Fuzzy Separation Axioms via Fuzzy α̂g-Kernel Set in Fuzzy Topological Spaces

European Journal of Scientific Research ISSN 1450-216X / 1450-202X Vol. 128 No 3 January, 2015, pp. 223-232 http://www.europeanjournalofscientificresearch.com

Fuzzy Separation Axioms via Fuzzy α ̂g-Kernel

Set in Fuzzy Topological Spaces

Qays Hatem Imran Department of Mathematics, College of Education for Pure Science

AL-Muthanna University, AL-Muthanna, Iraq E-mail: [email protected]

Murtadha M. Abdulkadhim

Department of Mathematics, College of Education for Pure Science AL-Muthanna University, AL-Muthanna, Iraq

E-mail: [email protected]

Abstract

In this paper, we introduce a new class of fuzzy closed sets called fuzzy α ̂g-closed sets also we introduce the concept of fuzzy α̂g-kernel set in a fuzzy topological space. We also investigate some of the properties of weak fuzzy separation axioms like fuzzy α̂g-Ri-space, i = 0,1,2,3 and fuzzy α̂g-Ti-space, i = 0,1,2,3,4.

Keywords: Fuzzy α ̂g-closed sets, fuzzy α̂g-kernel set, fuzzy α̂g-Ri-space, i = 0,1,2,3 and fuzzy α ̂g-Ti-space, i = 0,1,2,3,4.

1. Introduction The concept of fuzzy set and fuzzy set operations were first introduced by L. A. Zadeh in 1965 [7]. After Zadeh's introduction of fuzzy sets, Chang [3] defined and studied the notion of fuzzy topological space in 1968. In 1997, fuzzy generalized closed set (Fg-closed set) was introduced by G. Balasubramania and P. Sundaram [6]. In 2014, V. Senthilkumaran, R. Krishnakumar and Y. Palaniappan [14], introduced α̂g-closed set in topological spaces.

In this paper, we introduced the concept weakly ultra fuzzy separation of two fuzzy sets in fuzzy topological spaces using fuzzy α ̂g-closed set and using this concept to define the fuzzy α ̂g-kernel set of a fuzzy set A of a fuzzy topological space , . We also investigate some of the properties of weak fuzzy separation fuzzy α̂g-Ri-space, i = 0,1,2,3 and fuzzy α̂g-Ti-space, i = 0,1,2,3,4. 2. Preliminaries Fuzzy sets theory, introduced by L. A. Zadeh in 1965 [7], is the extension of classical set theory by allowing the membership of elements to range from 0 to 1. Let X be the universe of a classical set of objects. Membership in a classical subset Aof X is often viewed as a characteristic function from Xinto {0,1}, where

Fuzzy Separation Axioms via Fuzzy α̂g-Kernel Set in Fuzzy Topological Spaces 224

for any ∈X.

{0,1} is called a valuation set (see [13]). If the valuation set is allowed to be the real interval [0,1], A is called a fuzzy set in X. (or simply A( )) is the membership value (or degree of membership) of in A. Clearly, A is a subset of X that has no sharp boundary. A fuzzy set A in Xcan be represented by the set of pairs: A ={( , A( )), ∈X}.

Let A:X→[0,1] be a fuzzy set. If A( ) =1, for each ∈X, we denote it by 1 and if A( ) =0, for each ∈X, we denote it by 0 . That is, by 0 and 1 , we mean the constant fuzzy sets taking the values 0 and 1 on X, respectively [2]. Let 0,1 .The set of all fuzzy sets in X, denoted by [9].

Throughout this paper, the fuzzy closure and the fuzzy interior of A are denoted by cl(A) and int(A), respectively.

Definition 2.1:[4] Let A be a fuzzy set of a set X. The support of A is the elements whose membership value is greater than 0, i.e., supp(A)={ ∈X:A( ) 0}.

Definition 2.2:[5] Let A and B be any two fuzzy sets in X. Then we define AB:X[0,1] as follows: (AB)( ) =max {A( ),B( )}. Also, we define AB:X[0,1] as follows:

(AB)( ) = min{A( ),B( )}. By AB(AB), we mean the union (intersection) between two fuzzy sets A and B of X.

Definition 2.3:[4] Let A be any fuzzy set in a set X. The complement of A, is denoted by 1 or and defined as follows: =1A( ), for each ∈X.

Remark 2.4: From definition (2.2) and definition (2.3), we have, if A,B∈ , then AB, AB and 1 ∈ .

Definition 2.5:[8] A fuzzy point in a set X is a fuzzy set defined as follows:

Where 0 < λ 1. Now, supp( ) = : 0, but

supp( )= , so the value at is , and call the point its support of fuzzy point and is the height of . That is, has the membership degree 0 for all ∈X except one, say ∈X.

Definition 2.6:[3] A fuzzy topology on a set X is a family of fuzzy sets in X which satisfies the following conditions:

(i) 0 ,1 ∈ ,(ii) If A,B∈ , then AB∈ ,

(iii) If Ai:iJ is a family in , then ∈∨ ∈ .

is called a fuzzy topology for X and the pair , (or simply X) is a fuzzy topological space or fts for short. Every element of is called -fuzzy open set (fuzzy open set, for short ). A fuzzy set is -fuzzy closed (or simply fuzzy closed), if its complement is fuzzy open set. As ordinary topologies,

the indiscrete fuzzy topology on X contains only 0 and 1 (i.e., , X), while the discrete fuzzy topology on X contains all fuzzy sets in X.

0 , for ∉A ( see [4] )

1 , for ∈A

=

0 otherwise,

if

=

0otherwise,and0<1.Then,

if

=

225 Qays Hatem Imran and Murtadha M. Abdulkadhim

3. Fuzzy α ̂g-Closed Sets Definition 3.1:[1] A fuzzy subset Aof a fuzzy topological space , is called a fuzzy α-open set (briefly Fα-open set) if A int(cl(int(A))) and a fuzzy α-closed set (briefly Fα-closed set) if cl(int(cl(A))) A.

Definition 3.2:[6] A fuzzy subset A of a fuzzy topological space , is called a fuzzy generalized closed set (briefly Fg-closed set) if cl(A) U whenever A U and U is fuzzy open in X.

Definition 3.3:[11] A fuzzy subset A of a fuzzy topological space , is called a fuzzy αg-closed set (briefly Fαg-closed set) if αcl(A) U whenever A U and U is fuzzy α-open in X.

Remark 3.4:[12] Every fuzzy open (resp. fuzzy closed) set is a fuzzy α-open set (resp. fuzzy α-closed) set.

Proposition 3.5:[6,10] In a fuzzy topological space , , the following hold and the converse of each statement is not true:

(i) Every fuzzy closed set is Fg-closed. (ii) Every fuzzy α-closed set is Fαg-closed.

Definition 3.6: A fuzzy subset A of a fuzzy topological space , is called fuzzy α̂ generalized closed set (briefly Fα̂g-closed set) if int( cl( int(A))) U whenever A U and U is fuzzy open in . The complement of fuzzy α ̂g-closed set in X is fuzzy α ̂g-open in X, the family of all fuzzy α ̂g-open (fuzzy α̂g-closed) sets of an fuzzy topological space , is denoted by Fα̂g-O( ) (Fα ̂g-C( )).

Definition 3.7: The intersection of all fuzzy α̂g-closed sets in X containing A is called fuzzy α ̂ generalized closure of A and is denoted by α̂g-cl(A), α̂g-cl(A) B : A B, B is fuzzy α ̂g-closed.

Theorem 3.8: If a fuzzy subset A of X is fuzzy α̂g-closed in X, then int( cl( int(A))) –A does not contain any non empty fuzzy open subset of X.

Proof: Let A be fuzzy α̂g-closed. Let U be fuzzy open set such that int( cl( int(A))) –A U and U ≠ 0 , U int( cl( int(A))) –A 1 – A. That is A 1 – U, int(A) int(1 – U) 1 – U, cl( int(A)) 1 – U, int( cl( int(A))) 1 – A, U 1 – [int( cl( int(A)))]. Also U int( cl( int(A))), U (1 – [int( cl( int(A)))]) int( cl( int(A))) = 0 , a contradiction. This completes the proof. Theorem 3.9: If A is fuzzy α̂g-closed and A B int( cl( int(A))), then B is fuzzy α ̂g-closed. Proof: Let B U, where U is fuzzy open. Then int( cl( int(A))) U, B int( cl( int(A))). That is B cl( int(A)), int(B) B cl( int(A)), cl( int(B)) cl(cl( int(A))) = cl( int(A)), int( cl( int(B))) int( cl( int(A))) U. Hence B is fuzzy α ̂g-closed. Theorem 3.10: If A is both fuzzy open and Fg-closed in X, then it is fuzzy α̂g-closed in X. Proof: Let A be fuzzy open and Fg-closed in X. Let A U and U be fuzzy open in X. Now A A, cl(A) A by assumption. That is cl(A) U, int(A) cl(A). Thus cl(int(A)) cl(A),

int( cl( int(A))) cl(A) U. Hence A is fuzzy α̂g-closed in X. 4. Fuzzy α̂g-Kernel and Fuzzy α ̂g-Ri-Spaces, i = 0,1,2,3:Definition 4.1: The intersection of all fuzzy α̂g-open subset of X containing A is called the fuzzy α ̂g-kernel of A (briefly α̂g-ker(A)), this means α ̂g- ∧ ∈Fα ̂g-O( ): .

Definition 4.2: In a fuzzy topological space , , a fuzzy set A is said to be weakly ultra fuzzy α ̂g-separated from B if there exists a fuzzy α̂g-open set G such that ∧ 0 or ∧ α ̂g-

0 . By definition (4.2), we have the following: For every two distinct fuzzy points and of ,

(i) α ̂g- ∶ is not weakly ultra fuzzy α̂g-separated from . (ii) α ̂g- ∶ is not weakly ultra fuzzy α̂g-separated from .


Corollary 4.3: Let , be a fuzzy topological space, then ∈α ̂g- { }) iff ∈α̂g-({ }) for each ≠ ∈ .

Proof: Suppose that ∉α̂g- { }). Then there exists a fuzzy α ̂g-open set U containing such that ∉ . Therefore, we have ∉α ̂g- ({ }). The converse part can be proved in a similar way.

Definition 4.4: A fuzzy topological space , is called fuzzy α̂g-R0-space (Fα ̂g-R0-space, for short) if for each fuzzy α̂g-open set Uand ∈ , then α ̂g- .

Definition 4.5: A fuzzy topological space , is called fuzzy α̂g-R1-space (Fα ̂g-R1-space, for short) if for each two distinct fuzzy points and of with α̂g- ({ }) ≠ α̂g- ({ }), there exist disjoint fuzzy α ̂g-open sets U,V such that α̂g- and α ̂g- .

Theorem 4.6: Let , be a fuzzy topological space. Then , is Fα̂g-R0‐spaceif and only if α ̂g- α̂g- , for each ∈ .

Proof: Let , be a Fα̂g-R0‐space. If α̂g- α̂g- , for each ∈ ,then there exist anther fuzzy point such that ∈α ̂g- { }) and ∉α ̂g- { }) this means there exist an fuzzy α̂g-open set, ∉ implies α̂g- ({ } ≰ this contradiction. Thus α̂g- α ̂g- .

Conversely, let α̂g- α̂g- , for each fuzzy α̂g-open set U, ∈ , then α̂g- α ̂g- [By definition (4.1)]. Hence by definition (4.4), , is a Fα ̂g-R0-

space. Theorem 4.7: A fuzzy topological space , is an Fα ̂g-R0‐space if and only if for eachG

fuzzyα ̂g-closed set and ∈G , thenα̂g- ({ }) . Proof: Let for eachG fuzzy α ̂g-closed set and ∈G, then α̂g- ({ }) and letU be a

fuzzy α ̂g-open set, ∈ then for each ∉ implies ∈ is a fuzzy α ̂g-closed set implies α ̂g-({ }) [By assumption ]. Therefore ∉α ̂g- ({ }) implies ∉α ̂g- ({ })[By corollary

(4.3)]. Soα ̂g- .Thus , is anFα ̂g-R0‐space. Conversely, leta fuzzy topological space , beFα ̂g-R0‐space andGbe fuzzyα̂g-closed set

and ∈G. Then for each ∉G implies ∈ is fuzzy α̂g-open set, then α̂g- ({ }) [since, isFα̂g-R0‐space], soα ̂g- α̂g- . Thusα ̂g- ({ }) .

Corollary 4.8: A fuzzy topological space , isFα̂g-R0‐space if and only if for eachUfuzzy α ̂g-open set and ∈ , thenα ̂g- (α̂g- ({ })) .

Proof: Clearly. Theorem 4.9: Every Fα̂g-R1‐space is a Fα ̂g-R0‐space. Proof: Let , be a Fα̂g-R1‐space and let Ube a fuzzy α ̂g-open set, ∈ , then for each

∉U implies ∈ is an fuzzy α̂g-closed set and α ̂g- implies α̂g- ({ }) ≠ α ̂g-({ }). Hence by definition (4.5), α̂g- ({ }) . Thus , is a Fα̂g-R0‐space.

Theorem 4.10: A fuzzy topological space , is Fα ̂g-R1‐space if and only if for each ≠ ∈ with α̂g- ({ }) ≠ α ̂g- ({ }), then there exist fuzzy α̂g-closed sets , such that α̂g-

({ } , α ̂g- ({ } ∧ 0 and α ̂g- ({ }) , α ̂g- ({ })∧ 0 and ∨ 1 .

Proof: Let a fuzzy topological space , beFα ̂g-R1‐space. Then for each ≠ ∈ withα ̂g-({ }) ≠ α̂g- ({ }).Since everyFα ̂g-R1‐space is a Fα̂g-R0-space [by theorem (4.9)], and by

theorem (4.6), α̂g- ({ }) ≠ α̂g- ({ }), then there exist fuzzy α̂g-open sets , such that α̂g-({ }) and α ̂g- ({ }) and ∧ 0 [since , isFα ̂g-R1‐space], then and

are fuzzy α̂g-closed sets such that ∨ 1 .Put = and = . Thus ∈ and ∈ so that α̂g- ({ }) and α ̂g- ({ }) .

Conversely, let for each ≠ ∈ withα ̂g- ({ })≠ α ̂g- ({ }),there exist fuzzy α̂g-closed sets , such that α̂g- ({ }) , α̂g- ({ }) ∧ 0 and α̂g- ({ }) , α ̂g-

({ })∧ 0 and ∨ 1 , then and are fuzzy α̂g-open sets such that


∧ 0 . Put and . Thus, α ̂g- ({ }) and α ̂g- ({ }) and∧ 0 , so that ∈ and ∈ implies ∉α ̂g- ({ })and ∉α ̂g- ({ }), then α ̂g-

({ }) and α ̂g- ({ }) . Thus, , is aFα̂g-R1‐space.Corollary 4.11: A fuzzy topological space , isFα ̂g-R1‐space if and only if for each ≠

∈ with α̂g- ({ })≠ α̂g- ({ }) there exist disjoint fuzzy α̂g-open setsU,V such that α̂g- (α̂g-({ })) and α ̂g- (α̂g- ({ })) .

Proof: Let , be aFα ̂g-R1‐space and let ≠ ∈ withα ̂g- ({ })≠ α̂g- ({ }), then there exist disjoint fuzzyα ̂g-open setsU,Vsuch thatα̂g- ({ }) and α ̂g- ({ }) . Also , isFα̂g-R0‐space [by theorem (4.9)] implies for each ∈ , thenα̂g- ({ })= α̂g- ({ }) [By theorem (4.6)], but α̂g- ({ })= α̂g- (α̂g- ({ }))= α ̂g-cl(α̂g- ({ })). Thus α̂g- (α̂g- ({ })) and α̂g-

(α ̂g- ({ })) . Conversely, let for each ≠ ∈ withα̂g- ({ })≠ α̂g- ({ }) there exist disjoint fuzzy α̂g-

open sets U,V such that α̂g- (α̂g- ({ })) and α ̂g- (α̂g- ({ })) . Since { } α ̂g-({ }), then α ̂g- ({ }) α ̂g- (α̂g- ({ })) for each ∈ . So we get α̂g- ({ }) and α ̂g-

({ }) .Thus, , is a Fα̂g-R1‐space. Definition 4.12: Let , be a fuzzy topological space. Then is called:

(i) fuzzy α ̂g-regular space (Fα ̂gr-space, for short), if for each fuzzy point and each fuzzy α ̂g-closed set F such that ∈1 – F, there exist disjoint fuzzy α̂g-open sets Uand Vsuch that ∈U and F V.

(ii) fuzzy α ̂g-normal space (Fα̂gn-space, for short) iff for each pair of disjoint fuzzy α̂g-closed sets A and B, there exist disjoint fuzzy α̂g-open sets U andV such that A U and B V.

(iii) fuzzy α ̂g-R2-space (Fα̂g-R2-space, for short) if it is property fuzzy α̂g-regular space. (iv) fuzzy α ̂g-R3-space (Fα̂g-R3-space, for short) iff it is Fα̂g-R1-space and Fα̂gn-space.

Remark 4.13: Every Fα̂g-Rk-space is a Fα ̂g-Rk-1-space, 2,3. Proof: Clearly. Theorem 4.14: A fuzzy topological space , is Fα ̂gr-space (Fα ̂g-R2-space) if and only if for

each fuzzy α̂g-closed subset of and ∉ with α ̂g- α ̂g- ) then there exist fuzzy α ̂g-closed sets , such that α̂g- , α ̂g- ∧ =0 and α ̂g- ) , α ̂g-

) ∧ =0 and ∨ 1 . Proof: Let a fuzzy topological space , be Fα̂gr-space (Fα̂g-R2-space) and let be a fuzzy

α ̂g-closed set, ∉ , then there exist disjoint fuzzy α̂g-open sets , such that , ∈and ∧ 0 , then and are fuzzy α̂g-closed sets such that ∨ 1 . Put and

, so we get α̂g- , α ̂g- ∧ 0 and α ̂g- ) , α ̂g-) ∧ =0 and ∨ 1 .

Conversely, let for each fuzzy α ̂g-closed subset G of and ∉ with α ̂g- α ̂g-), then there exist fuzzy α ̂g-closed sets , such that α̂g- , α ̂g- ∧ =0

and α ̂g- ) , α ̂g- )∧ =0 and ∨ 1 . Then and are fuzzy α ̂g-open sets such that ∧ 0 and α ̂g- ∧ =0 , α ̂g- ∧ 0 . So thatand ∈ . Thus, , is Fα ̂gr-space (Fα̂g-R2-space).

Lemma 4.15: Let , be a Fα ̂gr-space and be a fuzzy α ̂g-closed set. Then α̂g- α ̂g- .

Proof: Let , be a Fα ̂gr-space and be a fuzzy α ̂g-closed set. Then for each ∉ , there exist disjoint fuzzy α̂g-open sets , such that and ∈ . Since α ̂g- , implies α ̂g-

∧ 0 , thus ∉α ̂g- (α̂g- . We showing that if ∉ implies ∉α̂g- (α̂g-


, therefore α̂g- (α̂g- α̂g- . As α ̂g- α ̂g- [By definition (4.1)]. Thus, α ̂g- α ̂g- .

Theorem 4.16: A fuzzy topological space , is Fα ̂gr-space (Fα̂g-R2-space) iff for each fuzzy α̂g-closed subset of and ∉ with α̂g- (α̂g- α̂g- (α ̂g- , then there exist disjoint fuzzy α̂g-open sets , such that α̂g- (α ̂g- and α ̂g- (α̂g- .

Proof: Let a fuzzy topological space , be Fα ̂gr-space (Fα̂g-R2-space) and let be a fuzzy α ̂g-closed set, ∉ . Then there exist disjoint fuzzy α̂g-open set , such that and ∈ . By lemma (4.15), α̂g- (α̂g- α ̂g- , in the other hand , is a Fα ̂g-R0‐space [By theorem (4.9) and remark (4.13)]. Hence, by theorem (4.6), α̂g- α̂g- , for each ∈

. Thus, α ̂g- (α ̂g- and α ̂g- (α̂g- . Conversely, let for each fuzzy α̂g-closed set and ∉ with α̂g- (α̂g- α̂g- (α ̂g-

, then there exist disjoint fuzzy α ̂g-open sets , such that α̂g- (α̂g- and α̂g-(α ̂g- . Then and ∈ . Thus, , is Fα̂gr-space (Fα̂g-R2-space).

Theorem 4.17: A fuzzy topological space , is Fα ̂gn-space if and only if for each disjoint fuzzy α ̂g-closed sets , with α̂g- α ̂g- then there exist fuzzy α̂g-closed sets , such that α̂g- , α ̂g- ∧ 0 and α ̂g- , α ̂g- ∧ 0 and ∨ 1 .

Proof: Let a fuzzy topological space , be Fα ̂gn-space and let for each disjoint fuzzy α̂g-closed sets , with α̂g- α ̂g- then there exist disjoint fuzzy α̂g-open sets , such that and and ∧ 0 , then and are fuzzy α̂g-closed sets such that ∨1 and α ̂g- ∧ 0 , α ̂g- ∧ 0 . Put and . Thus, α ̂g-

, α ̂g- ∧ 0 and α ̂g- , α ̂g- ∧ 0 . Conversely, let for each disjoint fuzzy α̂g-closed sets , with α̂g- α ̂g- , there

exist fuzzy α̂g-closed sets , such that α̂g- , α ̂g- ∧ 0 and α ̂g-, α ̂g- ∧ 0 and ∨ 1 implies and are fuzzy α̂g-open sets such that

∧ 0 . Put and , thus α ̂g- and α ̂g- , so that and . Thus , is Fα ̂gn-space.

Theorem 4.18: Every Fα ̂g-R3-space is Fα ̂gr-space. Proof: Let be a fuzzy α̂g-closed and ∉ . Then α̂g- α̂g- , then for each

∈ there exist fuzzy α̂g-closed sets , such that α̂g- ({ , α̂g- ({ ∧

0 and α̂g- ({ , α̂g- ({ ∧ 0 [since , is Fα̂g-R1-space and by theorem (4.10)], let ∧ : ∈ , so we have ∧ 0 . Hence , is Fα̂gn-space, then there exist disjoint fuzzy α̂g-open sets , such that and ∈ . Thus, , is Fα̂gr-space. 5. Fuzzy α̂g-Ti-Spaces, i = 0, 1, 2,3,4: Definition 5.1: Let , be a fuzzy topological space. Then is called:

(i) fuzzy α ̂g-T0‐space (Fα̂g-T0-space, for short) iff for each pair of distinct fuzzy points in , there exists a fuzzy α̂g-open set in containing one and not the other.

(ii) fuzzy α ̂g-T1-space (Fα̂g-T1-space, for short) iff for each pair of distinct fuzzy points and of , there exists fuzzy α̂g-open sets , containing and respectively such that ∉ and ∉H.

(iii) fuzzy α ̂g-T2-space (Fα̂g-T2-space, for short) iff for each pair of distinct fuzzy points and of , there exist disjoint fuzzy α̂g-open sets , in such that ∈ and ∈H.

(iv) fuzzy α ̂g-T3-space (Fα̂g-T3-space, for short) iff it is Fα̂g-T1-space and Fα̂gr-space. (v) fuzzy α ̂g-T4-space (Fα̂g-T4-space, for short) iff it is Fα̂g-T1-space and Fα̂gn-space.


Theorem 5.2: A fuzzy topological space , isFα̂g-T0‐space if and only if either ∉α ̂g-or ∉α ̂g- , for each ∈ .

Proof: Let , be aFα ̂g-T0‐space then for each ∈ , there exists a fuzzyα̂g-open setGsuch that ∈ , ∉ or ∉ , ∈ . Thus either ∈ , ∉ implies ∉α ̂g- { }) or∉ , ∈ implies ∉α̂g- ).

Conversely, let either ∉α ̂g- or ∉α ̂g- ), for each ∈ .Then there exists afuzzyα ̂g-open setGsuch that ∈ , ∉ or ∉ , ∈ .Thus , is aFα ̂g-T0‐space.

Theorem 5.3: A fuzzy topological space , is Fα ̂g-T0-space if and only if either α ̂g-({ }) is weakly ultra fuzzy α̂g-separated from or α ̂g- ) is weakly ultra fuzzy α̂g-

separated from { } for each ∈ . Proof: Let , be a Fα̂g-T0-space then for each ∈ , there exists a fuzzy α̂g-open set G such

that ∈ , ∉ or ∉ , ∈ . Now if ∈ , ∉ implies α̂g- ({ }) is weakly ultra fuzzy α̂g-separated from { }. Or if ∉ , ∈ implies α̂g- { }) is weakly ultra fuzzy α̂g-separated from { }.

Conversely, let either α ̂g- ({ }) be weakly ultra fuzzy α̂g-separated from { } or α̂g-{ }) be weakly ultra fuzzy α ̂g-separated from { }. Then there exists a fuzzy α ̂g-open set G such

that α̂g- and ∉ or α ̂g- , ∉ implies ∈ , ∉ or ∉ , ∈ . Thus, , is a Fα ̂g-T0-space.

Theorem 5.4: A fuzzy topological space , is Fα ̂g-T1-space if and only if for each ∈ , α̂g- ({ }) is weakly ultra fuzzy α̂g-separated from { } and α̂g- { }) is weakly ultra

fuzzy α ̂g-separated from { }.Proof: Let , be a Fα̂g-T1-space then for each ∈ , there exist fuzzy α̂g-open sets U,V

such that ∈U, ∉U and ∉V, ∈V. Implies α ̂g- ({ }) is weakly ultra fuzzy α̂g-separated from { } and α ̂g- { }) is weakly ultra fuzzy α̂g-separated from { }.

Conversely, let α̂g- ({ }) be weakly ultra fuzzy α̂g-separated from { }and α ̂g- ({ }) be weakly ultra fuzzy α̂g-separated from { }. Then there exist fuzzy α̂g-open sets U,V such that α ̂g-

, ∉U and α ̂g- , ∉V implies ∈U, ∉U and ∉V, ∈V. Thus, , is a Fα̂g-T1-space.

Theorem 5.5: A fuzzy topological space , is Fα̂g-T1-space if and only if for each ∈ , α ̂g-.

Proof: Let , be a Fα ̂g-T1-space and let α̂g- . Then α ̂g- contains anther fuzzy point distinct from say . So ∈α ̂g- implies α ̂g- is not weakly ultra fuzzy α̂g-separated from { }. Hence by theorem (5.4), , is not a Fα ̂g-T1-space this is contradiction. Thus α̂g- .

Conversely, let α̂g- , for each ∈ and let , be not a Fα̂g-T1-space. Then by theorem (5.4), α̂g- is not weakly ultra fuzzy α̂g-separated from { }, this means that for every fuzzy α̂g-open set G contains α̂g- { }) then ∈ implies ∈ ∧ ∈Fα̂g-O(X ) : ∈G} implies ∈α̂g- , this is contradiction. Thus, , is a Fα̂g-T1-space.

Theorem 5.6: A fuzzy topological space , isFα ̂g-T1‐space if and only if for each ∈ , ∉α ̂g- and ∉α ̂g- .

Proof: Let , be aFα̂g-T1‐space then for each ∈ ,there exists fuzzyα̂g-open sets , such that ∈U, ∉ and ∈ , ∉ .Implies ∉α ̂g- and ∉α ̂g- .

Conversely, let ∉α̂g- and ∉α ̂g- ), for each ∈ .Then there exists fuzzy α ̂g-open sets , such that ∈U, ∉ and ∈ , ∉ .Thus, , isFα ̂g-T1‐space.

Theorem 5.7:A fuzzy topological space , is Fα̂g-T1‐space if and only if for each ∈ implies α̂g- ∧ α ̂g- 0 .

Proof: Let a fuzzy topological space , be Fα ̂g-T1‐space. Thenα ̂g- and α ̂g- [By theorem (5.5)]. Thus α̂g- ∧ α ̂g- 0 .


Conversely, let for each ∈ implies α̂g- ∧ α ̂g- 0 and let , be not Fα ̂g-T1‐space then for each ∈ implies ∈α ̂g- or ∈α̂g- ) [by theorem (5.6)], thenα ̂g- ∧ α ̂g- 0 this is contradiction. Thus, , is aFα̂g-T1‐space.

Theorem 5.8: A fuzzy topological space , is Fα̂g-T1‐space if and only if , is Fα̂g-T0‐space and Fα ̂g-R0‐space.

Proof: Let , be Fα̂g-T1‐space and let ∈Ube fuzzy α ̂g-open set, then for each ∈ , α ̂g- ∧ α̂g- 0 [by theorem (5.7)] implies ∉α ̂g- ) and ∉α ̂g- ) this means α ̂g- , hence α̂g- . Thus, , is a Fα ̂g-R0‐space.

Conversely, let , be Fα̂g-T0‐space and Fα ̂g-R0-space, then for each ∈ there exists afuzzy α ̂g-open setUsuch that ∈U, ∉Uor ∉U, ∈U.Say ∈U, ∉Usince , is a Fα ̂g-R0‐space, then α ̂g- , this means there exists a fuzzy α̂g-open setVsuch that ∈V, ∉V.Thus,, is a Fα̂g-T1‐space.

Theorem 5.9: A fuzzy topological space , is Fα ̂g-T2‐space if and only if (i) , is Fα̂g-T0‐space and Fα̂g-R1‐space.

(ii) , is Fα̂g-T1‐space and Fα̂g-R1‐space. Proof (i): Let , be a Fα̂g-T2‐space then it is a Fα̂g-T0‐space. Now since , be Fα̂g-T2‐

space then for each ∈ , there exist disjoint fuzzy α̂g-open sets , such that ∈U and ∈V implies ∉α ̂g- and ∉α ̂g- , therefore α̂g- and α̂g-

. Thus, , isFα̂g-R1‐space. Conversely, let , be a Fα̂g-T0‐space and Fα ̂g-R1‐space, then for each ∈ , there exists

a fuzzy α̂g-open setUsuch that ∈U, ∉U or ∈U, ∉U, impliesα ̂g- α ̂g- ,since , is Fα̂g-R1‐space [By assumption], then there exist disjoint fuzzy α̂g-open sets , such that ∈Gand ∈H [Definition (4.5)].Thus, , is a Fα̂g-T2‐space.

Proof (ii): By the same way of part (i) a Fα̂g-T2-space is Fα̂g-T1-space and Fα ̂g-R1-space. Conversely, let , be a Fα ̂g-T1‐space and Fα ̂g-R1‐space, then for each ∈ ,there exist

fuzzy α ̂g-open setsU,V such that ∈U, ∉U and ∈V, ∉V implies α̂g- α ̂g- ,since , is Fα̂g-R1‐space, then there exist disjoint fuzzy α ̂g-open sets , such that ∈Gand ∈H. Thus, , is a Fα̂g-T2‐space.

Corollary 5.10:A fuzzy topological Fα ̂g-T0‐space isFα̂g-T2‐space if and only if for each ∈ with α̂g- α ̂g- then there exist fuzzy α̂g-closed sets , such that α̂g-

, α̂g- ∧ 0 and α ̂g- , α ̂g- ∧ 0 and ∨ 1 .

Proof: By theorem (4.10) and theorem (5.9). Corollary 5.11: A fuzzy topological Fα ̂g-T1‐space isFα̂g-T2‐space if and only if one of the

following conditions holds: (i) For each ∈ withα ̂g- α ̂g- ,then there exist fuzzy α ̂g-open setsU,V

such that α̂g- α ̂g- andα ̂g- α ̂g- .(ii) For each ∈ withα ̂g- α ̂g- ,then there exist fuzzy α ̂g-closed

sets , such thatα̂g- ,α ̂g- ∧ 0 andα ̂g- ,α ̂g- ∧ 0 and ∨ 1 .

Proof (i): By corollary (4.11) and theorem (5.9). Proof (ii): By theorem (4.10) and theorem (5.9). Remark 5.12: Every Fα̂g-Tk‐space is a Fα̂g-Tk‐1‐space, 1,2,3,4. Proof: Clearly. Theorem 5.13: A fuzzy topologicalFα ̂g-R1‐space is Fα̂g-T2‐space if and only if one of the

following conditions holds: (i) For each ∈ ,α ̂g- .


(ii) For each ∈ , α ̂g- α ̂g- )implies α̂g- ∧ α ̂g- 0 .

(iii) For each ∈ , either ∉α ̂g- or ∉α̂g- .(iv) For each ∈ then ∉α ̂g- and ∉α ̂g- ).

Proof (i): Let , be Fα̂g-T2‐space. Then , is a Fα̂g-T1‐space and Fα ̂g-R1‐space [By theorem (5.9)]. Hence by theorem (5.5), α̂g- for each ∈ .

Conversely, let for each ∈ , α̂g- , then by theorem (5.5), , is aFα̂g-T1‐space.Also , is aFα ̂g-R1‐space by assumption. Hence by theorem (5.9), , is a Fα ̂g-T2‐space.

Proof (ii): Let , be aFα ̂g-T2‐space. Then , is Fα̂g-T1‐space [By remark (5.12)].Hence by theorem (5.7),α̂g- ∧ α ̂g- 0 for each ∈ .

Conversely, assume that for each ∈ , α ̂g- α ̂g- ) implies α ̂g-∧ α̂g- 0 . So by theorem (5.7), , is aFα ̂g-T1‐space, also , is a Fα̂g-R1‐

space by assumption. Hence by theorem (5.9), , is aFα ̂g-T2‐space. Proof (iii): Let , be aFα̂g‐T2‐space. Then , is aFα ̂g-T0‐space [By remark (5.12)].

Hence by theorem (5.2), either ∉α ̂g- or ∉α ̂g- for each ∈ . Conversely, assume that for each ∈ , either ∉α ̂g- or ∉α ̂g- ) for

each ∈ . So by theorem (5.2), , is a Fα ̂g-T0-space, also , is Fα ̂g-R1‐space by assumption. Thus , is a Fα ̂g-T2‐space [By theorem (5.9)].

Proof (iv): Let , be a Fα ̂g-T2‐space. Then , is a Fα ̂g-T1‐space andFα ̂g-R1‐space [By theorem (5.9)]. Hence by theorem (5.6), ∉α̂g- and ∉α ̂g- ).

Conversely, let for each ∈ then ∉α̂g- and ∉α ̂g- ). Then by theorem (5.6), , is a Fα ̂g-T1‐space. Also , is a Fα ̂g-R1‐space by assumption. Hence by theorem (5.9), , is aFα̂g-T2‐space.

Remark 5.14: Each fuzzy separation axiom is defined as the conjunction of two weaker axioms: Fα ̂g-Tk-space = Fα̂g-Rk-1-space and Fα ̂g-Tk-1-space = Fα̂g-Rk-1-space and Fα ̂g-T0-space,

1,2,3,4. Theorem 5.15: Let , be a fuzzy topological space and α̂g- for each ∈

then , is Fα ̂g-T3‐space if and only if it is a Fα̂g-R2-space. Proof: Let , be a Fα ̂g-T3‐space. Then, , is a Fα̂g-R2-space [By remark (5.14)]. Conversely, let , be a Fα ̂g-R2-space then it is a Fα̂gr-space [Definition (4.12)(iii)]. By

assumption, α̂g- for each ∈ , then , is a Fα ̂g-T1‐space [by theorem (5.5)]. Hence by remark (5.14), , is a Fα̂g-T3‐space.

Theorem 5.16: Let , be a fuzzy topological space and let ∈ , implies α̂g-∧ α̂g- 0 , then , is a Fα ̂g-T3‐space if and only if it is a Fα̂g-R2-space.

Proof: Let , be a Fα ̂g-T3‐space. Then , is a Fα̂g-R2-space [By remark (5.14)]. Conversely, let , be a Fα ̂g-R2-space then it is a Fα̂gr-space [Definition (4.12)(iii)]. By

assumption, α ̂g- ∧ α̂g- 0 , for each ∈ , then by theorem (5.7), , is a Fα ̂g-T1‐space. Hence by remark (5.14), , is a Fα̂g-T3‐space.

Theorem 5.17: Let , be a fuzzy topological space and for each ∈ either ∉α ̂g- or ∉α ̂g- , then , is a Fα̂g-T3‐space if and only if it is a Fα̂g-R2-space.

Proof: Let , be a Fα ̂g-T3‐space. Then , is a Fα̂g-R2-space [By remark (5.14)]. Conversely, let , be a Fα̂g-R2-space then it is a Fα̂gr-space [Definition (4.12)(iii)]. By

assumption, for each ∈ either ∉α̂g- or ∉α̂g- . This means either α̂g-({ }) is weakly ultra fuzzy α̂g-separated from { } or α̂g- { }) is weakly ultra fuzzy α̂g-separated

from { }, so by theorem (5.3), , is Fα̂g-T0‐space. Hence by remark (5.14), , is a Fα̂g-T3‐space. Theorem 5.18: Let , be a fuzzy topological space and let ∈ , then ∉α ̂g-

and ∉α ̂g- , , is a Fα̂g-T3‐space if and only if it is a Fα̂g-R2-space.


Proof: Let , be a Fα ̂g-T3‐space. Then , is a Fα̂g-R2-space [By remark (5.14)]. Conversely, let , be a Fα̂g-R2-space then it is a Fα̂gr-space [Definition (4.12)(iii)]. By

assumption, for each ∈ then ∉α̂g- and ∉α̂g- . Therefore, α̂g- ({ }) is weakly ultra fuzzy α̂g-separated from { } and α̂g- { }) is weakly ultra fuzzy α̂g-separated from { }, so by theorem (5.4), , is Fα̂g-T1‐space. Hence by remark (5.14), , is a Fα̂g-T3‐space.

Remark 5.19: The relation between fuzzy α̂g-separation axioms can be representing as a matrix. Therefore, the element refers to this relation. As the following matrix representation shows:

and Fα̂g-T0 Fα̂g-T1 Fα̂g-T2 Fα̂g-T3 Fα̂g-T4 Fα̂g-R0 Fα̂g-R1 Fα̂g-R2 Fα̂g-R3 Fα̂g-T0 Fα̂g-T0 Fα̂g-T1 Fα̂g-T2 Fα̂g-T3 Fα̂g-T4 Fα̂g-T1 Fα̂g-T2 Fα̂g-R3 Fα̂g-T4 Fα̂g-T1 Fα̂g-T1 Fα̂g-T1 Fα̂g-T2 Fα̂g-T3 Fα̂g-T4 Fα̂g-T1 Fα̂g-T2 Fα̂g-R3 Fα̂g-T4 Fα̂g-T2 Fα̂g-T2 Fα̂g-T2 Fα̂g-T2 Fα̂g-T3 Fα̂g-T4 Fα̂g-T2 Fα̂g-T2 Fα̂g-R3 Fα̂g-T4 Fα̂g-T3 Fα̂g-T3 Fα̂g-T3 Fα̂g-T3 Fα̂g-T3 Fα̂g-T4 Fα̂g-T3 Fα̂g-T3 Fα̂g-T3 Fα̂g-T4 Fα̂g-T4 Fα̂g-T4 Fα̂g-T4 Fα̂g-T4 Fα̂g-T4 Fα̂g-T4 Fα̂g-T4 Fα̂g-T4 Fα̂g-T4 Fα̂g-T4 Fα̂g-R0 Fα̂g-T1 Fα̂g-T1 Fα̂g-T2 Fα̂g-T3 Fα̂g-T4 Fα̂g-R0 Fα̂g-R1 Fα̂g-R2 Fα̂g-R3 Fα̂g-R1 Fα̂g-T2 Fα̂g-T2 Fα̂g-T2 Fα̂g-T3 Fα̂g-T4 Fα̂g-R1 Fα̂g-R1 Fα̂g-R2 Fα̂g-R3 Fα̂g-R2 Fα̂g-T3 Fα̂g-T3 Fα̂g-T3 Fα̂g-T3 Fα̂g-T4 Fα̂g-R2 Fα̂g-R2 Fα̂g-R2 Fα̂g-R3 Fα̂g-R3 Fα̂g-T4 Fα̂g-T4 Fα̂g-T4 Fα̂g-T4 Fα̂g-T4 Fα̂g-R3 Fα̂g-R3 Fα̂g-R3 Fα̂g-R3

Matrix Representation

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Fuzzy Separation Axioms via Fuzzy α̂g-Kernel Set in Fuzzy Topological Spaces

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