Introduction : Abstract : م . م .يناسرخلا نيسحلا دبع رديح ناوزغ ،ةيرصنتسملا ةعماجلاتايضايرلا مسق ،ةيبرتلا ةيلك ,قارعلا ،دادغب Assistant.Teacher.Gazwan Haider AL-Khorsani A STUDY ABOUT FUZZY EC-COMPACT SPACE BY USING FUZZY NET

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A STUDY ABOUT FUZZY EC-COMPACT SPACE BY USING FUZZY NET

Assistant.Teacher.Gazwan Haider AL-Khorsani

Department of Mathematics, College of Education, AL-Mustinsiryah University, Baghdad, Iraq

م . م . يناسرخلا نيسحلا دبع رديح ناوزغ

،ةيرصنتسملا ةعماجلا تايضايرلا مسق ،ةيبرتلا ةيلك

, قارعلا ،دادغب

Abstract :

The purpose of this paper is to study a new class of fuzzy e-open set is called fuzzy EC-open Setand study the relationships with some fuzzy open set (fuzzy regular open set, fuzzy -open set, fuzzy -open set, fuzzy open set) in fuzzy topological space and construct the concept of fuzzy EC-compact space in fuzzy topological space and we give some characterization on fuzzy EC-compact space by using fuzzy net and we obtained several properties.

Introduction :

The concept, which we will be considered in this thesis, is the so called “fuzzy sets” which is totally different from the classical concept which is called “a crisp set”. The recent concept is introduced by [13] Zadeh in 1965, in which he defines fuzzy sets as a class of objects with a continuum of grades of membership and such a set is characterized by a membership function that assigns to each object a grade of membership ranging between zero and one. In (1968) [4]

Chang introduced the definition of fuzzy topological spaces and extended in a straight forward manner some concepts of crisp topological spaces to fuzzy topological spaces. While [12]

Wong in 1974 discussed and generalized some properties of fuzzy topological spaces. [10]

Ming, p.p. and Ming, L.Y. in 1980 used fuzzy topology to define the neighborhood structure of fuzzy point. On the other hand [1] introduced the notion fuzzy net and fuzzy filter base and some other related concepts. In this paper we introduce the concepts of fuzzy EC-compact in fuzzy topological spaces, we give some characterization. Moreover the study also included the relationship between have been studied and basic properties for these concept

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1- Preliminaries :

Definition 1-1 [13]:

Let X be a non empty set, and let I be the unit interval i.e I=[0,1], a fuzzy set in X is a function from X into the unit interval I (i.e : X [0,1] be a function).

A fuzzy set in X can be represented by the set of pairs: = {(x, ): x X} the family of all fuzzy sets in X is denoted by I^X.

Definition1-2 [7]:

A fuzzy point is a fuzzy set such that :

= r 0 if x = y , y X and = 0 if x y , y X

The family of all fuzzy points of will be denoted by ) . Definition 1-3 [2]:

A fuzzy point is said to belong to a fuzzy set in X (denoted by : ) if and only if ≤

proposition 1-4 [12]:

Let and be two fuzzy sets in with membership functions and respectively, then for all x X: -

1.  ≤ . 2. = = .

3. = C( = min{ , }.

4. = D( = max{ , }.

5. the complement of with membership function = 1 – . Definition 1-5 [3,10]:

A fuzzy set in X is called quasi-coincident with a fuzzy set in X denoted by if and only if + 1 for some x X. if A is not quasi-coincident with B then

+ 1 for every x X and denoted by . Proposition 1-6 [7]:

Let and are fuzzy sets in X then 1- If min{ , }= then B 2- B if and onl if ≤ . Proposition 1-7 [7]:

If is a fuzzy set in X, then if and only if

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collection T of a fuzzy subsets of X, such that T  P(X) is said to be fuzzy topology on X if it satisfied the following conditions

1. X , T

2. f , T , then T

3. If T , then T ,

(X , T is said to be uzzy topolo ical space and e ery member of T is called fuzzy open set in X and its complement is a fuzzy closed set .

Let be a fuzzy set in a fuzzy topological space (X ,T ) then :

 The closure of is denoted by (cl( )) and defined by

(x) = { is a fuzzy closed set in X , } .

 The interior of is denoted by (int( )) and defined by

(x) = max { is a fuzzy open set in X , }.

A fuzzy set in a fuzzy topological space (X ,T ) is called quasi-neighborhood of a fuzzy point in X if and only if there exist T such that and ≤ .

Let (X ,T ) be a fuzzy topological space and be a fuzzy point in X. then the family consisting of all quasi-neighborhood (q- neighborhood) of is called the system of quasi- neighborhood of

Proposition 1-12:

Let (X ,T ) be a fuzzy topological space and be a fuzzy set in X.

A fuzzy point ≤ (x) if and only if for every fuzzy open set in X, if then .

proof:

( ) suppose that be a fuzzy set in X, such that and ,

Then ≤ , but and be a fuzzy closed set in X, thus . ( ) let then there exist a fuzzy closed set in X such that ≤ And hence by proposition (1-7) we have

Since ≤ then by proposition (1-6: 2) Definition 1-13 [11,6,5]:

fuzzy set of a fuzzy topolo ical space (X ,T ) is said to be :- 1) Fuzzy regular open set if :

= , .

The family of all fuzzy regular open sets in X will be denoted by FRO(X)

2) Fuzzy – open set if for each there exist fuzzy open set such that

, The family of all Fuzzy – open set in X will denoted by F O(X)

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max{ , }, x X The family of all fuzzy – open sets in X will be denoted by F (X) .

A fuzzy filter base on X is a nonempty subset of I^X such that i. 0

ii. If A1 , A2 then there exist A3 such that A3 A1 A2

A mapping :D FB(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) = for each n D where x X, n D and (0,1] then the fuzzy net S is denoted as { , n D} or simply{ }.

A fuzzy net = { , m E} in X is called a fuzzy subnet of fuzzy net S = { , n D}

If and only if there is a mapping f : E D such that i. = S o f, that is = for each i E

ii. For each n D there exist some m E such that f(m) n.

We shall denote a fuzzy subnet of a fuzzy net { , n D} by { , m E}.

2

-

Fuzzy -open Set

: Definition 2-1:

fuzzy set of a fuzzy topolo ical space (X ,T ) is said to be fuzzy -open set (F O(X)) if for each fuzzy -open set and there exist a fuzzy closed set such that

The complement of fuzzy EC-open set is fuzzy EC-closed set Definition 2-2 [6]:

Let be a fuzzy set in a fuzzy topological space (X ,T ) then :

 The EC-closure of is denoted by (EC-cl( )) and defined by

(x) = { is a fuzzy EC-closed set in X , } .

 The EC-interior of is denoted by (EC-int( )) and defined by

(x) = max { is a fuzzy EC-open set in X , }.

A fuzzy set in a fuzzy topological space (X ,T ) is called EC-quasi-neighborhood of a fuzzy point in X if and only if there exist (F O(X)) such that

and ≤ . Definition 2-4 [6]:

Let (X ,T ) be a fuzzy topological space and be a fuzzy point in X. then the family consisting of all EC- quasi-neighborhood (EC- q- neighborhood) of is called the system of EC- quasi-neighborhood of

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Let (X ,T ) be a fuzzy topological space and let S = { , n D} be a fuzzy net in X And (F O(X))^,then S is said to be :

i. EC-Eventually with if and only if m D such that , n m.

ii. EC-Frequently with if and only if n D m D, m n and . Definition 2-6 [6]:

Let (X ,T ) be a fuzzy topological space and let S = { , n D} be a fuzzy net in X And FP(X) then S is said to be ^:

i. EC-Convergent to and denoted by S , if S is EC-Eventually with , ii. Has a EC-Cluster point and denoted by S , if S is EC-Frequently with ,

A fuzzy topological space ( X , T ) is said to be Fuzzy EC- space (FEC- ) if for every pair of distinct fuzzy points , in X , there exists fuzzy EC-open sets , such that

, and , . Definition 2-8:

A fuzzy topological space (X, T ) is said to be Fuzzy EC- space (FEC- )

((Hausdroff space)) if for every pair of distinct fuzzy points , in X there exist two , FECO(X) such that , and .

Proposition 2-9 :

Let (X , T ) be a fuzzy topological space then : 1) Every fuzzy -open set is fuzzy EC-open set 2) Every fuzzy open set is fuzzy EC-open set 3) Every fuzzy -open set is fuzzy EC-open set 4) Every fuzzy EC-open set is fuzzy -open set

Proof : Obvious .

Remark 2-10 :

The converse of proposition (1.3.3) is not true in general as following examples show:- Examples 2-11 :

 let X = { a , b } and , , , , , , , are fuzzy subset in where = { ( a , 1.0 ) , ( b , 1.0 ) }, = { ( a , 0.5 ) , ( b , 0.4 ) }

= { ( a , 0.2 ) , ( b , 0.5 ) }, = { ( a , 0.5 ) , ( b , 0.5 ) } = { ( a , 0.2 ) , ( b , 0.4 ) }, = { ( a , 0.8 ) , ( b , 0.8 ) } = { ( a , 0.2 ) , ( b , 0.2 ) }, = { ( a , 0.3 ) , ( b , 0.3 ) }

et T = { , , , , , , , } be a fuzzy topology on , Then the fuzzy set is a fuzzy EC– open set but not fuzzy – open set (Resp.not fuzzy open set, not fuzzy – open set).

 let X = { a , b } and , , , , , are fuzzy subset in where = { ( a , 1.0 ) , ( b , 1.0 ) }, = { ( a , 0.5 ) , ( b , 0.6 ) }

= { ( a , 0.7 ) , ( b , 0.5 ) }, = { ( a , 0.5 ) , ( b , 0.5 ) } = { ( a , 0.7 ) , ( b , 0.6 ) }, = { ( a , 0.2 ) , ( b , 0.2 ) }

et T = { , , , , , } be a fuzzy topology on ,

Then the fuzzy set is a fuzzy e – open set but not fuzzy EC– open set

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Figure - 1 – illustrates the relations between fuzzy EC-open set and the given types of fuzzy open set.

Theorem 2-13:

A fuzzy topological space (X , ) is a fuzzy EC- – space if and only if every fuzzy point is a fuzzy closed set .

Proof:

( ) Let , are two distinct fuzzy points in which are fuzzy closed sets Then , are fuzzy open sets.

Hence , are fuzzy EC-open sets

Let = (x) and = (x), then And , hence (X , ) is fuzzy EC- – space

( ) suppose that (X , ) is fuzzy EC- – space, let ,

Then there exist , FECO(X) such that , and , .

In part , We have .

Let = max { : }, one may easily verify that

= , hence is fuzzy open set, That is, is fuzzy closed set Figure (1)

fuzzy -open set

fuzzy open set

fuzzy -open set

fuzzy -open set fuzzy EC-open

set

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If a fuzzy space (X ,T ) is fuzzy EC- – space, then the fuzzy families FECO(X) = F O(X).

Proof:

Let be any fuzzy subset of a fuzzy topological space (X ,T ) and F O(X)

If = , then FECO(X), if , then for each . Since a fuzzy space (X ,T ) is fuzzy EC- , then by theorem(2-13) { } is fuzzy closed set Hence , Therefore FECO(X),

hence F O(X) FECO(X), but FECO(X) F O(X) generally , Therefore F O(X) = FECO(X).

Proposition 2-15:

A fuzzy point is a EC-Cluster point of a fuzzy net { , n D} where (D, ) is a directed set, in a fuzzy topological space (X ,T ) if and only if has a fuzzy subnet which EC-Convergent to

Proof :-

Let be a EC-Cluster point of the fuzzy net { , n D}

with the directed set (D, ) as the domain.

Then for any there exist n D such that .

Let = {(n, ) : n D , and }, then ( , ) is directed set where (m, ) (n, ) if and only if m n in D and in

Then : FP(X) given by (m, ) = is a fuzzy subnet of a fuzzy net { , n D}

To show that . let then there exists , n D such that (n, ) And . Thus for any (m, ) such that (n, ) (n, )

we have (m, ) = , hence .

If a fuzzy net { , n D} has not EC-Cluster point

Then every fuzzy point there is EC- q- neighborhood of and n D

such that for all m n . then obviously no fuzzy net EC-Convergent to Proposition 2-16:

Let (X ,T ) be a fuzzy topological space, FP(X) and FECO(X), then (EC-cl( )) if and only if there exists a fuzzy net in EC-Convergent to .

Proof :-

Let (EC-cl( )) then for every there exists

(y) =

Such that + 1 , notice that ( , ) is a directed set Then S : FP(X) is defined as S( ) = is a fuzzy net in To prove that S

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Since ) +( ) 1 and then ) +( ) 1 thus

Let then since ) +( ) 1 and then ) +( ) 1 thus therefore S .

Let { , n D} be a fuzzy net in where (D, ) is a directed set such that then for every there exists m D such that for all n m Since then by proposition (1-7)

thus therefore (EC-cl( )) Proposition 2-17:

If X is a fuzzy EC- space then EC-Convergent fuzzy net on X has unique limit point.

Proof :-

Let be a fuzzy net on X such that , and x y

Since we have m1 D such that n m1 also

we have m₂ D such that n m₂ Since D is directed set then there exist m D such that m1 mand m2 m Then ( ) n m thus 0 and contradiction.

let X be a not fuzzy EC- space

Then there exists distinct fuzzy points , in X and . , Put = { : , }

Thus there exists then

is a fuzzy net in X Prove that and , let then (since = X) Thus thus also, so

has two limit point.

3

-

Fuzzy -Compact Space

:

Definition 3-1 :

A collection { :  }of fuzzy EC-open sets in a fuzzy topological space (X , ) is called a fuzzy -open cover of a fuzzy set if

 

Definition 3-2 :

A fuzzy topological space (X , ) is said to be a fuzzy -compact space if every -open cover of X has a finite subcover.

Theorem 3-3:

(X,T) is fuzzy EC-compact space if and only if every collection of fuzzy EC-closed subsets of X which has the finite intersection property has a nonempty intersection

Proof :- ( ) Let X be fuzzy - compact and { :  } be a collection of fuzzy - closed

subsets of X with the finite intersection property Suppose

  (x) Then { :  }is a fuzzy -open cover of X

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Since X is fuzzy -compact it contains a finite subcover { : i =1, ….. , n} for X This implies that

(x) This contradicts that { :  }has the finite intersection property

( ) let { :  } be a fuzzy -open cover of X consider the collection { :  } of fuzzy -closed sets since { :  } is a cover of

X the intersection of all members of { :  } is empty Hence { :  } does not have the finite intersection property

In the other words there are finite number of fuzzy EC-open sets , , …. , such that min{ (x)

This implies that { , , …. , } is a finite subcover of X Hence X is a fuzzy EC- compact space

Theorem 3-4:

A fuzzy EC-closed subset of a fuzzy EC- compact space is fuzzy EC- compact.

Proof :-

Let be a fuzzy EC-closed subset of a fuzzy space X and let { :  } be any family of fuzzy EC-closed in with finite intersection property, since is fuzzy EC-closed in X, then { :  } also are fuzzy EC-closed in X, since X is fuzzy EC- compact space,

Then by theorem (3-3)

  (x) therefore is fuzzy EC- compact space Theorem 3-5:

A fuzzy topological space (X , ) is a fuzzy EC- compact space if and only if every fuzzy filter base on X has a fuzzy EC-Cluster point.

Proof :-

( ) Let X be a fuzzy EC- compact space and let ={ :  } be a filter base on X having no a fuzzy EC-Cluster point. Let x X corresponding to each n N (N denoted the set of normal numbers) there exist EC-q-neighborhood of the fuzzy point and an such that . Since 1- (x) , we have (x) = 1, where = { : n N}

Thus = { : n N , x X} is fuzzy EC-open cover of X. since X is fuzzy EC-compact then there exist finitely many members , , ….. , of

Such that = 1 since is filter base then there exist F

Such that F ……… but then F 1 Consequently F = 0 and this contradicts the definition of a fuzzy filter base.

( ) let every fuzzy filter base on X have a fuzzy EC-cluster point. We have to show that X is fuzzy EC-compact space , let = { :  } be a family of fuzzy EC-closed sets having finite intersection property condition has a fuzzy EC-Cluster point say

Thus , so _ = _

Thus { F , F } 0 , Hence by theorem (3-3) , X is fuzzy EC- compact space Theorem 3-6:

A fuzzy topological space (X , ) is a fuzzy EC- compact space if and only if every fuzzy net in X has a fuzzy EC-Cluster point.

Proof :-

( ) Let X be a fuzzy EC- compact space and let {S(n) : n } be a fuzzy net in X which has no a fuzzy EC-Cluster point. Then for each fuzzy point there is EC-q-neighborhood of

and an such that for all m with m . Since

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©2017 RS Publication, rspublicationhouse@gmail.com Page 21 Then 0 m . let denoted the collection of all where runs over all fuzzy points in X. now to prove that the collection = { 1- : } is a family of fuzzy EC- closed sets in X possessing finite intersection property.

First notice that there exists k , ….. , such that i = 1,2,3,….., m for all p k (p ). i.e 1- = for all p k.

Hence {1- i = 1,2,3,….., m} 0 , since X is fuzzy EC-compact space , Then by theorem (3-3) there exist a fuzzy point in X such that

{ 1- : }= 1- { 1- : }, thus 1- for all And hence in particular 1- , i.e.

But by construction, for each fuzzy point there exist such that and we arrive at a contradiction.

( ) To prove that converse by theorem (3-5), that every filter base on X has a EC-Cluster point Let be a fuzzy filter base on X , then each F is non empty set, we choose a fuzzy point

F, let S={ : F }, let a relation " " be defined in as follows if and only if in X for , then ( , ) is directed set, now S is fuzzy net with the directed set ( , ) as domain. By hypothesis the fuzzy net S has a EC-Cluster point ,

Then for every EC-q-neighborhood U of and for each F there exist G with G F Such that U , As G F. it follows that F U for each F

Then by proposition (1-12) EC-Cl(F), Hence is a EC-Cluster point of

Corollary 3-7 :- a fuzzy topological space (X , ) is fuzzy EC-compact space if and only if every fuzzy net in X has a EC-convergent fuzzy subnet.

Proof :- By proposition (2-15) and theorem (3-6)

Theorem 3-8:

Every fuzzy EC-compact subset of a fuzzy EC-Hausdroff topological space is fuzzy EC-closed

Proof :-

Let EC-Cl( ), then by proposition(2-16) there exist a fuzzy net such that Since is fuzzy EC-compact and X is fuzzy EC- space then by corollary (3-7)

And proposition (2-17) then hence is fuzzy EC-closed set Proposition 3-9

Let (X,T) be a fuzzy topological space if and are two fuzzy EC-compact subsets of X then is also fuzzy compact

Proof:

Let { :  } be a fuzzy EC-open cover of Then

  i.e. max { ,

}

  Hence

G



since then

Also then

It is follows that { :  } is a fuzzy EC-open cover of and fuzzy EC-open cover of Since and are two fuzzy EC-compact sets, then there exist a finite sub cover ( , ,….., ) which is covering belong to { :  }

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: i=1,2,….,n} hence 

i

n

i 1

G



and there exist a finite sub cover ( , ,….., ) which is covering belong to { :  }

Then

max{

j=1,2,….,m} hence, 

j

m

j 1

G



it is follows that

max{

k=1,2,….,n+m} then 

k

n m

k 1

G





Thus, is fuzzy EC-compact. 

Remark 3-10:

Let (X,T) be a fuzzy topological space if and are two fuzzy EC-compact subsets of X then need not to be fuzzy EC-compact

Theorem 3-11:

In any fuzzy space the intersection of a fuzzy EC-compact set with a fuzzy EC-closed set is fuzzy EC-compact.

Proof:

Let be a fuzzy EC-compact set and be a fuzzy EC-closed set, to prove that Is a fuzzy EC-compact set. Let be a fuzzy net in , then is a fuzzy net in , Since is fuzzy EC-compact then by corollary (3-7) for some FP(X) And by proposition (2-16) EC-Cl( ) , since is fuzzy EC-closed set, then Hence and , thus is fuzzy EC-compact 

Proposition 3-12 :

Let (X , T ) be a fuzzy topological space then :

1) Every fuzzy EC-compact space is fuzzy R-compact space 2) Every fuzzy EC-compact space is fuzzy compact space 3) Every fuzzy EC-compact space is fuzzy -compact space 4) Every fuzzy e-compact space is fuzzy EC-compact space Proof : (1).

Let { :  } be a fuzzy regular open cover of fuzzy space X and X= _{ } Since every fuzzy regular open set is fuzzy EC-open set and X is a fuzzy EC-compact, then there exists , ,……, such that X= ,

Thus X is fuzzy regular compact space.

Proof : (2)(3)(4) similar (1)

Remark 3-13 :

The converse of proposition (1.3.3) is not true in general as following example show:- Example 3-14 :

Let X={a,b} and T={0,1, } where : X [0,1] such that = 1 -

x X , n Z, notice that the fuzzy topological space (X,T) is fuzzy regular compact space but not fuzzy EC-compact space

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Figure - 2 – illustrates the relations between fuzzy EC-compact space and the given types of fuzzy compact space.

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Figure (2) Fuzzy R-compact Space

Fuzzy compact Space

Fuzzy -compact Space

Fuzzy e-compact Space Fuzzy

EC-compact Space

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