On fuzzy function spaces

Vol. 22, No. 4 (1999) 727–737 S 0161-17129922727-5 © Electronic Publishing House

ON FUZZY FUNCTION SPACES

GUNTHER JÄGER

(Received 9 December 1996 and in revised form 20 October 1997)

Abstract.In [3], we started the investigation of compactness in fuzzy function spaces in FCS, the category of fuzzy convergence spaces as defined by Lowen/Lowen/Wuyts [8]. This paper goes somewhat deeper in the investigation of fuzzy function spaces using the notion of splitting and conjoining structures on fuzzy subsets. We discuss the connection to the exponential law and give several examples of such structures. As a special case, we study a notion of fuzzy compact open topology.

Keywords and phrases. Fuzzy topology, fuzzy convergence space, fuzzy function space, splitting structure, conjoining structure, pointwise convergence, continuous convergence, compact open topology.

1991 Mathematics Subject Classification. 54A40.

1. Introduction. The theory of fuzzy topological spaces is meanwhile highly devel-oped. Especially, compactness and separation axioms have been thoroughly studied. Yet, one important field of classical topology has not yet attained wide attention in fuzzy topology: the theory of function spaces. Function spaces play an important role in functional analysis, in the theory of differential equations, in complex analysis, and in almost every other branch of modern mathematics, not to forget in topology itself. Therefore, it seems desirable to study function spaces also in fuzzy topology. In the meantime, three papers on this subject have appeared (Peng [11], Dang and Behera [2], and Alderton [1]). The fact that FTS, the category of fuzzy topological spaces (Lowen [9]), is not cartesian closed led Lowen and Lowen [7] to the definition of FCS, the cate-gory of fuzzy convergence spaces. This paper takes FCS as a starting point to discuss certain fuzzy function space structures via splitting and conjoining structures. It con-tinues a previous paper by the author, where compactness in fuzzy function spaces in FCS was considered [3] and also considers function spaces in FTS.

2. Preliminaries. LetXbe a nonvoid set. Fuzzy subsets ofXare denoted byA,B, C,...∈[0,1]X_{, (ordinary) subsets of X are denoted by small italics a,b,c,...⊂X.} Fora⊂X, we denote by1a thecharacteristic functionofaand in casea= {x}, we write1x. For the characteristic function of the whole set X,Y ,Z, we write for short againX,Y ,Z. The fundamental definitions of fuzzy set theory and fuzzy topology are assumed to be familiar to the reader. We especially take Lowen’s definition of fuzzy topology [9]. In order to make this paper self-contained, however, we summarize the main results of our papers [3, 4, 5]. Given a fuzzy subset A∈[0,1]X_{, we denote}

FX(A):= {B∈[0,1]X:B⊂A}and callA0:= {x∈X:A(x) >0}thesupportofA. For

A∈[0,1]X_{, we call a fuzzy subset of the formB=A∩1}

IfA=X, we regain the usual definition.

For a functionf:X→Y, we define its restrictiononAbyf|A(D):=f (D) (D∈ FX(A))and the correspondinginverse imageby(f|A)−1(E):=f−1(E)∩A (E∈FY(Y )) (cf. [4]). If, moreover,f (A)⊂B, we callg=f|A:A→Bafuzzy mappingfromAto

B[4].

A nonempty collectionF⊂FX(A)is called afuzzy filteronAif and only if it does not contain the empty fuzzy set∅:=1∅, is closed under finite intersections, and contains, forF∈F, every fuzzy supersetA⊃G⊃F.B⊂FX(A)is called afuzzy filter

basisonAif and only if it is not empty, does not contain the empty fuzzy set, and the intersection of two of its members contains a member ofB. For a fuzzy filter basisB onA,[B]A=[B]:= {F ⊂A:∃B∈Bsuch thatB⊂F}is a fuzzy filter onA. A fuzzy filter onAis called a prime fuzzy filterif and only if wheneverF∪G∈F,F ∈For

G∈F. For example, thefuzzy point filters[α1x]:= {G⊂A:G(x)≥α}(0< α≤A(x)) are prime fuzzy filters. The setF(A)of fuzzy filters onAis ordered by set inclusion. ForF∈F(A), the setP(F)of all prime fuzzy filters finer thanFis inductive and, by Zorn’s lemma, there exist minimal elements inP(F), the set of which is denoted by Pm(F)(cf. Lowen [10]). For a fuzzy filterF∈F(A), the systemı(F):= {F0:F∈F}is a filter onA0.Fis a prime fuzzy filter if and only ifı(F)is an ultrafilter. We further call for a fuzzy filterF∈F(A)

c(F):=inf

F∈Fsup_x∈XF(x) (2.1)

itscharacteristic value(Lowen/Lowen [6]). ForA∈[0,1]X_{, we call a mapping}

lim :

 

F_F(A)_→lim→F_FX(A) (2.2)

a fuzzy convergence onAif and only if the following conditions are satisfied: (PST) ∀F∈F(A): limF=G∈Pm(F)limG;

(F1p) ∀F∈F(A)prime fuzzy filters : limF≤c(F);

(F2p) ∀F,G∈F(A)prime fuzzy filters :F≤G⇒limG⊂limF; (C1) ∀x∈A0, 0< α≤A(x):α1x⊂lim[α1x].

(cf. [5, 7, 8]). The pair(A,lim)is then called afuzzy convergence space(fcs for short). A fuzzy topological space(X,∆)can be considered as an fcs if we put, forF∈F(X),

lim(∆)F:=

G∈Pm(F)

G∈G

G∆, (2.3)

which is the definition of limit of a fuzzy filter due to Lowen [10].

For two fuzzy convergences lim, lim on the same fuzzy set A∈[0,1]X_{, we say} that lim is finer than lim if and only if, for every prime fuzzy filter F∈F(A), we have lim_{F⊂limF. We then write lim≤lim. It is easily verified that, for two fuzzy}

have, for two fuzzy convergences lim, lim onA∈[0,1]X_{, lim≤lim if and only if}

idX|A:(A,lim_{)→(A,lim)is continuous.}

Now, let(gλ:A→(Aλ,limλ))λ∈Λ be a family of fuzzy mappings from a fuzzy set A∈[0,1]X _{to fcs’s(A}λ_,limλ)withAλ_∈[0,1]Xλ _{(λ∈Λ). If we put, for a prime fuzzy} filterF∈F(A),

initlimλ,gλF:=

λ∈Λ

g−1_limλg

λ(F), (2.4)

and derive init(limλ,gλ)for arbitrary fuzzy filtersF∈F(A)by (PST), then init(limλ,gλ) is the coarsest fuzzy convergence onAsuch that everygλis continuous [5].

IfB⊂A(A,lim)fcs, andıB:=idX|B:B→Ais the fuzzy inclusion, we call lim|B:= init(lim,ıB)thefuzzy convergenceonBinduced by lim and the pair(B,lim|B)afuzzy

subspaceof(A,lim). We have lim|BF=B∩lim[F]for a prime fuzzy filterF∈F(B). For more details on this subspace concept, see [3, 4, 5].

IfAλ_∈[0,1]Xλ _{(λ∈Λ)and, as usual,ΠA}λ_{((xλ)):=infλ∈ΛA}λ_{(xλ)for(xλ)∈ΠXλ,} the restrictions πµ :=prµ |ΠAλ of the projections prµ :ΠXλ →Xµ,(xλ)xµ are fuzzy mappings fromΠAλ_toAµ_{(µ∈Λ)(cf. [4]). If we denoteπ−lim :=init(limλ,π}

λ), then (ΠAλ_{,π−lim) is called theproduct spaceof the fcs’s((A}λ_{,limλ))λ∈Λ. For a prime} fuzzy filterF∈F(ΠAλ_{), we haveπ−limF=Π}

λ∈Λlimλπλ(F)=λ∈Λπλ−1(limλπλ(F)). For more details, we refer to [5].

Ifg=f|A:A→Bandh=k|C:C→Dare fuzzy mappings and if we define, as usual, the product-mappingf×k(a,c):=(f (a),k(c)), then it is easily verified that

f×k(A×C)⊂B×D. Hence, we can define thefuzzy product-mappingg×h:=f×k| A×C:A×C→B×D. The simple proofs of the next two propositions are left to the reader.

Proposition2.1. Let, in the situation above,F ⊂A×C. IfπA respectivelyπC, are

the fuzzy projections fromA×CtoArespectivelyC, andπB respectivelyπD, are the

fuzzy projections fromB×DtoBrespectivelyD, theng(πA(F))=πB(g×h(F))and

h(πC(F))=πD(g×h(F)).

Proposition2.2. Let(A,limA),(B,limB),(C,limC), and(D,limD) be fcs’s and let g:A→B andh:C →Dbe continuous fuzzy mappings. Then the product-mapping g×his(limA×limC,limB×limD)-continuous.

3. Continuous convergence on fuzzy subsets (The space C(A,B)). Let in this numberA∈[0,1]X_,B∈[0,1]Y_{, and always}

sup

x∈XA(X) 1B0⊂B. (3.1)

We then have BA00 = {g:A0→B0 |g(A)⊂B}, i.e., we do not need to distinguish

between mappings fromA0 toB0 and fuzzy mappings fromAtoB. For a mapping

g:A0→B0, we put

BA_{g=ηg:= inf} x∈A0B

i.e., BA_=Π

x∈A0B (identifying g and (g(x))x∈A0 in the natural way) is a fuzzy set onBA0

0 .

We consider the evaluation map

ev :

  

BA00 ×A0 →B0

g,x→g(x). (3.3)

If (3.1) is assumed, then ev :BA_{×A→B is a fuzzy mapping [3]. We further denote} byπBA :BA×A→BAthe restriction of the projection pr

BA₀0:(g,x)gand byπA:

BA_{×A→A the restriction of the projection prA}

0:(g,x)x onBA×A. Thefuzzy

convergence structure of continuous convergenceonBA_{is defined as follows (cf. [3,} 7, 8]). For(A,limA_),(B,limB_{)fcs’s,g:A→Ba fuzzy mapping andF∈F(B}A_{)a prime} fuzzy filter, we put

CF,g:=α∈[0,1]| ∀Θ∈F(BA_{×A)prime fuzzy filters such that}

πBA(Θ)≤F,∀x∈A0: limAπ_A(Θ)(x)∧α≤π_x−1limBev(Θ)g (3.4) and

c-limFg:=c(F)∧ηg∧supCF,g. (3.5) For an arbitrary fuzzy filterF∈F(BA_{), we derive c-limFby (PST). c-lim then satisfies} (PST), (F1p), and (F2p) and, in general, fails to satisfy (C1). Therefore, we speak of c-lim as a “ weak fuzzy convergence structure” (cf. [3]).

IfM⊂BA_{is a fuzzy subset ofB}A_{, we call c-lim|}

Mthe(weak) fuzzy convergence

struc-ture of continuous convergenceonMand denote this (weak) fuzzy convergence again by c-lim. The next proposition shows that we can calculate the fuzzy convergence of continuous convergence for certainM⊂BA_{“from inside.”}

Proposition3.1. Let(A,limA_),(B,limB_{)be fcs’s andM=B}A_∩1

M0be a crisp fuzzy

subset ofBA_{. If we put for a prime fuzzy filterF∈F(M)andg∈M0}

CMF,g:=

α∈[0,1]| ∀Θ∈F(M×A)prime fuzzy filter such that πM(Θ)≤F,∀x∈A0: limAπA(Θ)(x)∧α≤πx−1

limB_ev(Θ)g, (3.6)

then c-limF(g)=c(F)∧η(g)∧supCM(F,g). Here,πM=prM₀|M×Ais the restriction

of the mappingM0×A0(g,x)g∈M0andπA:M×A→Ais the fuzzy projection

andev :M×A→Bandπx=prx|M:M→Bis the restriction of the mappinggg(x).

Proof. We prove thatC([F],g)=CM(F,g). Letα∈C([F],g)and letΘ∈F(M×A) be a prime fuzzy filter such thatπM(Θ)≤Fand letx∈A0. Then[Θ]∈F(BA×A)is a prime fuzzy filter and[πM(Θ)]≤[F]. ForΨ⊂M×Aandg∈M0, we haveπM(Ψ)(g)= supx∈A0Ψ(g,x)=πBA(Ψ)(g)which yields[πM(Θ)]=πBA([Θ]). Obviously,πA([Θ])=

πA(Θ)and ev([Θ])=ev(Θ)(here, the fuzzy functions on the left sides are defined on

BA_{×Aand the ones on the right sides are defined onM×A). Hence, we get} limA_π

A(Θ)(x)∧α=limAπA([Θ])(x)∧α

≤π−1

x limBev([Θ])g

=π−1

x limBev(Θ)g,

(3.7)

Conversely, letα∈CM(F,g)and Θ∈F(BA×A)be a prime fuzzy filter such that

πBA(Θ)≤[F]and letx∈A0. Then, ası(π_BA(Θ))=ı([F])andMis a crisp fuzzy sub-set, we get thatM∈πBA(Θ)and, therefore, also M×A∈Θ. We putΘ:=ΘM×A=

{θ∩(M×A): θ ∈Θ}. Then [Θ_{]=Θ, and Θ is also a prime fuzzy filter. A}

sim-ple computation shows that πM(Ψ∩(M×A))=πBA(Ψ)∩M forΨ∈BA×A and so

πM(Θ)=(πBA(Θ))M. From this, we conclude thatπM(Θ)≤([F])M=F. As, further-more,πA(Θ)=πA(Θ)and ev(Θ)=ev(Θ)(where again the fuzzy functions on the left sides are defined onM×Aand those on the right sides are defined onBA_×A), this yieldsα∈C([F],g)which completes the proof.

Proposition3.2. Let(A,limA),(B,limB)be fcs’s andM=BA_∩1

M0be a crisp fuzzy

subset ofBA_{. The following hold:}

(i) ev :(M×A,c-lim×limA)→(B,limB)is continuous.

(ii) IfA=α1X, B=α1Y andlim∗is a fuzzy convergence onMsuch thatev :(M×

A,lim∗_×limA_)→(B,limB_{)is continuous, then c-lim≤lim}∗_.

Proof. Using Proposition 3.1, we can copy the corresponding proof of [3, Prop. 4.6].

We now put for two fcs’s(A,limA_)and(B,limB₎

C(A,B)0:=g:A,limA _→B,limB_{continuous (3.8)}

and define the fuzzy subsetC(A,B)ofBA_byC(A,B):=BA_∩1C(A,B)

0. In [3, Prop. 4.2], we showed that, for a continuous fuzzy mapping g:(A,limA)→(B,limB) and for 0< α≤η(g), we haveα1g⊂c-lim[α1g]. Hence,(C(A,B),c-lim)satisfies the axiom (C1), i.e., is an fcs.

We finally mention a result due to Lowen/Lowen [7]. LetX,Y ,Zbe nonvoid sets and

f:X×Y→Zbe a mapping. We define a mappingϕ(f ):X→ZY_{,ϕ(f )(x):=f (x,·).} The just-defined bijectionϕ:ZX×Y _→(ZY₎X _{is called an “exponential map” (Poppe} [12]).

Proposition3.3. Let(X,limX),(Y ,limY)and(Z,limZ)be fcs’s. Then

ϕC(X×Y ,Z)=CX,C(Y ,Z). (3.9)

Here,C(Y ,Z)is provided with the fuzzy convergence of continuous convergence and X×Y with the product fuzzy convergencelimX×limY.

Proof. ϕ(C(X×Y ,Z))⊂C(X,C(Y ,Z))is shown in [7, Thm. 5.2]. The reverse in-clusion follows using the continuity of the evaluation map (Proposition 3.2) and the continuity of the fuzzy product-mapping (Proposition 2.2) in exactly the same way as the proof of the corresponding “classical” theorem 2.2, (a)⇒(b) in Poppe [12].

4. Splitting and conjoining fuzzy convergences

Definition4.1. Let(A,limA),(B,limB)be fcs’s,M⊂BA_{and lim be a fuzzy} con-vergence onM.

(ii) lim is calledsplittingforMif and only if lim≤c-lim holds onM.

As an immediate consequence of Proposition 3.2, we obtain the following proposi-tion.

Proposition4.2. Let(A,limA_),(B,limB_{)be fcs’s andM⊂B}A_{. The following hold:} (i) Iflimis conjoining forM, thenev :(M×A,lim×limA)→(B,limB)is continuous.

(ii) IfA=α1X,B=α1Y andM=BA∩1M0 is a crisp fuzzy subset, then, from the

continuity ofev :(M×A,lim×limA_)→(B,limB_{), we get thatlimis conjoining.} Splitting and conjoining fuzzy convergences are closely related to the “exponential law” (for corresponding “classical” results (cf. Poppe [12])).

Proposition4.3. Let(Y ,limY),(Z,limZ)be fcs’s andlimbe a fuzzy convergence forC(Y ,Z). Then the following are equivalent:

(i) limis splitting forC(Y ,Z),

(ii) for each fcs(X,limX), we haveϕ(C(X×Y ,Z))⊂C(X,(C(Y ,Z),lim)).

Proof. Let first lim be splitting for C(Y ,Z). From Proposition 3.3, we get that, for a continuous fuzzy mapping f : X×Y →Z, the fuzzy mapping ϕ(f ): X → (C(Y ,Z),c-lim)is continuous. Hence, for a prime fuzzy filterF∈F(X), we have

ϕf(limX_{F)⊂c-limϕf(F)⊂limϕf(F), (4.1)}

i.e.,ϕ(f )is(limX,lim)-continuous.

Conversely, let condition (ii) hold.(C(Y ,Z),c-lim)is a fuzzy convergence space and, hence,

ϕC(C(Y ,Z),c-lim)×Y ,Z⊂C(C(Y ,Z),c-lim),(C(Y ,Z),lim). (4.2)

By Proposition 3.2, the evaluation map ev is continuous, i.e., ev∈C((C(Y ,Z),c-lim)× Y ,Z) and, hence, ϕ(ev) ∈ C((C(Y ,Z),c-lim), (C(Y ,Z),lim)). As ϕ(ev)(g)(y) =

ev(g,·)(y)=ev(g,y) =g(y)= idC(Y ,Z)(g)(y), i.e., ϕ(ev) is the identity map on

C(Y ,Z), we conclude that lim≤c-lim, i.e., lim is splitting forC(Y ,Z).

Proposition4.4. Let(Y ,limY),(Z,limZ)be fcs’s andlimbe a fuzzy convergence forC(Y ,Z). Then the following are equivalent:

(i) limis conjoining forC(Y ,Z),

(ii) for each fcs(X,limX), we haveC(X,(C(Y ,Z),lim))⊂ϕ(C(X×Y ,Z)).

Proof. Let first lim be conjoining forC(Y ,Z), and letf:(X,limX)→(C(Y ,Z),lim)

be continuous. As c-lim≤lim, then alsof:(X,limX_{)→(C(Y ,Z),c-lim)is continuous.} This yieldsC(X,(C(Y ,Z),lim))⊂C(X,(C(Y ,Z),c-lim)). Proposition 3.3 now implies condition (ii).

Conversely, let condition (ii) hold. As(C(Y ,Z),lim)is an fcs, we obtain

C(C(Y ,Z),lim),(C(Y ,Z),lim)⊂ϕ(C(Y ,Z),lim)×Y ,Z. (4.3)

As idC(Y ,Z) is(lim,lim)-continuous, we deduce, herefrom, that

is continuous. But, asϕ(ev)=idC(Y ,Z)andϕis a bijection, we getϕ−1_{(idC(Y ,Z))=ev}

and, hence, by Proposition 4.2, lim is conjoining forC(Y ,Z).

5. Examples for splitting and conjoining fuzzy convergences

5.1. The discrete and the indiscrete fuzzy convergences. LetA∈[0,1]X_{. If we} put for a prime fuzzy filterF∈F(A)

limıF:=c(F)1X∩A,

limδF(x):=

 

c(₀F) if and only if_else ı(F)=[x], (x∈X)

(5.1)

and derive limı respectively limδ for arbitrary fuzzy filters onAby (PST), then the following proposition holds.

Proposition5.1. LetA∈[0,1]X_{. Then} (i) limı,limδare fuzzy convergences onA,

(ii) for a fuzzy convergencelimonA, we havelimı≤lim≤limδ.

Proof. (i) That limıis a fuzzy convergence onAis obvious and, obviously, limδ satisfies axioms (F1p), (PST), and (C1). If F, G ∈F(A) are prime fuzzy filters and F≤Gand limδG(x) >0, thenı(G)=[x]and, consequently, alsoı(F)=[x]. Hence, limδF(x)=c(F)≥c(G)=limδG(x)and also (F2p) holds.

(ii) LetF∈F(A) be a prime fuzzy filter. As an immediate consequence of (F1p), we obtain limF⊂limıF. Now, let limδF(x)=c(F) >0. Thenı(F)=[x]and, hence,

A∩1x∈F. From this, we conclude that, forF∈F,

F(x)=F∩A∩1x(x)=sup

y∈XF∩A∩1x

y≥inf

G∈Fsupy∈XG(x)=c(F), (5.2) i.e.,F≤[c(F)1x]. Hence, it follows, by (F2p) and (C1), that

limF(x)≥lim[c(F)1x](x)≥c(F)1x(x)=c(F), (5.3)

i.e., limF⊃limδFand the proposition is proved.

limıis called theindiscrete fuzzy convergenceonAand limδ is called thediscrete fuzzy convergenceonA.

Corollary5.2. Let(A,limA),(B,limB)be fcs’s andM⊂BA_{. Then} (i) The indiscrete fuzzy convergence onMis splitting forM.

(ii) IfM⊂C(A,B), then the discrete fuzzy convergence onMis conjoining forM. 5.2. The fuzzy convergence of pointwise convergence. Let(A,limA_),(B,limB_)be fcs’s. We define a fuzzy convergence onBA _{by putting, for a prime fuzzy filterF∈} F(BA_),

p-limF:=

x∈A0

π−1

x limBπx(F) (5.4)

and derivep-limFfor an arbitrary fuzzy filter onBA _{by (PST), i.e.,p-lim is the} ini-tial fuzzy convergence onBA _{respectively the(π}

convergence of pointwise convergenceonBA_{[3]. In [3], we proved the following} propo-sition.

Proposition5.3. LetA∈[0,1]X_,B∈[0,1]Y _{be constant fuzzy sets of the same}

heightα >0and(A,limA),(B,limB)be fcs’s. Thenp-lim≤c-lim.

Corollary5.4. Under the assumptions of Proposition 5.3, the fuzzy convergence of pointwise convergencep-limis splitting forBA_.

5.3. The fuzzy convergence of strictly continuous convergence. Let (A,limA),

(B,limB_{)be fcs’s andM⊂B}A_{. We put for a prime fuzzy filterF∈F(M)and a fuzzy} mappingg:A→B

SCMF,g:=α∈[0,1]| ∀Θ∈F(M×A)prime fuzzy filters such that

πM(Θ)≤F,∀y∈B0: limBgπA(Θ)y∧α≤limBev(Θ)y, (5.5)

and define

sc-limFg:=c(F)∧ηg∧supSCMF,g, (5.6)

and derivesc-limFfor arbitrary fuzzy filters onMby (PST).

Proposition5.5. Under the assumptions mentioned above,sc-limis a fuzzy con-vergence onM.

Proof. By definition, sc-lim satisfies (PST) and (F1p). IfF≤Gfor two prime fuzzy filtersF,G∈F(M), we obviously have SCM(G,g)⊂SCM(F,g)and, hence, sc-limG⊂

sc-limF, i.e., (F2p) holds. So, all that remains to be shown is (C1): Let 0< α≤η(g)

andΘ∈F(M×A)be a prime fuzzy filter such thatπM(Θ)≤[α1g]. It then follows, in exactly the same way as in the proof of [7, Prop. 5.1], that ev(Θ)≤g(πA(Θ)). Using (F2p) for (B,limB), we deduce herefrom limBg(πA(Θ))⊂ limBev(Θ). Hence, 1∈SCM([α1g],g) and, therefore, sc-lim[α1g](g)=c([α1g])=αand sc-lim sat-isfies (C1). (sc-lim is called thefuzzy convergence of strictly continuous convergence

onM.)

Proposition5.6. IfA=α1XandB=α1Y, thensc-limis conjoining forC(A,B).

Proof. We putM:=C(A,B). LetF∈F(M)be a prime fuzzy filter andg∈C(A,B)0. Further, letα∈SCM(F,g)andΘ∈F(M×A)be a prime fuzzy filter such thatπM(Θ)≤F and letx∈A0. Theng(x)∈B0and the continuity ofgyields

limA_π

A(Θ)(x)∧α≤glimAπA(Θ)g(x)∧α

≤limB_gπ

A(Θ)g(x)∧α

≤limB_ev(Θ)g(x)

=π−1

x

limB_ev(Θ)g.

(5.7)

6. Splitting and conjoining fuzzy topologies. In this number, we only consider fuzzy topological spaces(X,∆),(Y ,Γ)in the sense of Lowen [9] and crisp subspaces ofYX_{. As in the (classical) theory of convergence space (Poppe [12]), we call a fuzzy} topologyTon a crisp subsetM=1M0⊂YXsplitting(resp.conjoining) forMif and only if the corresponding fuzzy convergence lim(T)is splitting (resp. conjoining) forM.

6.1. The discrete and the indiscrete fuzzy topologies. LetXbe a nonvoid set. If we put∆δ:=FX(X)and ∆ı:= {α: 0≤α≤1}, then, obviously,∆δ and∆ıare fuzzy topologies onX.∆δis calleddiscrete fuzzy topologyonXand∆ıis called theindiscrete

fuzzy topologyonX.

Proposition6.1. We havelim(∆δ)=limδandlim(∆ı)=limı.

Proof. First, we show that lim(∆δ)=limδ. Let F∈F(X)be a prime fuzzy fil-ter. By the definition of lim(∆δ), We obtain lim(∆δ)F=F∈FF. If lim(∆δ)F= ∅, as-sume that limδF(x) >0. Thenı(F)=[x]and, therefore,F≤[c(F)1x](see the proof of Proposition 5.1). (F2p) together with (C1) for lim(∆δ)then imply∅ =lim(∆δ)F⊃

c(F)1x, a contradiction to limδF(x) >0. Hence, also limδF= ∅. If lim(∆δ)F(x)=:α > 0< β:=lim(∆δ)F(y), thenF≤[α1x]andF≤[β1y]. Thus,ı(F)=[x]andı(F)=[y], i.e.,x=y. From this, we conclude that, lim(∆δ)F=α1x≤c(F)1x=limδF. But as

ı(F)=[x], we again getF≤[c(F)1x]and this yields, using (F2p), lim(∆δ)F⊃c(F)1x. Hence, lim(∆δ)=limδis established.

To prove lim(∆ı)=limıit suffices to remark that, for F ∈[0,1]X, we haveF∆ı = supx∈XF(x) and so lim(∆ı)F=infF∈Fsupx∈XF(x) =c(F) for a prime fuzzy filter F∈F(X).

Corollary6.2. Let(X,∆)and(Y ,Γ)be fuzzy topological spaces andM=1M0⊂YX

be a crisp fuzzy subset ofYX_{. Then}

(i) The indiscrete fuzzy topology onMis splitting forM.

(ii) IfM⊂C(X,Y ), then the discrete fuzzy topology is conjoining forM.

6.2. The fuzzy compact open topology. In [11], Peng defines a notion of fuzzy compact open topology using the notion ofN-compactness due to Wang [13]. Here, we alter his definition slightly using Lowen’s [9] definition of compactness. Let(X,∆),

(Y ,Γ)be fuzzy topological spaces. For a subsetk⊂XandG∈[0,1]Y_{, we put}

(1k,G)(g):=inf x∈kG

g(x). (6.1)

So,(1k,G)∈[0,1]Y. It is easily verified that the system

(1k,G):1kcompact in(X,∆),G∈Γ (6.2)

is a subbase for a fuzzy topology onYX_{, thefuzzy compact open topology∆}

coonYX.

If we restrict the functions(1k,G)onC(X,Y ), then the subspace topology of∆co is

also denoted by∆co.

Lemma1. LetXbe a nonvoid set and∆,Γbe fuzzy topologies onX. If, for everyD∈Γ and for everyx∈Xsuch thatD(x)=:αx>0, there isHx∈∆such thatHx∩αx⊂D

andHx(x)∧αx=D(x), then∆≥Γ.

Lemma2. Let(X,∆)and(Y ,Γ)be fuzzy topological spaces and let(X×Y ,∆×Γ)be their product space. ThenG∈∆×Γ if and only if, for every(x,y)∈X×Y and every , >0, there are fuzzy setsH,

x∈∆,Ky, ∈Γ such thatHx,×Ky,⊂GandH,x×Ky,(x,y)≥

G(x,y)−,.

Proposition6.3. Let(X,∆)and(Y ,Γ)be fuzzy topological spaces andTbe a con-joining fuzzy topology onYX _{(respectivelyC(X,Y )). ThenT≥∆co.}

Proof. Let1k∈[0,1]X be compact andG∈Γ and(1k,G)(g)=infx∈kG(g(x))=:

αg>0. Then, for everyx∈k, we haveαg≤G(g(x))=ev−1(G)(g,x). AsTis conjoin-ing, the evaluation map is continuous (Proposition 4.2 and [5, Prop. 4.3]) and, hence, ev−1_{(G)∈T×∆. Let, >0. Lemma 2 yields the existence of fuzzy setsH}

x∈T,Ox∈∆ such that

Hx×Ox⊂ev−1(G),

Hxg∧Ox(x)≥ev−1(G)g,x−₂,≥αg−,₂. (6.3)

From this, we see that the system{Ox:x∈k}is an open cover of(αg−(,/2))1k. As

1kis compact, there are finitely manyx,1,...,xn, ∈ksuch that

_n

k=1Oxk, ⊃(αg−,)1k. We putH:=nk=1Hxk,. ThenH∈T. We prove thatH∩αg⊂(1k,G)in two steps.

Step1. ev(H×(αg−,)1k)⊂G. Letb∈Y, then

evH×(αg−,)1k(b)= sup (f ,x):x∈k,f (x)=bH

f∧(αg−,). (6.4)

Letδ >0, then there arefδ,xδ∈ksuch thatfδ(xδ)=band

evH×(αg−,)1k(b)−δ≤Hfδ∧(αg−,). (6.5)

We choosek∈ {1,2,...,n}such thatO_x,k(xδ)≥αg−,. Then

evH×(αg−,)1k(b)−δ

≤Hfδ∧O_xk

,(xδ)≤ev

−1_(G)f

δ,xδ

=Gfδ(xδ)=G(b).

(6.6)

The arbitrariness ofδ >0 now yields the assumption.

Step2. For everyx∈k, we haveG(f (x))≥ev(H×(αg−,)1k)(f (x))≥H(f )∧

(αg−,). Hence,(1k,G)(f )≥H(f )∧(αg−,). The arbitrariness of, >0 now yields

References

[1] I. W. Alderton,Function spaces in fuzzy topology, Fuzzy Sets and Systems32(1989), no. 1, 115–124. MR 90i:54013. Zbl 691.54004.

[2] S. Dang and A. Behera,On fuzzy compact-open topology, Fuzzy Sets and Systems 80 (1996), no. 3, 377–381. CMP 96 13. Zbl 868.54003.

[3] G. Jäger,Compactness in fuzzy function spaces, Quaestiones Mathematica, to appear. [4] ,Compactness and connectedness as absolute properties in fuzzy topological spaces,

Fuzzy Sets and Systems94(1997), 405–410.

[5] ,Compactness in fuzzy convergence spaces, Fuzzy Sets and Systems90(1998), 341–348. MR 98j#54006. Zbl 917.54007.

[6] E. Lowen and R. Lowen,Characterization of convergence in fuzzy topological spaces, In-ternat. J. Math. Math. Sci.8(1985), no. 3, 497–511. MR 87b:54006. Zbl 593.54005. [7] ,A topological universe extension ofFTS, Applications of category theory to fuzzy subsets (Dordrecht), Theory Decis. Lib. Ser. B Math. Statist. Methods, vol. 14, Kluwer Acad. Publ., 1992, pp. 153–176. MR 93d:18011. Zbl 774.54002.

[8] E. Lowen, R. Lowen, and P. Wuyts,The categorical topology approach to fuzzy topol-ogy and fuzzy convergence, Fuzzy Sets and Systems40 (1991), no. 2, 347–373. MR 92f:54005. Zbl 728.54001.

[9] R. Lowen,Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl.56(1976), no. 3, 621–633. MR 55 13357. Zbl 342.54003.

[10] ,Convergence in fuzzy topological spaces, General Topology Appl.10(1979), no. 2, 147–160. MR 80b:54006. Zbl 409.54008.

[11] YuWei Peng,Topological structure of a fuzzy function space—the pointwise convergent topology and compact open topology, Kexue Tongbao (English Ed.)29(1984), no. 3, 289–292. MR 86g:54014. Zbl 551.54006.

[12] H. Poppe, Compactness in general function spaces, VEB Deutscher Verlag der Wis-senschaften, Berlin, 1974. MR 55 11199. Zbl 291.54012.

[13] GuoJun Wang,A newfuzzy compactness defined by fuzzy nets, J. Math. Anal. Appl.94 (1983), no. 1, 1–23. MR 85k:54005. Zbl 512.54006.

Jäger: Lessingstrasse13, D-76135Karlsruhe, Germany

Current address: Deutsches Zentrum Für Luft und Raumfahrt DLR, Institut Für Hochfrequen-ztechnik, Postfach 11 16, D-82230 Wessling, Germany