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Fuzzy uniformities induced by fuzzy proximities
Fuzzy Sets and Systems 31 (1989) 111-121 North-Holland
FUZZY U N I F O ~ E S PI~OxIMrHEs
111
INDUCED BY FUZZY
Giuliano ARTICO and Roberto MORESCO Dipartimento di Matematica Pura • Applicata, Universit~ di Padova, via Belzoni 7, 35131 Padova, Italy Received May 1987 Revised September 1987
Abstract: Given a fuzzy proximity 6, we construct a fuzzy uniformity (in the sense of Lo~en) which turns out to be the coarsest one among those which induce 6. We show that some ~asic theorems of general topology about precompact uniformities have a nice extension to fuzzy set topology.
AMS Subject Classification: 54A40. Keywords: Fuzzy uniform space and map; fuzzy proximity space and map; precompact uniformity; functor.
~educfion In [2] we have introduced the notion of fuzzy proximity as a map from I x × l X into I which gives a 'degree of nearness' between fuzzy sets, meaning in a certain sense how likely two fuzzy sets are of being near each other. We have shown that this concept generalizes the analogous one of classical topology and that it is quite compatible with the notion of fuzzy uniformity introduced by Lowen [5]. In particular we have seen that every fuzzy uniformity lI induces a fuzzy proximity 6u in such a way that the fuzzy topologies generated by the two structures coincide; moreover fuzzy uniform maps, fuzzy proximity maps and fuzzy continuous maps behave in the desired way, namely the given correspondences are functorial. To obtain a full analogy with the classical case we still needed to examine some questions which naturally arise: 1. Is every fuzzy proximity induced by a fuzzy uniformity? 2. If the answer to the previous question is affirmative, is there a coarsest uniformity among the ones which induce a given fuzzy proximity? 3. Given a fuzzy proximity 6, is it possible to provide an explicit method for constructing a fuzzy uniformity lI~ such that, of course, 6u~ = 67 f~,d is !!~ the uniformity requested in 2? 4. How can one describe the fuzzy uniformities of the form 1167 In this paper we give positive answers to the pi~evious questions. Moreover we show that the correspondence which assigns to a fuzzy proximity space (X, 6) the 0165-0114/89/$3.50 (~ 1989, Elsevier Science Publishers B.V. (North-Holland)
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G. Alnico, R. Moresco
fuzzy uniform space (,t, LIa) is functorial and that fuzzy uniform spaces of the type (,t", Ha) and uniformly continuous functions form a reflective full subcategory of the category of fuzzy uniform spaces and maps. As far as the fourth question concerns, it turns out that a fuzzy ~aiformity is of type Ha if and only if it admits a basis of elements of 'finite type' (Definition 3.2) (in the classical case this is the concept 0i totally bounded uniformity); tt~rthermore it is rather interesting to observe that these uniformities are exactly the precompact ones introduced and studied in [6]. We remark that a useful tool for our investigation is a theorem due to N. Morsi which says that a fuzzy proximity is completely determined by its behaviour on crisp subsets [7]; we thank the author for providing us with a preprint (,f his paper. Section 1
For definitions and results on fuzzy uniform spaces and fuzzy proximity spaces we refer the reader to [4], [5] sad [2]. However for convenience we recall the definitions of fuT~y uniform basis and of fuzzy proximity. By the way we point out that in the definition of fuzzy proximity we adopt the formulation of axiom FP4 proposed by Morsi in [7], since this formulation is equivalent to the analogous one provided in [2] and is quite simpler than it. As usual I denotes the real unit interval. X is a set and we use the same symbol to indicate both a subset of X and its characteristic function. Moreover, since no confusion can arise, if a belongs to I we denote the constant fuzzy subset with value a by the same symbol a. DeflJtion. A subset ~ c I x×x is called a fuzzy uniform basis if and only if the following conditions are fulfilled. F U B I . ~ is a prefilter basis. FUB2. For every p e ~ and for every x e X, p(x, x) = 1. FUB3. For every p e ~ and for every real number e > 0, there exists/~ e ~ such that .8~, - e ~< ,p. FUB4. For every fl¢ ~ and for every real number e > 0, there exists ,6, e ~ such that p+ o p, - e <. p.
Defin|don. A function ~" I x x I x--> I is called a fuzzy proximity if and only if for any/~, v, p e I x if fulfills the following conditions: ~ 1 . a(O, 1)=0. M Q a(., p) = a(p, FP3. p) v 6(v, p ) = v v, p). FP4. if 6(~u, p) = a, for every e > O, there exists C c_ X such that 6(~u, C) < a + e, 6(X\ C,p)~ (~ ^ p)(x) for every x e,t". FP6. if [p - p'[ ~
Fuzzy uniformitiesinduced by fuzzy proMmities
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define a map [A, BI :X × X ~ I as follows: 6(A, B) L1
IA, B[ (x, y ) = {
if (x, y) ¢ A × B; otherwise.
In other words [A, B[ is (X × X \ A × B) v 6(A, B). Notice that the function JA, B[ is the constant 1 in each of the following cases: A is empty; B is empty; A meets B. L I . Proposition. If 6 is a fuzzy proximity on a set X, then the collection $6 = {/~=1 JAi, BiI'(A~, Bi) ¢ 2x × 2x} is a fuzzy uniform basis on X. Proof. •UB2: trivial. FUBI: observe that ~6 is closed under finite meets and that 0 does not belong to ~6 by FUB2. FUB3: by FP2, it is immediate that slA, B[ = IB, AJ. FUB4: for given subsets A, B, by FP4 there exists C such that 6(A, C ) < ~(.4, B~ -~ ~, ~ ( X \ C, B) < ,~(A, B) + e. First observe that
IX\ C, Bjo [A, C[ (x, y)= zV ([A, CI (x, z) ^ IX\ C, BI (z, y)) cX IA, BI (x, y) + e; in fact if (x, y) e A x B, then either (x, z) ¢ A × C or (z, y) ~ ( X \ C) × B. Then denoting by ~ the element [X\ C, B[ ^ [A, C[, we have shown that o~o~< [A, BI + e. Finally, if we have an arbitrary element A~--~ [A~, B~[ of ~6, we take 0~isuch that o~o ~i ~<[A/, Bi[ + e and get (A ~ ) ° ( A of/) <~A~=l [A~, B,[ + e. [] Owing to this result, if we have a proximity 6 on a set X, it is clear in which way we can associate a fuzzy uniformity with it. Namely we consider the uniformity ~6 generated in the sense of Lowen [5] by the collection ~6. 1.2. Definition. The uniformity ~6 will be referred to as the fuzzy uniformity associated with the fuzzy proximity 6. It will be denoted by 118.
Section :2 It is not yet clear if the correspondence introduced above works well; indeed the aim of our next propositions is to show that this construction behaves in a way which is quite similar to the usual one. The first problem we tackle is to check that the assignation G given by G((X, 6 ) ) = (X, lie) is functorial. 2.1o Proposition° Let (~, ~), (Y, 7) be fuzzy proximity spaces, f :X--~ Y a fuzzy proximity map. Then f is a fuzzy uniform map from (X, 116) into (Y, 11,1).
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Therefore putting G ( f ) = f for every, fuzzy proximity map f, it turns out that the assignation G is a functor from the category FProx of fuzzy proximity spaces and maps into the category FUnif of fuzzy uniform spaces and maps. Iihreef. We have to show that for every/$ ¢ 11n the function (x, y)--,/~(f(x), f ( y ) ) belongs to lIo; clearly it is enough to check it for any/$ = IA, B I where A and B
are arbitrary subsets of Y. Denoting by
r-(A) the preimage of A, indeed
IA, BI (f(x), f ( y ) ) is r/(A, B) if (x, y) cf~'(A) × f ' - ( B ) , 1 otherwise. Since ~(A,B)>-.6(r-(A),r-(B)), this function is greater then or equal to Ir-(A), f"(B)l. E! Now we are going to describe the proximity associated in the sense of [2] with
the uniformity lla.
20.2. Lemma. If ~ is a basis for a fuzzy uniformity 11 on a set X, then for every l~, p ¢ I x we have
lboof. Trivially the first number is greater than or equal to the second one. To show the converse inequality, for every c~¢ 1~ and for every e > 0 take a~e ¢ ~) such that a~e - e ~< c~. Then
(~(~) A ~ ( p ) ) ( x ) ~ ( ( ~ - e)(~,) ^ ( ~ - e ) ( p ) ) ( x )
and the conclusion follows by the arbitrariness of e.
[21
2.3. Lenuna. Given a fuzzy proximity 6, denote by ~ the subset of ~6 consisting of those elements IA1, Bll ^ ' " ^ [A~, Bkl with the following properties: (1) A, O Ai #:fJ::~>A, = Ai;
B, n B ~ ~ B , = B j ; (3) Aj n ej = ~.
(2)
Then ~
is a basis for 116.
Pn.of. Since ~)~ is contained in ~6, it is enough to prove that, for every c~ ¢ ~6, there exists/J ¢ ~)~ such that/$ ~< c~. Let c~= IC~, Dtl A . . . A ICN, DNl. We may assume that 0r is not the constant 1 and that C~ N D~ = ~, otherwise IC~, D~I can be removed. For every (x, y ) ¢ I,.ff=~ C~ × D~ put
C~ •N {c,:x ~ C,)\U (c., :x e c,}
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and
Dy = N {D,:y e D , } \ U (D,:y ¢ D,}. Clearly the number of distinct C. as well as of distinct Dy is finite. Therefore PfA
IC~, D~I "(x, y) e U Q x D~ i=1
belongs to ~6. Clearly the properties (1) and (2) are verified since z e C~ implies C~ = C~, and similarly for the elements Dy; (3) is satisfied since Q and D~ are disjoint and if (x: y) belongs to a fixed C~ .x: D . then C~ x Dy ~_ Q × D~: hence p belongs to $~. ~mally we check that p(x, y ) ~ ~(x, y) for every (x, y): if (x, y) does not belong to [_,~/=~C~ × D. this is trivial since c~(x, y)ffi 1; otherwise, for every i such that £'~ x D~ contains (x, y), we have p(x, y) =/~(C~, Dy) <~~(C. D3; hence
P(x,y)<~A {,3(Q, D3:(x,y)eQ×D~}==(x,y).
El
In [7, Theorem 5.15] the author shows that every fuzzy proximity can be retrieved by its restriction to crisp subsets according to the formula 6(~, p ) = V ~ a ( e A 6 ( ~ , p~)), where p~ (p~) denotes the weak e-cut of/~ (p). 2.4. Theorem. With the notations introduced above, and denoting by cSu~ the fuzzy proximity induced by the fuzzy uniformity 116, as in [2], we have ~u6 = ~5. Proof. By [7, Theorem 5.15] it is enough to check this equality on crisp subsets. First we show that ~u~ ~ & Given A, B subsets of X and e > 0, take C such that ~(A, C) < ~(A, B) + e, ~(X\ C, B) < ~(A, B) + e and consider the element oc = [A, C[ A [B, X \ C[ of the uniformity Ha. Now if x e C, then
(ec(A ) ^ ec(B))(x)<~ IA, C[ (A )(x)= ~5(A, C ) < ~5(A, B) + e; if x ¢ X \ C, argue analogously. Hence we get the conclusion by the arbitrariness of e and the definition of ~u~. For the converse inequality, in view of Lemmas 2.2 and 2.3 it is enough to prove that for any a~e ~ and any A, B subsets of X, we have
A
a(A, B).
Suppose o~= A~=I [Ai, Bi[ with A , B~ satisying the properties of Lemma 2.3. If either A or B is empty the equality is trivial since the right side is 0. Let us suppose that both A and B are non-empty and observe that the left side is 1 if U~=t B~~ X. Again the left side is 1 if there exists i such that B~n B ~ ~ and U {Aj: Bj = B~} does not :ontain A; indeed for x e B~n B and y ~ A \ U {Aj: Bj = Bi) we have
(a(A) ^ o~(B))(x)=ol(A)(x)>>-A(y) ^ aq,y,x)= 1. Then, in case that A is contained in U {Aj'Bj = Bi) for every i such that B, meets
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B, we have
V
xeX
(a'(A) ^ o:(B))(x)~>
V ~(A)(x)=xya (,VAa'(y,x)) >~6(Ai,
Bi)
xeB
for every t such that A~ meets A and B~ meets B. Finally apF!ying FP3 twice and using FP2,
6(A, B)-< V {a(A, B,)'B, n <<-V {V {~5(Aj, Bi)'Bj= B, A f'IAJ*fJ}'B, NB *fJ} = V {6(Aj, Bi):A NAj *O, Bj N B *O} and we get the conclusion.
E!
2.$. Remark. For any non-empty subset A of X and any element of ~a, say ffi A~=t IA,, B,I, the following formula is worth noting:
ec(A)(x)= A {O(A, B,):A =_U {Aj:Bj ~x, ¢5(Aj, Bt)<~6(Ai, B,)}} (we recall that A ~ = 1). Indeed if the set on the right side of the equality is empty, there exists a point y belonging to A \ U {Aj:Bj ~ x}, which implies that ]At. Bi[ ( y , x ) - 1 Vi, hence e ( A ) ( x ) = 1. If the set on the right side of the equality is non-empty, then its infimum is a certain 6(Ak, Bk). Since
A =_U {Aj: ej
O(Aj,Bj)-< a0:X , then for every z such that A(z)= 1 the element A(z) ^ IAt, Bd (z, x) ^ . . . ^ IAN, BNI (z, x) is one of the numbers 6(A i, Bi) such that 6(Aj, Bj)<-6(Ak, Bk); hence e:(A)(x)<.,~(Ak, Bk). On the other hand, observe that there exists an index h such that 6(Ah, Bn)= ,5(Ak, Bk), Bh ~x and A r'lAh is not contained in the set U{Aj:Bj~x, 6(Aj, Bi)<6(Ak, Bk)}. Then there exists y(E(AFIAh)\ U{Aj:Bj~)x, 6(Ai, Bi)<6(Ak, Bk)}, and we have that ec(A)(x)~A(y) ^ o4y, x) = For a given fuzzy proximity 6 and fuzzy subsets t~, p, we may define a function from X × X into I by (x, y)--. 1 ^ (1 + 6(t~, p) - l~(x) ^ P(Y)). It is easily seen that this function coincides with [/~, p[ if ~, p are crisp subsets of x; hence we adopt the same notation to indicate this function also in the case that ~, p are arbitrary fuzzy subsets. 2.6. Proposition. If (X, 6) is a fuzzy proximity space and #, p e I x, then the function It~, P[ belongs to H6. ~of.
For every real number e > O, put
A, = {xeX'l~(x)>~6(l~, O)+ e} and B,= {x eX'p(x)>~6(~, p)+ e}. We show that It~, Pl (x, y) ~>IA,, B,I (x, y) - e. Indeed if (x, y) e A, x B,, by [7,
Fuzzy uniformities induced by fuzzy proximities
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Theorem 5.15] we have 6(~, p) ~ (~(~, p) + e) A 6(A~, B~); hence 6(~, p) ~> 6(A~, B~), so that I~, pl (x,
y) - 1 ^ (1 + ~(/~, p) - / ~ ( x ) ^ p(y)) ~>6(~, p) ~> 6(A~, B~) = IA~, B~i (x, y).
If (x, y) ¢ A, x Be, then hu, pl (x, y) is trivially greater than 1 - e. Finally I/~, Pl ~> V {IA~, B,l - e: e > 0} and since IA,, B,I belongs to ~ , then [~, p[ belongs to H~. I"l 2.7. Remark. We point out that indeed one can show that Ig, pl- V {IA~, B~Ie : e > 0}. To see that [g, Pl (x, y) ~ 0} argue as follows: if ~(/~, p)~>/~(x) A p(y), then [A~, B~i (x, y) = 1 for every e since (x, y ) ¢ A ~ x B~; if ~(~, p) 0 take e such that g(x) A p(y) -- ~(!~, p) < e <<.l~(X) A p(y) -- ~(/~, p) + e' and observe that: IA~, B~I (x, y ) -
e = 1 - e ~> 1 - ~(x) A p ( y ) ÷ ~(~, p) - e' ---I~, Pl (x, y ) -
e'.
Given a fuzzy uniformity II on a set X, it is natural now to consider the fuzzy uniformity U6n: from now on we shall denote it by t3lI. 2.8. Theorem. The fuzzy uniformity plI is coarser than lI, ~hat is ~!I is the coarsest fuzzy uniformity whose induced fuzzy proxir~;~ is 6u. Proof. It is enough to show that for A, B arbitrary subsets of X, the function IA, BI belongs to the fuzzy uniformity lI, where the fuzzy proximity which occurs in the definition of [A, BI is 6n, that is
For e > 0, take /~ ¢lI such that ( ~ (A) ^ ~ (B))(x) ~< 6u(A, B) * e for every x ¢X, and consider a point (x, y) belonging to A × B. The last inequality evaluated at y says that/$~ ( A ) ( y ) <~ 6u(A, B) + e, that Is VzEx (A(z) ^ [J~(z, y)) ~<6u(A, B) + e; for z = x this gives ~,(x, y)) ~< 6u(A, B) + e hence IA, B I is greater than or equal to V,>o ( ~ - e) so that it belongs to U by FU2 of [5, Definition 2.1]. L']
Section 3 The next step of our work consists of characterizing those fuzzy uniformities which ~come from' a fuzzy proximity, namely the ones of the form 116. For convenience of language in the following lemma, for every function ~" X x X .I, we define ~*:2x--~l x by ~ * ( A ) = ~ ( A ) . 3.1. [,emma. Let ~ : X x X--> L The following are equivalent: (i) the codomains of ~ and ~* are finite;
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(i~) the codomains of oc* and of every ¢r*(A) (as a member of I x) are finite; (iii) there exists a finit~ partition 91 = { A t , . . . , Ak} of X such that ¢r is constant on every At x A k ~f. (i)~(ii): This is trivial since o c ( A ) ( x ) f V y . A o~(y,x), SO that the codomain of o~(A) is contained in the codomain of o~for every crisp subset A. (ii) :~ (i): This is trivial since o~(x, y) = ~*({x })(y), so that the codomain of is contained in the union of the codomains of o~*(A), and there is only a finite number of distinct o~*(A). (iii)~(i): Clearly the codomain of o~ is finite. Then, since a~* preserves arbitrary suprema, we have only to show that the cardinality of the set {a,*({x}):x ¢ X } is finite. This is obvious since o~*({z}) and ¢*({y}) coincide whenever x and y belong to the same element of the partition. ( i ) ~ (iii): Put xRy iff a,*({x})--- a,*({y}) and xSy iff a,*({z})(x)= o~*({z})(y) for every z • X; clearly R and S are equivalence relations; denote by 91 the partition of X associated to the infimum of R and S. Since the codomain of o~* is finite, R has a finite number of classes; denote by Z a subset of X obtained picking up an element in each class. It is clear that xSy iff o~*({z})(x)= o~*({z})(y) for every z e Z, so that the number of classes of S is finite too, owing to the fact that the codomain of ~ is finite. Hence 91 is finite. Finally we show that o~ is constant on every At x A t if Ai, A~ belong to 91; in fact given x, x' e Ai, y, y' ¢ Aj we have ~(x, y ) = c~*({x})(y)- a'*({x'})(y)ffi a'*({x'})(y')= a'(x', y').
r'l
3.2. Definition. We say that a function ~ : X x X---, 1 is of finite type if it satisfies one of the equivalent conditions of Lamina 3.1. The proof of the following proposition is quite straightforward and we omit it. 3.3. Preposition. Let 6 be a fuzzy proximity on a set X. i'~e elements of the collection ~5 described in Proposition 1.1 are of finite type. Another lemma is needed before we are in a position to prove the theorem we are interested in.
3.4. Lem~a. Let lI be a fuzzy uniformity on a set X, oc any element of 11, A, B subsets of X. Then we have
au(A,
V
y): (x, y ) e A × e}.
Proof. Given e > O, take ~ ¢ U such that p =, p and p op ~
=
(sup(p(x, z) ^ p(z, y):x cA, y e B})
Fuzzy uniformities induced by fuzzy proximities
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--sup{zyx (fl(x, z) ^ [3(z, y)):x eA, y e B} =sup{flofl(x, y):x cA, y e B} ~
El
3.5. Theorem. Let H be a fuzzy uniformity on a set X, and denote by ~ the subset of its elements of.finite type. Then $ ffi ~1I. Proof. Clearly ~ is a prefilter basis, and #lI is coarser than ~ by Theorem 2.8 and Proposition 1.1. To show the converse, let ~ be an element of the collection and ~I = { A ~ , . . . , Ak} a finite partition sach that ~ is constant on every A~ × Aj. We claim that 0~~>A~.j IA, All. In fact if (x,y) e A,, x A~, we have
a'(x, .v)= V {o4z, w): (z, w) e A,,, x.4.} ~> 6u(,4,,,, A.) = Ai j 1,4,, ,4jl (x, .,v), where the inequality follows from Lemma ?.4 and the last equality from the definition of [A~,Aj[ (where the proximity used ~s ~u). Thh~ proves that ¢ e plI. El 3.6. Corollary. 1I = I~1I if and only if H admits a basis of elements of finite type. In [6] the authors have intro0uced and extensively studied the concept of precompact fuzzy uniform spaces; in particular they have investigated its connexions with fuzzy compactness (ultracompactness) and fuzzy completeness (ultracompleteness). Since in the classical case the precompact uniformities are exactly the ones induced by proximities, it is natural to wonder if the precompact fuzzy uniformities and the fuzzy uniformities induced by fuzzy proximities are related to each other. Although these notions are introduced in quite independent ways, we will show that indeed they concide. Let us recall that the operation o in 1x*x is associative so that if a ~ , . . . , a~v ¢ I x×x, the symbol ~,i o . - - o a~z is meaningful. Precisely we have: c N o . • .o ~ ( x , y ) ffi V
{A
~,(x,_~, x , ) : ( x ~ , . . . ,
}
x~_~) e x N - x X o - x, xN = y •
Recall that a fuzzy uniformity lI on X is precompact [6] if for every ~ e 1I and for every positive e there exists a finite number of points x ~ , . . . , XN e X such tha* V,"o, a,({x,})~>l
- e.
3.7. Theorem. Let lI be a fuzzy uniformity on a set X. Then II = pll if and only if
1I is precompact. Proof ~ : Let ~ be an element of lI; by Corollary 3.6 for every positive e there exists an element fle lI of finite type such that ~ ~>fl - e. Let ~ = {A1, • • •, Ak} be a finite partition such that fl is constant on every Ai x Aj, and for every i pick
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G. Artico, R. Moresco
up a point xieA~. Clearly ~ ( { x ~ } ) ( y ) = ~ ( x , y ) = 1 for every y eAi, therefore Vi~(~.~i})-" l and V i ~ ( { x i } ) >--Vi[3({xi}) - e = l - e. <=: let c~ be an element of R, e an arbitrary positive numbex, and take a symmetric ~ ¢ U such that / ~ s < ~ + e. There exist x l , . . . ,x~v such that V~=I ~({xi}) >~1 - e and for every i = 1 , . . . , N, put U~= {x'/~({xi})(x) ~> 1 e}. Observe that { U 1 , . . . , Us} is a cover of X and that if x, y e U~, then /$2(x, y) ~>/$(x, x~) ^/~(x~, y) ~ 1 - e. Let ~ = { A ~ , . . . , A~,} be a finite partition which refines { U ~ , . . . , U~} and define o : X x X ~ I in such a way that the value of cr is the supremum of {/](x, y):(x, y) ¢ Ai × A~} at any point of the rectangle A~ × A~. Trivially cr belongs to 1I and is of finite type, that is a belongs to 1~11.We claim that c¢~>o - 2e. Indeed if (x, y) ¢ A~ × A~ we have
,~(x, y) + e ~>V (/~(x, z) ^/3(z, w) ^ #~(w, y): (z, w)eX × x} >~V {/~(x, z) ^ ~(z, w) ^/~(w, y):(z, w)EA, ×"b} ~ V {0-e)^/~(z, w)^ 0 - e ) : ( z , w)~A~xA~}>~a(x,y)-e. The/:oncgusion is now clear by Corollary 3.6.
[2
Section 4 In the proposition below we collect some consequences of the study developed ~o far and provide them with a .o.ategorical setting. Its proof can be easily obtained; anyway let us remark that if (X, U) is a fuzzy uniform space and (Y, $ ) is a precompact fuzzy uniform space, then a function f : X ~ Y is fuzzy uniformly continuous from (X, R) into (Y~ ~ ) if and only if it is fuzzy uniformly continuous from (X, ~ll) into (Y, ~). Let us introduce some more notations. FPrec is the full subcategory of FUnif whose objects are the precompact fuzzy uniform spaces. F is the functor (see [2]) from the category FUnif into the categ,~ry FProx given by F(X, 11)--(X, 6u) on the objects, and F ( f ) = f on the maps. Moreover, we denote by ~ the correspondence from the category FUnif into the category FPrec given by ~(X, 1I)= (X, plI) on the objects, and ~ ( f ) = f on the morphisms. Finally let ¢~ be the corestriction to FPrec of the functor G defined in Proposition 2.1. 4.1. lhropos|fion. (i) ~ is a functor: indeed p = G o F. (ii) ~ is a reflector for FPrec. (iii) G is a full embedding and G is an isomorphism. (iv) F is a left ad]oint of G. Proof. The remark at the beginning of this section says that ~ is a reflector for FPrec; that is p is a k'ft adjoint of the embedding functor i of FPrec into FUnif; therefore F = G-~o ~ is a left adjoint of G = i o G. r-1
Fuzzy' t m i f o ~
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References [1] G. Artico and R. Moresco, Fuzzy proximities and totally bounded fuzzy uniformities, J. Math. Anal. Apnl. 99 (1984) 320-337. [2] G. Artico and R. Moresco, Fuzzy proximities compatible with Lowen fuzzy uniformities, Fuzzy Sets and Systems 21 (1987) 85-98. [3] H. Heniich and G.E. Strecker, Category Theory (A|lyn and Bacon, Boston, MA, 1973). [4] R. Lowen, Convergence in fuzzy topological spaces, Gen. Topology Appl. 10 (1979) 147-160. [5] R. Lowen, Fuzzy uniform spaces, J. Math. Anal. Appl. 82 (1981) 370-385. [6] R. Lowen and P. Wuyts, Completeness, compactness and precompactness in fuzzy uniform spaces, Part 1, J. Math. Anal. Appl. 90 (1982) 563-581. [7] N. Morsi~ [4eamess concepts in fuzzy neighbourhood spaces and in their proximity spaces, Fuzzy Sets and Systems 31 (1989) 83-109 (this issue).
FUZZY U N I F O ~ E S PI~OxIMrHEs
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INDUCED BY FUZZY
Giuliano ARTICO and Roberto MORESCO Dipartimento di Matematica Pura • Applicata, Universit~ di Padova, via Belzoni 7, 35131 Padova, Italy Received May 1987 Revised September 1987
Abstract: Given a fuzzy proximity 6, we construct a fuzzy uniformity (in the sense of Lo~en) which turns out to be the coarsest one among those which induce 6. We show that some ~asic theorems of general topology about precompact uniformities have a nice extension to fuzzy set topology.
AMS Subject Classification: 54A40. Keywords: Fuzzy uniform space and map; fuzzy proximity space and map; precompact uniformity; functor.
~educfion In [2] we have introduced the notion of fuzzy proximity as a map from I x × l X into I which gives a 'degree of nearness' between fuzzy sets, meaning in a certain sense how likely two fuzzy sets are of being near each other. We have shown that this concept generalizes the analogous one of classical topology and that it is quite compatible with the notion of fuzzy uniformity introduced by Lowen [5]. In particular we have seen that every fuzzy uniformity lI induces a fuzzy proximity 6u in such a way that the fuzzy topologies generated by the two structures coincide; moreover fuzzy uniform maps, fuzzy proximity maps and fuzzy continuous maps behave in the desired way, namely the given correspondences are functorial. To obtain a full analogy with the classical case we still needed to examine some questions which naturally arise: 1. Is every fuzzy proximity induced by a fuzzy uniformity? 2. If the answer to the previous question is affirmative, is there a coarsest uniformity among the ones which induce a given fuzzy proximity? 3. Given a fuzzy proximity 6, is it possible to provide an explicit method for constructing a fuzzy uniformity lI~ such that, of course, 6u~ = 67 f~,d is !!~ the uniformity requested in 2? 4. How can one describe the fuzzy uniformities of the form 1167 In this paper we give positive answers to the pi~evious questions. Moreover we show that the correspondence which assigns to a fuzzy proximity space (X, 6) the 0165-0114/89/$3.50 (~ 1989, Elsevier Science Publishers B.V. (North-Holland)
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fuzzy uniform space (,t, LIa) is functorial and that fuzzy uniform spaces of the type (,t", Ha) and uniformly continuous functions form a reflective full subcategory of the category of fuzzy uniform spaces and maps. As far as the fourth question concerns, it turns out that a fuzzy ~aiformity is of type Ha if and only if it admits a basis of elements of 'finite type' (Definition 3.2) (in the classical case this is the concept 0i totally bounded uniformity); tt~rthermore it is rather interesting to observe that these uniformities are exactly the precompact ones introduced and studied in [6]. We remark that a useful tool for our investigation is a theorem due to N. Morsi which says that a fuzzy proximity is completely determined by its behaviour on crisp subsets [7]; we thank the author for providing us with a preprint (,f his paper. Section 1
For definitions and results on fuzzy uniform spaces and fuzzy proximity spaces we refer the reader to [4], [5] sad [2]. However for convenience we recall the definitions of fuT~y uniform basis and of fuzzy proximity. By the way we point out that in the definition of fuzzy proximity we adopt the formulation of axiom FP4 proposed by Morsi in [7], since this formulation is equivalent to the analogous one provided in [2] and is quite simpler than it. As usual I denotes the real unit interval. X is a set and we use the same symbol to indicate both a subset of X and its characteristic function. Moreover, since no confusion can arise, if a belongs to I we denote the constant fuzzy subset with value a by the same symbol a. DeflJtion. A subset ~ c I x×x is called a fuzzy uniform basis if and only if the following conditions are fulfilled. F U B I . ~ is a prefilter basis. FUB2. For every p e ~ and for every x e X, p(x, x) = 1. FUB3. For every p e ~ and for every real number e > 0, there exists/~ e ~ such that .8~, - e ~< ,p. FUB4. For every fl¢ ~ and for every real number e > 0, there exists ,6, e ~ such that p+ o p, - e <. p.
Defin|don. A function ~" I x x I x--> I is called a fuzzy proximity if and only if for any/~, v, p e I x if fulfills the following conditions: ~ 1 . a(O, 1)=0. M Q a(., p) = a(p, FP3. p) v 6(v, p ) = v v, p). FP4. if 6(~u, p) = a, for every e > O, there exists C c_ X such that 6(~u, C) < a + e, 6(X\ C,p)
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define a map [A, BI :X × X ~ I as follows: 6(A, B) L1
IA, B[ (x, y ) = {
if (x, y) ¢ A × B; otherwise.
In other words [A, B[ is (X × X \ A × B) v 6(A, B). Notice that the function JA, B[ is the constant 1 in each of the following cases: A is empty; B is empty; A meets B. L I . Proposition. If 6 is a fuzzy proximity on a set X, then the collection $6 = {/~=1 JAi, BiI'(A~, Bi) ¢ 2x × 2x} is a fuzzy uniform basis on X. Proof. •UB2: trivial. FUBI: observe that ~6 is closed under finite meets and that 0 does not belong to ~6 by FUB2. FUB3: by FP2, it is immediate that slA, B[ = IB, AJ. FUB4: for given subsets A, B, by FP4 there exists C such that 6(A, C ) < ~(.4, B~ -~ ~, ~ ( X \ C, B) < ,~(A, B) + e. First observe that
IX\ C, Bjo [A, C[ (x, y)= zV ([A, CI (x, z) ^ IX\ C, BI (z, y)) cX IA, BI (x, y) + e; in fact if (x, y) e A x B, then either (x, z) ¢ A × C or (z, y) ~ ( X \ C) × B. Then denoting by ~ the element [X\ C, B[ ^ [A, C[, we have shown that o~o~< [A, BI + e. Finally, if we have an arbitrary element A~--~ [A~, B~[ of ~6, we take 0~isuch that o~o ~i ~<[A/, Bi[ + e and get (A ~ ) ° ( A of/) <~A~=l [A~, B,[ + e. [] Owing to this result, if we have a proximity 6 on a set X, it is clear in which way we can associate a fuzzy uniformity with it. Namely we consider the uniformity ~6 generated in the sense of Lowen [5] by the collection ~6. 1.2. Definition. The uniformity ~6 will be referred to as the fuzzy uniformity associated with the fuzzy proximity 6. It will be denoted by 118.
Section :2 It is not yet clear if the correspondence introduced above works well; indeed the aim of our next propositions is to show that this construction behaves in a way which is quite similar to the usual one. The first problem we tackle is to check that the assignation G given by G((X, 6 ) ) = (X, lie) is functorial. 2.1o Proposition° Let (~, ~), (Y, 7) be fuzzy proximity spaces, f :X--~ Y a fuzzy proximity map. Then f is a fuzzy uniform map from (X, 116) into (Y, 11,1).
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Therefore putting G ( f ) = f for every, fuzzy proximity map f, it turns out that the assignation G is a functor from the category FProx of fuzzy proximity spaces and maps into the category FUnif of fuzzy uniform spaces and maps. Iihreef. We have to show that for every/$ ¢ 11n the function (x, y)--,/~(f(x), f ( y ) ) belongs to lIo; clearly it is enough to check it for any/$ = IA, B I where A and B
are arbitrary subsets of Y. Denoting by
r-(A) the preimage of A, indeed
IA, BI (f(x), f ( y ) ) is r/(A, B) if (x, y) cf~'(A) × f ' - ( B ) , 1 otherwise. Since ~(A,B)>-.6(r-(A),r-(B)), this function is greater then or equal to Ir-(A), f"(B)l. E! Now we are going to describe the proximity associated in the sense of [2] with
the uniformity lla.
20.2. Lemma. If ~ is a basis for a fuzzy uniformity 11 on a set X, then for every l~, p ¢ I x we have
lboof. Trivially the first number is greater than or equal to the second one. To show the converse inequality, for every c~¢ 1~ and for every e > 0 take a~e ¢ ~) such that a~e - e ~< c~. Then
(~(~) A ~ ( p ) ) ( x ) ~ ( ( ~ - e)(~,) ^ ( ~ - e ) ( p ) ) ( x )
and the conclusion follows by the arbitrariness of e.
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2.3. Lenuna. Given a fuzzy proximity 6, denote by ~ the subset of ~6 consisting of those elements IA1, Bll ^ ' " ^ [A~, Bkl with the following properties: (1) A, O Ai #:fJ::~>A, = Ai;
B, n B ~ ~ B , = B j ; (3) Aj n ej = ~.
(2)
Then ~
is a basis for 116.
Pn.of. Since ~)~ is contained in ~6, it is enough to prove that, for every c~ ¢ ~6, there exists/J ¢ ~)~ such that/$ ~< c~. Let c~= IC~, Dtl A . . . A ICN, DNl. We may assume that 0r is not the constant 1 and that C~ N D~ = ~, otherwise IC~, D~I can be removed. For every (x, y ) ¢ I,.ff=~ C~ × D~ put
C~ •N {c,:x ~ C,)\U (c., :x e c,}
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and
Dy = N {D,:y e D , } \ U (D,:y ¢ D,}. Clearly the number of distinct C. as well as of distinct Dy is finite. Therefore PfA
IC~, D~I "(x, y) e U Q x D~ i=1
belongs to ~6. Clearly the properties (1) and (2) are verified since z e C~ implies C~ = C~, and similarly for the elements Dy; (3) is satisfied since Q and D~ are disjoint and if (x: y) belongs to a fixed C~ .x: D . then C~ x Dy ~_ Q × D~: hence p belongs to $~. ~mally we check that p(x, y ) ~ ~(x, y) for every (x, y): if (x, y) does not belong to [_,~/=~C~ × D. this is trivial since c~(x, y)ffi 1; otherwise, for every i such that £'~ x D~ contains (x, y), we have p(x, y) =/~(C~, Dy) <~~(C. D3; hence
P(x,y)<~A {,3(Q, D3:(x,y)eQ×D~}==(x,y).
El
In [7, Theorem 5.15] the author shows that every fuzzy proximity can be retrieved by its restriction to crisp subsets according to the formula 6(~, p ) = V ~ a ( e A 6 ( ~ , p~)), where p~ (p~) denotes the weak e-cut of/~ (p). 2.4. Theorem. With the notations introduced above, and denoting by cSu~ the fuzzy proximity induced by the fuzzy uniformity 116, as in [2], we have ~u6 = ~5. Proof. By [7, Theorem 5.15] it is enough to check this equality on crisp subsets. First we show that ~u~ ~ & Given A, B subsets of X and e > 0, take C such that ~(A, C) < ~(A, B) + e, ~(X\ C, B) < ~(A, B) + e and consider the element oc = [A, C[ A [B, X \ C[ of the uniformity Ha. Now if x e C, then
(ec(A ) ^ ec(B))(x)<~ IA, C[ (A )(x)= ~5(A, C ) < ~5(A, B) + e; if x ¢ X \ C, argue analogously. Hence we get the conclusion by the arbitrariness of e and the definition of ~u~. For the converse inequality, in view of Lemmas 2.2 and 2.3 it is enough to prove that for any a~e ~ and any A, B subsets of X, we have
A
a(A, B).
Suppose o~= A~=I [Ai, Bi[ with A , B~ satisying the properties of Lemma 2.3. If either A or B is empty the equality is trivial since the right side is 0. Let us suppose that both A and B are non-empty and observe that the left side is 1 if U~=t B~~ X. Again the left side is 1 if there exists i such that B~n B ~ ~ and U {Aj: Bj = B~} does not :ontain A; indeed for x e B~n B and y ~ A \ U {Aj: Bj = Bi) we have
(a(A) ^ o~(B))(x)=ol(A)(x)>>-A(y) ^ aq,y,x)= 1. Then, in case that A is contained in U {Aj'Bj = Bi) for every i such that B, meets
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B, we have
V
xeX
(a'(A) ^ o:(B))(x)~>
V ~(A)(x)=xya (,VAa'(y,x)) >~6(Ai,
Bi)
xeB
for every t such that A~ meets A and B~ meets B. Finally apF!ying FP3 twice and using FP2,
6(A, B)-< V {a(A, B,)'B, n <<-V {V {~5(Aj, Bi)'Bj= B, A f'IAJ*fJ}'B, NB *fJ} = V {6(Aj, Bi):A NAj *O, Bj N B *O} and we get the conclusion.
E!
2.$. Remark. For any non-empty subset A of X and any element of ~a, say ffi A~=t IA,, B,I, the following formula is worth noting:
ec(A)(x)= A {O(A, B,):A =_U {Aj:Bj ~x, ¢5(Aj, Bt)<~6(Ai, B,)}} (we recall that A ~ = 1). Indeed if the set on the right side of the equality is empty, there exists a point y belonging to A \ U {Aj:Bj ~ x}, which implies that ]At. Bi[ ( y , x ) - 1 Vi, hence e ( A ) ( x ) = 1. If the set on the right side of the equality is non-empty, then its infimum is a certain 6(Ak, Bk). Since
A =_U {Aj: ej
O(Aj,Bj)-< a0:X , then for every z such that A(z)= 1 the element A(z) ^ IAt, Bd (z, x) ^ . . . ^ IAN, BNI (z, x) is one of the numbers 6(A i, Bi) such that 6(Aj, Bj)<-6(Ak, Bk); hence e:(A)(x)<.,~(Ak, Bk). On the other hand, observe that there exists an index h such that 6(Ah, Bn)= ,5(Ak, Bk), Bh ~x and A r'lAh is not contained in the set U{Aj:Bj~x, 6(Aj, Bi)<6(Ak, Bk)}. Then there exists y(E(AFIAh)\ U{Aj:Bj~)x, 6(Ai, Bi)<6(Ak, Bk)}, and we have that ec(A)(x)~A(y) ^ o4y, x) = For a given fuzzy proximity 6 and fuzzy subsets t~, p, we may define a function from X × X into I by (x, y)--. 1 ^ (1 + 6(t~, p) - l~(x) ^ P(Y)). It is easily seen that this function coincides with [/~, p[ if ~, p are crisp subsets of x; hence we adopt the same notation to indicate this function also in the case that ~, p are arbitrary fuzzy subsets. 2.6. Proposition. If (X, 6) is a fuzzy proximity space and #, p e I x, then the function It~, P[ belongs to H6. ~of.
For every real number e > O, put
A, = {xeX'l~(x)>~6(l~, O)+ e} and B,= {x eX'p(x)>~6(~, p)+ e}. We show that It~, Pl (x, y) ~>IA,, B,I (x, y) - e. Indeed if (x, y) e A, x B,, by [7,
Fuzzy uniformities induced by fuzzy proximities
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Theorem 5.15] we have 6(~, p) ~ (~(~, p) + e) A 6(A~, B~); hence 6(~, p) ~> 6(A~, B~), so that I~, pl (x,
y) - 1 ^ (1 + ~(/~, p) - / ~ ( x ) ^ p(y)) ~>6(~, p) ~> 6(A~, B~) = IA~, B~i (x, y).
If (x, y) ¢ A, x Be, then hu, pl (x, y) is trivially greater than 1 - e. Finally I/~, Pl ~> V {IA~, B,l - e: e > 0} and since IA,, B,I belongs to ~ , then [~, p[ belongs to H~. I"l 2.7. Remark. We point out that indeed one can show that Ig, pl- V {IA~, B~Ie : e > 0}. To see that [g, Pl (x, y) ~
e = 1 - e ~> 1 - ~(x) A p ( y ) ÷ ~(~, p) - e' ---I~, Pl (x, y ) -
e'.
Given a fuzzy uniformity II on a set X, it is natural now to consider the fuzzy uniformity U6n: from now on we shall denote it by t3lI. 2.8. Theorem. The fuzzy uniformity plI is coarser than lI, ~hat is ~!I is the coarsest fuzzy uniformity whose induced fuzzy proxir~;~ is 6u. Proof. It is enough to show that for A, B arbitrary subsets of X, the function IA, BI belongs to the fuzzy uniformity lI, where the fuzzy proximity which occurs in the definition of [A, BI is 6n, that is
For e > 0, take /~ ¢lI such that ( ~ (A) ^ ~ (B))(x) ~< 6u(A, B) * e for every x ¢X, and consider a point (x, y) belonging to A × B. The last inequality evaluated at y says that/$~ ( A ) ( y ) <~ 6u(A, B) + e, that Is VzEx (A(z) ^ [J~(z, y)) ~<6u(A, B) + e; for z = x this gives ~,(x, y)) ~< 6u(A, B) + e hence IA, B I is greater than or equal to V,>o ( ~ - e) so that it belongs to U by FU2 of [5, Definition 2.1]. L']
Section 3 The next step of our work consists of characterizing those fuzzy uniformities which ~come from' a fuzzy proximity, namely the ones of the form 116. For convenience of language in the following lemma, for every function ~" X x X .I, we define ~*:2x--~l x by ~ * ( A ) = ~ ( A ) . 3.1. [,emma. Let ~ : X x X--> L The following are equivalent: (i) the codomains of ~ and ~* are finite;
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(i~) the codomains of oc* and of every ¢r*(A) (as a member of I x) are finite; (iii) there exists a finit~ partition 91 = { A t , . . . , Ak} of X such that ¢r is constant on every At x A k ~f. (i)~(ii): This is trivial since o c ( A ) ( x ) f V y . A o~(y,x), SO that the codomain of o~(A) is contained in the codomain of o~for every crisp subset A. (ii) :~ (i): This is trivial since o~(x, y) = ~*({x })(y), so that the codomain of is contained in the union of the codomains of o~*(A), and there is only a finite number of distinct o~*(A). (iii)~(i): Clearly the codomain of o~ is finite. Then, since a~* preserves arbitrary suprema, we have only to show that the cardinality of the set {a,*({x}):x ¢ X } is finite. This is obvious since o~*({z}) and ¢*({y}) coincide whenever x and y belong to the same element of the partition. ( i ) ~ (iii): Put xRy iff a,*({x})--- a,*({y}) and xSy iff a,*({z})(x)= o~*({z})(y) for every z • X; clearly R and S are equivalence relations; denote by 91 the partition of X associated to the infimum of R and S. Since the codomain of o~* is finite, R has a finite number of classes; denote by Z a subset of X obtained picking up an element in each class. It is clear that xSy iff o~*({z})(x)= o~*({z})(y) for every z e Z, so that the number of classes of S is finite too, owing to the fact that the codomain of ~ is finite. Hence 91 is finite. Finally we show that o~ is constant on every At x A t if Ai, A~ belong to 91; in fact given x, x' e Ai, y, y' ¢ Aj we have ~(x, y ) = c~*({x})(y)- a'*({x'})(y)ffi a'*({x'})(y')= a'(x', y').
r'l
3.2. Definition. We say that a function ~ : X x X---, 1 is of finite type if it satisfies one of the equivalent conditions of Lamina 3.1. The proof of the following proposition is quite straightforward and we omit it. 3.3. Preposition. Let 6 be a fuzzy proximity on a set X. i'~e elements of the collection ~5 described in Proposition 1.1 are of finite type. Another lemma is needed before we are in a position to prove the theorem we are interested in.
3.4. Lem~a. Let lI be a fuzzy uniformity on a set X, oc any element of 11, A, B subsets of X. Then we have
au(A,
V
y): (x, y ) e A × e}.
Proof. Given e > O, take ~ ¢ U such that p =, p and p op ~
=
(sup(p(x, z) ^ p(z, y):x cA, y e B})
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--sup{zyx (fl(x, z) ^ [3(z, y)):x eA, y e B} =sup{flofl(x, y):x cA, y e B} ~
El
3.5. Theorem. Let H be a fuzzy uniformity on a set X, and denote by ~ the subset of its elements of.finite type. Then $ ffi ~1I. Proof. Clearly ~ is a prefilter basis, and #lI is coarser than ~ by Theorem 2.8 and Proposition 1.1. To show the converse, let ~ be an element of the collection and ~I = { A ~ , . . . , Ak} a finite partition sach that ~ is constant on every A~ × Aj. We claim that 0~~>A~.j IA, All. In fact if (x,y) e A,, x A~, we have
a'(x, .v)= V {o4z, w): (z, w) e A,,, x.4.} ~> 6u(,4,,,, A.) = Ai j 1,4,, ,4jl (x, .,v), where the inequality follows from Lemma ?.4 and the last equality from the definition of [A~,Aj[ (where the proximity used ~s ~u). Thh~ proves that ¢ e plI. El 3.6. Corollary. 1I = I~1I if and only if H admits a basis of elements of finite type. In [6] the authors have intro0uced and extensively studied the concept of precompact fuzzy uniform spaces; in particular they have investigated its connexions with fuzzy compactness (ultracompactness) and fuzzy completeness (ultracompleteness). Since in the classical case the precompact uniformities are exactly the ones induced by proximities, it is natural to wonder if the precompact fuzzy uniformities and the fuzzy uniformities induced by fuzzy proximities are related to each other. Although these notions are introduced in quite independent ways, we will show that indeed they concide. Let us recall that the operation o in 1x*x is associative so that if a ~ , . . . , a~v ¢ I x×x, the symbol ~,i o . - - o a~z is meaningful. Precisely we have: c N o . • .o ~ ( x , y ) ffi V
{A
~,(x,_~, x , ) : ( x ~ , . . . ,
}
x~_~) e x N - x X o - x, xN = y •
Recall that a fuzzy uniformity lI on X is precompact [6] if for every ~ e 1I and for every positive e there exists a finite number of points x ~ , . . . , XN e X such tha* V,"o, a,({x,})~>l
- e.
3.7. Theorem. Let lI be a fuzzy uniformity on a set X. Then II = pll if and only if
1I is precompact. Proof ~ : Let ~ be an element of lI; by Corollary 3.6 for every positive e there exists an element fle lI of finite type such that ~ ~>fl - e. Let ~ = {A1, • • •, Ak} be a finite partition such that fl is constant on every Ai x Aj, and for every i pick
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up a point xieA~. Clearly ~ ( { x ~ } ) ( y ) = ~ ( x , y ) = 1 for every y eAi, therefore Vi~(~.~i})-" l and V i ~ ( { x i } ) >--Vi[3({xi}) - e = l - e. <=: let c~ be an element of R, e an arbitrary positive numbex, and take a symmetric ~ ¢ U such that / ~ s < ~ + e. There exist x l , . . . ,x~v such that V~=I ~({xi}) >~1 - e and for every i = 1 , . . . , N, put U~= {x'/~({xi})(x) ~> 1 e}. Observe that { U 1 , . . . , Us} is a cover of X and that if x, y e U~, then /$2(x, y) ~>/$(x, x~) ^/~(x~, y) ~ 1 - e. Let ~ = { A ~ , . . . , A~,} be a finite partition which refines { U ~ , . . . , U~} and define o : X x X ~ I in such a way that the value of cr is the supremum of {/](x, y):(x, y) ¢ Ai × A~} at any point of the rectangle A~ × A~. Trivially cr belongs to 1I and is of finite type, that is a belongs to 1~11.We claim that c¢~>o - 2e. Indeed if (x, y) ¢ A~ × A~ we have
,~(x, y) + e ~>V (/~(x, z) ^/3(z, w) ^ #~(w, y): (z, w)eX × x} >~V {/~(x, z) ^ ~(z, w) ^/~(w, y):(z, w)EA, ×"b} ~ V {0-e)^/~(z, w)^ 0 - e ) : ( z , w)~A~xA~}>~a(x,y)-e. The/:oncgusion is now clear by Corollary 3.6.
[2
Section 4 In the proposition below we collect some consequences of the study developed ~o far and provide them with a .o.ategorical setting. Its proof can be easily obtained; anyway let us remark that if (X, U) is a fuzzy uniform space and (Y, $ ) is a precompact fuzzy uniform space, then a function f : X ~ Y is fuzzy uniformly continuous from (X, R) into (Y~ ~ ) if and only if it is fuzzy uniformly continuous from (X, ~ll) into (Y, ~). Let us introduce some more notations. FPrec is the full subcategory of FUnif whose objects are the precompact fuzzy uniform spaces. F is the functor (see [2]) from the category FUnif into the categ,~ry FProx given by F(X, 11)--(X, 6u) on the objects, and F ( f ) = f on the maps. Moreover, we denote by ~ the correspondence from the category FUnif into the category FPrec given by ~(X, 1I)= (X, plI) on the objects, and ~ ( f ) = f on the morphisms. Finally let ¢~ be the corestriction to FPrec of the functor G defined in Proposition 2.1. 4.1. lhropos|fion. (i) ~ is a functor: indeed p = G o F. (ii) ~ is a reflector for FPrec. (iii) G is a full embedding and G is an isomorphism. (iv) F is a left ad]oint of G. Proof. The remark at the beginning of this section says that ~ is a reflector for FPrec; that is p is a k'ft adjoint of the embedding functor i of FPrec into FUnif; therefore F = G-~o ~ is a left adjoint of G = i o G. r-1
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References [1] G. Artico and R. Moresco, Fuzzy proximities and totally bounded fuzzy uniformities, J. Math. Anal. Apnl. 99 (1984) 320-337. [2] G. Artico and R. Moresco, Fuzzy proximities compatible with Lowen fuzzy uniformities, Fuzzy Sets and Systems 21 (1987) 85-98. [3] H. Heniich and G.E. Strecker, Category Theory (A|lyn and Bacon, Boston, MA, 1973). [4] R. Lowen, Convergence in fuzzy topological spaces, Gen. Topology Appl. 10 (1979) 147-160. [5] R. Lowen, Fuzzy uniform spaces, J. Math. Anal. Appl. 82 (1981) 370-385. [6] R. Lowen and P. Wuyts, Completeness, compactness and precompactness in fuzzy uniform spaces, Part 1, J. Math. Anal. Appl. 90 (1982) 563-581. [7] N. Morsi~ [4eamess concepts in fuzzy neighbourhood spaces and in their proximity spaces, Fuzzy Sets and Systems 31 (1989) 83-109 (this issue).