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Fuzzy preproximity spaces
Fuzzy Sets and Systems 35 (1990) 333-340 North-Holland
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FUZZY PREPROXIMITY SPACES A.S. M A S H H O U R Mathematics Department, Faculty of Science, Assiut University Assiut, Egypt
R. B A D A R D Dept. lnformatique, 1.N.S.A. de Lyon, 69621 Villeurbanne, France
A.A. R A M A D A N Mathematics Department, Faculty of Science, Beni-Suef, Egypt Received December 1986 Revised March 1988
Abstract: The concept of proximity is generalized by defining fuzzy preproximity. Some particular constructions of fuzzy preproximity spaces are given. A natural link is established between these concepts and the concept of fuzzy pretopological spaces and fuzzy preuniform spaces. In the last section the notion of product fuzzy preproximity is introduced.
Keywords: Fuzzy sets; fuzzy proximity; fuzzy pretopology; fuzzy preuniformity.
1. Introduction In what follows, we have used the Zadeh definition of fuzzy sets, but our results can be translated without difficulty to more general cases, particularly to L-fuzzy sets, in the sense of Goguen, for which the representation theorem of Negoita and Ralescu [5] holds. In the sequel X always denotes a non-empty set. Definition 1.1 [6]. A function A from X to the unit interval I = [0, 1] is called a fuzzy set on X, and I x denotes the family of fuzzy sets in X. If A takes only the values 0, 1, then A is called a crisp set on X. Particularly, the crisp set which always takes the value 1 on X is denoted by X, and the crisp set which always takes the value 0 on X is denoted by ~l. Let A be a fuzzy set on X and Xo ¢ X. If
A(x)={o(O
then A is called a fuzzy singleton and it is denoted by (x, 0c). Definition 1.2 [1]. A fuzzy pretopology on X is a function satisfies the following conditions: (1) a(0) = 0. 0165-0114/90/$3.50 ~) 1990, Elsevier Science Publishers B.V. (North-Holland)
a:IX--->l x which
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(2) a(A) = A for every A • I x. The pair (X, a) is said to be a fuzzy pretopological space. A fuzzy pretopological space (X, a) is called: (3) type I: For every A, B • I x such that A c B we have a(A) ~ a(B). (4) type D: For every A, B • I x we have a(A U B) = a(A) O a(B). (5) type S: For every A • I x we have a2(A) = a(A). A fuzzy pretopological space which is of type I, D and S is a fuzzy topological space and a is its Kuratowsky closure. Let (X, a) and (Y, b) be two fuzzy pretopological spaces. A function f : X - - * Y is said to be continuous if f (a(A )) c b ( f (A ) ), for every A • I x. Definition 1.3 [2]. A fuzzy preuniformity on X is a family U of fuzzy sets on X x X which satisfies the following conditions: (1) For every u • U, u(x, x) = 1, so u ~ A where ,4= {(x, x) • X x X : x • X } . (2) v = u • U implies that v is an element of U.
We will consider the following particular fuzzy preuniform structures: (3) Symmetrical: For every u • U, we have u -1 • U where u-l(x, y) = u(y, X). (4) Type D: For every u, v • U we have u t3 v • U. (5) Type S: For every u • U, there exists v • U such that v ® v c u, where (v ® v)(x, y) = V (v(x, z) ^ v(z, y)). zEX
2. Fuzzy prepro~fimity space Definition 2.1. Let 6 be a binary relation on I x, i.e., 6 ~ I x ® I x. The facts that A 6 B and A/3 B are denoted by the symbols 6(.4, B) and 3(A, B), respectively. A binary relation 6 on I x is said to be fuzzy preproximity on X if it satisfies: (Pt): 6(A, B) ~ A : g 0 a n d B : # ~ , (P2): A N B e f J ~ 6 ( A , B ) . The pair (X, 6) is called fuzzy preproximity space. We will consider the following particular fuzzy preproximity structures: (P3): 6(A, B) ~ 6(B, A); (X, 6) is symmetrical. (P4): A ~ B , C ~ D and 6(B, D ) ~ 6 ( A , C); (X, 6) is of type I. (Ps): 6(A t_JB, C ) ¢ , ~ ( A , C) or 6(B, C); (X, 6) is of type D. (P6): 3(A, B) ~ there is C • I x such that ~(A, C) and dt(C, B), where C is the complement of C; (X, 6) is of type S. A fuzzy preproximity which satisfies (P3), (Ps) and (P6) is a fuzzy proximity as defined by Katsaras [4]. In this case we remark that (P3) and (Ps) imply (P4). Definition 2.2 [4]. Let (X, 61) and (Y, 62) be fuzzy preproximity spaces. A function f : X---* Y is said to be a preproximally continuous if and only if it satisfies one of the following equivalent conditions: (1) For every A, B ~ I x, 61(A , B) ~ 62(f(A), f ( B ) )
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(2) For every C, D e I x, 62(C, D) ff tSI(A, B) for every A, B e I x such that f ( A ) = C and f ( B ) = D. We remark that when 61 is of type I then (2) becomes: t52(C, D) b,(f-'(C), f-'(O)). Definition 2.3. 6, is said to be finer than 6 2 when the identity map from (X, 6l) into (X, 62) is preproximal. So 8, is finer than 62 (8~ >/62) when the graphs of 6, and 62 in I x x I x satisfy G(6,) c G(62). Theorem 2.1. The set of preproximities on X with the relation >~ is a lattice with operations ^ and v defined by: 6, ^ 62(A, B) ¢:~ 6,(A, B) or 62(A, B), 6, v 62(A, B) ¢~ 6,(A, B) and 62(A, B).
Theorem 2.2. When 6, and 62 are symmetrical, so are 81 v 62 and 61 ^ 62. When 61 and 62 are of type I, so are 8, v 62 and 6, ^ 62. When 6, and b2 are of type D, SO is 61 A 6 2, When 61 and b2 are of type S, so is 81 v 62. Proof. The proof of symmetry (resp. type I) is obvious. If 61 and 62 are of type D, then 61(A U B, C) ¢~ 61(A, C) or 61(B, C) and 62(A U B, C) ¢~ 62(A, C) or 62(B, C). So, 6, ^ 62(A U B, C) ¢:> di,(A Y B, C) or 62(A U B, C) <:~ 61(A,C) or 6,(B,C) or 62(A,C) or 62(B,C) ¢~ 6 , ^ 8 2 ( A , C ) or 61^ 62(B,C). Hence 61^62 is of type D. If 61 and 62 are of type S, then 6, v 62 (A, B) ff /~,(A, B) or 62(A, B). Suppose that tSI(A, B) ~ there is C e I x such that tS,(A, C) and t51(C, B) ~ 6, v 62 (A, C) and 6, v 62 ((~, B). Hence 6, v 62 is of type S, which completes the proof.
3. Fuzzy preproximity and fuzzy pretopology It is interesting to discuss the links between fuzzy preproximity and fuzzy pretopology on X. Theorem 3.1. Let 6 be a fuzzy preproximity on X. We define: al(A ) = g { C 18(A, C)}. Then al is a fuzzy pretopology on X. If 6 satisfies (P4) (resp. (P3), (Ps) or (P4), (Ps) and (P4), (P6)), then al is of type I (resp. D and S).
Proof. al(0) = U {C I t~(O, C)}, but for every C e I x , we have 6($,C). So,
al(0)---X. Nowal(A)=U{CIS(A, C)}, but 8(A, C) ~ m n c = 0
~ CcA,
and then a l ( A ) c A ff A c al(A), which proves that al is a fuzzy pretopology. If 6 i s of type I a n d A ~ B , then for every C w e have tS(A,C) ~ tS(B,C) from which we conclude that U {C I tS(A, C)} c U {D I tS(B, D)}. Then al(A) aj(B) and al is of type I.
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If 6 is of type D, we must show that 61(A t3 B) = a~(A) U a~(B). First we show that al is of type I. Suppose that A, B, C • I x such that A ~ B and 6(B, C); we can write A --- B U D. Then 6(A, C) ¢:> 6(B U D, C) ¢:> 6(B, C) or 6(D, C), so 6(A, C). For every C we have iS(A, C) ~ t$(B, C) and as precedingly we have al(A) ~ al(B). So, al(A U B) ~ al(A) U al(B), since A td B ~ A, B. We prove now a1(A)Ual(B)=aI(AUB). Let G 3 = { C I 6 ( A , C ) and 6(B,C)}, G1 = {C I 6(A, C)} and Gz = {C I 6(B, C)}. We must show that U (33 = [_) G1 f3 I,_)(32. Suppose that V G1 A V G2(x) = or. Then for every fl < or there exists C e G1, D • (72 such that C(x) >-fl and D(x) >Ift. Since C • G1 and D e G2, it follows that t$(A, C) and 6(B, D). But C fq D c C, D and 6 is of type I. Hence 6(A, C N D) and 6(B, C N D). But 6 is of type D, and hence 6(A t.J B, C fl D). Therefore C fq D • G3. So, C N D (x) t> fl ~ V G3(x) I> ft. Since [0, 1] is a lattice, then for every fl < re, V G3(x) 1>fl f f V G3(x) >~or which proves that al(A) U al(B) (A U B). Suppose that (5 is of type S; we show that for every A, al(A)= a~(A). We only prove that al(A)~aZl(A), or equivalently al(al(A))Dal(A) or U (C I 6(al(A), C)} D (_J {C [ (5(A, C)}. The result is established if we prove that 6(A, C) ~ 6(al(A), C) or 61(al(A), C) ~ 6(A, C). Assume this is not the case. Then, there exists C such that 6(al(A), C) and 6(A, C), so we can find D such that (~(A,/5) and 6(D, C). But 61(al(A), C) and (5(D, C) implies D ~ al(A) (because if D ~ al(A) and 6(al(A), C), then from (P4) we have 6(D, C)). There is y • X such that D(y) ~(1) and for x :/:y, /)(x) = ~(D(x)) and C~(x) = 0. We prove that 6(A, Cy); suppose that on the contrary 6(A, Cy). Then by definition a~(A) ~ Cy. So, ~[a~(A)(y)]l> Cy(y) = ~(l) and a~(A)(y)~l, a contradiction. But from 6(,4, Cy) a n d / ) ~ Cy we deduce that 6(C,/)), by use of (P4), a contradiction. Many constructions of fuzzy pretopology from a fuzzy preproximity are possible. For instance: Definition 3.2. Let 6 be a fuzzy preproximity. We define:
az(A)(x) V (a [ 6(A, (x, =
Proposition 3.2. Let 6 be a type I fuzzy preproximity; then az is coarser than al. Proof. Let A • I x and x • support a2(A ). Hence a2(A)(x) > 0. So, there exists o~• I - (0} such that 6(A, (x, ~)). Since (x, ~) c (x, 1) and 6 is of type L then 6(A, (x, 1)), and hence for every fl ~> o~ we have 1 ~
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Proposition 3.4. When 61 is coarser than 62 then aa~ is coarser than aa~. Proof. For the first construction we have 61(A,B) ~ tSz(A,B), {B ] 31(A, B)} c {B I 32(A, B)}. Hence aa,(A) c aa~(B) and a6,(A) D aa~(B). For the second construction we have aa2(A)(x) = V {re 1 62(A, (x, o¢)} and but 62(A, (x, 00) ~
SO
aa,(A)(x) = V {~l 61(A, (x, c0},
61(A, (x, re)) and
{tr J 62(3, (x, o:))} c (tr J 6,(A, (x, or)} :ff aaz(A)(x)} c a~,(A)(x).
Theorem 3.5. Let (X, 61) and (Y, 62) be two fuzzy preproximity spaces and f :X--->Y be preproximally continuous. Then f is continuous for the fuzzy pretopologies induced by the second construction. Proof. Let a2 and a~ be the second construction pretopologies on X and Y respectively which are defined in terms of the preproximities 61 and 62 respectively. We want to show that f(aE(A)) c a~(f(A)), A ~ I x. Let y ~ Y and f(aE(A))(y) = re. We will show that a~(f(A))(y)~ 0:. Let fl e (0, o0. Then flt 7 > ft. Since this is true for every fle (0, t~), it follows that a~(f (A ) )(y ) >! oc. Theorem 3.6. Let (X, a) be a fuzzy pretopological space. We define 6~ as follows: For every. A, B ~ I x, 6a(A, B)C=>a(A) n a(B) ~0. Then 6 a is a symmetrical fuzzy preproximity. If a is of type I (resp. D), then so is 6o. Proof. (P1): Obvious, (P2): Let A, B e I x with A n B ~ 0. Then a(A) n a(B) =/=0 ~ 6(A, B). (P3): Obvious. (P4): Let A ~ B, C ~ D and 6a(B, D). Since a is type I, we have a(A) ~ a(B) and a(C) ~ a(D), which implies that a(A) n a(C) =/=0 ~ 6a(A, C). (Ps): 6~(A U B, C) ¢=> a(A U B) n a(C) q=ft. Since a is of type D, we have (a(A ) O a(C)) U (a(B) n a(C)) :/:0 a(A) n a(C) --/:fl or a(B) n a(C) =/:0 ¢:~ 6~(A, C) or tS~(B, C).
4. Frizzy prepro~fimity and fuzzy preuniformity Theorem 4.1. Let U be a fuzzy preuniformity. Define 6 as follows: 6(A, B) ¢~ for every u e U , ~ ( A , u ) n~(B,u)=/:O, where ~ ( A , u ) ( x ) = V ( u ( x , y ) ^
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A(y) l Y • X}. Then 6 is a symmetrical type I fuzzy preproximity. If U is of type D, then so is ~. Proof. (P0: ~(t~, u) = t~. We have A = ~ or B = ~ ~ iS(A, B). (P2): A A B 4 : ~ ~ ~ ( A , u ) n ~ ( B , u ) 4 ; t J , and we have A A B 4 : ~ t
8).
(P3): Obvious. (P4): C = A ~ ~ ( C , u ) m ~ ( A , u ) and D ~ B ~ ~ ( D , u ) = ~ ( B , u ) . We conclude that C = A, D = B, ~(A, u) O ~(B, u) :/: ~ f f ~(C, u) n ~(D, u) 4: t~. Then 6(A, B) ~ 6(C, O). ( P s ) : A U B D A , B; by (P4) we deduce that 6 ( A , C ) or 6 ( B , C ) t~(A U B, C). To show the converse, let A, B • I x such that t~(A, C) and t~(B, C). We will show that tS(A U B, C). Assume that 6(A U B, C); then for every u • U, ~(A U B, u) O ~(C, u) 4: ~ ¢:> for every u • U, (~(A, u) U ~(B, u)) O ~(C, u) :/: ~t ~ for every u • U, either ~(A, u) n ~(C, u) 4: ~ or ~(B, u) O ~(C, u):/:lJ. Since tS(A, C) and tS(B, C), then there are v, w • U such that ~(A, v) n ~(C, v) = ~ and ~(B, w) n ~(C, w) = ~. But v n w • U and v O w c v,w, and then ~(A, v n w ) o ~ ( C , v n w ) = ~ and ~(B, v n w ) n ~ ( C , v n w) = ~t, a contradiction.
5. Product of fuzzy preproximities Definition 5.1. Let f//:X---> (Y~, pi), i • / , be a family of functions, where Pi are fuzzy preproximities on Y~. The initial structure 6 on X is the coarsest one for which f/ is fuzzy preproximal. Let fii:(Xi, 6i)---> ¥, where i • l and 6i are fuzzy preproximities. The final structure p on Y is the finest one for which f,. is fuzzy preproximal. We can easily deduce that the initial and final structures are defined as:
6(A, B) ¢:> for every i • I, pi(ft.(A), f/(B)) and, when f//are surjective,
p(C, D) ¢~ there is i • L,
¢~i(m,B)
and fi(A) = C, f~(B) = D;
when 6i are of type I, then
p(C, D) ¢:> there is i • L 6i(F-:(C), f~-l(o)). Theorem 5.1. Let f~ :X---~ (Y,., Pi) and 6 be the initial structure on X. If for every i • I, Pi are symmetrical (resp. type I), then so is 6. Proof. (P3): 6(A, B) ~ for every i, pi(fi(A),fi(B)) ~ for every i, pi(fi(B), f,.(A))--> di(B, A). (P4): 6(A, B) ~ for every i, pi(fi(A),fi(B)). Let C ~ A and D = B; we have f~(C) mrs(A) and f~(D) =f(B). So, pi(f(C), f ( D ) ) ~ 6(C, D).
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Theorem 5.2. Let fii: ( X , 6 i ) ~ Y be bijective functions and 6 be the final structure on Y. If for every i • L ~ are symmetrical (resp. type I and D), then so is p. Proof. (P3): p(A, B) ~ there is an i • I such that 6i(H, G) and fii(H)=A, f//(G) = B. But ~i(H, G) ~ t~i(G,H), and we conclude that p(B, A). (e4): p(A, B) ~ there is i • I such that 6i(H, G) and fi(H) = A, f/(G) = B. Let C D A, D ~ B; we have f,:l(C) ~ H and f . ~ ( D ) ~ G, so 6~(f~(C), f ~ ( D ) ) , but fi(f~l(C)) = C and f~(f?~(D)) = D. Hence p(C, D). (Ps): p(A LJ B, C) ¢~ there i • I such ,~,~t b~(H, G), with f~(H) = A LJ B and f/(G) = C. Consider HA = H Nf~I(A) and HD = H Nf~-I(B); we have HA LI lib = H and f,(HA) = A, f~(Hs) = B. So 6i(HA t.J liB, G) <=> 6~(HA, G) or 6i(HR.G), and we deduce that p(A LJ B, C) <=~ p(A, C) or p(B, C). Definition 5.2. Let (X;, 6~), i • I, be fuzzy preproximity spaces and X = l-li Xi. The product structure t5 = t ~ 6 ~ on X is an initial structure with the functions being the projections P~ :X--> X~ are fuzzy preproximally continuous. From the results on initial structures we can deduce: Theorem 5.3. Let (Xi, 6i), i • I, be fuzzy preproximity spaces and X = II Xi.
Then 5(A, B) <:~ ~i ¢~i(A' B) <:~ for every i • I, p,(P~(A), P/(B)). If the 8i are symmetrical, type I, then the product structure is symmetric, type I respectively. Theorem 5.4. Let (Xi, ~), i • I, be a family of symmetric type I preproximity spaces. Let X = I-[ Xi and ~ = ~ i 6i. We define the relation p on I x x I x by: p(A, B) <=~ for every Ai (i = 1. . . . . n), Bj (j = 1 , . . . , m) such that [.-~i=1Ai= A, [.-J~j=lB~ = B there exist iv, Jo such that for every i, 6i(Pi(Aio), P/(Bj0)). Then the relation p is symmetric, type I and D, and it is the coarsest type D for which projections are preproximally continuous. Proof. Suppose that p(A, B). Then obviously we have 8(A, B). In fact p is related to iS, which is symmetric type I, in the following way: n A i = A and p(A, B) ¢:> for every finite covers of A and B, I._Ji=l I--J?=1 Bj = B, we can find iv and lo such that 6(Aio , B~o). It is then rather easy to show that such a construction gives us the coarsest fuzzy preproximity, finer than 6, which is type D.
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References [1] R. Badard, Fuzzy pretopological spaces and their representation, J. Math. Anal. Appl. 81 (1981) 378-390. [2] R. Badard, Fuzzy preuniform structures and the structures they induce, Part I, J. Math. Anal. Appl. 100 (1984) 530-548. [3] B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal AppL 58 (1977) 559-577. [4] A.K. Katsaras, Fuzzy proximity spaces, J. Math. Anal AppL 68 (1979) 100-110. [5] C.V. Negoita and D.A. Ralescu, Application of Fuzzy Sets to System Analysis, ISR Vol. 11 (Birkhauser, Basel-Boston, 1975). [6] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.
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FUZZY PREPROXIMITY SPACES A.S. M A S H H O U R Mathematics Department, Faculty of Science, Assiut University Assiut, Egypt
R. B A D A R D Dept. lnformatique, 1.N.S.A. de Lyon, 69621 Villeurbanne, France
A.A. R A M A D A N Mathematics Department, Faculty of Science, Beni-Suef, Egypt Received December 1986 Revised March 1988
Abstract: The concept of proximity is generalized by defining fuzzy preproximity. Some particular constructions of fuzzy preproximity spaces are given. A natural link is established between these concepts and the concept of fuzzy pretopological spaces and fuzzy preuniform spaces. In the last section the notion of product fuzzy preproximity is introduced.
Keywords: Fuzzy sets; fuzzy proximity; fuzzy pretopology; fuzzy preuniformity.
1. Introduction In what follows, we have used the Zadeh definition of fuzzy sets, but our results can be translated without difficulty to more general cases, particularly to L-fuzzy sets, in the sense of Goguen, for which the representation theorem of Negoita and Ralescu [5] holds. In the sequel X always denotes a non-empty set. Definition 1.1 [6]. A function A from X to the unit interval I = [0, 1] is called a fuzzy set on X, and I x denotes the family of fuzzy sets in X. If A takes only the values 0, 1, then A is called a crisp set on X. Particularly, the crisp set which always takes the value 1 on X is denoted by X, and the crisp set which always takes the value 0 on X is denoted by ~l. Let A be a fuzzy set on X and Xo ¢ X. If
A(x)={o(O
then A is called a fuzzy singleton and it is denoted by (x, 0c). Definition 1.2 [1]. A fuzzy pretopology on X is a function satisfies the following conditions: (1) a(0) = 0. 0165-0114/90/$3.50 ~) 1990, Elsevier Science Publishers B.V. (North-Holland)
a:IX--->l x which
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(2) a(A) = A for every A • I x. The pair (X, a) is said to be a fuzzy pretopological space. A fuzzy pretopological space (X, a) is called: (3) type I: For every A, B • I x such that A c B we have a(A) ~ a(B). (4) type D: For every A, B • I x we have a(A U B) = a(A) O a(B). (5) type S: For every A • I x we have a2(A) = a(A). A fuzzy pretopological space which is of type I, D and S is a fuzzy topological space and a is its Kuratowsky closure. Let (X, a) and (Y, b) be two fuzzy pretopological spaces. A function f : X - - * Y is said to be continuous if f (a(A )) c b ( f (A ) ), for every A • I x. Definition 1.3 [2]. A fuzzy preuniformity on X is a family U of fuzzy sets on X x X which satisfies the following conditions: (1) For every u • U, u(x, x) = 1, so u ~ A where ,4= {(x, x) • X x X : x • X } . (2) v = u • U implies that v is an element of U.
We will consider the following particular fuzzy preuniform structures: (3) Symmetrical: For every u • U, we have u -1 • U where u-l(x, y) = u(y, X). (4) Type D: For every u, v • U we have u t3 v • U. (5) Type S: For every u • U, there exists v • U such that v ® v c u, where (v ® v)(x, y) = V (v(x, z) ^ v(z, y)). zEX
2. Fuzzy prepro~fimity space Definition 2.1. Let 6 be a binary relation on I x, i.e., 6 ~ I x ® I x. The facts that A 6 B and A/3 B are denoted by the symbols 6(.4, B) and 3(A, B), respectively. A binary relation 6 on I x is said to be fuzzy preproximity on X if it satisfies: (Pt): 6(A, B) ~ A : g 0 a n d B : # ~ , (P2): A N B e f J ~ 6 ( A , B ) . The pair (X, 6) is called fuzzy preproximity space. We will consider the following particular fuzzy preproximity structures: (P3): 6(A, B) ~ 6(B, A); (X, 6) is symmetrical. (P4): A ~ B , C ~ D and 6(B, D ) ~ 6 ( A , C); (X, 6) is of type I. (Ps): 6(A t_JB, C ) ¢ , ~ ( A , C) or 6(B, C); (X, 6) is of type D. (P6): 3(A, B) ~ there is C • I x such that ~(A, C) and dt(C, B), where C is the complement of C; (X, 6) is of type S. A fuzzy preproximity which satisfies (P3), (Ps) and (P6) is a fuzzy proximity as defined by Katsaras [4]. In this case we remark that (P3) and (Ps) imply (P4). Definition 2.2 [4]. Let (X, 61) and (Y, 62) be fuzzy preproximity spaces. A function f : X---* Y is said to be a preproximally continuous if and only if it satisfies one of the following equivalent conditions: (1) For every A, B ~ I x, 61(A , B) ~ 62(f(A), f ( B ) )
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(2) For every C, D e I x, 62(C, D) ff tSI(A, B) for every A, B e I x such that f ( A ) = C and f ( B ) = D. We remark that when 61 is of type I then (2) becomes: t52(C, D) b,(f-'(C), f-'(O)). Definition 2.3. 6, is said to be finer than 6 2 when the identity map from (X, 6l) into (X, 62) is preproximal. So 8, is finer than 62 (8~ >/62) when the graphs of 6, and 62 in I x x I x satisfy G(6,) c G(62). Theorem 2.1. The set of preproximities on X with the relation >~ is a lattice with operations ^ and v defined by: 6, ^ 62(A, B) ¢:~ 6,(A, B) or 62(A, B), 6, v 62(A, B) ¢~ 6,(A, B) and 62(A, B).
Theorem 2.2. When 6, and 62 are symmetrical, so are 81 v 62 and 61 ^ 62. When 61 and 62 are of type I, so are 8, v 62 and 6, ^ 62. When 6, and b2 are of type D, SO is 61 A 6 2, When 61 and b2 are of type S, so is 81 v 62. Proof. The proof of symmetry (resp. type I) is obvious. If 61 and 62 are of type D, then 61(A U B, C) ¢~ 61(A, C) or 61(B, C) and 62(A U B, C) ¢~ 62(A, C) or 62(B, C). So, 6, ^ 62(A U B, C) ¢:> di,(A Y B, C) or 62(A U B, C) <:~ 61(A,C) or 6,(B,C) or 62(A,C) or 62(B,C) ¢~ 6 , ^ 8 2 ( A , C ) or 61^ 62(B,C). Hence 61^62 is of type D. If 61 and 62 are of type S, then 6, v 62 (A, B) ff /~,(A, B) or 62(A, B). Suppose that tSI(A, B) ~ there is C e I x such that tS,(A, C) and t51(C, B) ~ 6, v 62 (A, C) and 6, v 62 ((~, B). Hence 6, v 62 is of type S, which completes the proof.
3. Fuzzy preproximity and fuzzy pretopology It is interesting to discuss the links between fuzzy preproximity and fuzzy pretopology on X. Theorem 3.1. Let 6 be a fuzzy preproximity on X. We define: al(A ) = g { C 18(A, C)}. Then al is a fuzzy pretopology on X. If 6 satisfies (P4) (resp. (P3), (Ps) or (P4), (Ps) and (P4), (P6)), then al is of type I (resp. D and S).
Proof. al(0) = U {C I t~(O, C)}, but for every C e I x , we have 6($,C). So,
al(0)---X. Nowal(A)=U{CIS(A, C)}, but 8(A, C) ~ m n c = 0
~ CcA,
and then a l ( A ) c A ff A c al(A), which proves that al is a fuzzy pretopology. If 6 i s of type I a n d A ~ B , then for every C w e have tS(A,C) ~ tS(B,C) from which we conclude that U {C I tS(A, C)} c U {D I tS(B, D)}. Then al(A) aj(B) and al is of type I.
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If 6 is of type D, we must show that 61(A t3 B) = a~(A) U a~(B). First we show that al is of type I. Suppose that A, B, C • I x such that A ~ B and 6(B, C); we can write A --- B U D. Then 6(A, C) ¢:> 6(B U D, C) ¢:> 6(B, C) or 6(D, C), so 6(A, C). For every C we have iS(A, C) ~ t$(B, C) and as precedingly we have al(A) ~ al(B). So, al(A U B) ~ al(A) U al(B), since A td B ~ A, B. We prove now a1(A)Ual(B)=aI(AUB). Let G 3 = { C I 6 ( A , C ) and 6(B,C)}, G1 = {C I 6(A, C)} and Gz = {C I 6(B, C)}. We must show that U (33 = [_) G1 f3 I,_)(32. Suppose that V G1 A V G2(x) = or. Then for every fl < or there exists C e G1, D • (72 such that C(x) >-fl and D(x) >Ift. Since C • G1 and D e G2, it follows that t$(A, C) and 6(B, D). But C fq D c C, D and 6 is of type I. Hence 6(A, C N D) and 6(B, C N D). But 6 is of type D, and hence 6(A t.J B, C fl D). Therefore C fq D • G3. So, C N D (x) t> fl ~ V G3(x) I> ft. Since [0, 1] is a lattice, then for every fl < re, V G3(x) 1>fl f f V G3(x) >~or which proves that al(A) U al(B) (A U B). Suppose that (5 is of type S; we show that for every A, al(A)= a~(A). We only prove that al(A)~aZl(A), or equivalently al(al(A))Dal(A) or U (C I 6(al(A), C)} D (_J {C [ (5(A, C)}. The result is established if we prove that 6(A, C) ~ 6(al(A), C) or 61(al(A), C) ~ 6(A, C). Assume this is not the case. Then, there exists C such that 6(al(A), C) and 6(A, C), so we can find D such that (~(A,/5) and 6(D, C). But 61(al(A), C) and (5(D, C) implies D ~ al(A) (because if D ~ al(A) and 6(al(A), C), then from (P4) we have 6(D, C)). There is y • X such that D(y)
az(A)(x) V (a [ 6(A, (x, =
Proposition 3.2. Let 6 be a type I fuzzy preproximity; then az is coarser than al. Proof. Let A • I x and x • support a2(A ). Hence a2(A)(x) > 0. So, there exists o~• I - (0} such that 6(A, (x, ~)). Since (x, ~) c (x, 1) and 6 is of type L then 6(A, (x, 1)), and hence for every fl ~> o~ we have 1 ~
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Proposition 3.4. When 61 is coarser than 62 then aa~ is coarser than aa~. Proof. For the first construction we have 61(A,B) ~ tSz(A,B), {B ] 31(A, B)} c {B I 32(A, B)}. Hence aa,(A) c aa~(B) and a6,(A) D aa~(B). For the second construction we have aa2(A)(x) = V {re 1 62(A, (x, o¢)} and but 62(A, (x, 00) ~
SO
aa,(A)(x) = V {~l 61(A, (x, c0},
61(A, (x, re)) and
{tr J 62(3, (x, o:))} c (tr J 6,(A, (x, or)} :ff aaz(A)(x)} c a~,(A)(x).
Theorem 3.5. Let (X, 61) and (Y, 62) be two fuzzy preproximity spaces and f :X--->Y be preproximally continuous. Then f is continuous for the fuzzy pretopologies induced by the second construction. Proof. Let a2 and a~ be the second construction pretopologies on X and Y respectively which are defined in terms of the preproximities 61 and 62 respectively. We want to show that f(aE(A)) c a~(f(A)), A ~ I x. Let y ~ Y and f(aE(A))(y) = re. We will show that a~(f(A))(y)~ 0:. Let fl e (0, o0. Then fl
4. Frizzy prepro~fimity and fuzzy preuniformity Theorem 4.1. Let U be a fuzzy preuniformity. Define 6 as follows: 6(A, B) ¢~ for every u e U , ~ ( A , u ) n~(B,u)=/:O, where ~ ( A , u ) ( x ) = V ( u ( x , y ) ^
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A(y) l Y • X}. Then 6 is a symmetrical type I fuzzy preproximity. If U is of type D, then so is ~. Proof. (P0: ~(t~, u) = t~. We have A = ~ or B = ~ ~ iS(A, B). (P2): A A B 4 : ~ ~ ~ ( A , u ) n ~ ( B , u ) 4 ; t J , and we have A A B 4 : ~ t
8).
(P3): Obvious. (P4): C = A ~ ~ ( C , u ) m ~ ( A , u ) and D ~ B ~ ~ ( D , u ) = ~ ( B , u ) . We conclude that C = A, D = B, ~(A, u) O ~(B, u) :/: ~ f f ~(C, u) n ~(D, u) 4: t~. Then 6(A, B) ~ 6(C, O). ( P s ) : A U B D A , B; by (P4) we deduce that 6 ( A , C ) or 6 ( B , C ) t~(A U B, C). To show the converse, let A, B • I x such that t~(A, C) and t~(B, C). We will show that tS(A U B, C). Assume that 6(A U B, C); then for every u • U, ~(A U B, u) O ~(C, u) 4: ~ ¢:> for every u • U, (~(A, u) U ~(B, u)) O ~(C, u) :/: ~t ~ for every u • U, either ~(A, u) n ~(C, u) 4: ~ or ~(B, u) O ~(C, u):/:lJ. Since tS(A, C) and tS(B, C), then there are v, w • U such that ~(A, v) n ~(C, v) = ~ and ~(B, w) n ~(C, w) = ~. But v n w • U and v O w c v,w, and then ~(A, v n w ) o ~ ( C , v n w ) = ~ and ~(B, v n w ) n ~ ( C , v n w) = ~t, a contradiction.
5. Product of fuzzy preproximities Definition 5.1. Let f//:X---> (Y~, pi), i • / , be a family of functions, where Pi are fuzzy preproximities on Y~. The initial structure 6 on X is the coarsest one for which f/ is fuzzy preproximal. Let fii:(Xi, 6i)---> ¥, where i • l and 6i are fuzzy preproximities. The final structure p on Y is the finest one for which f,. is fuzzy preproximal. We can easily deduce that the initial and final structures are defined as:
6(A, B) ¢:> for every i • I, pi(ft.(A), f/(B)) and, when f//are surjective,
p(C, D) ¢~ there is i • L,
¢~i(m,B)
and fi(A) = C, f~(B) = D;
when 6i are of type I, then
p(C, D) ¢:> there is i • L 6i(F-:(C), f~-l(o)). Theorem 5.1. Let f~ :X---~ (Y,., Pi) and 6 be the initial structure on X. If for every i • I, Pi are symmetrical (resp. type I), then so is 6. Proof. (P3): 6(A, B) ~ for every i, pi(fi(A),fi(B)) ~ for every i, pi(fi(B), f,.(A))--> di(B, A). (P4): 6(A, B) ~ for every i, pi(fi(A),fi(B)). Let C ~ A and D = B; we have f~(C) mrs(A) and f~(D) =f(B). So, pi(f(C), f ( D ) ) ~ 6(C, D).
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Theorem 5.2. Let fii: ( X , 6 i ) ~ Y be bijective functions and 6 be the final structure on Y. If for every i • L ~ are symmetrical (resp. type I and D), then so is p. Proof. (P3): p(A, B) ~ there is an i • I such that 6i(H, G) and fii(H)=A, f//(G) = B. But ~i(H, G) ~ t~i(G,H), and we conclude that p(B, A). (e4): p(A, B) ~ there is i • I such that 6i(H, G) and fi(H) = A, f/(G) = B. Let C D A, D ~ B; we have f,:l(C) ~ H and f . ~ ( D ) ~ G, so 6~(f~(C), f ~ ( D ) ) , but fi(f~l(C)) = C and f~(f?~(D)) = D. Hence p(C, D). (Ps): p(A LJ B, C) ¢~ there i • I such ,~,~t b~(H, G), with f~(H) = A LJ B and f/(G) = C. Consider HA = H Nf~I(A) and HD = H Nf~-I(B); we have HA LI lib = H and f,(HA) = A, f~(Hs) = B. So 6i(HA t.J liB, G) <=> 6~(HA, G) or 6i(HR.G), and we deduce that p(A LJ B, C) <=~ p(A, C) or p(B, C). Definition 5.2. Let (X;, 6~), i • I, be fuzzy preproximity spaces and X = l-li Xi. The product structure t5 = t ~ 6 ~ on X is an initial structure with the functions being the projections P~ :X--> X~ are fuzzy preproximally continuous. From the results on initial structures we can deduce: Theorem 5.3. Let (Xi, 6i), i • I, be fuzzy preproximity spaces and X = II Xi.
Then 5(A, B) <:~ ~i ¢~i(A' B) <:~ for every i • I, p,(P~(A), P/(B)). If the 8i are symmetrical, type I, then the product structure is symmetric, type I respectively. Theorem 5.4. Let (Xi, ~), i • I, be a family of symmetric type I preproximity spaces. Let X = I-[ Xi and ~ = ~ i 6i. We define the relation p on I x x I x by: p(A, B) <=~ for every Ai (i = 1. . . . . n), Bj (j = 1 , . . . , m) such that [.-~i=1Ai= A, [.-J~j=lB~ = B there exist iv, Jo such that for every i, 6i(Pi(Aio), P/(Bj0)). Then the relation p is symmetric, type I and D, and it is the coarsest type D for which projections are preproximally continuous. Proof. Suppose that p(A, B). Then obviously we have 8(A, B). In fact p is related to iS, which is symmetric type I, in the following way: n A i = A and p(A, B) ¢:> for every finite covers of A and B, I._Ji=l I--J?=1 Bj = B, we can find iv and lo such that 6(Aio , B~o). It is then rather easy to show that such a construction gives us the coarsest fuzzy preproximity, finer than 6, which is type D.
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References [1] R. Badard, Fuzzy pretopological spaces and their representation, J. Math. Anal. Appl. 81 (1981) 378-390. [2] R. Badard, Fuzzy preuniform structures and the structures they induce, Part I, J. Math. Anal. Appl. 100 (1984) 530-548. [3] B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal AppL 58 (1977) 559-577. [4] A.K. Katsaras, Fuzzy proximity spaces, J. Math. Anal AppL 68 (1979) 100-110. [5] C.V. Negoita and D.A. Ralescu, Application of Fuzzy Sets to System Analysis, ISR Vol. 11 (Birkhauser, Basel-Boston, 1975). [6] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.