Download Fuzzy Semi Pre Compact Space on fuzzy topological spaces

3102 ‫ العدد األول‬................................................ ‫جملة كلية الرتبية‬
Fuzzy Semi Pre Compact Space on Fuzzy Topological Spaces
Saleem Yaseen Mageed and Prof. Dr. Nadir George Mansour
Department of Mathematics, College of Education, Al-Mustansiryah University,
Baghdad, Iraq.
1311/11/03:‫تقديم البحث‬
1313/13/10:‫قبول نشر البحث‬
Abstract
In this paper, we will study and introduce some types of compactess of fuzzy
topological spaces, such as fuzzy -compact sets, fuzzy semi compact sets and semi pre
compact sets defined on topological spaces which are also fuzzy. Then we study the
relationship between such types of compact fuzzy topological spaces.
Keyword: Fuzzy topological spaces, fuzzy semi compact spaces, fuzzy -compact spaces,
fuzzy semi pre compact spaces.
1. Introduction
The fuzzy concept has invaded almost all branches of mathematics ever since the
introduction the fundamental concept of fuzzy sets by Zadeh since the usual notion of a
set was generalization with that of fuzzy sets , the study of a set can be regarded as a
special case of fuzzy sets where all fuzzy sets take values 0 and 1 only.
Chang [2] in 1968, introduced the definition of fuzzy topological spaces and
extended in a straight forward manner some concepts of crisp topological spaces to
fuzzy topological spaces. Fuzzy topology was originating with Chang's article [2] in
1968, so it may be considered as a new branch of mathematics, then many additional
structures were studied by using fuzzy sets and the related problems in pure and applied
mathematics.
While Wong [15] in 1974 discussed and generalized some properties of fuzzy
topological spaces. Ming, p.p. and Ming, L.Y. [10] in 1980 used fuzzy topology to
define the neighborhood structure of fuzzy point.
Some different types of fuzzy compact spaces of fuzzy topological spaces are also
studied in literatures such as (fuzzy -compact space, fuzzy semi compact space and
fuzzy pre compact space).[1] [3] [5] [6]
In this paper we will study the fuzzy semi pre compactness of fuzzy topological
spaces.
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2. Preliminaries
In this section, some fundamental concepts and some well know results on fuzzy
topological spaces and fuzzy compact topological space are giving for completeness,
wish are necessary for the work of this paper.
2.1 Definition [2]:
A fuzzy topological on X is a family  of fuzzy sets in X which satisfies the
following conditions:
(1)
 and X   .
(2)
If A1 , A 2   , then A1  A 2   .
(3)
If A i   , for each i  , then
i
Ai   .
 is called a fuzzy topological for X, and the pair (X,  ) is a fuzzy topological
space (or f.t.s for short). Every member of  is called fuzzy open set in X and its
complement is called fuzzy closed set in X.
2.2 Definition [2], [10]:
A fuzzy set A in a f.t.s (X,  ) is called a fuzzy neighborhood (or f-nbd for short)


of a fuzzy point p x if and only if there exists a fuzzy open set B , such that p x  B
0
0
 A . A (f-nbd) A is said to be open if and only if A is fuzzy open set. The
neighborhood system ( N
px 0

) of a fuzzy point p x is the family of all f-nbds of the
0

fuzzy points p x .
0
2.3 Definition [9]:

Let (X,  ) be a f.t.s, A be a fuzzy set in X. A fuzzy point p x  A is said to be a
0
fuzzy limit point of A , if for every fuzzy open set G , we have:

( G  px )  A   .
0
2.4 Definition [10], [12]:
Let A be a fuzzy set in f.t.s (X,  ). The closure A (or cl( A )) and interior A
(or int( A )) of A are defined respectively be:
cl( A )  { F : F is fuzzy closed set and A  F }
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3102 ‫ العدد األول‬................................................ ‫جملة كلية الرتبية‬
i.e., the intersection of all fuzzy closed sets containing A , and:
int( A )  { G : G is fuzzy open set and G  A }
i.e., the union of all fuzzy open sets contained in A .
2.5 Proposition [8], [16]:
Let (X,  ) be a f.t.s and A , B  IX, then:
(1)
cl(  )   .
(2)
A  cl( A ).
(3)
cl(cl( A ))  cl( A ).
(4)
If A  B then cl( A )  cl( B ).
(5)
cl( A  B )  cl( A )  cl( B ).
cl( A  B )  cl( A )  cl( B ).
2.6 Proposition [8], [16]:
(1)
int(X)  X.
(2)
int( A )  A .
(3)
int(int( A ))  int( A ).
(4)
If A  B then int( A )  int( B ).
(5)
int( A )  int( B )  int( A  B ).
(6)
int( A  B )  int( A ) int( B ).
2.7 Definition [7], [13]:
A function f from a f.t.s (X,  ) to a f.t.s (Y,  ) is said to be:
(1)
"a fuzzy continuous function" if and only if the inverse image of any fuzzy open
set in Y is fuzzy open set in X.
(2)
"a fuzzy open function" if and only if the image of each fuzzy open set in X is a
fuzzy open set in Y.
2.8 Theorem
If f : X  Y is a fuzzy open and fuzzy continuous function then f1(cl( B )) 
cl(f1( B )) for every B  Y.
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3102 ‫ العدد األول‬................................................ ‫جملة كلية الرتبية‬
2.9 Theorem:
If f : X  Y is fuzzy continuous function then f1(int( B ))  int(f1( B )), for
every B  Y.
2.10 Definition [2]:
A family V of fuzzy sets has the finite intersection property if and only if the
intersection of the members of the each finite subfamily of V is nonempty.
3. Some Types of Fuzzy Compact Space of Fuzzy Topological Space
In this section some types of fuzzy compact of fuzzy topological space are giving
which are (fuzzy -compact space, fuzzy semi compact space and fuzzy pre compact
space),as well as, some fundamental results concerning such types of fuzzy
compactness.
3.1 Definition [11]:
Let (X, T) be a fuzzy topological space and let A be any fuzzy set in X, A is
called:
1) Fuzzy -open set if A  int(cl(int( A ))).
2) Fuzzy semi open set if A  cl(int( A )).
3) Fuzzy pre open set if A  int(cl( A )).
4) Fuzzy semi pre open set if A  cl(int(cl( A ))).
The complement of a fuzzy -open (resp. fuzzy semi-open, fuzzy pre open and
fuzzy semi pre open set is called fuzzy -closed (resp. fuzzy semi-closed, fuzzy pre
closed and fuzzy semi pre closed) set.
3.2 Remark:
1) Every fuzzy open (resp.closed) set is a fuzzy -open (-closed) set. But the
converse is not true in general.
2) Every fuzzy -open (-closed) set is a fuzzy semi open (resp.semi closed) set and a
fuzzy pre open (pre closed) set. But the converse is not true in general.
3) Every fuzzy semi open (resp.semi closed) set and a fuzzy pre open (pre closed) set is
a fuzzy semi pre open (semi pre closed) set. But the converse is not true in general.
In the next definition we will give the definition of some types of fuzzy coverings
and relationships between these types.
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3102 ‫ العدد األول‬................................................ ‫جملة كلية الرتبية‬
3.3 Definition [2], [4]:
Let (X,  ) be a f.t.s a family W of fuzzy sets is open cover of a fuzzy set A if
and only if A  { G : G  W} and each member of W is an open fuzzy set. A sub
cover of W is a sub family which is also cover.
3.4 Definition:
Let (X,  ) be a f.t.s a family W of fuzzy sets is -open cover of a fuzzy set A if
and
only
if
A  { G : G  W} and each member of W is an -open fuzzy set. A sub cover of
W is a sub family which is also cover.
3.5 Definition:
Let (X,  ) be a f.t.s a family W of fuzzy sets is semi open cover of a fuzzy set A
if and only if A  { G : G  W} and each member of W is semi open fuzzy set. A
sub cover of W is a sub family which is also cover.
3.6 Definition:
Let (X,  ) be a f.t.s a family W of fuzzy sets is pre open cover of a fuzzy set A if
and
only
if
A  { G : G  W} and each member of W is pre open fuzzy set. A sub cover of W
is a sub family which is also cover.
3.7 Definition:
Let (X,  ) be a f.t.s a family W of fuzzy sets is semi pre open cover of a fuzzy set
A if and only if A  { G : G  W} and each member of W is a semi pre open
fuzzy set. A sub cover of W is a sub family which is also cover.
3.8 Remark:
1) Every fuzzy open cover is a fuzzy -open (semi open, pre open and semi pre
open)cover. But the converse is not true in general.
2) Every fuzzy -open cover is a fuzzy semi open cover and a fuzzy pre open cover.
But the converse is not true in general.
3) Every fuzzy semi open cover and fuzzy pre open cover is a fuzzy semi pre open
cover. But the converse is not true in general.
In the next definition we will give the definitions of some types of fuzzy
compactness of fuzzy topological spaces and relationships between these types.
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3102 ‫ العدد األول‬................................................ ‫جملة كلية الرتبية‬
3.9 Definition [3], [15]:
A f.t.s (X,  ) is fuzzy compact space if and only if every fuzzy open cover of X
has a finite sub cover.
3.10 Definition [6]:
A f.t.s (X,  ) is fuzzy -compact space if and only if every fuzzy -open cover of
X has a finite sub cover.
3.11 Definition [1]:
A f.t.s (X,  ) is fuzzy semi compact space if and only if every fuzzy semi open
cover of X has a finite sub cover.
3.12 Definition[5]
A f.t.s (X,  ) is fuzzy pre compact space if and only if every fuzzy pre open cover
of X has a finite sub cover.
3.13 Proposition:
1) Every fuzzy semi compact space is a fuzzy -compact space and afuzzy compact
space.
2) Every fuzzy pre compact space is a fuzzy -compact space and afuzzy compact
space.
3) Every fuzzy -compact space a fuzzy compact space.
4. Fuzzy Semi Pre Compact Space
This section is devoted to present the definitions of fuzzy semi pre compact space
in fuzzy topological space. Moreover, in this section some of their remarks and
propositions are studied, also the relationships among the concepts of weak fuzzy
continuous functions and this type are discussed.
We start this section by giving a definition of fuzzy semi pre compact space.
4.1 Definition:
A f.t.s (X,  ) is fuzzy semi pre compact space if and only if every fuzzy semi pre
open cover of X has a finite sub cover.
4.2 Proposition:
Every fuzzy semi pre compact space is fuzzy compact space.
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3102 ‫ العدد األول‬................................................ ‫جملة كلية الرتبية‬
Proof:
Let (X,  ) be a fuzzy semi pre compact space, and let the collection V  { G  : 
 } be a fuzzy open cover of X
Since, sup{G (x)}  1 . Then X 



G
By remark(3.8),(1). Then the collection V  { G  :   } is fuzzy semi pre open cover
of a fuzzy semi pre compact space X. Therefore, X has a finite sub cover which belongs
to V{ G  :}
Hence, X is a fuzzy compact space.

4.3 Proposition:
Every fuzzy semi pre compact space is fuzzy semi compact space.
Proof:
Let (X,  ) be a fuzzy semi pre compact space, and let the collection V  { G  : 
 } be a fuzzy semi open cover of X
Since, sup{G (x)}  1 . Then X 



G
By remark(3.8),(3) . Then the collection V  { G  :   } is a fuzzy semi pre open
cover of a fuzzy semi pre compact space X. Therefore, X has a finite sub cover which
belongs to V  { G  :   }
Hence, X is a fuzzy semi compact space.

4.4 Proposition:
Every fuzzy semi pre compact space is fuzzy pre compact space.
Proof:
Let (X,  ) be a fuzzy semi pre compact space, and let the collection V  { G  : 
 } be a fuzzy pre open cover of X
Since, sup{G (x)}  1 . Then X 



G
By remark(3.8),(3). Then the collection V  { G  :   } is a fuzzy semi pre open
cover of a fuzzy semi pre compact space X. Therefore, X has a finite sub cover which
belongs to V  { G  :   }
Hence, X is a fuzzy pre compact space.

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3102 ‫ العدد األول‬................................................ ‫جملة كلية الرتبية‬
4.5 Remark:
A fuzzy compact (resp. -compact, semi compact, pre compact) space need not be
fuzzy semi pre compact space.
4.6 Theorem:
A f.t.s (X,  ) is fuzzy semi pre compact space if and only if every collection of
fuzzy semi pre closed sets in X with the finite intersection property has a nonempty
intersection.
Proof:
Suppose the X is fuzzy semi pre compact space and
W  { A  :   }
be collection of fuzzy semi pre closed sets in X with the finite intersection property
To show the collection W  { A  :   } has a nonempty intersection
i.e., to show the inf{A (x)}  0


Suppose inf{A (x)}  0 . Then



{ A  :   }  
c
 {A :   }  c  X . Therefore
c
Hence,
{ A  :   }  X and each

 


A c is fuzzy semi pre open set. Which implies that { A c :   } is a semi pre open
cover of a fuzzy semi pre compact space X
Hence, there exists a finite sub cover of the space X which belongs to { A  :   },
c
such that Max{Ac (x)}  1
i
n
i.e., X  {Ai : i  1,2,...,n}
c
i 1
n
X     {Aci : i  1,2,...,n} 
i1

c
c
n
  {Ai : i  1,2,...,n}
i 1
Therefore, { A  :   } does not satisfy the property of finite intersection which is
contradiction
Hence, we must have inf{A (x)}  0 . Then we have:


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{ A  :   }  

Conversely, suppose every collection of fuzzy semi pre closed sets in X with the
finite intersection property has a nonempty intersection.
To prove (X,  ) be a fuzzy semi pre compact space let V  { G  :   } be fuzzy semi
pre open cover of X
Since sup{G (x)}  1. Then:


X
G  { G  :   } X     {G  :   }
 


c
c

{ G  :   }
c

and each G  is fuzzy semi pre closed sets which implies that { G  :   } is a
c
c
collection of fuzzy semi pre closed sets with empty intersection and so by hypothesis
this collection does not have the finite intersection property.
Hence, there exists a finite member of fuzzy sets G i , i  1, 2, …, n, such that
c
Min{Gc (x)}  0
i
n
{ G i : i  1, 2, …, n}  
c
Then,
i 1
c
 n {G c : i  1,2,...,n}   c  X
i1 i

n
i 1
{ G i : i  1, 2, …, n}  X
Therefore, { G i : i  1, 2, …, n} is finite sub cover of the space X which belong to
fuzzy semi pre open cover { G  :   }, such that Max{G (x)}  1
i
Hence, (X,  ) is fuzzy semi pre compact space.

4.7 Theorem:
Let (X,  ) be a f.t.s which is a semi pre compact space then every infinite fuzzy
set in X has fuzzy semi pre limit point in X.
Proof:
Let A be an infinite fuzzy set in X
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

To prove that there exists fuzzy point p x  X, x  X,   (0, 1] such that p x is a fuzzy
semi pre limit point of A .
Suppose that A has no fuzzy semi pre limit point in X. Then for every fuzzy point
p x  X there exists a fuzzy semi pre open neighborhood N of p x which contains no

fuzzy point of A other than p x
Since, N is fuzzy semi pre open neighborhood then N is a fuzzy semi pre open set.
Then, the collection of fuzzy semi pre open sets { N  :   } is fuzzy semi pre open
cover of X
Hence, sup{ N (x)}  1


Since (X,  ) is fuzzy semi pre compact space. Then there are finitely many fuzzy points
p xi , i  1, 2, …, n, in X such that Max{ N (x)}  1
i
Hence, X 
n
i 1
{ N i : i  1, 2, …, n}
Since, A X Then A (x)  Max{ N (x)} , i.e., A 
i
n
i 1
{ N i :i  1,2,…,n}
Therefore, A has a finite points and so A is finite which is contradiction that A is
infinite fuzzy set in X.
Hence, A has fuzzy semi pre limit point in X. 
In the next proposition explaining the fuzzy semi pre compact space is hereditary
property.
4.8 Proposition:
Every fuzzy semi pre closed subset of a fuzzy semi pre compact space is fuzzy
compact.
Proof:
Let (X,  ) be a fuzzy semi pre compact space and F be a fuzzy semi pre closed
subset of X.
To prove F is fuzzy compact
Let V  { G  :   } be a fuzzy open cover of F in (X,  )
Since F is a subset of a collection V. Then F (x)  sup{G (x)}

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Hence, V  { G  :   } is a fuzzy semi pre open cover of F
c
Since F is a fuzzy semi pre closed subset of X then F is a fuzzy semi pre open subset
of X
Therefore, the collection { G  :   }  { F } is fuzzy semi pre open cover of X,
c
which is fuzzy semi pre compact space. Then there exist finitely many members of 
say 1, 2, …, n such that X
n
i 1
G i { Fc }
c
i.e., X has two finite sub cover say { G 1 , G  2 , …, G  n , F }
Since, F (x)  1 Then F  X and F cover no part of F
c
Hence, F (x)  Max{ G (x) } Therefore, F 
i
Hence, F is fuzzy compact.
n
i 1
G i

The following corollaries may be giving and aresults to the above the proposition.
4.9 Corollary:
Every fuzzy semi pre closed subset of a fuzzy semi pre compact space is fuzzy
semi pre compact.
Proof: It is clear. 
4.10 Corollary:
Every fuzzy closed subset of fuzzy semi pre compact space is fuzzy semi pre
compact.
Proof: It is clear. 
4.11 Corollary:
Every fuzzy closed subset of fuzzy semi pre compact space is fuzzy compact.
Proof: It is clear. 
4.12 Remark:
A fuzzy semi pre closed subset of fuzzy compact need not be compact.
In the next proposition we will explain that the union of any two fuzzy semi pre
compact is also fuzzy semi pre compact.
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4.13 Proposition:
Let (X,  ) be a f.t.s, if A and B are two fuzzy semi pre compact subsets of X,
then A  B is also fuzzy semi pre compact.
Proof:
Let { G  :   } be a fuzzy semi pre open cover of A  B
Then, AB (x)  sup{G (x)} , i.e., Max{ A (x) , B (x) }  sup{G (x)}


Hence, A  B 



G
Since, A (x)  AB (x) . Then A  A  B
Also B (x)  AB (x) . Then B  A  B
It is follows that { G  :   } is a fuzzy semi pre open cover of A and a fuzzy semi
pre open cover of B
Since A and B are two fuzzy semi pre compact sets, then there exists a finite sub
cover ( G 1 , G  2 , …, G  n ), which covering A belong to { G  :   }
Then A (x)  Max{ G (x) }
i
Hence A 
n
i 1
G i and there exists a finite sub cover ( G 1 , G 2 , …, G m ) which
covering B belongs to { G  :   }
Then, B (x)  Max{ G (x) }
j
Hence, B 
m
j1
Gj
It is follows that AB (x)  Max{ G  k (x)}, k  1, 2, …, n + m
Then, A  B 
nm
k 1
G k
Thus, A  B is fuzzy semi pre compact.

4.14 Proposition:
Let (X,  ) be a f.t.s if A and B are two fuzzy semi pre compact subsets of X,
then A  B need not to be fuzzy semi pre compact.
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Now, we give several definitions of some types of weak fuzzy continuous
function, by using the concept of fuzzy semi pre open sets.
4.15 Definition [14], [17]:
A function f from a f.t.s (X,  ) to a f.t.s (Y,  ) is said to be:
(1)
"a fuzzy semi pre continuous function" if the inverse image of any fuzzy open set
in Y is fuzzy semi pre open set in X.
(2)
"a fuzzy semi pre irresolute function" if the inverse image of any fuzzy semi pre
open set in Y is fuzzy semi pre open set in X.
(3)
"a fuzzy semi pre continuous function" if the inverse image of any fuzzy semi pre
open set in Y is a fuzzy open set in X.
4.16 Proposition:
(1)
Every fuzzy semi pre continuous function is fuzzy continuous function.
(2)
Every fuzzy continuous function is fuzzy semi pre continuous function.
(3)
Every fuzzy semi pre continuous function is fuzzy semi pre continuous function.
(4)
Every fuzzy semi pre continuous function is fuzzy semi pre irresolute function.
(5)
Every fuzzy semi pre irresolute function is fuzzy semi pre continuous function.
4.17 Remark
The following diagram explains the relationship among the different types of
weakly fuzzy continuous function.
Fuzzy continuous
Fuzzy semi pre
continuous
Fuzzy semi pre
continuous
Fuzzy semi pre
irresolute
In the next we will discuss the relationship among the concepts of weak fuzzy
continuous functions and concepts of fuzzy semi pre compact space.
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3102 ‫ العدد األول‬................................................ ‫جملة كلية الرتبية‬
4.18 Proposition:
The fuzzy semi pre continuous image of a fuzzy semi pre compact space is fuzzy
compact space.
Proof:
Let (X,  ) be a fuzzy semi pre compact space, (Y,  ) any f.t.s and f : (X,  )
 (Y,  ) be a fuzzy semi pre continuous function
To prove (Y,  ) is a fuzzy compact space let { G  :   } be any fuzzy open cover of
Y since Y(y)  

G
(y)  sup{G (y)} Then Y 


G
in view of f is a fuzzy semi pre continuous function
Then, {f1( G  ):   } is fuzzy semi pre open cover of X
Hence, 

f 1 (G  )
(x)  sup{f 1 (G ) (x)}


 sup{G (f (x))}  1


Since X is fuzzy semi pre compact space then there exists a finite sub cover {f1( G i ): i
 1, 2, …, n}, which covering X
Hence, X(x)   n
i 1
Therefore, X 
n
i 1
f 1 (G i )
(x)  Max{f 1 (G ) (x)}
i
f 1 (G i )
 n f 1 (G ) 
i 
 i1

Hence, f(X)  f 
Then, f(X) 
n
i 1
Therefore, Y 
n
f (f 1 (G i ))  G i
i 1
n
i 1
G i i.e., there exist a finite sub cover ( G 1 ,G 2 ,...,G n ) which
covering Y, then Y is a fuzzy compact space.

The following corollaries may be obtain as adirect results to the above
proposition.
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3102 ‫ العدد األول‬................................................ ‫جملة كلية الرتبية‬
4.19 Corollary:
The fuzzy continuous image of a fuzzy semi pre compact space is fuzzy compact
space.
Proof: It is clear. 
4.20 Corollary:
The fuzzy semi pre irresolute image of a fuzzy semi pre compact space is fuzzy
compact space.
Proof: It is clear. 
4.21 Corollary:
The fuzzy semi pre continuous image of a fuzzy semi pre compact space is fuzzy
compact space.
Proof: It is clear. 
4.22 Corollary:
The fuzzy semi pre irresolute image of a fuzzy semi pre compact space is fuzzy
semi pre compact space.
Proof: It is clear. 
4.23 Corollary:
The fuzzy semi pre continuous image of a fuzzy semi pre compact space is fuzzy
semi pre compact space.
Proof: It is clear. 
4.24 Corollary:
The fuzzy semi pre continuous image of fuzzy compact space is fuzzy compact
space.
Proof: It is clear. 
4.25 Corollary:
The fuzzy semi pre continuous image of a fuzzy compact space is fuzzy semi pre
compact space.
Proof: It is clear. 
4.26 Lemma:
If f : (X, ) 
(Y, ) is fuzzy open and fuzzy continuous function, then f is
fuzzy semi pre irresolute function.
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3102 ‫ العدد األول‬................................................ ‫جملة كلية الرتبية‬
Proof:
Let A be a fuzzy semi pre open set in Y Then A  cl(int(cl( A )))
therefore f1( A )  f1(cl(int(cl( A ))))
Since, f is fuzzy open and fuzzy continuous function, then by theorem (2.8) and theorem
(2.9)
f1(cl(int(cl( A ))))  cl(f1(int(cl( A ))))
 cl(int(f1(cl( A ))))
 cl(int(cl(f1( A ))))
Therefore, f1( A )  cl(int(cl(f1( A ))))
Hence, f is fuzzy semi pre irresolute function.

4.27 Theorem:
Let f : (X, )  (Y, ) be a fuzzy open and fuzzy continuous function, if
(X, ) is fuzzy semi pre compact space. Then (Y, ) is fuzzy semi pre compact space.
Proof: It is clear. 
4.28 Remark:
The following diagram explain, the relationship among different strong types of
fuzzy compact space.
Fuzzy semi
compact space
Fuzzy semi pre
Fuzzy -compact
Fuzzy compact
compact space
space
space
Fuzzy pre
compact space
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3102 ‫ العدد األول‬................................................ ‫جملة كلية الرتبية‬
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