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On Some results of Fuzzy - connected space in Fuzzy Topological Space on Fuzzy Set
Assist .Prof. Dr.Munir Abdul Khalik AL-Khafaji , Heba Mahdi Mousa
AL-Mustinsiryah University \ Collage of Education / Department of Mathematics
_________________________________________________________________
Abstract:
The main aim of this paper is to introduce and recall Some preliminary definitions and results of fuzzy
-
connected space and some kind of it . and to show the relationships between this kind.
I. Introduction
The concept of fuzzy set was introduced by Zadeh in his classical paper in [8] 1965.The fuzzy Topological space was introduced by chang [4]in 1968.Chakrabarty and Ahsanullah[3] introduced the notion of fuzzy topological space on fuzzy set
.Pu and Liu defind connectedness by using the concept fuzzy closed set .Lowen also defined an extension of a connectedness in a restricted family of fuzzy topologies [6].Fatteh and Bassam studied further the notion of fuzzy super connected and fuzzy strongly connected space[5].
In this paper we give some results and some kind of
fuzzy- connected space and study some relation between them.
1.Fuzzy topological space on fuzzy set
Definition( 1.1) [4] :A coll P( ) is said to be fuzzy topology on
1. ,
2. , then 3. If , then ,
its complement is a fuzzy closed set
Definition (1.2 ) [2] :A fuzzy set in a fuzzy topological space ( ) is said to be a fuzzy neighborhood of a fuzzy point in
such that,
Definitions (1.3) [1,7] :Let , be a fuzzy set in a fuzzy topological space ) then:
A fuzzy point is said to be quasi coincident with X such that +
and denote by , if then is not quasi coincident with a fuzzy set and is denoted by .
A fuzzy set is said to be quasi coincident (overlap) with a fuzzy set if there exist x X such that + anddenoted by , if + X then is not quasi coincident with a fuzzy set and is denoted by .
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Definition (1.4) : ( ) is said to be
Fuzzy -open set if for each there exist an fuzzy open set containing such that , [Fuzzy -closed ] set if its complement is Fuzzy -open set.The family of all Fuzzy -open (Fuzzy -closed) sets in will be denoted by F O( ) ( F C( )).
Definition (1.5): If ( , ) is a fuzzy topological space and , are fuzzy sets in Then and are f u z z y ɵ - s e p a r a t e d i f a n d o n l y i f = a n d = . Definition( 1.6 )[4]: If ,then the family = { : } is a fuzzy topology on , where is the restriction of to . Then ( , ) is called the fuzzy subspaceof the fuzzy topological space with underlying set .
Definition (1.7): If , then is said to be a fuzzy ɵ - connected subset of if is fuzzy ɵ -connected space as a fuzzy subspace of .
it is easy to see that if ,then is fuzzy ɵ -connected subset of the fuzzy topological space iff it is a fuzzy ɵ -connected subset of the fuzzy subspace of .
2.Fuzzy ɵ -Connected Spaces in Fuzzy Topological Spaces On Fuzzy Set
Definition(2.1): A fuzzy topological space ( , ) is said to be fuzzy ɵ-connected if there is no proper non-empty maximal fuzzy ɵ -separated sets and in such that = . If ( , ) is not fuzzy ɵ-connected then it is said to be fuzzy ɵ-disconnected space.
Example(2.2): Let ( , ) be a nonempty set, be a fuzzy set in and ={ } then ( , ) is fuzzy ɵ-connected space.
Theorem (2.3): Let and be fuzzy subsets in fuzzy topological space ( , ) such that If is fuzzy ɵ-connected then is fuzzy ɵ-connected
.
Proof:
If is fuzzy ɵ-disconnected , then there exist two fuzzy ɵ -separated subsets and subspace of . Such that = max{ , }
Then or
Let .As .
Also =min{ , }= .
This implies to = so and are not fuzzy ɵ -separated and is fuzzy ɵ-connected Theorem (2.4): If is fuzzy ɵ-connected ,then is fuzzy ɵ-connected .
Proof:
By contradiction, suppose that is fuzzy ɵ-disconnected.Then there are two nonempty fuzzy ɵ -
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separated sets and in such that = max{ , }.
Since =max{
,
} and
) ,
) and min { , }= , then min
, }= . Hence min
),
)=
Similarly min
),
)= ) therefore is fuzzy ɵ-disconnected.
3.On some kinds of fuzzy ɵ -connected space
Definition(3.1): A fuzzy topological space is called fuzzy super ɵ -connected if there is no proper fuzzy regular ɵ -open subset in it.
Theorem (3.2):
If is fuzzy topological space then the following statements are equivalent : (1) is fuzzy super ɵ -connected
(2) closure of every non-zero fuzzy ɵ -open set in is .
(3) Interior of every non-zero fuzzy ɵ -closed set in ,different from , is zero
(4) does not have non-zero fuzzy ɵ -open set and such that )+ ) . (5) does not have non-zero fuzzy sets and satisfying
)+ )= )+ )= . (6) does not have non-zero fuzzy ɵ- closed sets and Satisfying )+ )= )+ )= . Proof:
(1) (2) if has a non-zero fuzzy ɵ -open set such that ) ,then is a proper fuzzy regular ɵ -open set.
(2) (3) let be a fuzzy ɵ- closed set in different from . Now ) = )= ),as is a non-zero ɵ -open.
(3) (4) if has non-zero fuzzy ɵ -open sets and such that )+ ) ,then )+ ) .
So ) ) implies ) .
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Since ) ) , ) ) ,which contradicts (3).
(4) (1) if has a proper fuzzy regular ɵ -open set then and )= - ) are non-zero fuzzy ɵ -open
sets satisfying )+ )
(1) (5) if is not fuzzy super ɵ -connected ,then it has a proper fuzzy regular ɵ -open set .If we put )= - ),then ) ) and
)+ )= .Also )= )= )= ) as is fuzzy regular ɵ -closed set. Therefore )+ )= . So (5) is violated.
Conversely , if has non –zero fuzzy ɵ -open sets and such that )+ )= )+ )= , then = )= )= )
Since ) ) and )+ )= ,
) .also ) ) is given .Therefore is a proper fuzzy regular ɵ -open set. Therefore can not be fuzzy super ɵ -connected.
(5) (6) . (5) (6) follows if we take )= )
and )= ).Revers implication can be proved similarly .
Definition(3.3): subset of a fuzzy topological space is called a fuzzy super ɵ -connected of if it is a fuzzy super ɵ -connected topological space as a fuzzy subspace of .
Theorem (3.4): If is fuzzy topological space and then is a fuzzy super ɵ -connected subset of if and only if it is a fuzzy super ɵ -connected subset of the fuzzy subspace of .
Proof: Obvious
Theorem (3.5): Let be a fuzzy super ɵ -connected subset of a fuzzy topological space if there exist fuzzy ɵ- closed sets and in such that + = + = then = or =
Proof:
If and for some
then + )= and + )=
imply that ) and ) ) Thus and are non-zero fuzzy ɵ -open sets in such that + ,which contradicts
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the fact that is fuzzy super ɵ -connected subset of .
Theorem (3.6): Let ( , ) be a fuzzy topological space and , be a fuzzy super ɵ - connected subset of . such that is fuzzy ɵ -open set in . if is a fuzzy regular ɵ -open set in then either or .
Proof:
If = ) or )=
Suppose that ) and )
Let = and = Then and are such that
+ = + =
By the previous theorem or .So
or ,
as is fuzzy regular ɵ -open. Therefor = or = .
Theorem(3.7 ):If is fuzzy super ɵ -connected , then is fuzzy ɵ- connected.
proof:
If is fuzzy ɵ- disconnected , then there exist ɵ-open sets and for which
,
} , , } ) , , } ) and
, , } ) then , }
but and which contradicts being fuzzy ɵ-super connected.
Definition(3.8): A fuzzy topological space ( , ) is said to be fuzzy strongly ɵ -connected if it has no non- zero fuzzy ɵ-closed sets and such that .If is not fuzzy strongly ɵ - connected then it will be called fuzzy ɵ-weakly disconnected
.
Theorem (3.9): A fuzzy topological space ( , ) is fuzzy strongly ɵ -connected if and only if it has no (nonzero )fuzzy ɵ-open sets and such that , and
+
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Proof:
A fuzzy topological space ( , ) is fuzzy ɵ-weakly disconnected
if it has non-zero fuzzy ɵ-closed sets and such that +
if it has non-zero fuzzy ɵ-open = , =
such that , and + .
Theorem (3.10): If ( , ) is fuzzy topological space and , is a fuzzy strongly ɵ - connected subset of if and only if for any fuzzy ɵ-open sets and in , +
implies either or . Proof:
If is not a fuzzy strongly ɵ -connected subset of , then there exist fuzzy ɵ-closed sets and in . such that
(i) ) (ii) )
and (iii) + .If we put = and =
then = = .So (i),(ii),and (iii) imply that
+ , but and
Conversely if there exist fuzzy ɵ-open sets and , such that +
but and ,then , and + .So is not fuzzy strongly ɵ -connected.
Theorem (3.11): If is a subset of such that is fuzzy ɵ- closed in ,then is fuzzy strongly ɵ -connected implies that is a fuzzy strongly ɵ -connected subset of .
Proof:
Suppose is not fuzzy ɵ-closed in .Then there exist fuzzy ɵ-closed sets and in ,such that (i) ), (ii) ) and (iii) + . (iii) implies that min { , }+ min { , } ,
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where by (i) and (ii) min { , } ), min { , } ).
So is not fuzzy strongly ɵ -connected ,which is contradiction.
Remark (3.12):Every fuzzy strongly ɵ -connected implies fuzzy ɵ -connected Proof: obvious
Remark (3.13):
The convers of remark (3.12) is not true as the following example shows:-
Example ( 3.14):
Let X = { a , b } and , , , , are fuzzy subset in where = { ( a , 0.8 ) , ( b , 0.7 ) }
= { ( a , 0.0 ) , ( b , 0.7 ) } , = { ( a , 0.5 ) , ( b , 0.0 ) } = { ( a , 0.5 ) , ( b , 0.7 ) } , = { ( a , 0.8 ) , ( b , 0.0 ) }
Let = { , , , , } be a fuzzy topology on ,then F ɵ C = { , , }, then ( , ) is fuzzy ɵ -connected space since ) ) , ) )
min { ), ) } ) but not fuzzy ɵ -strong connected since ) ) ) .
Remark (3.15): fuzzy strong ɵ -connectedness and fuzzy super ɵ -connectedness are independent and the following example show that .
Example (3.16):
If X = { a , b } and let
T ~
={ , , }be a fuzzy topologies on ,
= {(a , 0.9) (b , 0.9) and = {(a , 0.2) (b , 0.3) } then ( , ) is fuzzy super ɵ -connected but not fuzzy ɵ-strongly connected . If X = { a , b } and let
T ~
={ ,
A ~
, , }be a fuzzy topologies on
= {(a , 0.8) (b , 0.8) , = {(a , 0.4) (b , 0.4) } and
= {(a , 0.3) (b , 0.3) then ( , ) is fuzzy ɵ strongly -connected but not fuzzy super ɵ- connected . Remark (3.17):
We will explain the relationship between some kinds of fuzzy ɵ -connected space
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