact set in X .
n
Then there exists a finitely many indices i1 , i2 ,..., in such that A   f 1 (Cij ) , So that
j 1
n
n
n
j 1
j 1
j 1
f ( A)  f (  f 1 (Cij ))   ( f ( f 1 (Cij )))   Cij .
Then f ( A) is a fuzzy b-compact set in Y □.
3.19 Definition
Let X be a fuzzy space , a fuzzy subset W of X is called a compactly fuzzy bclosed set if W  K is fuzzy b-compact set for every fuzzy b- compact set K in X .
3.20 Example
Every fuzzy subset A of indiscrete fuzzy space is compactly fuzzy b-closed set .
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Journal of Babylon University/Pure and Applied Sciences/ No.(7)/ Vol.(22): 2014
3.21 Proposition
Every fuzzy b- closed subset of a fuzzy space X is compactly fuzzy b- closed
set .
Proof :Let A be a fuzzy b- closed subset of X , and let K be a fuzzy b- compact subset in
X.
Then by theorem (3.10) , we have A  K is fuzzy b- compact set .
Then A is compactly fuzzy b- closed set □.
The converse of above proposition is not true in general as the following
example .
3.22 Example
Let X  {a, b} and let T be the indiscrete fuzzy space on X .
Then A : I  X which is defined by A(a)  0.1, A(b)  0.2 is compactly fuzzy bclosed set, but it is not fuzzy b-closed set .
3.23 Proposition
Let f : X  Y be a fuzzy b-irresolute , fuzzy st-b-compact, bijective function,
then A is compactly fuzzy b-closed subset in X if and only if f ( A) is compactly
fuzzy b-closed set in Y .
Proof : Let A be a compactly fuzzy b-closed set in X , and let K be a fuzzy b-compact
set in Y .
Since f is a fuzzy st-b-compact function, then f 1 (k ) is fuzzy b- compact set in X .
Since A is compactly fuzzy b-closed set in X
Then A  f 1 ( K ) is fuzzy b- compact set in X , see theorem (3.10)
Since f is fuzzy b-irresolute function , Then by proposition (3.18) we have
f ( A  f 1 ( K )) is fuzzy b-compact set in Y .
Since f is bijective function, then f ( A  f 1 ( K ))  f ( A)  K .
Then f ( A) is compactly fuzzy b- closed set in Y .
 Let f ( A) be a compactly fuzzy b-closed set in Y , and let K be a fuzzy b-compact
set in X
Since f is fuzzy b-irresolute function, then by proposition (3.18),we have f (K ) is
fuzzy b-compact set in Y .
Since f ( A) is a compactly fuzzy b-closed set in Y .
Then f ( A)  f ( K ) is fuzzy b-compact set in Y .
Since f is fuzzy st-b-compact function , then f 1 ( f ( A)  f ( K )) is fuzzy b-compact
set in X .
Since f is bijective function , then f 1 ( f ( A)  f ( K ))  A  K .
Then A  K is fuzzy b-compact set in X .
Then A is compactly fuzzy b-closed set in X □.
3.24 Definition
Let X be a fuzzy space, then a fuzzy subset A of X is called compactly fuzzy
b-k-closed if A  K is fuzzy b-closed set for every fuzzy b-compact set K in X .
3.25 Example
Every fuzzy subset of a fuzzy discrete space is compactly fuzzy b-k-closed set .
3.26 Proposition
Every compactly fuzzy b-k-closed subset of a fuzzy space X is compactly
fuzzy
b-closed .
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Proof :Let A be a compactly fuzzy b-k-closed subset of X , and let K be a fuzzy
b- compact set in X
Then A  K is fuzzy b- closed set .
Since A  K  K , and K fuzzy is b-compact set , then by theorem (3.10), we have
A  K is fuzzy b-compact set .
Therefore A is compactly fuzzy b-closed set .
3.27 Theorem
Let X be a fuzzy T2 - space , and let A be a fuzzy subset of X , then the following
statements are equivalent:i) A is compactly fuzzy b-closed set.
ii) A is compactly fuzzy b-k- closed set .
Proof:i  ii
Let A be compactly fuzzy b-closed subset of X , and let K be a fuzzy b- compact
set in X .
Then A  K is fuzzy b-compact set .
Since X is fuzzy T2 - space , then A  K is fuzzy b- closed set by theorem ( 3.6) .
Then A is compactly fuzzy b-k-closed .
ii  i
The prove is trivial from proposition (3.26) □.
3.28 Proposition
Let
Y
be a fuzzy T2 - space , and let f : X  Y be a function ,if the only fuzzy
subsets of Y which are compactly fuzzy b-k-closed are the whole space and the
empty set, and if f is fuzzy st-b-compact, and fuzzy b-irresolute function, then f is
surjection .
Proof:Let f : X  Y be a fuzzy st-b-compact, b-irresolute function.
( we must prove that f ( X ) is compactly fuzzy b-k-closed set in Y ).
Let K be a fuzzy b-compact subset of Y .
Since f is a fuzzy st-b-compact function , then f 1 ( K ) is fuzzy b-compact set in
X.
Since f is fuzzy b-irresolute function, then we have f ( f 1 ( K )) is fuzzy b-compact in
Y , see proposition (3.18).
Since Y is fuzzy T2 -space .
Then by theorem (3.6 ) ,we have f ( f 1 ( K )) is b- closed in Y .
But f ( f 1 ( K ))  f ( X  f 1 ( K ))  f ( X )  K .
Then f ( X )  K is fuzzy b-closed set in Y .
Then f ( X ) is compactly fuzzy b-k-closed set in Y , but f ( X ) is not empty set, then
f (X )  Y .
Therefore f : X  Y is surjection function □.
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