The general fuzzy filter approach to fuzzy topology, II

iti+¥

sets and systems ELSEVIER

Fuzzy Sets and Systems 76 (1995) 225 246

The general fuzzy filter approach to fuzzy topology, II Werner Giihler Institut fiir Mathematik. Universitiit Potsdam, Am Neuen Palais 10, Postfach 601553, 14415 Potsdam. Germany

Abstract

In this paper basic results on fuzzy topology are presented which are obtained in applying the general theory of fuzzy filters developed in Part I. Keywords: Fuzzy filters; Principal fuzzy filters; Fuzzy topologies; Monadic topologies; Fuzzy compactness

Introduction

This second paper continues the investigations of the first one. We also continue the numbering of sections and begin therefore with Section 6. Of course, in this paper, we also use the same terminology. As in Part I, throughout this paper let L be a completely distributive complete lattice with different least and last element 0 and 1, respectively. By means of the notion of interior of a fuzzy set, the notion of a fuzzy neighborhood filter is defined. Notice that in our approach several notions are related to usual points, e.g. the notion of fuzzy neighborhood filter. Fuzzy points are not used in our approach. For each fuzzy topological space, the mapping which assigns to each point x the fuzzy neighborhood filter at x can be considered itself as the fuzzy topology. There are several ways to describe these mappings axiomatically not requiring a given fuzzy topology at the beginning. In particular, we show that these mappings can be characterized as the monadic topologies with respect to the partially ordered fuzzy filter monad. The notion of monad is meant in sense of category theory and is

basically, especially for general algebra (cf. [11]). The more reach notion of partially ordered monad, introduced in [5], is useful especially in general topology. One of the main results in [5] consists in showing that the stratified fuzzy topologies are the monadic topologies with respect to the homogeneous partial ordered fuzzy filter monad. The characterization of arbitrary fuzzy topologies as monadic topologies presented here, is in some sense an extension of this result. The general theory of monadic topologies presented in I-5,6], leads here to a lot of notions and results in the special case of fuzzy topology. For instance, in this way the notions of neighborhood of a fuzzy filter, of open fuzzy filter and interior of a fuzzy filter arise. By means of each of these three notions an additional characterization of fuzzy topologies is given. Further notions which are treated here and arise from the general theory of monadic topologies are e.g. those of trace point, inner point, adherence point and of closure of a fuzzy filter. Notice, that these notions do not depend on an order-reversing

0165-0114/95/$09.50 © 1995 - Elsevier ScienceB.V. All rights reserved SSDI 0165-01 14(95)00057-7

W. Giihler / Fuzzy Sets and Systems 76 H995) 225-246

226

involution of L. They are in some sense internal notions. The last section of this paper deals with problems of compactness, also using results from the general theory of monadic topologies. As in the case of adherence points and the closure of a fuzzy filter, compactness is shown to be the same with respect to a fuzzy topology and its stratification.

6. Fuzzy topologies and their characterizations by interiors and by fuzzy neighborhoods

6.1. Fuzzy topologies By an L-fuzzy topology, or simply, a fuzzy topology on a set X we mean a subset of L x which is closed with respect to all suprema and all finite infima and contains the constant fuzzy sets 0 and T [1, 9]. A set X equipped with an L-fuzzy topology t on X is called an L-fuzzy topological space, or simply, a fuzzy topological space. For each fuzzy topological space (X, t), the elements of t are called the open fuzzy subsets of this space. If s and t are L-fuzzy topologies on a set X, s is said to be finer than t and t is said to be coarser than s provided t ___s holds. A fuzzy topological space (X, t) and also t are said to be stratified provided ~ ~ t holds for all ~ e L, that is, all constant fuzzy sets are open [10]. For each fuzzy topology t on a set X,

t*={flf=

(O,~t)EFV (gA~)forsomeFc_txL} (6.1)

is the coarsest stratified fuzzy topology finer than t, called the stratification of t. (X, t*) is also called the stratification of (X, t).

6.2. Interiors of fuzzy sets In the sequel let a fuzzy topological space (X, t) be fixed. The interior of a fuzzy set f : X --* L is the mapping i n t f : X --* L defined by intf=

V g~t,g~f

g.

(6.2)

i n t f is the greatest open fuzzy set less than or equal to f Clearly,

t = {fe LXlintf=f}

(6.3)

and the following conditions are fulfilled: (I1) intT -- i. (I2) i n t f ~ < f and int(intf) = i n t f hold for all

f ~ L x. (I3) int f A int 9 = i n t ( f A 9) for all f O ~ LX. Independently on fuzzy topologies, the notion of an interior operator for fuzzy sets can be defined as a mapping i n t : L x ~ L x which fulfill (I1) to (I3). It is well-known that (6.2) and (6.3) give a one-to-one correspondence between the fuzzy topologies and these operators, that is, fuzzy topologies can be characterized by interior operators. Obviously, fuzzy topologies are stratified if and only if instead of (I1) we even have that int ~ -- ~ for all ~ ~ L. For the interior operator int* of the stratification t* of t we have int*f=

V

(int9 A ~)

(6.4)

gAd <~f

for all f e L L, which follows easily by means of (6.1) and (6.2).

6.3. Fuzzy neighborhood filters An important notion in fuzzy topology is that of a fuzzy neighborhood filter at a point of the space. Let (X, t) be a fuzzy topological space. As follows by (I1) to (I3), for each x e X, the mapping JV(x):L x L, defined by JV(x)(f) = (int f)(x)

(6.5)

for all f 6 L x, is a fuzzy filter, called the fuzzy neighborhood filter at x. The fuzzy neighborhood filters fulfill the following conditions: (N1) 2 ~< JV(X) holds for all x ~ X. (N2) X(x)(y ~--,JV(y)(f)) = X ( x ) ( f ) for all

x ~ X and f~ L x. Clearly, y ~ JV(y)(f) is the fuzzy set int f.

Proposition 6.1 (Characterization of fuzzy topologies by fuzzy neighborhood filters). (6.5) gives a one-to-one correspondence between the fuzzy

W. Giihler / Fuzzy Sets and Systems 76 H995) 225 246

topologies and the mappings x v--~Jff(x) of X into ~ L X , where for each x ~ X, Jff (x) fulfills (N1) and (N2). Proof. Immediate, taking into account that the fuzzy topologies can be characterized by the interior operators. [] As follows from (6.5), a fuzzy topological space is stratified if and only if all fuzzy neighborhood filters in this space are homogeneous.

Proposition 6.2. Let (X, t) be a fuzzy topological space and x a point of this space. Then the homogenization JF (x)* of .At(x) is the fuzzy neighborhood filter at x with respect to the stratification t* of t. Proof. Easy (6.5). []

consequence

of

(2.5),

(6.4)

and

According to the two notions of fuzzy filter bases there are two notions of fuzzy neighborhoods.

6.4. Valued fuzzy neighborhoods In the sequel let (X, t) be any fuzzy topological space. Then for each a > 0, the mappings f E a-pr JV(x) are called a-fuzzy neighborhoods at x. Because of Proposition 2.1 a fuzzy subset f of X is an a-fuzzy neighborhood at x if and only if a ~< (int f)(x)

227

(II) a <~f(x) holds for each a-fuzzy neighborhood at x. (III) If f and g are a-fuzzy neighborhoods at x, then also f A g is an a-fuzzy neighborhood at x. (IV) If f is an a-fuzzy neighborhood at x, then 0 < fl ~< a and f ~< g imply that g is a r-fuzzy neighborhood at x. (V) If f is an a-fuzzy neighborhood at x, then there is an open fuzzy set g ~
Proposition 6.3. The valued fuzzy neighborhoods can be defined independently on a fuzzy topology as fuzzy sets which fulfill the above conditions (I) to (V), where the property in (V) of g to be open is to define by means of the valued fuzzy neighborhoods according to (6.7). Clearly, such an independent definition at the end is given by a mapping which assigns to each x ~ X the set of all valued fuzzy neighborhoods at x. (6.6) and (6.7) give a one-to-one correspondence between the fuzzy topologies and these mappings.

Proof. Easily seen.

[]

Remark. The notion of a valued fuzzy neighborhood is closely related to that of fuzzy neighborhood at a fuzzy point, used in fuzzy topology (cf. [12]). By a fuzzy point is meant a fuzzy subset of X which has the value 0 except at exactly one element of X. Denoting for each x e X and a > 0 by x, the fuzzy point of X with value a at x and 0 otherwise, a fuzzy neighborhood at x, is nothing else than an a-fuzzy neighborhood at x.

(6.6)

6.5. Superior fuzzy neighborhoods holds. By a valued fuzzy neighborhood at x we mean an a-fuzzy neighborhood at x for some a > 0. The open fuzzy sets can be characterized by the valued fuzzy neighborhoods as follows:

f e L x is open ¢*- for each x e X with f ( x ) > O there is an f(x)-fuzzy neighborhood g at x with g ~
In case Y ( x ) is homogeneous, each mapping f e base JV(x), not equal to 0, is called a superior fuzzy neighborhood at x. Because of Proposition 2.7 a fuzzy subset f of X is a superior fuzzy neighborhood at x if and only if 0 < sup f = (int f)(x)

(6.8)

holds. Hence, a superior fuzzy neighborhood at a point x is nothing else than a sup f - fuzzy neighborhood at x. In the following assume that (X, t) is stratified. Then the open fuzzy sets can be characterized by

IJ( G~ihler / Fuzzy Sets and Systems 76 (1995) 225-246

228

the superior fuzzy neighborhoods as follows:

6.6. Convergence

f e L x is open ,**- for each x e X with f ( x ) > O there is a superior fuzzy neighborhood g at x such that g ~
By means of the fuzzy neighborhood filters, convergence of fuzzy filters can be defined as follows: Let (X, t) be any fuzzy topological space, x a point of this space and ~t' a fuzzy filter on X. ~t' is said to converge to x, written ~ / - ~ x, provided ./¢' is finer than the fuzzy neighborhood filter ~hr(x).

Remark. In order to obtain (6.9) we use that t is stratified. Notice that in (6.9), in general, g cannot be assumed to coincide with f. The superior fuzzy neighborhoods fulfill the following conditions: (i) For each ct > 0, ~ is a superior fuzzy neighborhood at each point x. (ii) Each superior fuzzy neighborhood at a point x differs from 0 and takes on the maximum at x. (iii) If f is a superior fuzzy neighborhood at a point x, then also each fuzzy set g with f ~< g and s u p f = sup g is a superior fuzzy neighborhood at x. (iv) If F is any set of superior fuzzy neighborhoods at a point x, then also V s ~ r f i s a superior fuzzy neighborhood at x. (v) I f f and g are superior fuzzy neighborhoods at a point x, then either f A g = 0 or f A g is also a superior fuzzy neighborhood at x. (vi) For each superior fuzzy neighborhood f at a point x there exists an open superior fuzzy neighborhood g at x such that g ~
Proposition 6.5. The following are equivalent: (1) ~ ' 7-, x. (2) For each ~ > 0 and ~-fuzzy neighborhood f a t x we have f e :¢-pr J / . In case (X, t) is stratified and J¢ is homogeneous, then (1), (2) and the following condition are equivalent: (3) Each superior fuzzy neighborhood at x is contained in base Jr'.

Proof. Straightforward.

[]

6. 7. Continuity In the following let a mapping f between fuzzy topological spaces (X, t) and (Y,s) be fixed, f is said to be continuous provided g e s implies g o f ~ t.

Proposition 6.6. The following are equivalent: Remark. In case L is a complete chain, i.e. the ordering of L is linear, with f and g also f A g is a superior fuzzy neighborhood at x. Then s u p ( f A g) = s u p f A sup g > 0 always holds.

Proposition 6.4. The superior fuzzy neighborhoods can be defined independently on a stratified fuzzy topology as fuzzy sets f : X ~ L which fulfill the conditions (i) to (vi), where the property in (vi) of g to be open is to define by means of the superior fuzzy neighborhoods according to (6.9). Analogously as in the case of valued fuzzy neighborhoods, such an independent definition is given by a mapping which assigns to each x ~ X the set of all superior fuzzy neighborhoods at x. (6.8) and (6.9) give a one-to-one correspondence between the stratified fuzzy topologies and these mappings.

Proof. Easily seen.

[]

(1) f is continuous. (2) x ~ X and J [ -~ x imply g/rLf(~Ct) ~ f ( x ) . (3) For each x, ~ > 0 and ~-fuzzy neighborhood g at f(x), g o f is an ~-fuzzy neighborhood at x. If(X, t) and (Y, s) are stratified, then even (1), (2), (3) and the following condition are equivalent: (4) For each x and superior fuzzy neighborhood g at f(x), g o f i s a superior fuzzy neighborhood at x. Proof. Straightforward.

[]

7. The fuzzy neighborhood operator of a fuzzy topology and open fuzzy filters 7.1. The fuzzy neighborhood operator In the following let a fuzzy topological space (X, t) be fixed and let p: X ~ ~ L X be the mapping

W. Giihler / Fuzzy Sets and Systems 76 (1995) 225 246

which assigns to each x ~ X the fuzzy neighborhood filter A/'(x) at x. Moreover, let /Zx be the mapping of ~ L ( ~ L X ) into ~-LX defined by (5.3). The mapping nb : ~ L X ~ ~ L X , defined by

229

As a special case of (7.7), for each fuzzy subset f of X we have base nb [ f ] = { g If - - int g and sup g = sup int 9} (7.9)

nb = Px ~LP,

(7.1)

°

is called the fuzzy neighborhood operator of the fuzzy topology t. Moreover, for each fuzzy filter ~ ' on X, n b J t ' is called the fuzzy neighborhood of J / . From (5.2) and (6.5) it follows that e x ( f ) ° p = int f

(7.2)

for all f e L x. Hence, by means of (5.3) we get (nb J # ) ( f ) = ~#(int f )

(7.3)

for all fuzzy filters ~ ' on X and all f e L x. Therefore, for each a > 0 and ~ E ~ L X , cc-pr n b ~ ' = { f l int f E ~-pr J# }.

(7.4)

(7.4) implies that (~,),>0 with ~ = { f ~ ~-pr ~ ' I f open} for all ~ > 0 is a valued base of nb ~//. Hence, from Proposition 4.2 for each fuzzy filter ~ ' on X it follows

nbJ//=

A

[f~].

(7.5)

f m a - p r .~, f o p e n

For all ~ > 0, fl > 0 a n d f ~ L x with ~ ~< s u p f b e cause of (4.2) as special case of (7.4) we have fl_prnb[f,~]=~!o_lf<~intg}_

U1}

if fl~<~, otherwise.

(7.6)

Together with (4.1) and (7.5) this implies

nb[f~] =

A f ~< ff, ff o p e n

(7.7)

Hence, from Proposition 4.7 for each homogeneous fuzzy filter J / o n X it follows

nb,Al =

A f E base,4/, f o p e n

If]'

=

A

[f],

(7.10)

f ~o,gopen

where =_ is defined in Section 4. p can be determined back from nb as follows: p = nboqx.

(7.11)

(5.1) and (5.5) namely imply p = I~x°rl~,x°p = /~x ° ;~LP ° r/x. Of course, (7.1) and (7.11) give a oneto-one correspondence between the fuzzy topologies and the related fuzzy neighborhood operators. In the following the fuzzy neighborhood operators are characterized independently on a given fuzzy topology.

Proposition 7.1. A mapping h : ~L X ---}~L X is the fuzzy neighborhood operator of a fuzzy topology on X if and only if the following conditions are fulfilled. (1) h is a hull operator, i.e. ~ l ~ h(Jg) holds for all Jg e J~L X . (2) h is idempotent, i.e. h o h = h. (3) h = I~X°~Lh°~LqX. The mapping p : X ~ ~ L X which represents the associated fuzzy topology, of course, is then given by p = horlx. Proof. Analogously as in the homogeneous case in [53. []

r/x ~< p

(7.8)

(7.12)

and yxO.~Lpo p = p,

{ f l i n t f e base J / a n d sup f = sup int f } .

nb[f]

Conditions (N1) and (N2) of the fuzzy neighborhood filters can be rewritten as

[g,~]-

For each homogeneous fuzzy filter J / / o n X we have base nb J / =

and therefore

(7.13)

respectively. Clearly, (N1) and (7.12) are equivalent. By means of (7.1) and (7.3) it follows that also (N2) and (7.13) are equivalent. According to the general definition of a monadic topology given in [5], a mapping q : X - - , ~ L X is a monadic topology with respect to the partial ordered fuzzy filter

W.. Giihler / Fuzzy Sets and Systems 76 (1995) 225-246

230

monad provided r/x ~< q and Px o ~ L q o q = q hold. Hence we have the following.

Proposition 7.2. By means of the correspondence given in Proposition 6.1, the f u z z y topologies can be identified with the monadic topologies with respect to the partial ordered f u z z y filter monad.

Proposition 7.4. A mapping q : X -~ FLX is a monadic topology with respect to the homogeneous partial ordered f u z z y filter monad (FL, ~<, q', #') /f and only if the composition ix °q of q with the inclusion mapping ix : FLX ~ ~L X is a monadic topology with respect to the partial ordered f u z z y filter monad (~:r, <<-,~,P).

The following proposition shows some relations between the properties of the fuzzy topologies represented as monadic topologies and the properties of the related fuzzy neighborhood operators.

Proof. Easily seen by means of (5.7) and (5.8) not-

Proposition 7.3. Let q" X ~ ~ L X be a mapping and

have

let h = Px ° ~Lq. Then

nb.A(=

l f qx <%q holds, then condition (1) of Proposition 7.1 is fulfilled. I f h o q = q, that is Px o ~ L q ° q = q holds, then condition (2) of Proposition 7.1 is fulfilled. (7.14) implies condition (3) of Proposition 7.1.

Proof. From (5.4) and (5.6) it follows ° ~L h

= I.tx ° , ~ L P X =

=

°

~L~L q

]1X o ]A~Lx o ~ L , ~ L q I.tx ° J ' ~ L q o l t x

=

h o #x,

that is, (7.14). Because of (5.5) and condition (M2) of Proposition 5.3, r/x ~< q implies

[]

Proposition 7.5. For each fuzzy filter J[[ on X we

/k

nbEf~].

(7.15)

fEa-pr

(7.14)

Px o ~ L h = h o #x.

I~x

ing that ~ L q ° ix = iFLX° FLq.

Moreover, for each non-empty set A of fuzzy filters on X we have

./g~A

I f ( X , t) is stratified, then the related homogeneous f u z z y neighborhood operator nb': FLX ~ FLX, defined with the range-restriction p ' : X ~ FLX of p: X ~ ~ L X by nb' = #] o FLp', is a domain-rangerestriction of nb : ~,~LX ~ ~ x X . That is, for each homogeneous f u z z y filter Jg on X then nb'J/¢ = nb J[. Moreover, for each homogeneous f u z z y filter Jr[ on X we have

nbJ[=

A

nb[f].

(7.17)

f ~ base.4g

l~Lx = Px ° ~LrlX <<-Px ° ~ L q

=

h,

that is, condition (1) of Proposition 7.1. Because of (7.14), h o q --- q implies

Proof. Let ~ ' be a fuzzy filter on X. From Proposition 3.4 it follows that (:~#)#>o is a valued base of A/~-pr~ n b [ f ~], where for each fl > 0

h o h = h o l2x O,~L q = p x O~ L h o J~L q = # x ° ~ L ( h o q) = h ,

hence condition (2) of Proposition 7.1. By means of (5.5) it follows that (7.14) implies condition (3) of Proposition 7.1. [] Note that in [5] the stratified fuzzy topologies are characterized analogously as the monadic topologies with respect to the homogeneous partial ordered fuzzy filter monad, that is, by mappings q : X ~ FLX, for which q~ ~< q a n d / ~ o FLq o q = q hold.

~B = { g l A

-~

"'"

A gn [{gl . . . . , gn}

U f ~ct-pr nb,4t'

f l - p r n b [ f ~ ] , n > 0 I. )

(7.6) as well as condition (I3) of the interior operator and condition (F2) of a fuzzy filter imply ~# = {9[f<~ intg for some f ~ fl-pr J / }

= {g[intg e fl-pr ~/} = fl-pr nb J [ . Thus, (7.15) holds.

W. Giihler / Fuzzy Sets and Systems 76 (1995) 225-246

Because of Proposition 5.3,/~x preserves suprema of non-empty sets. Since (~-L, ~<) is ranging in aeSLAT and therefore ~-Lp also preserves suprema of non-empty sets, by means of (7.1) we get that (7.16) holds for each non-empty set A of fuzzy filters on X. The last part of the proposition follows easily by means of(5.8). []

Proposition 7.6. In general, Eq. (7.15) and (7.17) cannot be generalized to r l b J = / ~ a r l b J t ' , where A is any subset of ~L X and FLX, respectively, for which the infimum ~ = A~¢~,4 ~ exists. Proof (by an example). Let L = {0,½,1}, X = { 1, 2, 3, 4} and f, g :X ~ L be the mappings defined by f ( l ) = f ( 2 ) = 0, f(3) = f ( 4 ) = 1 and g(1) = 0, 9(2) 9(3) = g(4) = ½. Let t be the stratified fuzzy topology on X consisting of all constant fuzzy subsets of X as well as of f A g. Then [ f A g] is open and coincides therefore with nb [ f A g]. Since n b [ f ] = n b [ g ] = [0], we have n b [ f A g] :~ n b [ f ] A nb[g]. [] 7.2. Open fuzzy filters

231

(1) ~ ' is open. (2) For each ~t > 0 andf e ~-pr J¢ there is an open fuzzy set g E ~-pr ~¢/such that 9 <~f holds. (3) (~'~)~>o with ~ = { f 6 ~-pr~Clf open} for all ~ > 0 is a valued base of ~l. Proof. Clearly, the family (~,),> o in condition (3) is a valued fuzzy filter base. (7.5) implies that this family generates nb~¢/. Hence, (1) and (3) are equivalent. If J// is open, then because of (7.4) we have ~ - p r ~ ' = { f l i n t f e c t - p r J / } for all ~ > 0 , from which follow that (2) is fulfilled. Assume now that (2) is fulfilled. For each f ~ L x we have J / ¢ ( f ) = Vge~t.pr.lC,o<~fO( and because of condition (2) therefore J / / ( f ) -- Vg~'.9 ~ s ~. Hence, (~,),> 0 generates J¢'. Thus, ~ / i s open. []

Proposition 7.9. / f (X, t) stratified, then for each homogeneous fuzzy filter ~ the following are equivalent: (1) Jt[ is open. (2) { f e base J / I fopen} is a superior base of~¢[.

A fuzzy filter ~ ' on X is said to be open provided nb Jg = J[. Since nb is a hull operator, for a fuzzy filter ~// to be open we only need to show that nb J/~< . / / h o l d s .

Proof. If J / i s open, then from Proposition 4.7 and (7.8) it follows that condition (2) is fulfilled. If condition (2) is fulfilled, then also { f l int f E base J¢ and sup f = sup int f } is a base of ~/g. By means of (7.7) therefore then n b J / / = ./// follows. []

Proposition 7.7. For each fuzzy subset f on X and

Proposition 7.10. For each fuzzy topological space

each ~ > 0 with ~ < s u p f equivalent. (1) f is open. (2) [ f, :t] is open.

(X, t) the following conditions are fulfilled. (O1) The supremum of each non-empty set of open fuzzy filters on X is an open fuzzy flter. (02) If A is a set of open fuzzy filters on X for which the infimum exists, then it is also an open fuzzy filter. (03) Each open fuzzy filter ~¢[ is the infimum of all open valued principal fuzzy filters coarser than °11.

the following

are

Proof. Assume at first that f is open, that is, f = int f. Then f ~< g and f ~< int g are equivalent. (4.2) and (7.6) therefore imply [ f ~ ] = nb[fc~], that is, [ f ~] is open. Assume now that [ f ~] is open. Because of (4.2) and (7.6) therefore {glf<~ intg} = {glf<~ 9}, from which f e {glf<<.intg} and therefore f = i n t f follows. []

Proposition 7.8. For each fuzzy filter Jl[ on X the following are equivalent.

Proof. (7.16) implies that for each non-empty set of open fuzzy filters the supremum also is open. In the following let a set A of open fuzzy filters be given such that the infimum J of A exists. Because of (7.16) nb is isotone. Hence, n b J ~< n b J ¢ = ~// for all J¢ ~ A. Therefore nb J ~< J follows. Thus, J is open.

232

W. Giihler / Fuzzy Sets and Systems 76 (1995) 225-246

Finally fix an open fuzzy filter ,It'. Because of Eq. (7.4), for each ~ > 0 a n d f e ct-pr nb ~//we have that i n t f e ~-pr~'. From Proposition 7.7, it follows that lint f ~] is open. Since [int f, ~] ~< [ f 0~] holds, by means of (4.4) it follows that ,,// is the infimum of all open valued principal fuzzy filters coarser than Jg. []

tained from (O1) to (03) by replacing the fuzzy filter by the homogeneous fuzzy filter and the valued principal fuzzy filter by superior principal fuzzy filter and where instead of to be open is said to be element of T.

Remarks. Of course, taking in (02) as A the empty set, implies that the coarsest fuzzy filter on X is open. Condition (02) does not contradict the property of a fuzzy topology that, in general, only all finite infima of open fuzzy sets are open. This can be seen by means of the valued base of a fuzzy filter infimum defined by (3.8), in which only finite infima of fuzzy sets appear. Condition (02) shows again that for each fuzzy filter ~/g on X there exists its open hull, that is, the finest open fuzzy filter coarser than ~//, which of course is nb J//.

Clearly, (O2') implies that the coarsest homogeneous fuzzy filter on X is open.

Proposition 7.11. A fuzzy topology on a set X can be characterized as a set T of fuzzy filters on X which fulfill the conditions (01)-(03), where instead of J [ ~ T is said that ~[ is open. Clearly, then the open fuzzy sets are defined by means of the equivalence in Proposition 7.7. Proof. Easy consequence of Propositions 4.3, 4.4, 7.7 and 7.10. [] Proposition 7.12. I f (X,t) is stratified and J¢ is an open fuzzy filter, then also ~¢[* is open.

Proof. Easy consequence of (3.5), (3.9) and Propositions 4.9, 7.11, 7.12 and 7.13. []

8. The interior fuzzy filter operator Let (X, t) be a fuzzy topological space. For each fuzzy filter ~ ' on X for which { Y l n b X ~< J / } is non-empty, we define

int~ =

V

X.

n b .#" ~< ~¢,/

i n t J [ is called the interior of the fuzzy filter J / . Since for each fuzzy filter X on X, nb X is open, i n t ~ ' exists if and only { Y l J V ~< J¢' and JV open} is non-empty.

D = {J-/~ ~ L X I nb X ~< J / / f o r some X ~ ~LX}

= {Jg ~ ~ L X I ~

<~~ l for some

open Jff ~ ~ L X } is the set of all fuzzy filters J / f o r which int J/¢ exist. For each de' E D, i nt J / i s the supremum of all open fuzzy filters finer than ~ ' and is hence the coarsest open fuzzy filter finer than Jr'. In particular, for each J g ~ D we have i n t J l E D . Hence, we may define the mapping

Proof. Easy consequence of Propositions 2.12, 7.8 and 7.9. []

int:D ~ D

Proposition 7.13 (Giihler [6]). In a stratified fuzzy topological space a fuzzy set f is open if and only if the fuzzy filter I - f ] is open.

which assigns to each fuzzy filter ~ ' ~ D its interior. int will be called the interior fuzzy filter operator of the fuzzy topology t. Of course, for each fuzzy filter ~'eD,

Proof. Given in [6] and easy consequence of Propositions 7.7 and 7.12. [] Proposition 7.14. A stratified fuzzy topology can be characterized as a set T of homogeneous fuzzy filters which fulfill the three conditions (01')-(03') ob-

int.//~< ~f¢

(8.1)

holds. Hence, if a fuzzy filter J¢' e D is homogeneous, int J¢ also is. Clearly, the interior fuzzy filter operator is idempotent, that is, int o int = int.

(8.2)

233

W. Giihler / Fuzzy Sets and Systems 76 (1995) 225-246

Proposition 8.1. For a fuzzy filter J [ on X, int ~¢/

Proposition 8.3. For each ct ~ L and f e L x, int [ f, ct]

exists if and only if

exists if and only if

~ / ( f ) ~ sup(int f )

(8.3)

~< sup(int f )

holds for all f e L x. For each fuzzy filter .tl ~ D, the family (M,),> o with

holds and then

~

Proof. From (4.1) and Proposition 8.1 it follows

= {int f i f e

~-pr ~ ' }

(8.4)

for all ct > 0 is a valued base of int ~'.

Proof. Assume at first that i n t ~ ' exists. Hence, there is a fuzzy filter Jff such that nbJV" ~< J / holds. By means of (7.3) for each f ~ L x it follows that ~//(f) ~< JV'(int f ) ~< ~C(sup(int f ) ) ~< sup(int f ) and hence (8.3) holds. Assume now that (8.3) holds for all f s L x. Then for all ~ > 0 a n d f e ~-pr J / w e have ~t <~ sup(int f). Hence, then (:~)~>o fulfills condition (V1) of a valued base. As is easily seen, also condition (V2) is fulfilled. Thus (~)~> o is a valued base of a fuzzy filter .h". Hence, then ,/~(int f ) =

V i n t g ~< int f, g~-pr,Ct'

>~

V

g <~f, ge~z-prJ/

~ = ,///(f)

for all f ~ L x, i.e. n b ~ s ~< J¢ holds. Thus, then int~/g exists. For any fuzzy filter £0 with n b ~ ~< ~gt' because of (7.3) we get £~(int f ) >~ ~¢/(f) for a l l f ~ L x. Hence, for each ~ > 0, f r o m f ~ ~-pr J / i t follows i n t f ~ ~-pr L~ and therefore ~ , __ ~-pr L~'. Thus, £e is finer than ,A~ and therefore ~V'= int.#. []

Proposition 8.2. For each fuzzy filter ~ ~ D and all f e L x we have (intJl)(f)=

V

J/(g)-

(8.5)

int g ~< f

Proof. Proposition 8.1 implies that for all ~/' e D a n d f E L x we have (int J t ' ) ( f ) =

i nt [ f, ~] = [int f, ~].

(8.6)

that i n t [ f , ~ ] exists if and only if ~ ~< sup(intf) holds. Assume in the following that int [ f, ~] exists. For each fuzzy filter J / o n X because of (7.3) we have nbJ/~< [f,~] if and only if ~'(intg)/> [f,~](g) holds for all g e LX. By means of (4.1) and Proposition 4.1 therefore n b J l ~< [f,~] is equivalent to each of the following conditions: ~'(intf)~> :~, i n t f e ~-pr J / a n d J/~< lint f,~]. Hence, int[f,~] =

V

J/=

[intf,~].

[]

n b . @ ~< [ f , ~t]

Proposition 8.4 (cf. G~ihler [6]). Suppose that (X, t) is stratified. Then for each homogeneous fuzzy filter jar on X the following are equivalent: (1) int~/exists. (2) sup f = sup(int f ) for a l l f e base J / . l f Jt[ ~ D is a homogeneous fuzzy filter and :~ is a superior base of J/l, then { i n t f l f e ~ } is a superior base of int ~ .

Proof. The equivalence of (1) and (2) has been already proved in [6], Proposition 7.9. There also has been shown that { i n t g l g e b a s e d t } is a superior base of int Jg. Noting that (X, t) is stratified, for any superior base ~ of J / it is easily seen that {intglg~:~} is a superior fuzzy filter base. By means of Proposition 2.7 it follows that {intglg ~ ~} and {intglg ~ base J/g} generate the same fuzzy filter, that is, intJg. [] In the sequel we present a relation between the interiors of a fuzzy filter ~ / a n d of its homogenization J/I*.

Proposition 8.5. Suppose that (X, t) is stratified. V

Then for each fuzzy filter J [ on X, i n t ~ ' exists if and only if int ~/* exists. Moreover, then

~,

i n t g ~ f,g~-prdt'

form which (8.5) immediately follows.

[]

int ~ ' * = (int ~')*.

(8.7)

W. Giihler / Fuzzy Sets and Systems 76 (1995) 225-246

234

Proof. From Proposition 7.12 it follows that int ~ ' exists if and only if i n t J / * exists. Assume now Jg ~ D. By means of Propositions 2.12 and 8.1 we get that

Proposition 8.10. For each subset A of D for which the infimum J = A,a~A ~[ exists and is an element of D, also A.a~a int s//exists and we have intj

is a superior base of (intd/)*. Since i n t ( f A ~) = (int f ) A ~ for a l l f e L x and ~ e L, Propositions 2.12 and 8.4 imply that ~ also is a superior base of int~#*. Hence, (8.6) holds. []

Proposition 8.6. I f (X, t) is stratified, then for each f e L x, i n t [ f ] exists if and only if sup f = sup(int f ) and then int[f] = [intf].

Proof. Because

of Proposition 4.9, [ f ] = [f, s u p f ] * and [ i n t f ] = [ i n t f , s u p f ] * . Hence, (8.5) and (8.6) imply i n t [ f ] = (int[f, s u p f ] ) * = [intfsupf]* = [intf].

[]

Proposition 8.7. For all fuzzy filters J/¢ e ~,~LX and sF ~ D the following are equivalent. (1) J//~< i n t X . (2) nb J¢ .%
Proof. From (2) follows (1) because of the definition of int. From (1), the isotony of nb, the openess of intJff and of (8.1) it follows nb~/¢ ~< intA/" ~< Y , hence (2). []

Proposition 8.8. For each fuzzy filter J4 on X we have

nbJ[ =

V

=

A

int~'.

(8.8)

,g~A

~¢ = {0} w{(intf) A ~ l f • ~-pr~'}

sff.

.~PeD,,/t'~< intoM

Proof. Immediate from Proposition 8.7.

[]

Proposition 8.9. A fuzzy filter J/[ on X is open if and only if int J# exists and coincides with J[.

Proof. If~//is open, then nb J / / = Jr' e D and from (8.1) and Proposition 8.7 it follows that i n t J / = J / . Clearly, if J C e D and i n t ~ ' = ~ ' , then J¢ is open. []

Proof. Since int is isotone, the set {int ~'1 de' • A} has int J as lower bound. Hence its infimum exists. As follows by means of Proposition 7.10, A a~A i n t ~ ' is an open fuzzy filter finer than J . Since i n t J is the coarsest open fuzzy filter finer than J , (8.8) holds. [] In the following the interior fuzzy filter operators are characterized independently on a given fuzzy topology.

Proposition

8.11. A mapping k : D ---,D with D a non-empty subset o f ~ L X is the interior fuzzy f l t e r operator of a fuzzy topology on X if and only if the following conditions are fulfilled. (K1) ~hr ~ D and X <~ J [ imply J¢ e D. (K2) k is a kernel operator, that is, k(J[) <~Jg holds for all de/• D. (K3) k is idempotent, that is, k o k = k. (K4) For each subset A of D for which the infimum j = /~,4[f:A ~/[ exists and is an element of D, also A.a~a k(~l) exists and we have k(7)=

A k(~). ~'eA

(K5) I f J¢ • D is a valued principal fuzzy filter, then k(s/¢) is also one. For each mappin 9 k: D ~ O which fulfills conditions (K1)-(K6), kiD] consists of all open fuzzy filters of the related fuzzy topology. Proof. Let at first k be the interior fuzzy filter operator of the fuzzy topology t. From a remark after Proposition 7.10 it follows that the coarsest fuzzy filter on X is open. Hence, D is non-empty. Obviously, (K1) is satisfied. From (8.1), (8.2) and Propositions 8.3 and 8.10 it follows that conditions (K2) to (K5) are also fulfilled. In the sequel assume that k: D ---,D with D a nonempty subset of ~ L X is a mapping which fulfills conditions (K1) to (K5). Instead of ~/¢ e k [D] let us say that Jg is open.

W. Giihler / Fuzzy Sets and Systems 76 (1995) 225-246

At first let A be a non-empty subset of kiD]. Because of (K3), for all ~ e A we have k(Jg) = J [ . Hence, together with (K2), it follows k(V.~,~a J/¢') ~< V~,~a ~ = V~,~a k(~'). Because of (K4), k is isotone. Hence, V.~,~a k(J/g) ~< k(V.~,~a J¢) also holds. Thus, k ( V a ~ A ~() = V.,~,a J¢'. This shows that V,a~a Jg is open. Hence, (O1) is fulfilled. Clearly, (K3) and (K4) imply (02). In the following fix a fuzzy filter J/¢ ~ k [D]. Since Jr' is the infimum of all valued principal fuzzy filters coarser than J//, by means of the isotony of k and by (K5) it follows that J / i s the infimum of all open valued principal fuzzy filters coarser than Jh', that is, (03) is fulfilled. Because of Proposition 7.11 then k[D] consists of all open fuzzy filters with respect to a uniquely defined fuzzy topology t on X. Because of (K1) and (K2) a fuzzy filter ~ ' on X is an element of D if and only if there is an open fuzzy filter finer than J / . If ~ is an open fuzzy filter finer than J¢, by means of the isotony of k and condition (K2) it follows that .~" is finer than k(J//), hence k(.//) is the coarsest open fuzzy filter finer than ~ ' , that is, k ( ~ ' ) is the interior o f ~ / w i t h respect to the fuzzy topology t. []

9. Inner points and projective openess 9.1. Trace points

At first let a set X, an element x of this set and a fuzzy filter ~ ' on X be given, x is said to be a trace point of ./¢/provided 2, defined by (1.5), is finer than .//4', that is, ~gl(f) <.f(x)

holds for all f ~ L x.

235

Because of Proposition 2.1, x is an e-trace point of f if and only if e ~f(x)

holds. Let e-trc f d e n o t e the set of all c~-trace points o f f If J [ is a principal fuzzy filter I f ] , a trace point of ~ ' is nothing but a maximal point o f f i.e. an element x of X such that sup f = f (x).

Let m x f d e n o t e the set of all maximal points o f f As is easily seen, x is a trace point of [ f ] if and only if x is a trace point of [ f, sup f ]. Let trc J / d e n o t e the set of all trace points of ~///. Of course, trc [ f, e] = trc [ f A ~] = e-trc f for a l l f e L x and e ~< s u p f Moreover, trc[f]

= mxf

for all f e L x. Let t r c : ~ L X -+PX, e - t r c : { f e LXl ~ <~ supf} -+ PX and mx : L x -+ P X be the mappings, which assign to each element of the resp. domain the related set of trace points. Here PX mean the power set of X, which we assume to be equipped with the inclusion as partial ordering. Whereas trc and e-trc are isotone with respect to the finer relations of fuzzy filters and fuzzy sets, respectively, mx is isotone with respect to the preordering ~ . The coarsest fuzzy filter on X and also the coarsest homogeneous fuzzy filter on X have each point of X as trace point. For each set A of fuzzy filters on X for which the infimum J = A~,~A ~ / exists, obviously we have t r c J = (~ t r c ~ .

(9.1)

.¢¢~A

Proposition 9.1. Let J¢ be a f u z z y filter on X and (~,),>o a valued base of J / . Then x is a trace point of Jg if and only iffor each e > 0 and g E ~ we have ~ g(x).

Proposition 9.2. Let ~ l be a f u z z y filter on X. Then trcJ//=

~ fe~t-pr,~

Proof. Easily seen from (2.1).

{xLJg(f)<~f(x)}

f eL r

(9.2)

[]

In case Jg is a principal fuzzy filter [ f, el, a trace point of Jr' will also be called an e-trace point o f f

c~-trcf= ~

and

trc Jg = trc J¢*.

(9.3)

W. Giihler / Fuzzy Sets and Systems 76 (1995) 225-246

236

Proposition 9.3. Let J[ be a fuzzy filter on X. Then

I f Jl[ is homogeneous, then also tre ~ =

("]

mx f .

(9.4)

ipJt'=

f ~ base,.gt'

~

a-ipf=

(] i p [ f ~ / ( f ) ] . f ~L x

f ~ ~-pr.//

Proof. Easy consequence of (4.4), (4.6), (4.10) and Proposition 2.12. []

In case (X, t) is stratified,

9.2. Inner points

I f ( X , t) is stratified and ~ is homogeneous, then

In the following let an L-fuzzy topological space (X, t), a point x of this space and a fuzzy filter J/¢ on X be fixed, x is said to be an inner point of ~t' provided the fuzzy neighborhood filter JV'(x) at x is finer than J¢', that is,

ip~#=

J r ( f ) <~./V(x)( f ) holds for all f ~ L x. In case ~ is a principal fuzzy filter [f, a], an inner point of ~ / w i l l also be called an a-inner point off. [ f, 0] has each element of X as inner point. If a > 0 holds, x is an a-inner point of f if and only if f i s an a-fuzzy neighborhood at x. For e a c h f E L x and a ~< sup f l e t a-ip f d e n o t e the set of all a-inner points o f f We have a - i p f = {x E X l a ~< (intf)(x)}. In case . / / i s a principal fuzzy filter I f ] , an inner point of J / w i l l also be called an inner point o f f x is an inner point o f f if and only if either f = 0 o r f i s a superior fuzzy neighborhood at x. Let i p f d e n o t e the set of all inner points o f f We have i p f = {x ~ X [ ( i n t f ) ( x ) = sup f } .

ip[fa] = a-ipf and in case (X, t) is stratified, also i p [ f , a ] = i p [ f A ~] for all f E L x and a ~< sup f. Moreover, ip[f] = ipf for all f ~ L x. Obviously, if A is a set of fuzzy filters on X for which the infimum J = / k ~ A ~t' exists, then ,/¢EA

ip ~t'.

~

(9.6)

ipf.

f ~ base

Proof. The first part of the assertion follows by means of (4.4) and (4.6), the second part by means of Proposition 2.12. The last part was already stated in [6]. []

9.3. Projective openess and projectively open hulls Clearly,

ipJ¢ _~ trc J l . If even i p ~ / = trc J / , then Jr' is said to be projectively open. Since 2 <~ J / i m p l i e s JV(x) <~ nb ~¢t' = ~/¢, each open fuzzy filter .1/I is projectively open. In particular, the coarsest fuzzy filter is projectively open.

Proposition 9.4. Let A be a set of projectively open fuzzy flters on X for which the infimum J = /~./g~a J l exists. Then Af also is projectively open. Proof. Immediate from (9.1) and (9.5).

For each fuzzy filter J/¢ on X, let ip J/¢ denote the set of all inner points of Jr'. Clearly, we have

ip J = ~

ip J¢ = ip J//*.

(9.5)

[]

For each fuzzy filter ~t', there exists the infimum of all projectively open fuzzy filters coarser than ~'. Because of Proposition 9.4 this infimum is the finest projectively open fuzzy filter coarser than ~g. It will be called the projectively open hull of Jr', written poh Jg. Of course, Jg is projectively open if and only if ~ / / = poh .~g. The mapping poh : ~-LX ~ ~-LX which assigns to each fuzzy filter on X its projectively open hull, is an idempotent and isotone hull operator. Of course, for each fuzzy filter ~ on X

poh Jr' ~< nb J / holds.

(9.7)

W. Giihler / Fuzzy Sets and Systems 76 (1995) 225-246

Proposition 9.5. Assume that (X,t) is stratified. Then for each homogeneous fuzzy filter .t1 on X, also poh ~ ' is a homogeneous fuzzy filter. Proof. Since Jr' is homogeneous we have ~/<~ (poh~/)*. From (9.3) and (9.6) it follows that ( p o h J / ) * is projectively open. Hence, from the definition of projectively open hull we get poh .///= (poh ~t')*. [] For [f, e] to be projectively open we also say that f is e-projectively open. Obviously, f is a-projectively open if either • = 0 or f is an a-fuzzy neighborhood at all x with e <~f(x). Hence, f is 0tprojectively open if and only if ~ <.f(x) implies e ~<(int f)(x). For [ f ] to be projectively open we also say that f is projectively open. f is projectively open if and only i f f is sup f-projectively open. The following gives an important relation between openess and projective openess for fuzzy sets in the general and in the stratified case.

Proposition 9.6. A fuzzy s e l f is open if and only iff is c(-projectively open for all a <~sup f If(X, t) is stratified, then a fuzzy subset f of X is open if and only if f A ~ is projectively open for all • <~sup f Proof. The first part of the assertion is easily seen. The second part follows by means of (6.9) and has been already stated in [7]. []

10. Adherence points and projective closedness 10.1. Adherence points An element x of X is said to be an adherence point of a fuzzy filter ~ ' if the infimum ~/¢ A Y ( x ) of ~ / and the fuzzy neighborhood filter Jff(x) at x exists.

237

(3) There is a fuzzy filter finer than J4 which converges to x. (4) For all f, g • L x, ~(f)

A .#'(x)(g) ~< s u p ( f A g)

(10.1)

holds. Proof. Follows from Propositions 2.12, 3.4 and 6.2. [] An adherence point of a valued principal fuzzy filter [ f, e] will also be called an a-adherence point o f f Moreover, an adherence point of a superior principal fuzzy filter [ f ] will also be called an adherence point o f f x is an adherence point of f if and only if x is a sup f-adherence point off.

Proposition 10.2. An element x of X is an e-adherence point of a fuzzy s e l f if and only iffor each fl > 0 and fl-fuzzy neighborhood g at x we have e A fl ~< s u p ( f A g).

(10.2)

In case (X, t) is stratified, x is an adherence point of a fuzzy set f if and only if s u p f A supg = s u p ( f A 9)

(10.3)

for each superior fuzzy neighborhood g at x. Proof. The first part of the assertion follows by means of (6.6). The second part was already stated in [6]. [] For each f • L x and e ~< sup f let e-ad f denote the set of all 0t-adherence points o f f Moreover, for each f • L x let a d f d e n o t e the set of all adherence points off.

Proposition 10.3. For each f •

L x and • <~sup f we

have x•e-adf

¢~ e ~ < ~ ( f )

for s o m e J f ~ x . (10.4)

Moreover, for each f • L x we have

Proposition 10.1. For each x • X and fuzzy filter .t[ on X the followin9 are equivalent. (1) x is an adherence point of Jig with respect to t. (2) x is an adherence point of J [ with respect to the stratification t* of t.

xeadf

,~ ~ ( f ) = s u p f

forsome~l ~x.

(lO.5) In both cases, ~[ may additionally be assumed to be homogeneous.

W. G~ihler / Fuzzy Sets and Systems 76 (1995) 225-246

238

Proof. x • ct-adfmeans that [f, ct] A JV(x) exists, or equivalently, that there is a fuzzy filter .~/, for which J//~< [ f, ~] and J//~< JV(x) hold. J¢ ~< [ f , ~ ] is equivalent to ct ~< ~ ' ( f ) and J/~< JV(x) means J / ~ x. Hence (10.4) holds. (10.5) follows analogously. []

I f =]g is homogeneous, then also

adJ/=

(-]

adf.

(10.8)

f • base ,//

ad[f,~] = ~-adf

Proof. Because of (10.1) the following conditions are equivalent: (1) x • ad J / . (2) For all f • L x, fl > 0 and fl-fuzzy neighborhood g at x we have ~ [ ( f ) A fl ~< s u p ( f A g). Because of (10.2), condition (2) means that x • Ny~Lx a d [ f ~ ' ( f ) ] . Noting the isotony of ad shows that (10.6) is fulfilled. (10.7) is an easy consequence of (3.9). The last part of the assertion has been already stated in [6] under the assumption of (X, t) to be stratified. Because of Proposition 10.1, it also holds in the general case. []

for all f • L x and ~ ~< sup f and

Corollary 10.2.

ad[f]

a - a d f = a d ( f A ~).

Corollary 10.1.

For each ct • L, ad ~ = X.

Proof. Follows immediately from (10.5) with d/=£ [] Let ad ~ ' denote the set of all adherence points of v#. Of course,

= adf

for all f • L x. As trc and ip, the mapping a d : ~ L X - * PX, which assigns to each fuzzy filter on X the set of its adherence points, is isotone, that is, Jr' ~< Jff implies ad J / / ~ ad Jff. The analogously defined mappings ~-ad and ad are isotone with respect to ~< and _~, respectively. Note that ad does not fulfill, in general, an analogous equation as (9.1) and (9.5), not even in the filter case. In general, ad is not isotone with respect to ~<, as will be shown by the following.

Example. Let L = {0,½,1}, X = {1,2,3} and f g and h be the fuzzy subsets on X defined by f(1) = f(2) = ½, f(3) = 1, g(1) = 0, g(2) = ½, g(3) = 1 and h(1)---h(2) = 1, h(3)= 0. Let t be the stratified fuzzy topology on X consisting of all constant fuzzy subsets of X and of h and h A 5- We have f ~< g, a d f = X and a d g = {3}. Proposition10.4. adJ//=

N f eat-prJf

Let ~¢¢ be a fuzzyfiher on X. Then

a-adf= (-] a d [ f J / ( f ) ]

(10.6)

f ~L x

and

ad ~ = ad ~ ' * .

(10.7)

For each f • L x and ct <~sup f

(10.9)

Proof. Follows from Proposition 4.9 and (10.7). [] 10.2. Projective closedness and projectively closed hulls

We have

trc~/___ a d J / . If even trc J¢ = ad de, then J¢ is said to be projectively closed with respect to t. Since each element of X is an adherence point of the coarsest fuzzy filter on X, the coarsest fuzzy filter is projectively closed. (9.3) and (10.7) imply that ~/' is projectively closed if and only if ~¢* is projectively closed. Proposition 10.1 implies that a fuzzy filter .// is projectively closed with respect to t if and only if ~¢ is projectively closed with respect to t*.

Proposition 10.5 (G/ihler [6]). Let A be a set of projectively closed fuzzy flters on X for which the infimum j = A ~ E a J ( with respect to the finer relation exists. Then J also is projectively closed. Proof. Analogously as in the homogeneous case given in [6]. []

W. Giihler / Fuzzy Sets and Systems 76 (1995) 225-246

Because of Proposition 10.5 for each fuzzy filter J / t h e r e exists the finest projectively closed fuzzy filter coarser than J¢. It is called the projectively closed hull of Jr, written pch ~ . Clearly, ~/' is finer than pch Jr' and ~ = pch d¢ means that J / is projectively closed. The mapping pch:ffLX --, ~ L X which assigns to each fuzzy filter on X its projectively closed hull, is an idempotent and isotone hull operator. It will be called the projectively closed hull operator with respect to r Because of Proposition 10.1 the projectively closed hull operators with respect to t and t* coincide. In the following let a fuzzy subset f of X be fixed and let f + be the fuzzy subset of X defined by supf f +(x) = [ f(x)

if x e a d f , otherwise,

Proposition 10.6. For each fuzzy subset f on X we have

As in [6] in a more general case, for each fuzzy filter ~ on X, J / + will be called the projective closure of J / , also written pcl J / , that is, pcl ,~' = ~ / + .

J / i s projectively closed if and only if pcl ~ = J[. Analogously, for each fuzzy subsetfof X, f + will be called the projective closure of f, also written pclf, that is,

For each fuzzy subsetfon X, pch fwill be called the projectively closed hull o f f Clearly, f~< p c h f holds. If pch f = f, then f will be said to be projectively closed. From (10.10)it follows t h a t f i s projectively closed if and only if mx f = ad f f is also projectively closed if and only if p c l f = f pch f i s the least projectively closed fuzzy subset of X greater than or equal to f such that sup pch f = sup f

Proposition 10.7 (G~ihler [7]).

I f L is a complete chain, then for each fuzzy subset f of X, we have

ad(pcl f ) = ad f

(10.10)

Proof. Because of f ~ _ p c h f we have [ f ] <~ [pchf]. For each fuzzy filter J / o n X we define ,/#+

for each fuzzy subset g of X. If ~ is a limit ordinal and for each ~ < 4 we have [ f ] ~ = [ f ; ] , then by means of Proposition 4.8 it follows I f ] C = V~<¢ [ f ] ; = [f¢]" Thus, [ f ] ¢ = [ f ~ ] holds for all ordinals 4. Hence (10.10) is proved. []

pclf=f ÷ .

for all x e X . Clearly, f<~f+ and a d f ~ < m x f + hold. By transfinite induction we obtain for each ordinal 4 a fuzzy subset f~ of X defining f o = f, for each ordinal ~ which has a predecessor ~, f ~ = (f~)+, and for each limit ordinal 4, f¢ = V;<~f ~. There is a least ordinal 4 such t h a t f ~ = f ¢ + i . For this ~,f¢ will also be denoted p c h f It is the least fuzzy subset greater than or equal to f such that supf=suppchf and [ p c h f ] is projectively closed.

pch[f] = [pchf].

{;V

V

x

ifad~'=~O,

x ~ a d J,f

(10.11)

otherwise. J [ + is the finest fuzzy filter Jff which is coarser than Jg and for which ad J ¢ c trc X holds. This follows by means of Proposition 52 in 1-6] noting that the assumption in this proposition is fulfilled because of Proposition 3.3. (4.11) and Proposition 4.8 imply that [g]+ = [g+]

239

(10.12)

and p c h f = pclf.

(10.13)

Proof. The proof of (10.12) is omitted here, since this equation will follow immediately from (11.17) and (11.24). Clearly, (10.12) implies that in case L is a complete chain the generation of pch f by transfinite induction is already finished after the first step, that is, (10.13) holds. [] The mappings pch : ~ L X --* ~ L X and pch : L x L x which assign to each fuzzy filter and each fuzzy set its projectively closed hull, are isotone with respect to the finer relation of fuzzy filters and the preordering _ , respectively.

240

W. Giihler / Fuzzy Sets and Systems 76 (1995) 225-246

11. The closure operator of a fuzzy topology and closed fuzzy filters

Let (X, t) be a fuzzy topological space. In the following we make use of the set of all pairs of fuzzy filters J¢ and points x to which ~ converges, that is, the set cvt = { ( ~ g , x ) ~ L X x X l J l C ~ x } . Let tl : cv t --* ~ L X and t2 : cv t --* X be both the projections (J/, x) ~ . 4 ¢ and (J//, x) ~-*x, respectively. Since ~ ,--*x holds for all x e X, t2 is a surjection.

11.1. A characterization of the fuzzy neighborhood operator Before turning to the notion of closure of a fuzzy filter, let us present at first a characterization of the fuzzy neighborhood of a fuzzy filter which in some sense is complementary to the definition of closure of a fuzzy filter, given later. For the following see Fig. 3. Proposition 11.1. For each fuzzy filter ~¢¢ on X the preimage ~ { t2(~¢ ) exists and

n b J¢ = (ltx o ,~Lt 1) ('-~'L t2 ('-~')) •

(1 1.1)

Proof. Since t 2 is surjective, from Corollary 3.2 and Proposition 3.7 it follows that ~-Z tE(J[) exists and

~{t2(~/l)(k)=

V

Jl(g)

for all f e L x. The condition gotz<~ex(f)otl means that g(x) <~~ f ( f ) holds for all (:~(, x) e cvt. We may take here Jg as the coarsest fuzzy filter converging to x, namely the fuzzy neighborhood filter vV'(x). Because of (6.5), therefore g°t2 <<. e x ( f ) °tl may be replaced in (11.2) by g ~< intf. Hence, from (7.3) and (11.2) it follows that (11.1) is fulfilled. [] Recall that for each fuzzy filter J /

on X,

~ L t2(.//'/) is the coarsest fuzzy filter Z# on cvt for which ~Lt2(Z#) = ~/¢. In the following the role of both the mappings t~X°~Ltl and ~-z.t2 will be changed.

11.2. The closure operator As it follows from [5] in a more general case, for each fuzzy filter Jr' on X there exists the coarsest fuzzy filter L# on cvt for which ( / t x O ~ t . t 0 ( . L a ) = J t '. The image ,~-/.t2(~) of 5e with respect to the mapping t2 is called the closure of Jr', written el ~t' (see Fig. 3). The mapping el : ~ - L X

---~L X

which assigns cl ./¢' to each fuzzy filter ~ ' on X, is called the closure operator of the fuzzy topology t. el is isotone and is a hull operator, that is, for all fuzzy filters Jr' and JV" on X, ~ ' ~< JV" implies n b . ~ ' ~< n b , / V and

g°t2 <~k

for all k e L cv'. By means of (5.3) this implies holds. This also has been shown in [5] in a more general case.

(#x ° ~L t x)(~-Z t2 (J¢')) ( f ) =

V

J/(9)

(11.2)

@°t2 <~ex(f)otl

Proposition 11.2. For each fuzzy filter .14 on X and each mapping f e L x we have

(cl

=

V ex(g) °tl < f °t2

..~'LCV t

cv t

=

// ~-LX

V

~//(g) •

(11.3)

v~V(g) <~f(x) if ~,: 7~x

S%~LX X ~LX Fig. 3.

Proof. Let a fuzzy filter ~ on X be fixed and define a mapping .~: L c"t _, L in taking ~f(h) =

V ex(g)otl ~
.A/(g)

(11.4)

W. G?ihler / Fuzzy Sets and Systems 76 (1995) 225 246

for all mappings h e Lcvt. For each g ~ L x and (Jff, x ) e c v t we have (ex(g)otO(Jff, x ) = J V ( g ) . With (JV, x) ~ cv t also (Jff*, x) e cv t holds. Hence, for each • ~ L the greatest fuzzy subset G of X with ex(g) ° t~ <~£t is ~, where ~ and ~ are here the constant fuzzy subset of X and cv t, respectively, with value ct. Hence, for each ~ e L we have L-~(~ =

J/(~)

(11.5)

and therefore ~ ( ~ ~< ~. Since L is completely distributive and for all mappings f , g ~ L x and h,k ~ L ~'' from e x ( f ) o t l <. h and ex(g)o tl ~ k it follows e x ( f A g)ot~ ~< h A k, we get ~ ( h ) A .~(k) ~< ~q~(h A k) for all h, k ~ L ~ t. Clearly, ~ is isotone. Hence, ~ is a fuzzy filter on cv t. Because of(5.3) for e a c h f ~ L x it follows

(t~x ° .,~Ltl)(£e)(f) = :Lh(~')(ex(f)) = - ~ ( e x ( f ) ° tO.

Proposition 11.4. For each fuzzy filter .1t on X cl(~'*) = (cl Jg)*.

(1 1.7)

Hence, the closure of each homogeneous fuzzy filter on X is also a homogeneous fuzzy filter. Proof. Because of Proposition 1 1.3 we may assume that t is stratified. Let o f and Z,e be the coarsest fuzzy filters on cv t which are mapped by Px ° :-L t to J¢* and J / , respectively. It follows from the proof of Proposition 11.2 that these fuzzy filters exist. Since J / * is finer than ~¢/, o f is finer than L,e. Analogous to (11.5), here we obtain, of(07)= J¢'*(~) for all ~ e L. Therefore off is homogeneous and thus, o f ~< L,¢* holds. Since t is stratified, tl has a factorization tl = ix ot'~ with ix the inclusion mapping of FLX into :-LX. Hence ~ L t l °icvt = ~,~Lix°iFLx°FLt'~. Noting (5.8) it follows that I~xo~Lt~ maps £,e* onto J[*. Hence, o f = L,a*. (2.7) implies therefore (1 1.7). []

Proposition 11.5. For each fuzzy filter J [ on X,

Since ex(g) o t~ <~e x ( f ) ° t~ implies ~t'(g) ~< J t ( f ) , we have (Px ° ffLt~)(~)( f ) = ~g( f ) and therefore (#x ° ~ L t , ) ( ~ ) = . / / .

241

(11.6)

Let now o f be any fuzzy filter on cv t such that (/~xO~-Ltl)(of) = ~3'. For each g ~ L x it follows of(ex(g) ° t~) = ~ ( a ) and because of(11.4) for each h ~ L c** therefore ~(h)~< of(h). Hence, .~ is the coarsest fuzzy filter on cv t such that (11.6) holds. Noting that ex(g)ot: <
Proposition

11.3. The closure of a fuzzy filter with respect to t coincides with the closure of J¢ with respect to the stratification t* of t.

Proof. Since t and t* have the same converging ultra fuzzy filters, from Corollary 11.1 the assertion follows. []

ad ~ ' _ trc(cl J//)

(1 1.8)

and pcl ~ ' ~< cl ~ '

(1 1.9)

hold. Proof. (11.8) has been shown already in [6] in a more general case. (1 1.9) follows easily from (1 1.8) and the definition of pc l ~ ' given in (10.1 1). []

Proposition 11.6. Let f : (X, t) --, (Y, s) be a continuous mapping between fuzzy topological spaces. For each fuzzy filter J / o n X then :-Lf(CI .//t) ~< cl o~Lf(.1[ )

(11.10)

holds. Moreover, for each fuzzy filter JV on Y for which the preimage : - ~ f ( J V ) exists, : - t S f ( c l - 4 : ) also exists and

cl :-if(W) ~< :-Lf(Cl ,A:)

(I1.11)

holds. Proof. Let at first a mapping there is a mapping k e L r with (JV, y) e cvs, then for g = k o f (hof)(x) for all (o~f',x)~ cvt.

h e L r be fixed. If JV'(k) ~< h(y) for all we have of(g) ~< This follows since

W. Giihler / Fuzzy Sets and Systems 76 (1995) 2 2 5 - 2 4 6

242

9ff,--,x implies X ~ y for JV = ~Lf(aff) and y = f (x) and hence JV(k) ~< h(y). This shows that

for all x ~ X, is called the closure o f f (10.4) implies (clf)(x) =

V o f (g) ~< (h ,. f ) (x) if of ~.x

V

•

(11.15)

x e ~t-ad f

for all x e X. In case t is stratified we even have >~

V

jg(ko f)

. ~ ( k ) ~ h(y) if ,A/ s~y

(cl f)(x) =

and therefore (11.10) holds. In the following, let JV be a fuzzy filter on Y for which the preimage with respect tofexists. Because of JV ~< cl Jff and Proposition 3.5 then also the preimage of el Jff with respect to f exists. Fix any g e L x. Because of Propositions 3.7 and 11.2 then V

sup9

(11.16)

for all x e X, as follows from [6], and also from (10.5).

(cl J[)(h of) = ~Lf(cl dt')(h)/> el ~ L f ( J [ ) ( h )

el ~ L f ( J f f ) ( g ) =

V g ~< f, x e a d f f

holds, which because of (11.3) means that

~[f(JV)(h)

Propositions 11.7. For all f e L x and ct ~ L we have f~< pclf~< clf,

(11.17)

sup f = sup(cl f ) ,

(11.18)

c18 = 8,

(11.19)

c l ( f A 8) = c l f A 8.

(11.20)

~[(h) ~ (g)(x) if ./¢ 7*x

and ~{f(Jff)(h) =

V

For all f g e L x, fro_ g implies cl f _ cl g and f <~g implies cl f ~< cl g.

W(k),

kof<~h

hence cl o~/f(Jff)(g) =

V

X(k).

(11.12)

Proof. Easily seen.

[]

.~Lf(~K) ~ g(x) if ~t' 7,x

Because of Propositions 3.7 and 11.2 on the other hand, ~ { f ( c l JV')(g) =

V

(el Jff)(h)

h o f
and (el Jff)(h) =

V

As it follows from the next proposition, by means of the closures of fuzzy sets the closures of fuzzy filters can be characterized.

Jff(k)

3if(k) ~
V

~<

Remark. Notice that pch and pcl are, in general, not isotone with respect to ~<. In the example before Proposition 10.4 we h a v e f ~ g and neither pch f <~ pch g nor pcl f <~ pcl g.

W(k),

~Lf(~ll)(k) ~ (h of )(x) if ~¢/7*x

Proposition 11.8. For each fuzzy filter J [ on X and

thus,

all f e L x we have

V

~ / f ( c l W)(g) ~<

W(k)

(11.13)

~ L f (JI) <~g(x) if . ~ r x

holds. From (11.12) and (11.13) follows (11.11).

[]

V

~'(g).

(11.21)

clg<~f

I f ( ~ ) ~ >o is a valued base of J/g consisting of prefi lters, then (cg~)~>o with

11.2. The closure o f fuzzy sets

For each fuzzy s e t f e L x, the fuzzy set c l f e L x defined by (clf)(x)= V J [ ( f )

(cIJC')(f)=

(11.14)

rd~--- {clgIg e :~} is a valued base of cl ~t. l f Jg is a homogeneous fuzzy filter and g$ is a superior base of .Al, then ~ = {clg[g e ~} is a superior base of el ~g.

243

W. Giihler / Fuzzy Sets and Systems 76 H995) 225-246 For each f e L x and ~ <~s u p f

(11.22)

cl[f,~] = [clf,.].

Moreover, for each f e L x

(11.23)

clEf]=[clf].

Proof. (11.21) follows from (11.3) and (11.14). For all f, g e L x and x e X the condition that Jff(g) <~f(x) holds for all J f f ~ x means that (cl g)(x) <~f(x) holds. Let now (~,),>o be a valued base of Jr' consisting of prefilters. Clearly, then (c~,),> o defined as in the proposition, is a valued fuzzy filter base. Since V

ct=

clg ~< f , h <~g, heY$~

~/

{

Proposition 11.11. For each fuzzy subset f of X the following are equivalent: (1) f is closed. (2) J / I ( f ) <~f (x) holds for all fuzzy filters ~ 7-, x. Proof. Immediate from definition (11.14).

[]

Proposition 11.12. Assume that L is a complete chain. Then for each f e L x we have

ad(cl f ) = ad f.

(11.24)

:t

c l g ~< f , ge.#~

for e a c h f e L x, from (11.21) it follows that (cg~),>o generates el Jr'. The analogous result in the homogeneous case has been proved already in [6]. For all f,g e L x and ~t ~< sup f, from (4.1) and (11.21) it follows cl[f,~](g)=

A fuzzy subsetfof X is said to be closed provided f= elf

o~ if c l f ~ < g CT, 1 if g = l , 0 otherwise,

because of (4.1) therefore el [ f :t] (9) = Eel f, ct] (g). Hence, (11.22) is fulfilled. (11.23) has been already proved in [6]. [] 11.3. Closed fuzzy filters and closed fuzzy sets

Proof. Since ad and cl remain invariant in replacing t by t*, we may assume that t is stratified. Fix an x e a d ( c l f ) and assume x C a d f Because of Proposition 10.2, there is an open superior fuzzy neighborhood g at x such that sup g ~< s u p f and s u p ( f A g) < sup g hold. By means of Proposition 10.2 and (11.20) it follows s u p ( c l f A g ) - - s u p g . Hence there is an element y of X such that s u p ( f A 9) < (clf)(y) A g(y)

(11.25)

holds. (11.16) and (11.25) imply that there is a fuzzy set h e L x with h ~
(11.26)

A fuzzy filter ~ ' on X is said to be closed provided Jr' = c1~/¢. (11.9) implies that each closed fuzzy filter is projectively closed.

k = g A g(y) is a superior fuzzy neighborhood at y. Since y is an adherence point of h, Proposition 10.2 implies supk A suph = sup(k A h) and hence g(y) A sup h = sup(g A h) A g(y). Because of (11.25), (11.26) and h ~
Proposition 11.9 (G/ihler [6]). The infimum of each

s u p ( f A g) < g ( y ) / \ suph = sup(g A h) A g(y)

set of closed fuzzy filters on X is a closed fuzzy filter, provided it exists.

Proposition 11.10. Let f : ( X , t ) ~ ( Y , s )

be a continuous mapping between fuzzy topological spaces. Then for each closed fuzzy filter Y on Y, the preimage ~ Z f ( J V ' ) is a closed fuzzy filter on X, provided it exists.

Proof. Immediate from Proposition 11.6. []

~< sup(g A f ) , which is a contradiction. Hence, a d ( c l f ) _ a d f holds. Noting that sup f = sup clf, also a d f ad(clf) and therefore (11.24) holds. [] Corollary 11.2. Assume that L is a complete chain. Then for each fuzzy filter J [ on X we have ad(clJ¢') = a d ~ ' .

(11.27)

W. Giihler / Fuzzy Sets and Systems 76 (1995) 225-246

244

Proof. Because of (10.7) and (11.7) we have ad J4 = a d J ¢ * and ad cl J/4 = ad(cl de)* = ad c1(~¢4"). Hence we may assume that .//4 is homogeneous. From (10.8) and (11.24) it, therefore, follows adJ4 =

(~

a d ( c l f ) = N adg = ad(cl J4),

f c b a s e ~(

g~

From the definition of projective closure of a fuzzy set it follows that x e a d ( f A ~) and ~ <<.f(x) imply p c l ( f A ~)(x) = s u p ( f A g) = ~. Hence, (clf)(x) ~< ~/, ~
where ~ is the superior base { c l f [ f E b a s e ~ ' } of cl~'. []

Proposition 11.13. I f L is a complete chain, then for each fuzzy subset f of X, c l f is closed, that is, the mapping cl:LX-~ L x which assigns to each fuzzy subset of X its closure, is idempotent.

Proof. Immediate from (11.15) and Proposition 11.12.

[]

Proposition 11.14. I l L is a complete chain, then for each fuzzy filter .1[ on X, cl J¢ is closed, that is, the closure operator cl : ~L X ~ ~:L X which assigns to each fuzzy filter on X its closure, is idempotent. Proof. From (11.21) and Proposition 11.13 for each fuzzy filter ~g on X and e a c h f e L x we obtain

12. Compactness of fuzzy filters, fuzzy sets and fuzzy topological spaces Let an L-fuzzy topological space (X, t) be fixed. A fuzzy filter ~ on X is said to be compact in (X, t), if for each fuzzy filter ~ finer than a~ff there is a fuzzy filter Jg finer than Ae that converges to a trace point of ~ .

Proposition 12.1. Let oF be a fuzzy filter oF on X. Then the following are equivalent: (1) o,ugis compact in (X, t). (2) The associated homogeneous fuzzy filter ~'ff* is compact in (X, t). (3) ~ is compact in the stratification (X, t*).

Proof. Immediate from Propositions 2.12 and 6.2. --

V

"AC(g)=(cIJ/')(f) • []

clclg~
The following result gives an important relation between the projectively closed hulls and the closures of fuzzy sets.

Proposition 11.15. Assume that L is a complete chain. For each fuzzy subset f of X then clf=

~/

pch(fAa).

(11.28)

~
Hence, f is closed if and only if f A ~ is projectively closed for all ~ % supf.

Proof. Fix any x e X. Corollary 10.2 and (11.15)

[]

Corollary 12.1. A principal fuzzy filter [ f, ~] is compact if and only if [ f A Et] is compact. Proof. Follows by means of Proposition 4.9.

[]

Instead of [ f ] to be compact we also say that the fuzzy set f is compact. The fuzzy topological space (X,t) is said to be compact if i is a compact fuzzy set. Because of Proposition 12.1, (X, t) is compact if and only if the coarsest fuzzy filter on X is compact, hence, if and only if each fuzzy filter on X has a finer converging fuzzy filter. Recall that the coarsest fuzzy filter has each point of X as trace point.

imply (cl f)(x) =

V x ~ a d ( f ^ ~),~t ~
~.

Proposition 12.2. If(X, t) is Hausdorff (that is, fuzzy flters on X converge to at most one point), then each

w. Giihler/ Fuzzy Sets and Systems 76 (1995) 225 246 compact fuzzy filter is projectively closed. On the other hand, each projectively closed fuzzy filter finer than a compact fuzzy filter is compact.

Proof. Special case of Propositions 62 and 63 in [6].

[]

It follows from Proposition 4.6. that this proposition also holds if instead of fuzzy filters fuzzy sets are taken.

Proposition 12.3 (G/ihler [6]). For each continuous mapping f: (X, t) ~ (Y, s) between fuzzy topological spaces, the image ~ L f (3[r) of a compact fuzzy filter on X is compact. Proof. Special case of Proposition 65 in [6]. The extension principle assumed in this proposition is here fulfilled because of Proposition 3.9. [] Proposition 12.4 (Generalized Tychonoff Theorem). Let I be a non-empty set and for each i ~ I let (Xi, ti) be a fuzzy topological space and J~f'i a fuzzy filter on Xi. Let ~T" denote the product of ()Ci)i~1 and (X, t) the product of the fuzzy topological spaces (Xi, ti). Then the following are equivalent: (1) ,~, is compact. (2) Each fuzzy filter 3f"i is compact in (Xi, ti). Proof. Special case of Proposition 66 in [6] noting that because of Propositions 3.9 and 3.10 the assumptions used in this proposition are here fulfilled, []

Corollary 12.2. Let I be a non-empty set and for each i ~ I let (Xi, ti) be a non-empty fuzzy topological space. Then the product of these spaces is compact if and only if each of these spaces is compact. Proof. Easy consequence of Corollary 3.3 and Proposition 12.4.

[]

A set F of L-fuzzy subsets of a set X will be said to have the finite intersection property provided for each non-empty finite subset { f l . . . . . f.} of F it

245

holds sup(fl A -.. Af.) = sup f l A ... A s u p f . . The next result generalizes one of the classical standard characterization of compactness in topological spaces. Recall that for each fuzzy set f, m x f d e n o t e s the set of all maximal points off.

Proposition 12.5 (cf. G/ihler [7]). Assume that L is a complete chain. Then a homogeneous fuzzy filter on X is compact if and only if the following condition is fulfilled: If F is a set of projectively closed fuzzy subsets of X such that F ~ base ~ has the finite intersection property, then all mappings .f e F w base J f have a common maximal point. Moreover, then a fuzzy set g : X ~ L is compact if and only if the following condition is satisfied: For each set F of projectively closed fuzzy subsets of X for which F ~ { g } has the finite intersection property, ((~f~ m x f ) c ~ m x g is non-empty. Proof. Easy consequence of Proposition 70 in [6] and of Propositions 2.12, 10.7 and 12.1.

[]

Corollary 12.3 (Giihler [7]). In case L is a complete chain, the fuzzy topological space (X, t) is compact if and only iffor each set F of projectively closed fuzzy subsets of X which has the finite intersection property, Nf~r m x f is non-empty. Proof. Follows 12.5.

immediately

from

Proposition

[]

References [1] C.L.Chang, Fuzzytopologicalspaces,J. Math. Anal. Appl. 24 (1968) 182-189. [2] P. Eklund and W. Giihler, Fuzzy filter functors and convergence, in: Applications of Category Theory to Fuzzy Subsets (Kluwer, Dordrecht, 1992) 109-136. [3] P. Eklund and W. Gahler, Contributions to fuzzyconvergence,in: Recent Developmentsof General Topologyand its Applications, Internat. Conf. in Memory of Felix Hausdorff (1868-1942), Math. Res. 67 (Akademie Verlag, Berlin, 1992) 118-123.

246

W. Giihler / Fuzzy Sets and Systems 76 (1995) 225-246

I-4] P. Eklund and W. G~ihler, Completions and compactifications by means of monads, in: Fuzzy Logic: State of Art (Kluwer, Dordrecht, 1993) 39-56. [5] W. G~ihler, Monadic topology a new concept of generalized topology, in: Recent Developments of General Topology and its Applications, Internat. Conf. in Memory of Felix Hausdorff (1868-1942), Math. Res. 67 (Akademie, Berlin, 1992) 136-149. [6] W. G~ihler, Convergence, Seminarberichte aus dem Fachbereich der Fernuniversit'at Hagen 46 (1993) 31 73; also: Fuzzy Sets and Systems, to appear. [7] W. G~ihler, The canonic point-based approach to fuzzy topology, 5th IFSA Congress, Seoul, 4 pages.

[8] W. G~ihler, The general fuzzy filter approach to fuzzy topology. Part I, to appear in: Fuzzy Sets and Systems. I-9] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. lg (1967) 145 174. [10] R. Lowen, Convergence in fuzzy topological spaces, General Topology and Appl. 10 (1979) 147-160. 1-11] S. Mac Lane, Categories For the Working Mathematician (Springer, Berlin, 1972). [12] Pu Pao-Ming and Liu Ying-Ming, Fuzzy topology, I, Neighborhood structures at a fuzzy point and MooreSmith convergence, J. Math. Anal. Appl. 76 (1980) 571 599.