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Fuzzy nearness structure
Fuzzy Sets and Systems 44 (1991) 295-302
295
North-Holland
Fuzzy nearness structure S.K. Samanta Department of Mathematics, Visva-Bharati, Santiniketan-731235, India Received February 1988
Revised April 1989 Abstract: In this paper the notion of fuzzy nearness structure is introduced and some investigations are made towards the unification of several structures; namely, fuzzy topological structure, fuzzy
proximity structure, fuzzy contiguity structure and fuzzy uniform structure. Keywords: Fuzzy topology; fuzzy proximity; fuzzy nearness.
1. Some notations and preliminaries Following [1] and [2], we first introduce s o m e notation. L e t X d e n o t e a n o n e m p t y set. A m a p p i n g f r o m X to [0, 1] is called a fuzzy subset of X. F o r or e [0, 1], the constant function o v e r X whose value is or, is a fuzzy subset and this is d e n o t e d by 6l. L e t us d e n o t e I ( X ) , the class of all fuzzy subsets of X, P I ( X ) , the class of all subsets of I ( X ) , ~ , ~ . . . . usually the subsets of I ( X ) . For A, B e I ( X ) , A c B if A ( x ) <- B(x),
(A t3 B)(x) = m a x { A ( x ) , B(x)}, (A f3 B)(x) = m i n { A ( x ) , B(x)},
Vx e X.
F o r subsets ~¢, ~ of I ( X ) ,
M ^ ~ = {A N B ; A esg, B e ~ } , 5g v ~3= {A tJ B ; A e M, B e ~3}, ~t < ~ ¢~ Y A e sg, 3 B e ~3 s.t. A c B. ~¢<~ ¢:> V A e s g , 3 B e ~ s . t . BcA. ~¢c = {A¢; A e M}, w h e r e A c = 1 - A. For A e I ( X ) , let SA = {x e X ; A ( x ) > 0.5}. Definition 1.1. A family M of fuzzy subsets is said to be quasi-coincident (briefly q.c.) if there is x e X such that A ( x ) + A ' ( x ) > 1, Vii, A ' e sg. Definition 1.2. (see [4]). A family M of fuzzy subsets is said to be a or*-shading (where 0 < or ~< 1) if for all x e X t h e r e exists A e 5¢ such that A ( x ) >1 or. 0165-0114/91/$03.50 (~) 1991--Elsevier Science Publishers B.V. All rights reserved
S.K. Samanta
296 Note
1.3. M is q.c. iff (-')z~a Sa ~ 0, and f-]a,~ SA = 0 iff M c is a 0.5*-shading.
Definition 1.4. A subset ~ of PI(X) satisfying the following conditions is said to be a fuzzy-nearness structure (briefly FN-structure) on X: (FN1) if M < ~3 and ~3 e ~, then M ~ ~, (FN2) if M is q.c., then M ~ ~, (FN3) 0 4= ~ :# PI(X), (FN4) i f M v S ~ , thenMe~or ~e~, (FN5) if { c I ~ A ; A e M } e ~ , then M e ~ , where c l ~ A = l - s u p { B e I ( X ) ; {B, A} $ ~}. If ~ is a FN-structure on X, then the pair (X, ~) is called a fuzzy nearness space (briefly FN-space). 1.5. If (X,~) is a FN-space, then cl~:l(X)-+l(X), defined by cleA = 1 - s u p { B e l ( X ) ; {B, A} ¢ ~}, is an operator on I(X) and satisfies the following properties: VA,B e l(X). (FT1) A c cI~A, (FT2) cl~0 = 0, (FT3) A c B ~ cl~A ccl~B, (FT4) cl~(A U B) = cl~A U cl~ B, (FT5) cl~(cl~A) = cl~A. Theorem
(FT1) If possible, let A ~6cl~A. Then there is x o e X such that A(xo)> cl~A(xo), that is 1-sup[p(xo); {p, A} $ ~] (cl~A U cl~B)(xo). So, there are Pl, P2 e l(X) such that Proof.
{A, pl} ~ ~,
{B, p2} ~ ~,
p,(xo), Pz(Xo) > sup{qO(Xo); {A U B, p} ~ ~}.
(1)
Put P ' = P l N p 2 . Then { A , p ' } ~ , { B , p ' } ~ and thus { A , p ' } v { B , p ' } = {A U B, A U p', p' U B, p' U p'}, which implies {A U B, p'} ~ ~ (since {A, p'} v {B, p'} < {A U B, p'}). So, sup{p(x0); {A U B, p} ~ ~} I> p(xo), a contradiction to (1). Hence, cl~(A U B) = cl~A U cl~ B. (F~5) By (FT1), cl~A c cl~(cl~A). Since for any p el(X), {p, A} ¢ ~ {cl~p, cl~A} ~ ~ (by FN5)), it follows that sup{p(x); {p, A} $ ~} ~sup{o(x); {o, cleA} $ ~}, Vx e X, that is cl~A i> cl~(cl~A). Hence cl~A = cl~(cl~A).
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Thus any fuzzy nearness structure ~ on X induces on X a fuzzy topological closure operator. The fuzzy topological space so obtained from ~ is called the underlying fuzzy topological space of (X, ~).
Definition 1.6. A FN-space ~ is called a topological FN-space if it satisfies the condition ~ {clgA; A • M} is q.c. (T) M • Theorem 1.7. Let (X, cl) be a fuzzy topological space and ~ {clA; A • M} is a q.c. family} and let for A • I ( X ) , cl~A(x) = 1 I ( X ) and {p, A} ~ ~}. Then cl~A = clA iff the following condition for any fuzzy open set A. ( * ) p .< A(x) ~ there exists B • I ( X ) such that cl B c A and p
= {M • P I ( X ) ; sup{p(x); p • ( * ) is satisfied
< B(x).
Proof. Suppose that condition ( * ) holds. Now, for x • X , cl~A(x)=lsup{p(x); p • I ( X ) and cl p(y) + cl A ( y ) ~< 1, Vy • X}/> cl A(x). If cl A(x) = 1, then clearly cl~A(x)=clA(x). On the other hand if c l ( A ) x < l , take 0 < p < 1 - c l A ( x ) . Using condition ( * ) , we get B • I ( X ) such that p < B ( x ) and cl B c 1 - cl A. So, cl~A(x) ~< 1 - p . Since p < 1 - cl A(x) is arbitrary, it follows that, cl~A(x) ~< clA(x). Thus, cl~A(x) = clA(x), Vx • X. Conversely, suppose cl~A = c l A . For any fuzzy open set A, let A c=- 1 - A . ThenclA c=l-A. So, A = I - c l A c. Now, sup{p(x); p • I ( X ) and cl p ( y ) + cl A t ( y ) ~< 1, Vy • X} = 1 - cl AC(x)
= A(x). Take 0 < p < A(x). Then there exists B • I ( X ) such that cl B ( y ) ~< 1 - cl AC(y) = A(y), Yy • X and p < B(x). Hence the condition ( * ) holds. The following example illustrates a concrete situation where condition ( * ) is not satisfied and in which case cl A :/: cl~A.
Example 1.8. Let X = { 0 , 1,2}. Define fuzzy subsets A , B by A ( 0 ) = 0 . 2 , A(1) = 0 . 4 , A(2) = 0 . 1 , B(0) = 0 . 8 , B ( 1 ) = 0 . 5 , B(2) =0.9. Let i f = {A, B, 0, i} be a fuzzy topology on X. Then cl A = B e < 1 = cl~A. Theorem 1.9. / f (X, cl) is a fuzzy topological space satisfying condition ( * ), then = {M • PI(X); { c l A ; A • M} is a q.c. family} is a topological FN-structure on X with cl~ = cl. Before we prove T h e o r e m 1.9, let us first prove a lemma.
Lemma 1.10. If none of M and ~ is q.c., then M v ~ is not q.c. Proof. Since M and ~ are not q.c., for each x c X, there exists, A~, B1, Bz e 90 such that
A,(x)+Ae(x)<~l
and
B~(x)+Bz(x)<-l.
A 2 E
,5~ and
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S.K. Samanta
Consider the class {A1 U B1, A1 U B2, A21.3 B1, A2 U B2}. Without loss of generality suppose Al(X)>~AE(X), BI(X)I> B2(X). NOW the following cases arise: (1) Az(x) >I Ba(x),
(2) a d x ) <-B,(x), a d x ) >t Bdx), a,(x) >t Bl(X), (3) a2(x) ~ BI(X),
(5) A2(x) <~Bl(X), B2(x) >~Adx), B:(x) <~Ai(x), Am(x)~ Bl(X), (6) A d x ) <~hi(x), Bdx) >~Adx), B~(x) >~Adx). In case (1), (A1 tO B1)(x) + (A2 tO BO(x) = Al(x) + A2(x) ~< 1. Similarly, considering all the cases we see that in each case, there is a pair of members of {A1 tO B1, A1 tO B2, A2 U B1, A2 tO B2} the sum of whose values is less than or equal to 1. Thus the class M v ~ is not q.c.
Proof of Theorem 1.9. (FN1) Let M < ~ and 93 • ~. Then {cl B; B ~ ~} is q.c. and every A • M contains some B e ~. So, {clA;A e M } is also q.c., that is (FN2) M i s q . c . ~ { c l A ; A e M } i s q . c . ~ M e ~ . (FN3) For any fixed A ( ~ . 6 ) e I(X), M = {A} =/=0 and {clA} is q.c. and hence M • ~, that is ~4:0. Further, take 0, i e l ( X ) . Now c l 0 = 0 and cl i = i. Let M = {0, i). Now {cl 0, cl i) is not q.c. and hence M • ~, that is 0 4= ~ 4= PI(X). (FN4) Follows from Lemma 1.10. (FN5) Follows from the fact that cl(cl) = cl & cl~ = cl. Thus ~ is a topological FN-structure on X with cl, = cl.
2. Fuzzy proximity spaces If (X, ~) is a FN-space, then it can be easily shown that the relation 6 on I(X) defined by A 5B iff (A, B) e ~ is a fuzzy proximity on X, that it is satisfies for all A, B, C • I(X). (FP1) A O B ~ B S A , (FP2) A c B & A S C ~ BSC, (FP3) ( A , B } i s q . c . ~ AOB, (FP4) A 6 B ~ A 4 : 0 , B 4 : 0 , (FP5) A S ( B U C ) ~ A 6 B o r A 6 C , (FP6) A 5B and B c cl~ C ~ A 5C, where cl~ C = 1 - sup(P e I(X); P~C). Thus 6 is a fuzzy proximity (briefly, F-proximity) on X, which is called the underlying F-proximity on X. Remark 2.1. A partial relation is obtained between the notions of fuzzy proximity as considered by us and the fuzzy proximity as considered by Katsaras [3]. In fact, if 5 is a fuzzy proximity on X in the sense of Katsaras [3], then it satisfies, for all A, B, C • I(X). (FP1)' A O B iff B S A , (FP2)' (A O B ) 5 C iff A SC or B6C,
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(FP3)' A 6 B ~ A~O, B¢6, (FP4)' A 4 B ~ B ~ I - A , (FP5)' if A~B, then there exists C • I ( X ) with A ~ C and (1 - C) fiB. Then obviously, 6 satisfies (FP1), (FP3), (FP4) and (FP5). (FP2) also follows from (FP2)'. Now, from (FP5)', A ~ B implies the existence of C • I ( X ) such that A ~ C and 1 - C~B. So, 1 - cluB = sup{p • I ( X ) ; p~B} ~ 1 - C, that is d a B c C, and hence A ~cl B. Thus (FP6) holds. However, it is not known whether the converse is true.
3. Fuzzy contiguity spaces If we modify the axioms (FNi) by requiring all collections s/, ~ in question to be finite then the structure ~f so formed is called a fuzzy contiguity (briefly, F-contiguity) structure on X. The space (X, ~r) will be called the underlying F-contiguity space of (X, ~).
Definition 3.1. A FN-space (X, ~) is said to be a contigual FN-space if it satisfies the following condition: (c) if every finite subset of ~ / b e l o n g s to ~ then .d belongs to ~. Theorem 3.2. If (X, fl) is a F-contiguity space, then ~ = {A c I ( X ) ; V ~ c ~ / ( ~ is finite ~ ~ • r / ) } is a contigual FN-structure on X with ~ f = { A • ~ ; ~ is finite} = ft. Proof. (FN1) Let ~ / < ~ and ~ e ~. Let 0-//= {ul, • • . , un} be a finite subset of s/. Since z g < ~ , there exists B i e ~ such that B i c U i , l<~i~
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S.K. Samanta
Axioms for fuzzy farness. Let ~ be a subset of PI(X). ~ ~•~, (FF1) ( M < ~ a n d M • ~ ) (FF2) M • ~ ~ M i s n o t q . c . , (FF3) 0 * ~ ~ PI(X), (FF4) (FFS) M • ~ ::> {cl A; A • M} • ~, where cl A = 1 - sup{p • I(X); {p, A } • ~}. Axioms for fuzzy uniform cover. Let # be a subset of PI(X). (FU1) ( M < ~ a n d M • # ) :~, ~ • # , (FU2) M • / ~ ~ for e a c h x • X , there is a p a i r A , A ' • M s u c h thatA(x)+ A'(x) ~> 1, (FU3) 0 :#/* ~ PI(X), (FU4) M, ~ • / ~ =), M A ~ • # , (FU5) M e g ~ { i n t u A ; A • M } • #, where intuA=l-inf{oePI(X); {o, A} • ~}.
Remark 3.4. (a) If ~ is a FN-structure on X, then #~ = {M c l(X); M c $ ~} is a covering structure (i.e. satisfies (FU1)-(FU5)) on X induced by ~. Also, if # is a F-covering structure on X (i.e. if # satisfies (FU1)-(FU5) then ~ defined by ~, = {M c I(X); M c $ #} is a nearness structure on X induced by #. (b) If ~ is a F-nearness structure and # is a F-covering structure on X, then ~,~ = ~ and #¢, = #. Theorem 3.5. If ~ is a FN-structure and Iz is a F-covering structure on X, then (i) s ~ • # ~ i f f V ~ • ~ , ~ n s e c ~ 4 : 0 , (ii) M e ~ i f f V ~ e #, ~ n s e c M ~ 0 . Here sec M = {p • I(X); VA • M, {p, A} is q.c. }. Proof. Suppose M • #~. If possible, let there exists ~ • ~ such that M N sec ~3 6: 0. So, for each A • M, there exists B • ~3 such that {A, B} is not q.c., and hence A c B c. Therefore M < ~ ¢ . Thus M • # ~ ~ ~ • # ~ =), ~ $ ~ , ~ = ~ , a contradiction. Conversely, suppose V:~ • ~, M N sec ~ =~0. If possible, let M $ #~. Then M c • ~,~ = ~. But M rl sec M c = 0. Hence the proof of (i). The proof of (ii) is similar.
4. Fuzzy uniform spaces Definition 4.1. A F-covering structure # on X is called a fuzzy uniformly (briefly, F-uniformity) if # satisfies the following condition: (FU)' if M • #, then there exists ~ • # such that for each B e ~ there exists A • M with I,_J{ C • ~ ; B tq C = 0 } c A . Definition 4.2. A FN-space is called a uniform FN-space if it satisfies the following condition: (FU) If M • ~, then there exists ~ • ~ such that for each B • ~ there exists A e M with A c O { C e ~; C U B = i).
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Theorem 4.3. If (X, ~) is a FN-space then the conditions (FU) and (FU)' are equivalent. Proof. The proof is trivial. T h e o r e m 4.4. If (X, #) is a F-uniform space the ~ = {M • PI(X); V ~ •/z, fq sec M :/: O} is a uniform FN-structure on X with I*~ =/~. Proof. By the definition of ~, ~ = ~,. Since/~e = "~, = # and/~ is a F-uniformity and since (FU) and (FU)' are equivalent, it follows that ~ is a uniform FN-structure on X w i t h / ~ = / , .
Definition 4,5. Let M, ~ • PI(X). Then ~ is said to be a star-refinement of M (symbolically ~ < , M) is for each B • ~ there exists A • M with (._J (C e ~ ; B N C:#O} c A . Theorem 4.6. For any FN-structure ~ on X with the corresponding F-covering
structure I~, the set I~. consisting of all M • I~ for which there exists a sequence • " "<, ~o2<, ~ < , M of star-refinements in It, is a F-uniform structure on X. The corresponding FN-structure ~o is a uniform FN-structure on X. Proof. Clearly /L, satisfies the conditions ( F U 1 ) - ( F U 3 ) . For (FU4) take M, • /~,,. Then M ^ ~ • / ~ . T o show M A ~ • / 4 , , consider the sequences of star-refinements • • • < , M e < , M~ < , M and • • • < , ~ 2 < , ~1 < , ~ of M and ~ , respectively, Then, the sequence {M, A ~ , } is a sequence of star-refinements of M ^ ~. So, M A ~ •/~,. Thus (FU4) is true. The condition (FU5) can readily be verified by using the implication M < , ~ ~ int M < , i n t ~ . Thus #, is a F-covering structure. Clearly, it satisfies the condition (FU)'. So, ~t,, is a F-uniformity, so that by the equivalence of (FU) and ( F U ) ' , ~,, = ~,,, is a uniform FN-structure.
Definition 4.7. A F-topological space is said to be compact if every family of F-closed sets having finite q.c. property (i.e. every finite subfamily is q.c.), is q.c. Definition 4.8 [4]. A F-topological space is said to be oc*-compact if each tr*-shading family of fuzzy open sets has a finite tr*-shading subfamily. Theorem 4.9. A f u z z y topological space (X, b) is compact in our sense iff it is 0.5*-compact. Proof. (X, 6) is not compact in our sense ¢:> there is a family M of F-closed sets having finite q.c. property but not q.c. ¢~ {SA}A~* has finite intersection property but f-'Lt~aSA = 0 ¢:> { X \ S A } A ~ a is a covering of X having no finite subcovering ¢~ M c is a 0.5*-shading family of F-open sets having no finite 0.5*-shading subfamily ¢:> (X, 6) is not 0.5*-compact.
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S.K. Samanta
With this definition of compactness of a F-topological space we prove the following theorem. Theorem 4.10. A topological FN-space is contigual iff its associated fuzzy topological space is compact. Proof. Let (X, ~) be a topological FN-space which is contigual. Then (i) M • ~ ~ { c l ~ A ; A • M } i s q . c . , (ii) if every finite subset of M belongs to ~ then M • ~. Consider the F-topological space (X, cl~). Let {F~}~Ea be a family of F-closed sets of (X, cle) having finite q.c. property. By (FN2) and (ii), {F~},Ea • ~. Since F~'s are F-closed sets, by (i), {F~)~a is q.c. Hence (X, cl~) is compact. Conversely, suppose (X, ~) is a topological FN-space whose associated F-topology cl~ is compact. To show ~ is contigual, suppose every finite subset of sit belongs to ~. Since (X, ~) is topological, for every finite subfamily ~ of M, {cleB; B • ~} is q.c. So, {cl~A; A • M} has finite q.c. property and hence by compactness of (X, c1¢), (cle A; A • A } is q.c. Thus A • ~, that is ~ is contigual. Following [2], we now give the definition of precompactness as follows. Definition 4.11. A F-uniform space (X, #) is called precompact if M •/~ there exists a finite subset ~ c M with ~ •/,.
Theorem 4.12. A FN-space is uniform and contigual iff it is a pre-compact uniform space. The proof is obvious.
Acknowledgements The author is thankful to Dr. K.C. Chattopadhyay, Burdwan University, for some valuable discussions with him during the preparation of this paper. The author is also thankful to the referee for his important suggestions in rewriting the paper in the present form.
References [1] H. Herrlich, Topological structures, Math. Centre Tracts 52 (1974) 59-122. [2] J.R. Isbeil, Uniform spaces, Math. Surveys No. 12. [3] A.K. Katsaras, On fuzzy syntopogenous structures, Rev. Roumaine Math, Pures Appl. 30 (1985) 419-431. [4] R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446-454. [5] Pu Pao-Ming and Liu Ying-Ming, Fuzzy topology, I. Neighbourhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980) 571-599.
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North-Holland
Fuzzy nearness structure S.K. Samanta Department of Mathematics, Visva-Bharati, Santiniketan-731235, India Received February 1988
Revised April 1989 Abstract: In this paper the notion of fuzzy nearness structure is introduced and some investigations are made towards the unification of several structures; namely, fuzzy topological structure, fuzzy
proximity structure, fuzzy contiguity structure and fuzzy uniform structure. Keywords: Fuzzy topology; fuzzy proximity; fuzzy nearness.
1. Some notations and preliminaries Following [1] and [2], we first introduce s o m e notation. L e t X d e n o t e a n o n e m p t y set. A m a p p i n g f r o m X to [0, 1] is called a fuzzy subset of X. F o r or e [0, 1], the constant function o v e r X whose value is or, is a fuzzy subset and this is d e n o t e d by 6l. L e t us d e n o t e I ( X ) , the class of all fuzzy subsets of X, P I ( X ) , the class of all subsets of I ( X ) , ~ , ~ . . . . usually the subsets of I ( X ) . For A, B e I ( X ) , A c B if A ( x ) <- B(x),
(A t3 B)(x) = m a x { A ( x ) , B(x)}, (A f3 B)(x) = m i n { A ( x ) , B(x)},
Vx e X.
F o r subsets ~¢, ~ of I ( X ) ,
M ^ ~ = {A N B ; A esg, B e ~ } , 5g v ~3= {A tJ B ; A e M, B e ~3}, ~t < ~ ¢~ Y A e sg, 3 B e ~3 s.t. A c B. ~¢<~ ¢:> V A e s g , 3 B e ~ s . t . BcA. ~¢c = {A¢; A e M}, w h e r e A c = 1 - A. For A e I ( X ) , let SA = {x e X ; A ( x ) > 0.5}. Definition 1.1. A family M of fuzzy subsets is said to be quasi-coincident (briefly q.c.) if there is x e X such that A ( x ) + A ' ( x ) > 1, Vii, A ' e sg. Definition 1.2. (see [4]). A family M of fuzzy subsets is said to be a or*-shading (where 0 < or ~< 1) if for all x e X t h e r e exists A e 5¢ such that A ( x ) >1 or. 0165-0114/91/$03.50 (~) 1991--Elsevier Science Publishers B.V. All rights reserved
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1.3. M is q.c. iff (-')z~a Sa ~ 0, and f-]a,~ SA = 0 iff M c is a 0.5*-shading.
Definition 1.4. A subset ~ of PI(X) satisfying the following conditions is said to be a fuzzy-nearness structure (briefly FN-structure) on X: (FN1) if M < ~3 and ~3 e ~, then M ~ ~, (FN2) if M is q.c., then M ~ ~, (FN3) 0 4= ~ :# PI(X), (FN4) i f M v S ~ , thenMe~or ~e~, (FN5) if { c I ~ A ; A e M } e ~ , then M e ~ , where c l ~ A = l - s u p { B e I ( X ) ; {B, A} $ ~}. If ~ is a FN-structure on X, then the pair (X, ~) is called a fuzzy nearness space (briefly FN-space). 1.5. If (X,~) is a FN-space, then cl~:l(X)-+l(X), defined by cleA = 1 - s u p { B e l ( X ) ; {B, A} ¢ ~}, is an operator on I(X) and satisfies the following properties: VA,B e l(X). (FT1) A c cI~A, (FT2) cl~0 = 0, (FT3) A c B ~ cl~A ccl~B, (FT4) cl~(A U B) = cl~A U cl~ B, (FT5) cl~(cl~A) = cl~A. Theorem
(FT1) If possible, let A ~6cl~A. Then there is x o e X such that A(xo)> cl~A(xo), that is 1-sup[p(xo); {p, A} $ ~]
{A, pl} ~ ~,
{B, p2} ~ ~,
p,(xo), Pz(Xo) > sup{qO(Xo); {A U B, p} ~ ~}.
(1)
Put P ' = P l N p 2 . Then { A , p ' } ~ , { B , p ' } ~ and thus { A , p ' } v { B , p ' } = {A U B, A U p', p' U B, p' U p'}, which implies {A U B, p'} ~ ~ (since {A, p'} v {B, p'} < {A U B, p'}). So, sup{p(x0); {A U B, p} ~ ~} I> p(xo), a contradiction to (1). Hence, cl~(A U B) = cl~A U cl~ B. (F~5) By (FT1), cl~A c cl~(cl~A). Since for any p el(X), {p, A} ¢ ~ {cl~p, cl~A} ~ ~ (by FN5)), it follows that sup{p(x); {p, A} $ ~} ~sup{o(x); {o, cleA} $ ~}, Vx e X, that is cl~A i> cl~(cl~A). Hence cl~A = cl~(cl~A).
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Thus any fuzzy nearness structure ~ on X induces on X a fuzzy topological closure operator. The fuzzy topological space so obtained from ~ is called the underlying fuzzy topological space of (X, ~).
Definition 1.6. A FN-space ~ is called a topological FN-space if it satisfies the condition ~ {clgA; A • M} is q.c. (T) M • Theorem 1.7. Let (X, cl) be a fuzzy topological space and ~ {clA; A • M} is a q.c. family} and let for A • I ( X ) , cl~A(x) = 1 I ( X ) and {p, A} ~ ~}. Then cl~A = clA iff the following condition for any fuzzy open set A. ( * ) p .< A(x) ~ there exists B • I ( X ) such that cl B c A and p
= {M • P I ( X ) ; sup{p(x); p • ( * ) is satisfied
< B(x).
Proof. Suppose that condition ( * ) holds. Now, for x • X , cl~A(x)=lsup{p(x); p • I ( X ) and cl p(y) + cl A ( y ) ~< 1, Vy • X}/> cl A(x). If cl A(x) = 1, then clearly cl~A(x)=clA(x). On the other hand if c l ( A ) x < l , take 0 < p < 1 - c l A ( x ) . Using condition ( * ) , we get B • I ( X ) such that p < B ( x ) and cl B c 1 - cl A. So, cl~A(x) ~< 1 - p . Since p < 1 - cl A(x) is arbitrary, it follows that, cl~A(x) ~< clA(x). Thus, cl~A(x) = clA(x), Vx • X. Conversely, suppose cl~A = c l A . For any fuzzy open set A, let A c=- 1 - A . ThenclA c=l-A. So, A = I - c l A c. Now, sup{p(x); p • I ( X ) and cl p ( y ) + cl A t ( y ) ~< 1, Vy • X} = 1 - cl AC(x)
= A(x). Take 0 < p < A(x). Then there exists B • I ( X ) such that cl B ( y ) ~< 1 - cl AC(y) = A(y), Yy • X and p < B(x). Hence the condition ( * ) holds. The following example illustrates a concrete situation where condition ( * ) is not satisfied and in which case cl A :/: cl~A.
Example 1.8. Let X = { 0 , 1,2}. Define fuzzy subsets A , B by A ( 0 ) = 0 . 2 , A(1) = 0 . 4 , A(2) = 0 . 1 , B(0) = 0 . 8 , B ( 1 ) = 0 . 5 , B(2) =0.9. Let i f = {A, B, 0, i} be a fuzzy topology on X. Then cl A = B e < 1 = cl~A. Theorem 1.9. / f (X, cl) is a fuzzy topological space satisfying condition ( * ), then = {M • PI(X); { c l A ; A • M} is a q.c. family} is a topological FN-structure on X with cl~ = cl. Before we prove T h e o r e m 1.9, let us first prove a lemma.
Lemma 1.10. If none of M and ~ is q.c., then M v ~ is not q.c. Proof. Since M and ~ are not q.c., for each x c X, there exists, A~, B1, Bz e 90 such that
A,(x)+Ae(x)<~l
and
B~(x)+Bz(x)<-l.
A 2 E
,5~ and
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Consider the class {A1 U B1, A1 U B2, A21.3 B1, A2 U B2}. Without loss of generality suppose Al(X)>~AE(X), BI(X)I> B2(X). NOW the following cases arise: (1) Az(x) >I Ba(x),
(2) a d x ) <-B,(x), a d x ) >t Bdx), a,(x) >t Bl(X), (3) a2(x) ~
(5) A2(x) <~Bl(X), B2(x) >~Adx), B:(x) <~Ai(x), Am(x)~ Bl(X), (6) A d x ) <~hi(x), Bdx) >~Adx), B~(x) >~Adx). In case (1), (A1 tO B1)(x) + (A2 tO BO(x) = Al(x) + A2(x) ~< 1. Similarly, considering all the cases we see that in each case, there is a pair of members of {A1 tO B1, A1 tO B2, A2 U B1, A2 tO B2} the sum of whose values is less than or equal to 1. Thus the class M v ~ is not q.c.
Proof of Theorem 1.9. (FN1) Let M < ~ and 93 • ~. Then {cl B; B ~ ~} is q.c. and every A • M contains some B e ~. So, {clA;A e M } is also q.c., that is (FN2) M i s q . c . ~ { c l A ; A e M } i s q . c . ~ M e ~ . (FN3) For any fixed A ( ~ . 6 ) e I(X), M = {A} =/=0 and {clA} is q.c. and hence M • ~, that is ~4:0. Further, take 0, i e l ( X ) . Now c l 0 = 0 and cl i = i. Let M = {0, i). Now {cl 0, cl i) is not q.c. and hence M • ~, that is 0 4= ~ 4= PI(X). (FN4) Follows from Lemma 1.10. (FN5) Follows from the fact that cl(cl) = cl & cl~ = cl. Thus ~ is a topological FN-structure on X with cl, = cl.
2. Fuzzy proximity spaces If (X, ~) is a FN-space, then it can be easily shown that the relation 6 on I(X) defined by A 5B iff (A, B) e ~ is a fuzzy proximity on X, that it is satisfies for all A, B, C • I(X). (FP1) A O B ~ B S A , (FP2) A c B & A S C ~ BSC, (FP3) ( A , B } i s q . c . ~ AOB, (FP4) A 6 B ~ A 4 : 0 , B 4 : 0 , (FP5) A S ( B U C ) ~ A 6 B o r A 6 C , (FP6) A 5B and B c cl~ C ~ A 5C, where cl~ C = 1 - sup(P e I(X); P~C). Thus 6 is a fuzzy proximity (briefly, F-proximity) on X, which is called the underlying F-proximity on X. Remark 2.1. A partial relation is obtained between the notions of fuzzy proximity as considered by us and the fuzzy proximity as considered by Katsaras [3]. In fact, if 5 is a fuzzy proximity on X in the sense of Katsaras [3], then it satisfies, for all A, B, C • I(X). (FP1)' A O B iff B S A , (FP2)' (A O B ) 5 C iff A SC or B6C,
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(FP3)' A 6 B ~ A~O, B¢6, (FP4)' A 4 B ~ B ~ I - A , (FP5)' if A~B, then there exists C • I ( X ) with A ~ C and (1 - C) fiB. Then obviously, 6 satisfies (FP1), (FP3), (FP4) and (FP5). (FP2) also follows from (FP2)'. Now, from (FP5)', A ~ B implies the existence of C • I ( X ) such that A ~ C and 1 - C~B. So, 1 - cluB = sup{p • I ( X ) ; p~B} ~ 1 - C, that is d a B c C, and hence A ~cl B. Thus (FP6) holds. However, it is not known whether the converse is true.
3. Fuzzy contiguity spaces If we modify the axioms (FNi) by requiring all collections s/, ~ in question to be finite then the structure ~f so formed is called a fuzzy contiguity (briefly, F-contiguity) structure on X. The space (X, ~r) will be called the underlying F-contiguity space of (X, ~).
Definition 3.1. A FN-space (X, ~) is said to be a contigual FN-space if it satisfies the following condition: (c) if every finite subset of ~ / b e l o n g s to ~ then .d belongs to ~. Theorem 3.2. If (X, fl) is a F-contiguity space, then ~ = {A c I ( X ) ; V ~ c ~ / ( ~ is finite ~ ~ • r / ) } is a contigual FN-structure on X with ~ f = { A • ~ ; ~ is finite} = ft. Proof. (FN1) Let ~ / < ~ and ~ e ~. Let 0-//= {ul, • • . , un} be a finite subset of s/. Since z g < ~ , there exists B i e ~ such that B i c U i , l<~i~
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Axioms for fuzzy farness. Let ~ be a subset of PI(X). ~ ~•~, (FF1) ( M < ~ a n d M • ~ ) (FF2) M • ~ ~ M i s n o t q . c . , (FF3) 0 * ~ ~ PI(X), (FF4) (FFS) M • ~ ::> {cl A; A • M} • ~, where cl A = 1 - sup{p • I(X); {p, A } • ~}. Axioms for fuzzy uniform cover. Let # be a subset of PI(X). (FU1) ( M < ~ a n d M • # ) :~, ~ • # , (FU2) M • / ~ ~ for e a c h x • X , there is a p a i r A , A ' • M s u c h thatA(x)+ A'(x) ~> 1, (FU3) 0 :#/* ~ PI(X), (FU4) M, ~ • / ~ =), M A ~ • # , (FU5) M e g ~ { i n t u A ; A • M } • #, where intuA=l-inf{oePI(X); {o, A} • ~}.
Remark 3.4. (a) If ~ is a FN-structure on X, then #~ = {M c l(X); M c $ ~} is a covering structure (i.e. satisfies (FU1)-(FU5)) on X induced by ~. Also, if # is a F-covering structure on X (i.e. if # satisfies (FU1)-(FU5) then ~ defined by ~, = {M c I(X); M c $ #} is a nearness structure on X induced by #. (b) If ~ is a F-nearness structure and # is a F-covering structure on X, then ~,~ = ~ and #¢, = #. Theorem 3.5. If ~ is a FN-structure and Iz is a F-covering structure on X, then (i) s ~ • # ~ i f f V ~ • ~ , ~ n s e c ~ 4 : 0 , (ii) M e ~ i f f V ~ e #, ~ n s e c M ~ 0 . Here sec M = {p • I(X); VA • M, {p, A} is q.c. }. Proof. Suppose M • #~. If possible, let there exists ~ • ~ such that M N sec ~3 6: 0. So, for each A • M, there exists B • ~3 such that {A, B} is not q.c., and hence A c B c. Therefore M < ~ ¢ . Thus M • # ~ ~ ~ • # ~ =), ~ $ ~ , ~ = ~ , a contradiction. Conversely, suppose V:~ • ~, M N sec ~ =~0. If possible, let M $ #~. Then M c • ~,~ = ~. But M rl sec M c = 0. Hence the proof of (i). The proof of (ii) is similar.
4. Fuzzy uniform spaces Definition 4.1. A F-covering structure # on X is called a fuzzy uniformly (briefly, F-uniformity) if # satisfies the following condition: (FU)' if M • #, then there exists ~ • # such that for each B e ~ there exists A • M with I,_J{ C • ~ ; B tq C = 0 } c A . Definition 4.2. A FN-space is called a uniform FN-space if it satisfies the following condition: (FU) If M • ~, then there exists ~ • ~ such that for each B • ~ there exists A e M with A c O { C e ~; C U B = i).
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Theorem 4.3. If (X, ~) is a FN-space then the conditions (FU) and (FU)' are equivalent. Proof. The proof is trivial. T h e o r e m 4.4. If (X, #) is a F-uniform space the ~ = {M • PI(X); V ~ •/z, fq sec M :/: O} is a uniform FN-structure on X with I*~ =/~. Proof. By the definition of ~, ~ = ~,. Since/~e = "~, = # and/~ is a F-uniformity and since (FU) and (FU)' are equivalent, it follows that ~ is a uniform FN-structure on X w i t h / ~ = / , .
Definition 4,5. Let M, ~ • PI(X). Then ~ is said to be a star-refinement of M (symbolically ~ < , M) is for each B • ~ there exists A • M with (._J (C e ~ ; B N C:#O} c A . Theorem 4.6. For any FN-structure ~ on X with the corresponding F-covering
structure I~, the set I~. consisting of all M • I~ for which there exists a sequence • " "<, ~o2<, ~ < , M of star-refinements in It, is a F-uniform structure on X. The corresponding FN-structure ~o is a uniform FN-structure on X. Proof. Clearly /L, satisfies the conditions ( F U 1 ) - ( F U 3 ) . For (FU4) take M, • /~,,. Then M ^ ~ • / ~ . T o show M A ~ • / 4 , , consider the sequences of star-refinements • • • < , M e < , M~ < , M and • • • < , ~ 2 < , ~1 < , ~ of M and ~ , respectively, Then, the sequence {M, A ~ , } is a sequence of star-refinements of M ^ ~. So, M A ~ •/~,. Thus (FU4) is true. The condition (FU5) can readily be verified by using the implication M < , ~ ~ int M < , i n t ~ . Thus #, is a F-covering structure. Clearly, it satisfies the condition (FU)'. So, ~t,, is a F-uniformity, so that by the equivalence of (FU) and ( F U ) ' , ~,, = ~,,, is a uniform FN-structure.
Definition 4.7. A F-topological space is said to be compact if every family of F-closed sets having finite q.c. property (i.e. every finite subfamily is q.c.), is q.c. Definition 4.8 [4]. A F-topological space is said to be oc*-compact if each tr*-shading family of fuzzy open sets has a finite tr*-shading subfamily. Theorem 4.9. A f u z z y topological space (X, b) is compact in our sense iff it is 0.5*-compact. Proof. (X, 6) is not compact in our sense ¢:> there is a family M of F-closed sets having finite q.c. property but not q.c. ¢~ {SA}A~* has finite intersection property but f-'Lt~aSA = 0 ¢:> { X \ S A } A ~ a is a covering of X having no finite subcovering ¢~ M c is a 0.5*-shading family of F-open sets having no finite 0.5*-shading subfamily ¢:> (X, 6) is not 0.5*-compact.
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With this definition of compactness of a F-topological space we prove the following theorem. Theorem 4.10. A topological FN-space is contigual iff its associated fuzzy topological space is compact. Proof. Let (X, ~) be a topological FN-space which is contigual. Then (i) M • ~ ~ { c l ~ A ; A • M } i s q . c . , (ii) if every finite subset of M belongs to ~ then M • ~. Consider the F-topological space (X, cl~). Let {F~}~Ea be a family of F-closed sets of (X, cle) having finite q.c. property. By (FN2) and (ii), {F~},Ea • ~. Since F~'s are F-closed sets, by (i), {F~)~a is q.c. Hence (X, cl~) is compact. Conversely, suppose (X, ~) is a topological FN-space whose associated F-topology cl~ is compact. To show ~ is contigual, suppose every finite subset of sit belongs to ~. Since (X, ~) is topological, for every finite subfamily ~ of M, {cleB; B • ~} is q.c. So, {cl~A; A • M} has finite q.c. property and hence by compactness of (X, c1¢), (cle A; A • A } is q.c. Thus A • ~, that is ~ is contigual. Following [2], we now give the definition of precompactness as follows. Definition 4.11. A F-uniform space (X, #) is called precompact if M •/~ there exists a finite subset ~ c M with ~ •/,.
Theorem 4.12. A FN-space is uniform and contigual iff it is a pre-compact uniform space. The proof is obvious.
Acknowledgements The author is thankful to Dr. K.C. Chattopadhyay, Burdwan University, for some valuable discussions with him during the preparation of this paper. The author is also thankful to the referee for his important suggestions in rewriting the paper in the present form.
References [1] H. Herrlich, Topological structures, Math. Centre Tracts 52 (1974) 59-122. [2] J.R. Isbeil, Uniform spaces, Math. Surveys No. 12. [3] A.K. Katsaras, On fuzzy syntopogenous structures, Rev. Roumaine Math, Pures Appl. 30 (1985) 419-431. [4] R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446-454. [5] Pu Pao-Ming and Liu Ying-Ming, Fuzzy topology, I. Neighbourhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980) 571-599.