E L S E V I E R Fuzzy Sets and Systems 85 (1997) 385-389

FUZ2Y sets and systems

On lower separation axioms

M.H. Ghanim*, O.A. Tantawy, Fawzia M. Selim Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

Received November 1993; revised October 1995

Abstract

In [3] Mashhour et al. introduced the concept of fuzzy disjointness. Two fuzzy sets A,B in X are said to be fuzzy disjoint if A ~< co B, where co B is the complement of B. Using this concept, they introduced FT~ separation axioms as a generalization for the basic separation axioms T~ (i = 1, ... ,4),

In [5] Singal and Rajvanshi used the concepts of regular open fuzzy sets [13 and fuzzy disjointness to introduce some fuzzy almost separation axioms which are stronger than those of Mashhour. Unfortunately, some results in [5] are incorrect.

In this paper, we use the concept of the quasi-coincident relation I-4] to restate lower fuzzy separation axioms and give a corrected version for some definitions and results in [5].

Keywords: Fuzzy topology; Separation axioms

1. Preliminaries and

Definition 1.1 (Chang [2]). Let X be a non-empty set, z c I x, the pair (X, ~) is an fts iff it satisfies the following condit ions:

(1) O, x e ~ . (2) If u, v s v, then u n v ~ r,

(3) I f u j s z f o r a l l j e J , then U { u j I j E J } e ~ . Let A be a fuzzy set in X. The closure and the

interior of A (denoted by cl A and int A resp.) are defined by

in tA = U { G e l X l G e r and G ~< A},

respectively.

Definition 1.2 (Azad [1]). Let (X,~) be an fts. A fuzzy set A in X is said to be regular-open iff A = int cl A. A fuzzy set u in X is said to be regular- closed iffco u is regular-open. Equivalently u ~ I x is regular-closed iff

c lA = O { F e l X I F e r ~ and A ~< F} u = cl int u.

* Corresponding author.

Thus every regular-open (regular-closed) fuzzy set is open (closed) but the converse does not hold El].

0165-0114/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved SSD1 0 1 6 5 - 0 1 1 4 ( 9 5 ) 0 0 3 4 3 - 6

386 M.H. Ghanim et al. / Fuzzy Sets and Systems 85 (1997) 385-389

Definition 1.3 (Azad [1]). A subfamily fi of I x (i.e. fl c I x) is a base for a fuzzy topology on X iff:

(1) X = U { B I B ~ f l } . (2) For each B1, B2 in fl and for each fuzzy point

xx e B1 c~B2 there exists B* ~ fl such that x;, ~ B* B 1 ( ~ B 2 .

Definition 1.4 (Azad [1]). Let (X,z) be an fls. The family fl = {A ~ IX lA is regular-open} is a base for a fuzzy topology z* on X. The fts (X, z*) is called the semi-regularization of (X,z). The fts (X,z) is said to be a semi-regular space iff r = z*.

Definition 1.5 (Pu Pao Ming and Liu Ying Ming [4]). Let A , B be two fuzzy sets in X. A is said to be quasi-coincident with B (denoted A qB) iff there exists x e X such that A ( x ) + B ( x ) > 1. Conse- quently, a fuzzy point xx is quasi-coincident with A i f f ) . + A ( x ) > 1.

Definition 1.6 (Singal and Rajvanshi [5]). Let (X, r) be an fts. A fuzzy set A in X is said to be b-open iff A s r * , that is A e I x is b-open iffA = (j~u~ where ui is a regular-open fuzzy set for each i. The fuzzy set A in X is said to be b-closed iff co A is b-open. Equivalently, A ~ I x is b-closed iff A = ~ F i where Fi is a regular-closed fuzzy set for each i.

Proposition 1.1 (Azad [1]). The closure of an open f u z z y set is regular-closed.

(2) The interior of a closed f u z z y set is regular- open.

Definition 1.7 (Azad [1]). Let (X,z) and (Y, U) be two fts. A mapping f : X ~ Y is called a fuzzy al- most-continuous mapping iff the inverse image of every regular-open fuzzy set in Y is open in X.

Definition 1.8 (Singal and Rajvanshi [5]). Let (X, r) and (Y, U) be two Its. A mapping f : X --+ Y is fuzzy almost open (closed) iff for every regular-open (regular-closed) fuzzy set u in X, f (u ) is open (closed) in Y.

For notions and results used but not defined or shown we refer to [4].

2. Fuzzy quasi-separation axioms

Using the concept of a quasi-coincident relation, we introduce the fuzzy quasi-separation axioms. These new axioms as well as those in [3] are re- duced, in the crisp case, to the ordinary basic separ- ation axioms.

Definition 2.1. An fts (X, z) is said to be: (i) Fuzzy quasi-To (in short FQT0) space iff for

every pair of fuzzy points x x , y , in X (x # y) there exists u ~ r such that x ~ q u <~ co y , or y u q u <. c o x 2 .

(ii) Fuzzy quasi-T1 (in short FQT1) space ifffor every pair of fuzzy points xx, y , in X (x # y) there exist u,v ~ z such that x zq u ~< coy , and y, qv ~< c o x 2 .

(iii) Fuzzy quasi-T2 (in short FQT2) space iff for every pair of fuzzy points xz, y, in X (x # y) there exist u, v ~ r such that x2 q u ~ co y,, Yu q v ~< co x~ and ugly.

(iv) Fuzzy quasi-Ts (in short FQTs) space iff every fuzzy point in X is closed.

(v) Fuzzy quasi-T21/2 (in short FQT21/2) space iff for every pair of fuzzy points xz, Yu in X (x 4= y) there exist u, v ~ z such that x;. q u ~< co y,, y, q v ~< co xa and (cl u)~((cl v).

(vi) Fuzzy quasi-regular (in short FQR) space iff for every fuzzy point xa in X and closed fuzzy set F in X such that xa q (co F) there exist u, v e z such that x~ q u, F q v and uc~v = O.

A fuzzy quasi-regular space which is FQTs is said to be FQT3-space.

(vii) Fuzzy quasi-normal (in short FQN) space iff for every pair of closed fuzzy sets F1, F2 in X such that FI q (co F2) there exist u, v e r such that F lqU, F 2 q v and uc~v = O.

A fuzzy quasi-normal space which is FQTs is said to be FQT4-space.

Lemma 1.1 (Singal and Rajvanshi [5]). I f f : X ~ Y is f u z z y almost continuous and f u z z y almost open, then the inverse image of every f u z z y regular-open (regular-closed) set is regular-open (regular-closed).

One may notice that, in the crisp case, the above FQTi axioms are reduced to the ordinary Ti-axiom for every i ~ {0, 1, 2, 2½, 3, 4}. Also FQTs is reduced to the ordinary Tl-axiom.

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Theorem 2.1. An f t s ( X , z ) is FQT1 /ff every crisp point in X is closed.

Proof. Let x~ be a crisp point in X. Then, for every fuzzy point p = y , in X with x v a y there exist u, up ~ z such that x:. q u <~ co p and p q up ~< co Xl.

Hence, for every fuzzy point p with pq(cox~) , there exist up e z such that p q up. Consequently, Pq( U up). Thus, pq (cox~) implies p q(~jup). Since U u~ ~< c o x , , then, for every fuzzy point p with pq(~j up), p q(co x~). Hence p q (/.j u~) i f fpq (co x~), i.e. ~ju~ = c o x , . Therefore, cox1 is open and x~ is closed.

Conversely, let xa, y, be a pair of fuzzy points in X with x 4: y. Since x~ and y~ are closed, co xa and coy~ are open. Since x: .qcoy~; y, qcox~ , cox~ <. cox:. and coy~ ~< coy , ; then (X,z) is FQT~.

Corollary 2.1. Every FQT~ is FQT1, but the con- verse is not true in general.

Example 2.1. Let X:{Xl,X2} and let A = { x ~ l x e X , 2~[½,1]} and B = { u e l X ] r a n g e u

[½,1]}. Let z = A w B w { O } . Then, z is a fuzzy topology on X. Since all crisp points are r-closed, (X, z) is FQTa. But the fuzzy point p with support x~ and value ¼ is not r-closed. Consequently, (X, z) is not FQT~.

Theorem 2.2. For every i ~ {0, 1, 2, 2½, 3, S } the cor- responding F Q T i property is hereditary.

Proof. Let us prove, for example, the case = 0. Let (X, z) be an FQT0 fts and let (Y, zy) be a subspace of (X, z). Let x~, y, be two fuzzy points in Y such that x # y. Since Y ~< X, then there exist u ~ z such that x x q u <~ coy , or y, q u <<, cox , . Let u* = Yc~ u~ rr , then x z q u * ~ < c o y , or yuqu*~<cox~ . Hence, (Y, zy) is FQT0. The proofs for other cases are similar.

E l , F 2 are closed in X and F~ q(coF2), then there exist u, vE z such that F l q u , F 2 q v and uc~v = O. Let u* = uc~ Y, v* = v~ Y. Hence (Y, zv) is FQN.

Remark 2.1. One may easily prove that for every iE{0,1,2 ,2½,3,4, S} the corresponding FQTi property is additive (see [-3] for the definition of an additive property).

3. Fuzzy almost-quasi-separation axioms

In this section we introduce the concepts of fuzzy almost-quasi-separation axioms. Using these new axioms we avoid some deviations in [5].

Definition 3.1. An fts (X, z) is said to be fuzzy al- most-quasi-To (in short FAQTo) ifffor every pair of fuzzy points x~ ,y , in X with x :~ y there exists a regular-open fuzzy set u such that x~ q u ~< co y, or y, q u <~ coxx. Equivalently, there exists a 6- open fuzzy set w such that x x q w <<, c o y , or y u q w <~ c o x , .

Remark 3.1. In [5] Singal et al. has defined an Its (X, z) to be FAT0 if it satisfies one of the following conditions:

(i) There exists a regular-open fuzzy set u in X such that x:. ~ u <~ co y~ or y, ~ u ~< co x~.

(ii) There exists a 6-open fuzzy set w in X such that x~ ~ w ~< c o y , or yu~w ~< c o x , .

Indeed, the two conditions are equivalent in the crisp case. But, in general, they are not equi- valent. The reason is the known fact that x~ ~ ~){ui[ i ~ J} ~ there exists i ~ J such that x ~ ui, where xz is a fuzzy point in X and ul is a fuzzy set in X for all i E J. Using the concept of a quasi-coincident relation, we avoid this deviation. Now, we give corrected versions for Theorems (4.3)-(4.5) in [5].

Theorem 2.3. Fuzzy quasi-normality is hereditary with respect to closed subspaces.

Proof. Let (X, z) be an F Q N space and let (Y, rr) be a closed subspace. Let F1 and F2 be Zy-closed fuzzy sets such that F1 qcoyF2 . Since Y is closed, then

Theorem 3.1. An f ts (X, z) is FAQTo iff for every pair o f fuzzy points xz, Yu in X (x ~ y), x~q~6-cl Yu or yuq~6-cl x: .

Proof. Let xx, y, be a pair of fuzzy points in X with x ~ y, then there exists a regular-open fuzzy set u

388 M.H. Ghanim et aL / Fuzzy Sets and Systems 85 (1997) 385 389

in X such that xa qu ~< coyu or yuqu ~< coxa. Let x~qu <. c oy , . Then xx(Ecou. Since y. ~< cou and cou is 6-closed, then xz¢6-cly . .

Conversely, let xz, y. be a pair of fuzzy points in X with x ¢ y, xa¢(b-cly.) or y.(s(b-clxz). Then x~qco(b-cly .) or y. qco(b-clx~). Since co(6- c lx~ )= Uiul, c o ( 6 - c ly , )= ~_)iVi, where Ui, V i are regular-open fuzzy sets in X for all i. Then, there exists a regular-open fuzzy set ui in X such that xaqui<~coy , or yuqui<<-c°x~. Thus (X,z) is FAQTo.

Corollary 3.1. A n f t s (X,z) is FAQTo /ff(X,z*) is FQT0.

Corollary 3.4. A fuzzy semi-regular space is FAQT~ iff it is FQT1.

Definition 3.3. An fts (X, z) is FAQT~ iff every fuzzy point in X is 6-closed.

Thus, every FAQT. is FAQT1, but the converse is, in general, not true. The fts in Example 2.1 is FAQT~ but it is not FAQT,.

Definition 3.4. An fts (X, ~) is FAQT2 space iff for every pair of fuzzy points x~, y, in X with different supports, there exists a pair of regular-open fuzzy sets u,v in X such that x~qu <~ coy , , y, qv ~< co x~ and u ~< co v.

Corollary 3.2. A fuzzy semi-regular space is FAQTo iff it is FQTo.

Definition 3.2. An fts (X, r) is FAQT~ iff for every pair of fuzzy points xa,yu in X with different supports, there exists a pair of regular-open fuzzy sets u,v in X such that x~qu ~< coy , and yuqv ~< CO X~.

Theorem 3.2. Ant i s (X,z) is FAQT1 iff every crisp point is 6-closed.

Proof. Let xl be a crisp point and let p be a fuzzy point in X such that supp p # x. Then, there exists a regular-open fuzzy set u~ in X such that pqup <% cox1. Let A = ~){up: p q c o x l } . One may easily verify that co Xl = A. Consequently, co Xl is 6-open and xl is f-closed.

Conversely, let x~, y, be a pair of fuzzy points in X with x -# y, then Xl and Yl are 6-closed fuzzy sets. Consequently, cox~ and coy~ are 6-open. Then, c ox 1 = ~fliUi and coy1 = ~Jjvj where u~,v~ are regular-open fuzzy sets in X for all i,j. Then, yuq(coxl ) ~< coxx and x) .q(coyl) ~< coy , . Hence, there exist two regular-open fuzzy sets ui, vj in X such that y, qug~<coxx and x a q v j ~ < c o y , . Consequently, (X, z) is FAQT~.

Corollary 3.3. An fts (X,z) is FAQT1 iff (X, z*) is FQT1.

Definition 3.5. An fts (X,z) is FAQT21/2 space iff for every pair of fuzzy points x~,y, in X with different supports, there exists a pair of regular- open fuzzy sets u, v in X such that xz q u ~ co y~, y, q v ~< co xz and (6-cl u) ~< c0(6-cl v). Thus, every FAQT2 1/2 is FAQT2.

Theorem 3.3. Let (X,z) and ( Y , U ) be fts. Let f : X ~ Y be injective fuzzy almost continuous and fuzzy almost open. Then X is FAQTi if Y is FAQTI, where i ~ {0, 1,2,2½}.

Proof. Let us prove the theorem in the case i = 0. Let x). and y, be two fuzzy points in X (x # y). Then .f(x~) and f ( y , ) are two fuzzy points in Y whose supports are f ( x ) , f ( y ) respectively, and f ( x ) ~ f ( y ) . Hence, there exists a regular-open fuzzy set u in Y such t h a t f ( x ~ ) q u <<, co(f (yu)) or f ( y u ) q u ~<co(f(xz)). Therefore, x ~ q ( f - l (u)) <~ c o ( f - l f ( y , ) ) or y , q ( f l(u)) <~ c o ( f - l f ( x D ) . Since f - l ( u ) is regular-open, from Lemma 1.1, t h e n x z q ( f l (u)) ~< co yu or y, q ( f - l ( u ) ) < ~ c o x ~ . Thus, (X,z) is FAQT0. The proofs of other cases are similar.

Theorem 3.4. Every open subspace of an FAQTI space is FAQTI, where i ~ {0, 1, 2, 2½}.

Proof. We prove the theorem when i = 2. Let (X, z) be FAQT2 and let (Y, zy) be an open subspace. Let x;,, y, be two fuzzy points in Y with different supports. Then, there exist two regular-open fuzzy

M.H. Ghanim et al. / Fuzzy Sets and Systems 85 (1997) 385 389 389

sets u,v in X such that x a q u <~ co y , , y u q v ~< coxa , and u~<cov . Let u * = u ~ Y , v * = v ~ Y . Then u*, v* are regular-open fuzzy sets in Y. Thus (Y, Zr) is FAQT2-space. The proofs of other cases are similar.

Corollary 3.6. A f u z z y semi-regular space is FAQTi i f f it is FQT/ , where i t {0, 1,2,2½}.

References

Corollary 3.4. Every open subspace o f an FAQTI space is FQTi , where i ~ {0, 1, 2, 2½}.

Theorem 3.5. Let (X,r), ( Y , U ) be two f ts . Le t f : X ~ Y be injective and f u z z y almost continuous. Then, (X,z) is FQTI i f ( Y , U is FAQT/ , where i ~ {0, 1,2,2½}.

Proof. In this case, the inverse image of a regular- open fuzzy set is an open fuzzy set. The proof can be completed using similar arguments as in Theorem 4.3.

Corollary 3.5. An f t s (X,z) is FAQTi /ff (X,z*) is FQTi , where i e {0, 1,2,2½}.

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