Fuzzy initial convergence structure

Fuzzy Sets and Systems 64 (1994) 241-244 North-Holland

241

Fuzzy initial convergence structure Bu Young Lee and Jin Han Park Department of Mathematics, Dong-A University, Pusan, 604-714, South Korea Received August 1992 Revised January 1994

Abstract:

We introduce the notion of fuzzy initial convergence structure induced by family of functions. The notion of fuzzy initial convergence structure is shown to be a generalization of fuzzy product convergence structure.

Keywords: Fuzzy convergence structure; fuzzy pretopoiogical structure and fuzzy topological structure; fuzzy initial convergence structure; fuzzy product convergence structure and fuzzy continuous function.

1. Introduction and preliminaries D.C. Kent [4] introduced the convergence structure and investigated some properties. Using this notion, we defined a fuzzy convergence structure which is a correspondence between a prefilter on a given set X and a fuzzy set in X in [5]. With each fuzzy convergence structure there is an associated fuzzy pretopological structure, the finest fuzzy pretopological structure which is coarser than the fuzzy convergence structure. In this paper, we introduce the notion of fuzzy initial convergence structure and investigate some properties. We recall some definitions and known results to be used in sequel (see [5, 8]) Let x be a nonempty set and I be the unit interval. A f u z z y set in X is an element of the set of all functions from X into L For each fuzzy set A and B in X, A ~ B if A(x) <~B(x) for all x ~ X. A fuzzy point in X is a fuzzy set in X defined by p ( x ) = Z for X=Xp ( 0 < Z ~ < l ) a n d p ( x ) = 0 for x ~ Xp, x e is called support of p and A the value o f p . Let A be a fuzzy set in X. Ifp(xp)<-A(xp), p is called f u z z y element of A, with p ~ A as notation. A fuzzy point p is said to be Correspondence to: Dr. B.Y. Lee, Department of Mathematics, Dong-A University, Pusan, 604-714 South Korea.

quasi-coincident with A, denoted by p q A, if p ( x p ) + A ( x p ) > l or p(Xp)>A'(xp), where A' denotes the complement of A, defined by A'=X-A. For a nonempty set X, F ( X ) denotes the set of all prefilters on X and P ( X ) the set of all fuzzy sets on X. For each fuzzy point p in X,/~ is denoted by {A e I x I P q A }. A fuzzy convergence structure on X is a function c from F ( X ) into P ( X ) satisfying the following conditions: (1) For each fuzzy point p in X, p e c(p). (2) For to, ~ e F ( X ) , if ( O ~ q t , then

c(to) ~_c(~'). (3) I f p e c(to), t h e n p ~ c ( t o A p ) . The pair (X, c) is said to be f u z z y convergence space. I f p ~ c(to), we say that to c-converges to p. The prefilter ~c(P) obtained by intersecting all prefilters which c-converge to p is said to be the c-neighborhood prefilter at p. If ~(p) cconverges to p for each fuzzy point p in X, then c is said to be a fuzzy pretopological structure, and (X, c) a fuzzy pretopological space. The fuzzy pretopological structure c is said to be a fuzzy topological structure and (X, c) is said to be a fuzzy topological space, if for each fuzzy point p in X, the prefilter ~ ( p ) has a prefilterbase ~ c ( p ) ~ ~ ( p ) with the following property: r q U e ~(p)

implies

U ~ ~(r).

Theorem 1.1 ([5]). Let ~ ( X ) be the set of all fuzzy convergence structures on X, and <~ a relation on ~ ( X ) defined by for each Cl, c2

~e(x), Cl~C 2

if and only

if

c2((/))czct(to )

for any to ~ F(X). Then <~ is a partially order on ~(X). Let Cl, c2 E (~(X). If cl <~c2, we say that c2 is finer than Cl, also that cl is coarser than c 2.

Theorem 1.2 ([5]). Let c be a fuzzy convergence

0165-0114/94/$07.00 (~ 1994---Elsevier Science B.V. All rights reserved SSDI 0165-0114(94)00034-5

242

B.Y. Lee, J.H. Park / Fuzzy initial convergence structure

structure on X, and ~ a function from F ( X ) to P ( X ) defined by p 6~(~)

if and only if

~U~(p)~_ cI)

for each q) • F(X), where p is a fuzzy point in X. Then ~ is the finest fuzzy pretopological structure on X coarser than c. Let f be a function from a fuzzy convergence space (X, Cx) onto a fuzzy convergence space (Y, cr). f is said to be fuzzy continuous at a fuzzy point p in X, if the prefilter f(tO) cr-converges to f ( p ) for every prefilter q) Cx-converging to p. If f is fuzzy continuous at every fuzzy point p in X, then f is said to be fuzzy continuous. Theorem 1.3 ([5]). (1) lf f is fuzzy continuous at fuzzy point p in X, then °r~(f(p)) = f C r ~ ( p ) ) . (2) If cr is a fuzzy pretopological convergence structure on Y and °U~(f(p)) = f ( ~ ( p ) ) , then f is fuzzy continuous at fuzzy point p in X. Let ( X ~ ] i • I } be a family of sets, ~i a prefilterbase on X~ for each i • I and ~ the family of fuzzy sets in the product set Ni~+Xi which are of the form Hi~iB,, where Bi =X~ except for a finite number of indices and Bi • ~ for each i • I such that B~ 4: Xi. The formula

shows that ~3 is a fuzzy prefilterbase on Hi~t X~. Let q)~ be a prefilter on Xi for each i • I. The product of { cI)i[i • I} is the prefilter on 1-[i~+Xi generated by the family of the form Hi~tBi, where B~• ~ for each i • 1 and B~=X~ for all but a finite number of indices. The product prefilter of { q>i ] i • I} is denoted by Hi+t q),.

2. Fuzzy initial convergence structure Let X be a set, (Xi, G) be a fuzzy convergence space for each i • l , f i : X - - > ( X , ci) be a surjection, c is a map from F ( X ) into P ( X ) satisfying the following condition: For any fuzzy point p in X and ~ • F ( X ) , p • c ( ~ ) if and only if f/(~) ci-converges to fi(P) for each i • I.

Then we obtain a fuzzy convergence structure c on X that is called the fuzzy initial convergence structure induced by the family { f ] i • I } (or {ci]i•I}. The following statements can easily be verified. Theorem 2.1. The initial convergence structure c is the coarsest fuzzy convergence structure with respect to all fuzzy convergence structures on X which allow every f to be fuzzy continuous for each i ~ I. Theorem 2.2. The map f from a fuzzy convergence space (S, r) onto the fuzzy initial convergence space (X, c) is fuzzy continuous if and only if fi of is fuzzy continuous for each i ~ I. Theorem 2.3. If c~ is a fuzzy pretopological structure for each i • 1, then the fuzzy initial convergence structure c induced by ( ci I i • I} is a fuzzy pretopological structure. Proof. Let p be a fuzzy point in X. Then, since f~ is fuzzy continuous for each i • I, by Theorem 1.3, ~_~,(fi(P))~f(of~(p)) for each i • I. Since c~ is a fuzzy pretopological structure for each i • 1, T'~,(f/(p)) ci-converges to fi(P). Thus, °fc(p) c-converges to p. Therefore c is a fuzzy pretopological structure. Let X be the product Ilia+X/ of the family {(Xi, ci) I i • l} of fuzzy convergence spaces. The fuzzy initial convergence structure c on X induced by {P~ [ ~ ; the canonical projection from X to Xi} is called fuzzy product convergence structure of {ci [ i • I}, and (X, c) is called fuzzy product convergence space of {(Xi, ci) [ i • I}, c = Hi~+ci as notation. Lemma 2.4. For each fuzzy point p = (Pi)i~i in fuzzy product convergence space (IIi~/Xi, c), (1) [~i~l ~/~ci(Pi) C E.(P), (2) P/(~.(p)) = ~r,(P,) for each i • I. Proof. (1) Let F be any element of IJi~t ~,(Pi)Then there exists a fuzzy set F0 = Hi~IF~ such that F0 c F, where Fi • ~c,(Pi) for each i • I and F~= Xi for all but a finite number of indices. Suppose that F0~t ~ ( p ) . Then there exists a prefilter ~ c-converging to p such that E)~ 4.

B.Y. Lee, J.H. Park / Fuzzy initial convergence structure

Since @ c-converges to p, P~(¢,) G-converges to P,(p) =p~ for each i e L Thus ~K~,(p~)~ P~(tb) for each i c I. Therefore i~l

i~l

This contradicts that F0¢ q). Therefore F0e °V~(p). Since F 0 ~ F , F 6 °V~(p). That is, IIiel °~ci(P,) C °t/'c(p ). (2) Since P~ is continuous, by T h e o r e m 1.3, ~,(Pi) c P/(°V~(p)) for each i ~ I. Also, we show that ~(°V,(p)) = °V,.(pi). If Gj is any element of P ~ ( ~ ( p ) ) , then there exists a F ~ °V~(p) such that P j ( F ) c G~. Let tp be an arbitrary prefilter which ci-converges to Pi. Take the prefilter Hi~t q~ on l]~t Xi, where {~ ~=

/~i

convergence structures on IIi,tXi of {ci l i ~ I) and {ci I i ~ I}, respectively. Then the following statements hold: (1) c' <~~. (2) If ILE, ~V,.(p,) = °Vc(p) for each f u z z y point P -- (Pi)i~t in [Ii~t X , then c' = & Proof. (1) By Theorem 1.2, 6 is the finest fuzzy pretopological structure coarser than c. Since ?i ~
(2) Let °V,.(p) = IL~, ~-,(pi). I f p = (p,),~, • c'(t0), then p~6e~(Pi(t0)) for each i ~ l . Therefore, °V,,.,(pi) c P/(q0, and so

= [I i61

otherwise.

Theorem 2.5. ci is a f u z z y pretopological structure for each i • I if and only if the f u z z y product convergence structure c of {ci I i e I) is f u z z y pretopological. Proof. If c~ is fuzzy pretopologicai for each i ~ I, then by T h e o r e m 2.3, c is also fuzzy pretopological. Conversely, let fuzzy product convergence structure c be fuzzy pretopologicai. Then for each fuzzy point p = (Pi)i~l in X, p • c ( ~ , ( p ) ) . Thus, by Lemma 2.4, Pi = e i ( p ) E ci( ei( ~/'c(p ) ) ) = ci(~l/'c, (Pi) )

for each i e I . Therefore, c~ is pretopological structure for each i c I.

a

fuzzy

c~ and c; be two f u z z y convergence structures on X~ such that c; <~c~ for each i ~ I, c and c' the f u z z y initial convergence structure on X induced by the family { S i f t : X---~(X~,c,), i • l } and {flfi:X---~ ( Xi, c ~), i • I}, respectively. Then c ' <~c. Lemma

T h e o r e m 2.7. Let c and c' be f u z z y product

if i = j ,

Then l q ~ (b~ c-converges to p = (P~)~t- Since F e °V,.(p) c Hi,t ~,, Pj(F) e ~(H,~, q~,) = t/t. Therefore Gi • ~. That is, Gi e °Vc,(p~).

2.6. Let

243

= I1 i~l

a,.

Thus p • 6(q0). T h e o r e m 2.8. Let (S, Cs) and (Xi, ci) be f u z z y

convergence spaces for each i • I, gi:(S, Cs)---~ (Xi, ci) a function and

g : (S, cs)--, (H X. c) the function defined by g(s) = (gi(s))i,i, where s is f u z z y point in S and c is a f u z z y product convergence structure of {ci ] i • 1}. Then the function g is f u z z y continuous at s if and only if each gi is continuous at s. Proof. Consider that & = p~og, where Pi is the canonical projection for each i c I. Let q0 be any prefilter in S Cs-converging to s. If g is fuzzy continuous at s, then g ( ~ ) c-converges to g(s). Thus

ga(s ) = P~(g(s ) ) • G(P/(g(q~))) = ci(gi( CI))) for each i • L Therefore g~ is fuzzy continuous at S.

Conversely, let each coordinate function g~ be fuzzy continuous at s and • a prefilter on S. If t0 Cs-converges to s, then gi(q~) c,-eonverges to gi(s), that is, gi(s) --- Pi(g(s)) E ci(gi( CI)))

Proof. If prefilter • on X c-converges to p, then

c c,(f/(a,)) Thus • c'-converges to p.

= G(P~(g(q0) ), thus g ( s ) • c ( g ( q ) ) ) . continuous at s.

Therefore

g

is

fuzzy

B.Y. Lee, J.H. Park ] Fuzzy initial convergence structure

244

2.9. Let g be the function Theorem 2.8. (1) l f g is fuzzy continuous at s, then

Theorem

as in

Thus

g(s ) = (gi(s ) )i~t

[I ~,(gi(s)) c g(~cds)).

ec (iel [I °l#ci(gi(s) ) )

ieI

(2) If ci is a fuzzy pretopological convergence structure for each i e I and

~(g(~r~(s ) ) ) = c(g( ~ ) ). T h e r e f o r e g is fuzzy c o n t i n u o u s at s.

[I ~'c,(gi(s)) c g ( ~ , s ( s ) ) , iel

then g is fuzzy continuous at s. References Proof. (1) Since g is fuzzy c o n t i n u o u s at s, by T h e o r e m 2.8, gi is fuzzy c o n t i n u o u s at s. T h u s °l/'ci(gi(s)) = gi(~cs(S)) for e a c h i c I. T h e r e f o r e

I ] ~Ag~(s)) ~ [ I g,(~s(s)) i~.l

iel

=

l-[ P,(g(~c~(s))) = g(%~(s)). i~l

(2) L e t ~ be a n y p r e f i l t e r in S a n d s e Cs(~). T h e n °V~(s) c ~ a n d

[I %,(g,(s)) ~ (°rc~(s)) = g(a,). ieI

Since

ci

is

fuzzy

pretopological,

ci(~c,(gi(s))), and so gi(s ) • Ci(e~ci(gi(s ) ) ) ~ ci( ei(i~l °l/'ci(gi(s) ) ) ).

gi(s)•

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