on fuzzy convergence

INFORMA TION SCIENCES 39,X9- 284 (1986) 269

On Fwzy Convergence

GIANGIACOMO GERLA*

Dipartimento di Mafematica ed Applicozioni “R. Caccioppoli, ” zlia Mezzocannone 8, 80134 Napoli, lt&

Communicated by Azriel Rosenfeld

ABSTRACT

We propose and examine two concepts of convergence for sequences of elements of a set X with respect to a fuzzy topology on X. Such concepts are obtained by interpreting in a multivalued logic two different first order formulas expressing the classical convergence.

1. INTRODUCTION

Recall that if X is a set and L a complete lattice, then an L-subser, or fuzzy subset, of X is a map from X to L [7, 131. Moreover, an L-topology or fuzzy ropologv is a class of L-subsets of X verifying closure properties analogous to those of a classical topology [l]. In order to relate a convergence concept to each fuzzy topology and to investigate other local properties, C. K. Wong [12], E. E. Kerre [S], and other authors have proposed two notions of fuzzy point, equipped with a corresponding notion of membership relation E between fuzzy points and fuzzy subsets. They did this in the case L = [0, 11.

In this paper we observe that these definitions and the related notions of convergence are criticizable from several points of view and that they cannot be extended in general to the case L + [O,l] (Section 3). Moreover we prove that every attempt to give another definition of point is necessarily unsatisfactory if, following the cited authors, we want E to be a nonfuzzy relation (Section 4).

This suggests that we reexamine the question of a suitable “fuzzification” of the concepts of subset, point, topology, and convergence. Multiv~ued first order logic is the heuristic tool for such an investigation. Then we show that Zadeh’s definition of fuzzy subset as a map from X to L, and of fuzzy point as

*This work has been supported by a contribution (60%. 1984) from M.P.I.

QElsevier Science Publishing Co.. Inc. 1986 52 Vanderbitt Ave.. New York, NY 10017

270 GIANGIACOMO GERLA

an element of X, seem to be the most natural and useful (Section 5). Moreover,

they do not exclude a suitable treatment of convergence in fuzzy topology

theory. This fact makes Wong’s and Kerre’s definitions of fuzzy point unneces-

sary for a treatment of local properties.

In fact, we propose two types of convergence for sequences of elements of X

related to a fuzzy topology: the p-convergence and the g-convergence. Some

basic facts about these convergences are examined. In particular we examine

their relationships with Kerre’s and Wong’s convergence and with fuzzy con-

tinuity (Section 6). Also we observe that it is possible to utilize p-convergence

and q-convergence in order to give convergences preserving operators in

functional spaces. For examples, we obtain convergences for sequences of fuzzy

subsets preserving entropy and energy operators as defined in fuzzy set theory

(Section 7). Finally we give some conditions for uniqueness of limits.

2. PRELIMINARIES

In the sequel L denotes a complete lattice whose greatest and least elements

are denoted by 1 and 0 and whose operations v and A are called disjunction

and conjunction, respectively. If X is a set, the direct power F( X, L) = { f 1 f : X + L } is a complete lattice whose elements are called L-subsets of X

[7, 131. L-sets on X, fuzzy subsets of X, and fuzzy sets on X are equivalent

expressions. We call union and intersection the lattice operations defined in the

direct power F( X, L). If (Y is an element of L, then f, denotes the L-subset of

X defined by setting f,(x) = OL for every x E X. An L-subset f such that f(x) E {O,l} for every x E X is called crisp. We identify the subsets of X with

the crisp L-subsets via the characteristic functions.

An L-topology or fuzzy topology on X is a class Y of L-subsets of X

containing fO and fi and closed with respect to unions and finite intersections

[l]. The elements of Y are called open. The L-topologies whose elements are

crisp coincide with the classical topologies.

Occasionally we have to assume that in L there are operations different from

the usual lattice operations. In the sequel an implication is any binary operation + of L such that x-+y=l if and only if x < y. Moreover, we define

Lukasiewicz negation as any involutory decreasing operation and intuitionistic negation as the map Y defined by setting

i 1 if a=O,

T(y= 0 if LY>O.

In L a convergence concept is definable as usual. Namely, let ((x,) be a sequence of elements of L. Then

lim.a,=V A (Y,, lim.ol,=A V (Y,. m n3m m n>m

ON FUZZY CONVERGENCE 271

- In general we have that lim. a, 6 lim. a,; in the case that lim. a, = lim. a, = a

- -

we write lim.a, = a.

3. WONG’S AND KERRE’S DEFINITIONS OF POINT AND

CONVERGENCE

In [12] Wong defines a fuzzy point for the space F(X) = F( X,[O, 11) as an

element of P, = X x [O,l), and he sets, for every (x, a) E X X [O,l) and f E

F( X)

(x,fi)Ewf w f(x)>a

Likewise, in [8] Kerre calls fuzzy point an element of Pk = X X (0, l] and sets

(x,a)Ekf - f(x) aa.

We define w-points and k-points as the points defined by Wong and Kerre,

respectively, and denote by E w and E k the corresponding membership

relations.

The w-convergence is defined as follows. Given a fuzzy topology y, a

sequence (p,,) of w-points of X, and a w-point p, we set sw-lim.p,, = p provided that, for every f E LT such that p E w f, there exists m E N such that

p, E w f for every n > m. In the same manner one defines the k-convergence.

The above definitions of point and of membership relation are justified by

the fact that the following very natural properties hold:

(a) pEf Vg - eitherpEf orpEg,

(b) pEf Ag - pEf andpEg, ;; :p_=gpIp EfoJ =a and {p EJ’lp =fil =P,

e { p E P 1 p E f } = { p E P 1 p E g} (extensionality principle),

(e) P = 4 - {f I p E f } = { f I q E f } (indiscernability principle),

where E denotes either E w or E k, P denotes either P, or Pk, f, g E F( X, L), and p, q E P. In spite of that, such definitions and the corresponding defini-

tions of convergence are, in my opinion, unsatisfactory from several points of view. Namely, given a family (f,) of elements of F( X, L), the very natural

property

(a’) p E Uf, * p E f, for a suitable i

does not hold for the w-points even if it is verified by the k-points. Likewise,

the property

(b’) PEW, Q pEf, foreveryi

does not hold for the k-points even if it is verified for the w-points.


Another criticism is that, in general, Wong’s and Kerre’s definitions cannot

be extended to the case L # [O,l], if we want to preserve properties (a)-(e). For

example, if L is not totally ordered, then from (x, a) E w f u g, i.e. f(x) v g(x) > a, we cannot infer that either f(x) > a or g(x) > a, i.e. (x, a) E ,f or

(x,a)E,g. Other criticisms are possible for w-convergence and k-convergence. For

example, if L = [O,l] but every element of the fuzzy topology 7 is crisp, then

the above convergences are not coincident with the ordinary convergence. This

contradicts a generally accepted principle of fuzzy set theory. Moreover,

whatever the L-topology may be, it is impossible to obtain uniqueness of the

limits. Indeed, for example, if (y, a) is a limit of a sequence and /3 < OL, then

(y, /3) is a limit of the same sequence. This is rather unsatisfactory.

4. IS ANY GOOD DEFINITION OF FUZZY POINT POSSIBLE?

On account of the above considerations, it is very natural to put forward the

following questions:

(1) Is it possible to define a new concept of fuzzy point for F(X) in such a

manner that both (a’) and (b’) hold? (2) If L # [O,l], is it possible to define a concept of point in such a manner

that (a), (b), (c), (d), and (e) hold [and eventually, (6) and (b’)]? (3) Is it possible to define a fuzzy convergence concept that coincides with

the classical one for the crisp topology and that does not preclude the

uniqueness of the limits?

As an attempt to answer these questions, in [5] we define a point space for

F( X, L) as any pair (P, E ) verifying (a), (b), (c), and (e). If also (a’) [respec-

tively (b’)] holds, then (P, E ) is called a disjunction point space or d-space [conjunction point space or c-space, respectively]. If both (a’) and (b’) hold, then

(P, E ) is called a goodpoint space for F( X, L). In this terminology, (P,, E ,) and ( Pk, E k) are respectively a d-space and a c-space for F( X, L), and these spaces are not good point spaces. Moreover, we can reformulate the first two questions as follows:

(1) Is there a good point space for F( X, L)? (2) If L is any lattice, are there point spaces (d-spaces, c-spaces, good

spaces) for F( X, L)?

Now, a simple modification of Stone’s representation techniques for lattices [5] enables us to prove that

F( X, L) admits a point space if and only if L is distributive.


This means that if L is nondistributive, a reasonable definition of point for

F( X, L) (and therefore a convergence concept for the L-topologies) seems to be

impossible. Instead, the existence of d-spaces and c-spaces is related to the V -irreduci-

bility concept. Recall that an element (Y of L is called V -irreducible (infinitely

v-irreducible) provided that, for every finite (either finite or infinite) family

(01,) of elements of L, from a < VLY, it follows that a d a, for a suitable i. Thus

(see [5, Proposition 5.61) it is possible to prove that

A c-space (a good point space) for F( X, L) exists if and only if every element of L is disjunction of v -irreducible (respectively infinitely V-irreducible) ele-

ments.

A dual result holds for the c-spaces. In particular, since the elements of (0, 11

are not infinitely v -irreducible,

There is no good point space for the class F(X) of the fuzzy subsets of a set X.

In other words, since properties (a’) and (b’) are very natural, it is impossible

to give a good definition of point for F(X).

The same considerations hold, in many cases, even if L is different from

[O,l]. Moreover, even if a good definition of point for F( X, L) is possible, the

associated convergence concept is subject to the same critiques expressed for

w-convergence and k-convergence. This is a simple consequence of Proposition 5.3 of [5].

In any case, the above considerations do not mean that Wong’s and Kerre’s

definitions, and the more general definition of point space, are devoid of

interest. For example, they enable us to define very interesting functors from

the category of the fuzzy topological spaces into the category of the classical

ones. This view is examined in [4], [lo], and other papers.

5. FUZZY MEMBERSHIP RELATION

The disadvantages observed in Sections 3 and 4, in my opinion, are conse-

quences of the basic contradiction between the concept of fuzzy set and the

concept of a crisp membership relation. Thus, it seems very natural to admit that points and sets are nonftuzy objects while the membership relation is a fuzzy relation.

Such a view is present in every attempt to give an axiomatization of fuzzy set theory via multivalued first order logic (see for example [ll]). In these attempts, the usual first order language 9 for Zermelo-Fraenkel set theory is assumed, namely a language with a binary relation symbol E . The interpretation of E

is a binary L-relation on a domain D, i.e. an L-subset of D2.


Now, the aim of this paper is not to utilize mathematical logic to give a

complete and rigorous foundation of fuzzy set theory. We will utilize it only as

a heuristic tool to extend, in a fuzzy setting, the principal topologic concepts.

Thus we do not hesitate to utilize a richer language and to (apparently) involve

second order concepts. In particular, symbols for the empty set, union and

intersection operations, and sequences of points are assumed in 3’.

Obviously, we have to assume that for every logical connective a correspond-

ing operation of L exists. We denote the connective and the corresponding

operations by the same symbols. The interpretation of the formulas proceeds as

usual. Moreover, we claim that a formula A holds provided that its valuation

IAl is 1. Thus, for example, a formula of the type B + C holds if and only if

PI d ICI. One assumes that a suitable set of axioms is verified. For example, it seems

reasonable to assume that, for every x, y, z E D,

XEYUZ f) XEYVYEZ, XEynz f) XEYAXEZ, -.XE0,

y=z 4-B Vx(xEy f) XEZ) (extensionality principle),

x=y - VZ(XEZ @ YEZ) (indiscemability principle).

Since E is interpreted as a binary L-relation E: D2 --) L, this is equivalent to

assuming that, for every x, y, z E D,

&(X,yUz)=&(X,y)VE(X,Z), E(X,ynz)=E(X,y)A&(X,z),

&(X,0) =o,

y=z * E(X,Y) =E(x,z) forevery XE D,

x=y - E(X,Z) =e(y,z) foreveryzED.

This suggests defining a multivalued version of the power set P(P) as a

structure of the type (P, S, E), where E: P x S + L is a binary L-relation and S an algebraic structure with two binary operations U, n and two distinguished elements 0 and p such that the following hold:

(3 e(p, s u r’) = E(P, s)V e(p, $9, 6) E(~,~~s’)=E(P,s)AE(~,s’),

(9 e(p,@)=O, c(p,P)=l, (d) s = s’ e E(~,s) = e(p,s’) for every s E S, (5) p=q - E(p,r)=e(q,s)foreverysES,

where p, q E P and s, s’ E S.


We call such a structure a fuzzy space. In classical set theory we have also

that, for every family (x,) of elements of D and x E D,

XEUX, ++ YixEx,, xEflx, ++ VixEx,

This is equivalent to assuming, in a multivalued model, that

E(dJX,) =vE(x,x,), E( X,fb,) = A&(X,X,)

Then it is very natural to assume that in S there are two infinitary operations U and fl, and that

(2) e(P,Us,) = Ve(p,s,), 6’) 4P,ns,) = h(P, s,),

where p E P and (3,) is any family of elements of S. A fuzzy space is a fuzzy

d-space [a fuzzy c-space] if (3) [respectively (b’)] is verified. A fuzzy space is a

good fuzzy space if both (a’) and (6’) are verified. If E is a crisp relation and

S = F( X, L), then fuzzy spaces, fuzzy d-spaces, and fuzzy c-spaces coincide

with the above defined point spaces, d-spaces, and c-spaces, respectively. In

particular, Wong’s and Kerre’s spaces are crisp fuzzy spaces.

Now it is immediate to observe that we can obtain a good fuzzy space by

setting S= F(X,L), P= X, P=f,, 0=f,,and .s(x,f)=f(x)foreveryx~ P and f E S. This space is called the Zadeh space and represents Zadeh’s original

definition of point and fuzzy set. This means also that whatever L may be, a

good fuzzy space for F( X, L) exists.

The following proposition shows that every point space is a subspace of the

Zadeh space. Thus Zadeh’s original definitions (and Goguen’s generalizations)

are, in a sense, the best ones if one requires conditions (a-(e).

PROPOSITION 5.1. Let S be a sublattice of F( X, L) such that for every

x, y E X with x # y there exists f E F( X, L) such that f(x) f f(y). Moreover,

define E: XXS+L by setting e(x,f)=f(x) for every XEX, f EF(X,L). Then (X, S, E) is a fuzzy space called a Zadeh subspace. If S is closed with respect

to infinite unions (intersections, intersections and unions) then (X, S, E) is a

d-space (c-space, good space). Conversely, every fuzzy space (d-space; c-space; good space) is isomorphic to a Zadeh subspace (with S closed with respect to the

infinite unions; intersections; unions and intersections, respectively).

Proof. The direct part of the proposition is matter of routine. To prove the converse, let (P, S, E) be any point space and s E S. Moreover, define g, E F(P,L) by setting g,(p)=e(p,s) for every pEP, and set H(s)=g,. Then from (a), (b), (?), and (d) it follows that the map H: S 4 F( P, L) is an

embeddingofthealgebraicstructure(S,U,n,Izr,P)into(F(P,L),U,n,f,,f,).


From the homomorphism theorems for the algebraic structures it follows that

H(S) is an algebraic substructure, and therefore a sublattice, of E( P, L). From (e) it follows that if p + q, then f(p) #f(q) for a suitable f E H(S).

Then H(S) individuates a Zadeh subspace. It is obvious that such a subspace is

isomorphic to (P, S, E ) via H.

Let (P, S, E ) be a d-space, (H(s,)) any family of elements of H(S),

f = UH(s,), and g = H(Us,). Then

f(P) =ww;) =Eb,Us,) =dd

and therefore UH(si) = H(Usi) E H(S). In the same manner one proceeds for the c-spaces and the good spaces.

COROLLARY 5.2. If (P, S, E ) is a fuzzy space, then (S, n , u ,0, p) is a lattice, in fact, a subdirect power of L.

Proof. It follows from the proof of Proposition 5.1.

6. FUZZY TOPOLOGY AND FUZZY CONVERGENCE

In the sequel we assume that (P, S, E ) is the Zadeh space for F( X, L). Now, recall that a topological space on a set X is a subset T of 9(X) such

that the following formulas hold:

JET, XET, sETAtET + sntET, Vi S;ET + US,ET.

An interpretation in multivalued logic of such formulas leads us to define a fuzzy topology or Ltopology as a fuzzy subset y of S such that

Y(fO) = F(fi) =I, S(f ng) >S(f)A~‘(g), WfJ a Wf,)

for every family (h) of elements of F( X, L) and f, g E F(X, L). If .7 is crisp, this definition coincides with the known definition of fuzzy topology [l]. If r is crisp and the elements of .!7 are crisp, then r is a classical topology.

To define the convergence of a sequence (n,) of elements of X to an element y of X, observe that, classically, this convergence is expressed by a formula of type

SET A YES + 3m Qn>m X,ES. (6.1)


This suggests assuming that (x,) p-converges to y with respect to 7, in

symbols %p-lkn. x, = y, if

or, equivalently,

l;m.f(x,) ~~T(fbvf(Y) QfU(X,L). (6.3)

Ob~ously, if ..T is a classical topology, the above convergence coincides with

the classical one.

PROPOSITION 6.1. Let L be totally and densely ordered. Then the following are equivalent :

(i) Sp-lim. x, = y.

(ii) For evevf E F(X, L) and a E L, if S(f) > a andf(y) > a, then there exists m E N such that f (x,) > a for every n 2 m.

(iii) lim.x, = y with respect to the classical topology generated by the subsets

{x E Xlf(x) ’ a) with a E L and S(f) > a.

Proof. Assume (i), i.e. (6.2), and that f(y) A .Tf( f) > a. Then since L is

totally ordered, from V, A, > mf(~n f > a it follows that there exists m E N such that A ,,~,,,f(x)~a,andthereforef(x,)~aforeveryn~m.Thisproves

(ii).

Assume (ii), and let a < f(y) A S(f). By hypothesis there exists m E N

such that f(x,) > a for every n > m. Then A,.,j(x,) 2 a and therefore

V~A~~~~(~~)~ > a. This means that

V A f(xn) ~V{aELIa<fty)A~(f)I =f(~)Ay(f), m n>P?nl

and this proves (i). The equivalence between (ii) and (iii) is obvious.

From (ii) we obtain the following interesting interpretation of p-conver-

gence. Suppose .7 crisp, and interpret the elements of 7 as fuzzy proper-

ties defined on X. Then Sp-lim.x, = y means that everywhere the property

f E 9 holds for y with a membership degree greater than a, f holds for x, with a degree greater than a for every n greater than a suitable integer m.

From (iii) it follows that p-convergence coincides with the ordinary convergence with respect to a suitable classical topology on X. Then all the ordinary properties of the classical convergence hold also for the fuzzy convergence.


We have obtained the p-convergence concept via a nonclassical interpretation of the formula (6.1) expressing the classical concept of convergence. Now, such a convergence can be expressed also by other, classically equivalent formulas-as an example, by the formula

SET + 3mQn2m - -(yEs --) x,Es). (6.4)

Now, suppose that in Y there are two different types of negation, - and -,, and that they are interpreted by a Lukasiewin and an intuitionistic negation. Then it is possible to rewrite (6.4) in several ways, for example as

SEY + 3mQnam 7-(yEs --, &ES). (6.5)

The inte~retation, in multiv~ued logic, of such a formula suggests calling a sequence (x”) q-convergent toy with respect to 9, in symbols Sq-lim. x, = y, if

The following proposition gives a more explicit meaning to q-convergence and shows that it is reducible to the classical notion.

PROPOSITION 6.2. The following are equivalent:

(i) .%q-lim. x, = y. (ii) If F(f) # 0, there exists m E N such that f (x,) 2 f(y) for every n > m. (iii) If F(f) f 0, 1y E L, and f(y) > CY, then there exists m E N such that

f(x,) a (Y Vn > m. (iv) lim.x, = y with respect to the classical topology on X generated by the sets

{x~Xlf(x)>a}, whereaELandF(f)+O.

Proof. To prove the equivalence between (i) and (ii), observe that, if F(f) > 0, since T - (f(y) -+ f(x,)) E {O,l), Equation (6.6) is equivalent to

VA -- (f(Y) -f(x,N =I. m n>m

This means that if F(f) > 0, then there exists m E N such that -, - (f(r) + f(x,))=l for every n&m. Since, for every trek, --a=1 if and only if cy = I, we have that (i) is equivalent to (ii).

(ii)~(iii): Let F(f)#Oand f(y)aa, then by (ii) there exists m E N such that f(x,,) >f(y) 3 a for every n 3 m.

(iii) * (ii): Let Y(f) # 0, and set a = f(y). Then by (iii) there exists m E N such that f( xn) > ff ,P) = a for every n 2 m.

(iii) IJ (iv): Obvious.


From Proposition 6.2 it follows that the same considerations given for

p-convergence hold also for q-convergence.

The following proposition relates p-convergence and q-convergence.

PROPOSITION 6.3. If .7 is crisp, then q-convergence implies p-convergence, while the converse does not hold. Moreover, if every element of L is infinitely

V -irreducible, then p-convergence and q-convergence coincide.

Proof. If 3q-lim. x, = y, then from Proposition 6.2 it follows that

v A f(xJ af(,:) m n>m

for every f such that r(f) > 0. Since r is crisp, this is equivalent to (6.2), and therefore .%p-lim. x, = y.

Assume that X=[O,l], L=[O,l], y= {fO,fi,i}, where i:X+L is the

identity map. Then the sequence (1 - l/n) is p-convergent to 1 with respect to

r-, while it is not q-convergent.

Finally, assume that every element of L is infinitely v -irreducible and that

Sp-lim,x, = y. Then, if Y-(f) = 1, from V,A,, > m f(x,,) > f(y) it follows that

A n.mf(xn) >/f(Y) f or a suitable m, and therefore that f (x,,) > f(y) for every

n > m. From (ii) of Proposition 6.2 it follows that 3q-lim.x, = y.

The following proposition relates q-convergence and p-convergence with

Kerre’s and Wong’s convergence, respectively.

PROPOSITION 6.4. Assume that L = [O,l] and that Y is crisp. Then

Sq-lim. x, = y * Xk-lim.(x,,cu) =(~,a) VaE(O,l],

3p-lim. x, = y e Sw-lim.(x,,a) =(.~,a) VdaE[O,l)

Proof. It follows from (iii) of Proposition 6.2 and (ii) of Proposition 6.1

Obviously, ‘many other definitions of fuzzy convergence are possible. For

example, we can interpret the formula

SET -+ 3m Vn>m -y(yES’X,ES)

or the formula

SET + 3m Vn>m -77(y~S+X”~S).

We limit ourselves to consider p-convergence and q-convergence.


We conclude this section by examining the relationship between the above-

defined convergences and the continuity.

Let X and X’ be two sets and T,T’ classical topologies on X and X’,

respectively. Moreover, recall that a map h: X --* X’ is continuous with respect

to T and T’ provided that

s~T’-+h-‘(s) ET,

where K’(S) is characterized by the formula

XEhP(S) H h(X)ES.

Then an interpretation by multivalued logic of these conditions induces one to

call a map h: X + X’ f-continuous with respect to the fuzzy topologies 7 and

F’ on X and X’ if

for every f E F( X’, L), where h-‘(f) is defined by setting

h-‘(f)(x) =f(W) VXEX.

In this manner we obtain an obvious generalization of the known concept of

fuzzy continuity.

The following proposition shows that p-convergence and q-convergence are

well defined with respect to the above concept of continuity.

PROPOSITION 6.5. If h: X-, X’ is f-continuous with respect to .Y and Y’, then

Sp-lim. x, = y * .Y’-p-lim.h(x,) = h(y),

3q-lim. x, = y * .Y’-q-lim.h(x,) = h(y).

Proof. Assume that Zp-lim. x, = y and that f E F( X’, L). Then, since

~(h-‘(f)) ap(f) and h-‘(f) EF(X,L),

we have that

l//?_f(h(x,)) >:(h-‘(f))Af(h(y)) ap(f)Af(h(x)).

This proves that .F’-p-lim. h( xn) = h(y). One proceeds likewise for q-convergence.


7. COPLEN FUZZY SETS

Assume that r is crisp and that - is a lukasiewicz negation. Then the

complement - f of a fuzzy set j is defined by setting (- j)(x) = - j(x) for

every x E X. Moreover j is called closed provided that - f E F-, and f is

copleniffEYmd-fELT. The interest of the coplen fuzzy sets is shown by the following proposition.

PROPOSITION 7.1. Let f he a coplen fuzzy set. Then from 9p-lim.x,, = y it

follows that lim.f( x,) = j(y), undfrom %q-lim.r,~ = _r it fo~~o~~ thut f( x,,) = f( y ) for eveq n > m, where m is a su~t~bie integer.

Proof. From Tp-lirn.~,~ = y it follows that

hm .f(xn) af(Y) and lim.-f(x,,)> -f(y).

Then

Iim.f(x,)= -I&.-f(x,)C - -f(_v)=f(y),

and therefore

f(y) Ql&.ffx,) Q lim.f(x,) <f(Y)

This proves that (j( x,)) is convergent to j( y ).

Assume that .Q-lim.x, = y. Then by Proposition 6.2 there exists m E N

such that j(q) 3 j(y) and - f(x,) a -f(y), and therefore f(x,) = f(y). for

every n2m.

It is possible to utilize Proposition 7.1, both in classical and in fuzzy

mathematics, to give convergences preserving a given set of operators. Namely,

let F be a functional space, T a topology on F, and H any set of operators on

F with values in L = R U { - 00, i 00 }. Moreover, suppose that - coincides

with the negation operation, that s= ( h E F( X, L) I- h E S 1, and that 3”’ is

the L-topology generated by T u S U 3. Then, for any sequence (f,) of

elements of S such that .9+lim.f, = f, we have that

lim.j, = f

hm.h(f,) =h(f >

with respect to T,

for every operator h E H.

In such a manner we can obtain convergences preserving the classical operators of integration and derivation. Also, if H is the set of all continuous


linear functionals on F, it is possible to obtain weak convergence. Proposition 6.1 enables us to obtain such convergences also by suitable classical topologies.

If F is a class of fuzzy sets, then we can suppose that in H there are operators of some interest for fuzzy set theory, for example the entropy and the energy operators 12, 31. Then we obtain convergence for sequences of fuzzy sets preserving entropy and energy.

8. UNIQUENESS OF THE LIMITS

We conclude by examining the question of the uniqueness of the limits. To this end, let us define an a-neighborhood of a point y as any L-subset f such that S(f) > OL and f(y) > (Y. Moreover, we call two L-subsets f and g a-disjoint if f A g < Q.

PROPOSITION 8.1. Let L be totally ordered, and assume that for every y, y’ E X with y # y’ there exist two a-disjoint a-neighborhooak of y and y’. Then we have

the uniqueness of the p-limits and q-limits.

Proof. Let us assume, by absurdity, that 5p-lim. x, = y and 5p-lim. x, = y’ with y f y’ and let f and g be two a-disjoint ~-nei~borh~s of y and y’, respectively. Then

~<s(f)r\f(Y)“~(g)hf(Y’)6(V A fW)A(V A m n>m m’ a’+ In’

= V (l\{f(x,)Ag(x,,)InZm,n’>m’})~a, m,m’

a contradiction. Since q-convergence implies p-convergence, we have also proven the unique-

ness of the q-limits.

PROPOSZIION 8.2. Suppose F crisp, and that for every y, y’ E X with y # y’, there exists a coplen fuzzy set f such that f(y) + f( y’). Then we have the uniqueness of the p-limits and of the q-limits.

Proof. It follows from Proposition 7.1.

REFERENCES

1. C. L. Chang, Fuzzy topological spaces, J. Math. Anal. ApI. 24:182-190 (1968). 2. A. De Luca and S. Tern-k& A definition of a non-probabi~stic entropy in the setting of

fuzzy sets, In@m. und Control Z&301-312 (1972).


3. G. Gerla, Nonstandard entropy and energy measures of a fuzzy set, J. Math. Anul. Appl.

104:107-116 (1984).

4. G. Gerla, Representation of fuzzy topologies, Fuzzy Sefs and Sysfem 11:103-113 (1983).

5. G. Gerla, On the concept of fuzzy point, Fuzz_r Sets and Sysrem 18:159-172 (1986).

6. G. Gerla, Generalized fuzzy points, J. Math. Anal. Appl.. to appear.

7. J. A. Goguen, L-fuzzy sets, J. Math Anal. Appl. 18:145-171 (1967).

8. E. E. Kerre, Fuzzy Sierpinski space and its generalizations. J. Math. Anal. .Appl.

74:318-324 (1980). 9. Pu Pao-Ming and Liu Ying-Ming, Fuzzy topology I. Neighborhood structure of a fuzzy

point and Moore-Smith convergence, J. Math. Anal. Appl. 76:571-599 (1980).

10. S. E Rodabaugh. The Hausdorff separation axiom for fuzzy topological spaces, Topology

A,@. 11:31Y-334 (1980).

11. G. Takeuti and S. Titani, Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. J.

Symbolic Logic 49:851-866 (1984).

12. C. K. Wong. Fuzzy points and local properties of fuzzy topology, J. Muth. Anul. Appl.

46:316-328 (1974).

13. L. A. Zadeh, Fuzzy sets, Inform. and Control 8:33X-353 (1965).

on fuzzy convergence

Documents