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On fuzzy compactness
FUZZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 80 (1996) 15-22
On fuzzy compactness M.W. Warner Department of Mathematics, Northampton Square, London ECIV OHB, UK
Received October 1994; revised January 1995
Abstract
The modelling of General Topology by Fuzzy Topology is illustrated by fuzzy compactness. This is done through a discussion of its history, leading up to the present position. Keywords: Modelling; Fuzzy compactness; Nice; Strong; Ultra
1. Introduction
The question of the extent to which Fuzzy Topology models General Topology is as old as Fuzzy Topology itself. It has even been argued, in particular by Lowen [12], that the only interesting fuzzy topologies are pathological ones, i.e. those which do not follow the same behaviour pattern as the classical. This discussion, on the other hand, will use fuzzy compactness to illustrate the attempt over the past 25 years to model positively the classical compactness of General Topology. The first criterion for satisfactory modelling in general was suggested by Lowen [9]. This was formulated in terms of fuzziness with respect to the unit interval I and relied on the fact that the set so(z) of lower semi-continuous functions from a topological space (X,r) to I forms a fuzzy topology. A fuzzy topological property P/, then, is said to be a 'good extension' of its general topological counterpart P as long as ~o(~) possesses
PI iff ~ possesses P. This criterion is now generally applied, although it is far from guaranteeing uniqueness. For instance, Lowen himself produced seven suggested versions of /-fuzzy compactness, five of which he proved to be good extensions. What other requirements should we therefore make? Here we must abandon generality and consider each topological property P separately. Sticking to compactness, we submit that the Tychonoff theorem plays a central role and should be used as another touchstone for the modelling. Other considerations are mentioned in the text. In general we shall consider compactness with respect to a complete completely distributive lattice L. Warner [19] has provided a goodness of extension criterion in this case based on the fact that if L is given the Scott topology the set of continuous functions from a topological space (X, z) to L forms an L-fuzzy topology. The definition of 'good extension' then follows word for word that given above for/-fuzzy theory.
0165-0114/96/$15.00 ~') 1996 ElsevierScienceB.V. All rights reserved SSDI 0165-01 14(95)00132-8
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M.V£ Warner / Fuzzy Sets and Systems 80 (1996) 15 22
2. Some definitions For the sake of completeness and the convenience of readers not familiar with the field we include some definitions of terms either already used or used without comment in the rest of the text.
Definition. For a topological space (X, z) a function f:(X,z) ~ I is said to be lower semi-continuous (1.s.c.) i f f f 1(~, 1] E z for all ~ ~ [0, 1), i.e. f is continuous with respect to the right ray topology on I. Definition. The Scott topology on a lattice (L, ~<) is defined as follows: 0 ~ L is open if (i) x e 0 and x ~< y imply y s O; i.e. 0 is an upper set. (ii) If the supremum of X is in 0 for X a directed set then X ~ 0 ~ O; i.e. there exists x e X such that x~O. Definition. A fuzzy topological space (X, ~) is called stratified if all constant fuzzy sets are in z. The Tychonofftheorem states that the product of compact spaces is compact. For local compactness the product of a family of spaces is locally compact iff each factor is locally compact and all but a finite number of them are compact.
3. The history of fuzzy compactness Fuzzy topology, like all fuzzy set theory, was originally studied with respect to the closed unit interval I. Compact spaces, and indeed fuzzy topological spaces in general, were introduced by Chang [2] who simply expressed in fuzzy terms the classical open covering definition of general topology. This compactness turned out to be unsatisfactory as a basis for a theory, not least because it is not a good extension of compactness. Also, although Goguen [4] proved a fuzzy Tychonoff theorem for finite products he showed that it is false for infinite products. As mentioned above Lowen (e.g. [9, 10]) addressed these problems, producing seven suggested versions of/-compactness, five of which he proved to be good extensions. We shall consider only three of these, namely ultra fuzzy
compactness, strong fuzzy compactness, and what Lowen called compactness. For the sake of clarity, this last version will sometimes be termed Lowencompactness. Now Lowen-compactness has given rise to fruitful work, for instance Lowen's [9] proof that a compact Hausdorff fuzzy space is topologically generated. But here fuzzy compactness seems to depend on some arithmetic properties of the real numbers. Of more interest are the definitions which depend only on the unit interval's lattice theoretic properties, including in particular Lowen's strong compactness theory with respect to a complete completely distributive lattice L equipped with an order-reversing involution (henceforth called a fuzzy lattice). Different degrees of compactness, called e-compactness [3], were introduced, and an e-compactness Tychonoff theorem was proved for arbitrary products and some restricted values of einL. Originally these fuzzy compactnesses were defined only for the whole fuzzy topological space rather than for arbitrary fuzzy subsets. Chadwick [1] recently rectified this for Lowen (I) fuzzy compactness. Wang [16], on the other hand, introduced a new theory based on the fuzzy nets of Pu and Liu [13] and called it nice compactness (N-compactness); its properties include being defined for arbitrary subsets, that it is a good extension, and that the general Tychonoff theorem holds. Wang also proved that N-compactness implies strong I-compactness, and he used fuzzy nets to give a definition of strong compact fuzzy subsets. Now N-compactness generalises with ease, as was shown by Zhao [20], to an N-compactness theory for L-fuzzy topological spaces. This has all the advantages of Wang's theory, as well as being a generalization of it, and has been extensively used (e.g. [8-]) to further the study of fuzzy compactness. Meanwhile Warner and McLean [18] have suggested a generalisation of strong compactness which, unlike that of Gantner et al. [3], fits neatly into the N-compactness picture. This will be called S-compactness and is implied by N-compactness. It is sufficient for an adequate compactness theory in fuzzy topology, being a good extension [18], defined on arbitrary fuzzy sets, and with a general Tychonoff theorem. An S-compact Hausdorff space is topologically generated [18]; indeed, in a
M.W. Warner /Fuzzv Sets and Systems 80 (1996) 15 22
Hausdorff space, just as in /-theory, N- and Scompactness coincide. G o o d extensions of weaker fuzzy S-compactness (e.g. almost, nearly) are also easily obtainable [7]. An S-local compactness also emerges satisfactorily [-6]. In this paper we shall recall briefly the relevant early fuzzy theories as a background to a discussion of L-fuzzy N- and S-compactness. We note that our discussion is not exhaustive either with respect to authors in the field or with respect to theories. Further references are to be found in the survey paper of Shostak [15].
4. Some early fuzzy compactnesses The following three versions of /-compactness were introduced by Lowen [10].
Definition 1. The fuzzy topological space (X, 6) is fuzzy compact ifffor each family fl c 3 and for each ~ I such that sup~p kt ~> ~ and for each e in (0, e] there exists a finite subfamily flo of fl such that s u p ~ o / ~ ~> e - e. When there is a possibility of confusion this will be called Lowen-compactness.
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space is topologically generated. This is therefore also true of strong and ultra because all three are equivalent for Hausdorff spaces. The term a-shading was originally coined by Gantner et al. [3] in the more general context of L-fuzzy theory, L a fuzzy lattice. In an L-fuzzy space (X,6) a collection ~# c 6 will be called an a-shading of X if, for each x ~ X , there exists a U ~ g with U ( x ) > e , e ~ L \ { l } . These same authors called a space (X,6) a-compact, e 6 L \ { 1 } , if each a-shading has a finite esubshading. A form of strong compactness emerges by requiring a-compactness for all ~ e L \ { 1 } . But the Alexander Subb~ise theorem and hence the Tychonoff property were proved only for some restricted values of ~, namely members of the subset L a = {e ~ L: e is comparable to each fl ~ L and iffl > e and 7 > e then fl A 7 > e}. This restriction was also required for their one-point compactification theorem. The theory has been pursued by Rodabaugh [-14] and others, and remained the main version of L-fuzzy compactness until the introduction of niceness by the Chinese school.
5. N-compactness Defining a family of fuzzy subsets fi ~_ I x to be an a-shading iff for all x e X there exists /~ e fl such that/~(x) > ~ we have
Definition 2. (X, 3) is strong fuzzy compact iff for each e ~ I \ { 1 } = [0,1) every a-shading family in 3 has a finite a-shading subfamily.
Definition 3. (X, 3) is ultra fuzzy compact iff(X, z(6)) is compact, where z(b) is the initial topology on X generated by subsets of X of the form f - 1 ( 2 , 1] for a l l f e 3 and 2 ~ [0, 1). The following implications hold and are not reversible [15]: ultra ~ strong ~ Lowen. All three compactnesses have the Tychonoff product property, and Lowen later proved [12] that a (Lowen-) compact Hausdorff fuzzy topological
Nice compactness was originally defined in terms of closed fuzzy sets, a-nets, and remote neighbourhoods [16]. Starting with/-fuzzy sets we have the following definitions.
Definitions. (1) In a fuzzy topological space (fls) (X, 6), a fuzzy point e = x~ has a remote neighbourhood P if P is a closed fuzzy set such that 2 ~ P(x) (e¢P). Remote neighbourhoods will be written as R-nhbds. Here a fuzzy point x~ has value ,~ at x and zero elsewhere, and 0 < 2 ~ < 1 . Also x ~ 6 P iff ~ P(x). (2) A fuzzy net S = {S(n),n ~ D} is a function S:D ~ P where D is a directed set with order relation >~ and P is the collection of all fuzzy points of X. If the crisp net of values of S(n) converges in (0, 1] to e e (0, 1], then S is called an a-net. And if all the values of S(n) are e,S is a constant a-net.
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m.W. Warner / Fuzzy Sets and Systems 80 (1996) 15 22
(3) A fuzzy point e is a cluster point of a fuzzy net S if for each R-nhbd P of e we have frequently (see e.g. [-1]) S(n)¢P. (4) Let (X, 6) be an fts and A a fuzzy set in X. Then A is N-compact if each s-net (a ~ (0, 1]) contained in A has in A at least one cluster point with value ~. And when A = 1 is N-compact, (X,3) is called an N-compacttis. Wang proves that a space is strong fuzzy compact iffevery constant s-net has a cluster point with value ~ e (0, 1], and is thus able to deduce immediately that N-compactness implies strong compactness. He also shows that ultra fuzzy compactness implies N-compactness, and produces a version of Lowen-compactness in terms of cluster points. In /-fuzzy, then, N-compactness is inserted in the table of implications between strong and ultra fuzzy compactness, i.e.
Denoting by fl(e), the greatest minimal set of e, let fl*(e) = fl(e)nM(L). Then if 7 6 fl*(e) we write 7<1e. Definition. A fuzzy set A of an fts (X, 6) is N-compact if for any e e M(L) and every family q~ of closed e-remote nbhds of A, there exists a finite subfamily 4o - ~b of v-remote nbhds of A for some 7
Theorem. In antis (X, 6), a fuzzy set A is N-compact
This is easily seen to be an extension of the /-version since for 1 7 <~ e iff 0 < 7 < e. Zhao also proves that it is exactly equivalent to N-compactness defined in terms of cluster points of fuzzy nets - the statement being word for word the same as Wang's for L = I, but with the definition of s-net adjusted in terms of the relation
iff each e-remote nhbd family of A has a finite subset which is an r-remote nhbd family of A for some r e (0, o0.
6. S-compactness
Ultra ~ nice ~ strong ~ Lowen. It was left to Zhao [20] to give a geometric characterisation of Wang's N-compactness. Defining a family cb of closed fuzzy sets to be an e-remote nhbd family of a fuzzy set A (e ~ (0, 1]) if each fuzzy point x~ E A has an R-nhbd in 4, he proved
Turning to L-fuzzy topological spaces, L a fuzzy lattice, Zhao needed to use minimal sets. Definition. A subset B of L is called a minimal set of an element e of L if VB = e and for each b e B and every subset ~ of L with V~o >~ e, there exists 7 ~ ~o such that 7 ~> b where VB is the height of B. Definition. An element c~ of L is called (union-) irreducible if whenever p V q/> e then p/> e or
q>~e,p,q~L. The set of all such non-zero irreducible elements of L is denoted by M(L). Putting M(X, L) = {x,: x ~ X , c ~ M ( L ) } , the elements of M ( X , L ) are called points.
The (geometric) formulations of N-compactness are based on closed sets as remote neighbourhoods and the relation ~ representing the negation of set membership. The dual of this relation was actually suggested by Warner [17] to describe fuzzy set membership itself. Identification of the points of the frame of L-fuzzy subsets of a set X with its prime elements led to the fuzzy points of X being given as pairs (x, 2), x ~ X, 2 prime in L. They are written xz and for a fuzzy set A, x~ ~ A iff A(x) ~ 2. Recall that an element p of L is called prime if p 4:1 and wherever x A y ~< p then x ~< p or y ~< p. Primes are thus dual to the (union)-irreducible elements of N-compactness. This concept led to the following definition of compactness [18].
M.W. Warner / Fuzzy Sets and Systems 80 (1996) 15-22
Definition. An L-fuzzy topological space (X,6) is S-compact if for every prime p of L and every collection (fj)j~j of open L-fuzzy sets with ( V ~ j fj)(x) ~; p for all x e X, there is a finite subset F of J with (Vj~vfj)(x) 4 p for all x e X.
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Reference has been made by Wang [16] and others to strong Q-compactness. It is probably the same as S-compactness, but translation from the Chinese is awaited to verify this.
When L = I this is simply strong compactness since in I all elements apart from 1 are prime. Warner and McLean [18] prove it to be a good extension and also prove that an S-compact Hausdorff space is topological. Wang [16] has shown that a fuzzy space is strong fuzzy compact iff every constant e-net has a cluster point with height e. It follows that N-compactness implies Scompactness. In any case a duality between S-compactness and a weakened form of the geometric version of Ncompactness is easy to detect since a family of eremote neighbourhoods gi of X satisfies ( / ~ i gi)(x) 2~ e for all x ~ X. Thus we have an equivalent version of S-compactness.
7. L-fuzzy subsets
Definition. (X, 6) is S-compact ifffor every e ~ M(L)
Theorem. If g is a compact fuzzy subset, then for
and every collection (gj)j~ of closed fuzzy sets with (/~j~s gj)(x) ~ e for all x ~ X there is a finite subset F of J with (/~j~F gfl(x) ~ e for all x e X. The difficulties encountered by Gantner et al. [3] in the Tychonoff theorem now vanish. The permitted values of e are precisely the primes, and the substitution of 2~ for < disposes of the comparability requirement. S-compactness of fuzzy subsets follows naturally, either by adapting the above definition or by using constant a-nets. It is inherited by closed fuzzy subsets, continuous images and finite suprema. Finally, since an S-compact Hausdorff space is topological and both N- and S-compactness are good extensions, it follows that on a Hausdroff L-fuzzy space these two compactnesses are equivalent. The difference in general between the two theories is only in the lack of necessity for an S-compact set to have a maximum; this, as we noted above, is a doubtful advantage for niceness. We note that L-fuzzy ultra-compactness can be formally defined as for/-fuzzy, and again implies L-fuzzy N-compactness.
Definition. For an L-fts (X, z) a fuzzy subset g is compact iff for every prime p e L and family (f)g~j of open L-fuzzy sets with (Vi~jfi)(x) ~ P for all fuzzy points xp with xp¢g' (i.e. for all x with g(x) >~p'), there exists a finite subset F of J such that (Vi~F f~)(x) ~ p for all x with g(x) >1p'.
Note: For g = X this is S-compactness. For L = I this is strong Q-compactness. We give an example of a proof of a typical theorem on compact L-fuzzy subsets.
every closed fuzzy subset h, h c~g is compact. Proof. Using the notation of the definition, we have {j~}i~jw{h'} a family of open sets with Vk~a k(x)g£ p for all x with g(x)>1 p'. Since if h(x) >>,p' then (hc~g)(x) >~p', so (~/k~a k)(x) ~ p. And if h(x) ~ p' then h'(x) ~ p, so (~/k~p k)(x) ~ p. Thus, if h(x) >>,p', no new point is added to hc~g, while if h(x) ~ p' then point p may be 'let in' but will be covered by h' (the open set). Hence h (via h') compensates for any points it lets in. Then by the compactness of g, there is a finite subfamily Y = {fbf2, ... ,f,,h'} with (VkE?, k)(x) ~ p ~TtX with g(x) >~p'. Then (Vi~ll ...... I f~)(x) 4 p for all x with (h~g)(x)>~p'. In fact, if (hc~g)(x)>~ p' then g(x) >~p' so (Vk~r k)(x) ~ p. So there exists 3k • 7 with k(x) 4p. But h'(x)<~p, so ~/i~11...... If~(x) ~;p. []
Note: This example shows that we need a 'Q'relation x~¢g' rather than xp ~ g. A 'tidier' formulation would occur in terms of closed sets and remote neighbourhoods, leading to every constant a-net in g having a cluster point of height e.
M.W. Warner / Fuzzy Sets and Systems 80 (1996) 15-22
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8. Some weakening of fuzzy compactness It is straightforward to obtain fuzzy versions of the classical weaker forms of compactness. We illustrate this briefly with a definition and goodness proof for almost compactness [7].
Definition. An fts (X,z) is almost compact iff Vp ~ pr(L) (where pr(L) is the set of primes of L) and every family (f~)i~1 of open sets with (V~lJ~) (x) 4; p Vx E X, there exists a finite family F __ I with (Vi~F cl(fi))(x) 4; P Vx ~ X. Lemma 4.1. Let (X,z) be a topological space and A ~_ X. Considering the L-fts (X,(co(z))), with e)(z) the set of all l.s.c, functions from x to L, e ~ L and f ( x ) -:
if x ~ A,
we have {0 cl(f)(x) =
if x ~ c l ( A ) , otherwise.
Proof. Let g(x) = { ;
if x E el(A), otherwise.
We shall prove that c l ( f ) = g. Since, for every b6L, g-~({t ~ L;t >~b})
=
X
if b = O ,
cl(A)
if e~>b
and b ~ O ,
0
ife~b
and b # O
is closed in (X, r), g is closed in (X, a)(z)). We also have g ~>f Then, f~< c l ( f ) ~< cl(g) = g. Thus, we have cl(f)(x) = 0 for all x~cl(A) and cl(f)(x) = e for all x ~ A. From c l ( f ) ~ < g we obtain ( c l ( f ) ) - l ( { t E L ; t ~ e}) ~_ g - l ( { t ~ L ; t ~ e } ) = (cl(A))'. Hence cl(f)(x) = e for all x ~ cl(A) and cl(f)(x) = 0 for all x¢cl(A) and e l ( f ) = g. []
Corollary 4.2. Let (X, z) be a topological space and A ~_ X. Considering the L-fts (X,(o~(z))), we have
~(cI(A) cl(ZA) where ZA is the characteristic function of A. ---~
Proof. This follows immediately from Lemma 4.1. [] Theorem 4.3. Fuzzy almost compactness is a good extension of almost compactness. Proof. (i) Let (Ai)i~s be an open cover of (X, 0. Then ()~Ai)iEJ is a family of open L-fuzzy sets in (X,o)(z)) with (Vi~j Za,)(x)= 1 4; p for all x ~ X and for all p ~ pr(L). From the almost compactness of (X,e)(z)) there exists a finite subset F of J with (Vi~rcl(zA,))(x) = 1 4; p for all x ~ X and for all p s pr(L). Since from Corollary 4.2 cl(zA,) = ,~cl(Ai) we have (Vi~F Zcl¢A,))(X)= 1 for all x s X. Hence Ui~F cl(Ai) = X and (X,r) is almost compact. (ii) Conversely, assume (X,z) almost compact. We use basic open sets in (X,¢o(z)) for this proof, namely a family fl = (fi)i~s with f~(x) = {;i
ifx~Ui~z, otherwise.
We sometimes write such an J~ as f ia,U~. So, taking p ~ pr(L) and such a family/3 with (Viii fi)(x) 4; p for all x e X, it is enough to prove that there is a finite subfamily whose closures have this property. Now for each x ~ X there is an i ~ J with f~'V'(x) 4; p, i.e. with a i 4; p. Let ~ = {Ui: 3i ~ J with ai 4; p and f.~,v,s/3}. Then cg is a family of open sets of (X,3) which covers the space. Take a finite subfamily ~¢ of off,say {U1.... , U, } such that Ui~,~l...... } cl(Ui) = X. Using Lemma 4.1, c l ( f i ) ( x ) = { ; i ifx~cl(U~),otherwise, and the result follows, since (Vi~fl ...... I cl(f/))(x) <~p for all x e X. []
9. Local compactness To model local compactness in L-fuzzy topology we should require goodness of extension and the fuzzy version of the 'Tychonoff theorem' for local
M.W. Warner / Fuzzy Sets and Systems 80 (1996) 15 22
compactness defined in Section 2. However, local compactness has another interesting property, namely that it implies the continuity of the locale of open sets of the topology. It would be interesting to look for an L-fuzzy local compactness which also implies the continuity of the fuzzy topological locale. For a discussion of local compactness and continuous locales see Johnstone [5]. Fuzzy local compactness immediately presents difficulties, however. Defining an fts to be locally compact iff for all fuzzy points Xp every open neighbourhood of Xp contains a compact neigbourhood of Xp, we have failed to verify even the goodness of extension criterion. A partial solution has been suggested as follows [6]:
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expect a unique fuzzy version to emerge. But the task is not complete, as is instanced by the difficulties encountered in modelling L-fuzzy local compactness. However, it would be reasonable to expect satisfactory progress in this area in the not too distant future. Some other problems such as that of RS-compactness, of which a good version has not yet emerged, will also probably be solved. It is then possible that activity in positive modelling in this field will decline in favour of a study of the pathologies mentioned in our first paragraph. The extent to which fuzzy topologies can diverge from the classical could be as interesting and instructive as the modelling of General Topology discussed in this paper.
Definition. A fuzzy set C is very compact if it is given by C(x) = {f0, x ~ D , otherwise, where { ~ L and D is fuzzy compact. Then, replacing 'compact neighbourhood' by 'very compact neighbourhood', in the above suggested definition of L-fuzzy locally compact, Kudri and Warner [6] have proved that (i) L-fuzzy local compactness is a good extension of local compactness. (ii) A stratified compact Hausdorff fts is locally compact. (iii) The Tychonoff theorem for local compactness has an L-fuzzy version. (iv) For a stratified locally compact /-fuzzy space (X, ~) the locale ~ is continuous. We have not yet succeeded in extending property (iv) to a general lattice L and the question of whether our version of fuzzy local compactness does indeed produce a continuous locale remains open.
10. Conclusion
It appears, then, that the modelling of compactness in fuzzy topology is well advanced, although it is clearly unrealistic and probably undesirable to
References [1] J.J. Chadwick, A generalised form of compactness in fuzzy topological spaces, J. Math. Anal. Appl. 162 (1991) 92-110. [2] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182-190. 1-3] T.E. Gantner, R.C. Steinlage and R.H. Warren, Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62 (1978) 547-562. [4] J.A. Goguen, The fuzzy Tychonofftheorem, J. Math. Anal. Appl. 43 (1973) 734 742. [5] P.T. Johnstone, Stone Spaces (Cambridge University Press, Cambridge, 1982). [6] S.R.T. Kudri and M.W. Warner, L-fuzzy local compactness, Fuzzy Sets and Systems 67 (1994) 337 345. [7] S.R.T. Kudri and M.W. Warner, Some good L-fuzzy compactness-related concepts and their properties I, Fuzzy Sets and Systems 76 (1995) 141-155. I-8] Y.M. Liu and M.K. Luo, Induced spaces and fuzzy Stone-Cech compactifications, Scientia Sinica (A) 30 (1987) 1034-1044. 1-9] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976) 621-633. [10] R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446 454. [11] R. Lowen, Compact Hausdorff fuzzy topological spaces are topological, Topology Appl. 12 (1981) 65-74. [12] R. Lowen, On the Existence of Natural Non-Topological Fuzzy Topological Spaces (Heldermann Verlag, Berlin, 1985). [13] P.M. Pu and Y.M. Liu, Fuzzy topology l, neighbourhood structure of a fuzzy point and Moore Smith convergence, J. Math. Anal. Appl. 76 (1980) 571 599.
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[14] S.E. Rodabaugh, The Hausdorff separation axiom for fuzzy topological spaces, Topology Appl. 11 (1980) 319-331. [15] A.P. Shostak, Two decades of fuzzy topology: basic ideas, notions and results, Russian Math. Surveys 44 (1989) 125-189. [16] G.J. Wang, A new fuzzy compactness defined by fuzzy nets, J. Math. Anal. Appl. 94 (1983) 1 23.
1-17] M.W. Warner, Frame-fuzzy points and membership, Fuzzy Sets and Systems 42 (1991) 335-344. [18] M.W. Warner and R.G. McLean, On compact Hausdorff L-fuzzy spaces, Fuzzy Sets and Systems 55 (1993) 103-110. [19] M.W. Warner, Fuzzy topology with respect to continuous lattices, Fuzzy Sets and Systems 35 (1990) 85 91. [20] D.S. Zhao, The N-compactness in L-fuzzy topological spaces, J. Math. Anal. Appl. 128 (1987) 64-79.
sets and systems ELSEVIER
Fuzzy Sets and Systems 80 (1996) 15-22
On fuzzy compactness M.W. Warner Department of Mathematics, Northampton Square, London ECIV OHB, UK
Received October 1994; revised January 1995
Abstract
The modelling of General Topology by Fuzzy Topology is illustrated by fuzzy compactness. This is done through a discussion of its history, leading up to the present position. Keywords: Modelling; Fuzzy compactness; Nice; Strong; Ultra
1. Introduction
The question of the extent to which Fuzzy Topology models General Topology is as old as Fuzzy Topology itself. It has even been argued, in particular by Lowen [12], that the only interesting fuzzy topologies are pathological ones, i.e. those which do not follow the same behaviour pattern as the classical. This discussion, on the other hand, will use fuzzy compactness to illustrate the attempt over the past 25 years to model positively the classical compactness of General Topology. The first criterion for satisfactory modelling in general was suggested by Lowen [9]. This was formulated in terms of fuzziness with respect to the unit interval I and relied on the fact that the set so(z) of lower semi-continuous functions from a topological space (X,r) to I forms a fuzzy topology. A fuzzy topological property P/, then, is said to be a 'good extension' of its general topological counterpart P as long as ~o(~) possesses
PI iff ~ possesses P. This criterion is now generally applied, although it is far from guaranteeing uniqueness. For instance, Lowen himself produced seven suggested versions of /-fuzzy compactness, five of which he proved to be good extensions. What other requirements should we therefore make? Here we must abandon generality and consider each topological property P separately. Sticking to compactness, we submit that the Tychonoff theorem plays a central role and should be used as another touchstone for the modelling. Other considerations are mentioned in the text. In general we shall consider compactness with respect to a complete completely distributive lattice L. Warner [19] has provided a goodness of extension criterion in this case based on the fact that if L is given the Scott topology the set of continuous functions from a topological space (X, z) to L forms an L-fuzzy topology. The definition of 'good extension' then follows word for word that given above for/-fuzzy theory.
0165-0114/96/$15.00 ~') 1996 ElsevierScienceB.V. All rights reserved SSDI 0165-01 14(95)00132-8
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M.V£ Warner / Fuzzy Sets and Systems 80 (1996) 15 22
2. Some definitions For the sake of completeness and the convenience of readers not familiar with the field we include some definitions of terms either already used or used without comment in the rest of the text.
Definition. For a topological space (X, z) a function f:(X,z) ~ I is said to be lower semi-continuous (1.s.c.) i f f f 1(~, 1] E z for all ~ ~ [0, 1), i.e. f is continuous with respect to the right ray topology on I. Definition. The Scott topology on a lattice (L, ~<) is defined as follows: 0 ~ L is open if (i) x e 0 and x ~< y imply y s O; i.e. 0 is an upper set. (ii) If the supremum of X is in 0 for X a directed set then X ~ 0 ~ O; i.e. there exists x e X such that x~O. Definition. A fuzzy topological space (X, ~) is called stratified if all constant fuzzy sets are in z. The Tychonofftheorem states that the product of compact spaces is compact. For local compactness the product of a family of spaces is locally compact iff each factor is locally compact and all but a finite number of them are compact.
3. The history of fuzzy compactness Fuzzy topology, like all fuzzy set theory, was originally studied with respect to the closed unit interval I. Compact spaces, and indeed fuzzy topological spaces in general, were introduced by Chang [2] who simply expressed in fuzzy terms the classical open covering definition of general topology. This compactness turned out to be unsatisfactory as a basis for a theory, not least because it is not a good extension of compactness. Also, although Goguen [4] proved a fuzzy Tychonoff theorem for finite products he showed that it is false for infinite products. As mentioned above Lowen (e.g. [9, 10]) addressed these problems, producing seven suggested versions of/-compactness, five of which he proved to be good extensions. We shall consider only three of these, namely ultra fuzzy
compactness, strong fuzzy compactness, and what Lowen called compactness. For the sake of clarity, this last version will sometimes be termed Lowencompactness. Now Lowen-compactness has given rise to fruitful work, for instance Lowen's [9] proof that a compact Hausdorff fuzzy space is topologically generated. But here fuzzy compactness seems to depend on some arithmetic properties of the real numbers. Of more interest are the definitions which depend only on the unit interval's lattice theoretic properties, including in particular Lowen's strong compactness theory with respect to a complete completely distributive lattice L equipped with an order-reversing involution (henceforth called a fuzzy lattice). Different degrees of compactness, called e-compactness [3], were introduced, and an e-compactness Tychonoff theorem was proved for arbitrary products and some restricted values of einL. Originally these fuzzy compactnesses were defined only for the whole fuzzy topological space rather than for arbitrary fuzzy subsets. Chadwick [1] recently rectified this for Lowen (I) fuzzy compactness. Wang [16], on the other hand, introduced a new theory based on the fuzzy nets of Pu and Liu [13] and called it nice compactness (N-compactness); its properties include being defined for arbitrary subsets, that it is a good extension, and that the general Tychonoff theorem holds. Wang also proved that N-compactness implies strong I-compactness, and he used fuzzy nets to give a definition of strong compact fuzzy subsets. Now N-compactness generalises with ease, as was shown by Zhao [20], to an N-compactness theory for L-fuzzy topological spaces. This has all the advantages of Wang's theory, as well as being a generalization of it, and has been extensively used (e.g. [8-]) to further the study of fuzzy compactness. Meanwhile Warner and McLean [18] have suggested a generalisation of strong compactness which, unlike that of Gantner et al. [3], fits neatly into the N-compactness picture. This will be called S-compactness and is implied by N-compactness. It is sufficient for an adequate compactness theory in fuzzy topology, being a good extension [18], defined on arbitrary fuzzy sets, and with a general Tychonoff theorem. An S-compact Hausdorff space is topologically generated [18]; indeed, in a
M.W. Warner /Fuzzv Sets and Systems 80 (1996) 15 22
Hausdorff space, just as in /-theory, N- and Scompactness coincide. G o o d extensions of weaker fuzzy S-compactness (e.g. almost, nearly) are also easily obtainable [7]. An S-local compactness also emerges satisfactorily [-6]. In this paper we shall recall briefly the relevant early fuzzy theories as a background to a discussion of L-fuzzy N- and S-compactness. We note that our discussion is not exhaustive either with respect to authors in the field or with respect to theories. Further references are to be found in the survey paper of Shostak [15].
4. Some early fuzzy compactnesses The following three versions of /-compactness were introduced by Lowen [10].
Definition 1. The fuzzy topological space (X, 6) is fuzzy compact ifffor each family fl c 3 and for each ~ I such that sup~p kt ~> ~ and for each e in (0, e] there exists a finite subfamily flo of fl such that s u p ~ o / ~ ~> e - e. When there is a possibility of confusion this will be called Lowen-compactness.
17
space is topologically generated. This is therefore also true of strong and ultra because all three are equivalent for Hausdorff spaces. The term a-shading was originally coined by Gantner et al. [3] in the more general context of L-fuzzy theory, L a fuzzy lattice. In an L-fuzzy space (X,6) a collection ~# c 6 will be called an a-shading of X if, for each x ~ X , there exists a U ~ g with U ( x ) > e , e ~ L \ { l } . These same authors called a space (X,6) a-compact, e 6 L \ { 1 } , if each a-shading has a finite esubshading. A form of strong compactness emerges by requiring a-compactness for all ~ e L \ { 1 } . But the Alexander Subb~ise theorem and hence the Tychonoff property were proved only for some restricted values of ~, namely members of the subset L a = {e ~ L: e is comparable to each fl ~ L and iffl > e and 7 > e then fl A 7 > e}. This restriction was also required for their one-point compactification theorem. The theory has been pursued by Rodabaugh [-14] and others, and remained the main version of L-fuzzy compactness until the introduction of niceness by the Chinese school.
5. N-compactness Defining a family of fuzzy subsets fi ~_ I x to be an a-shading iff for all x e X there exists /~ e fl such that/~(x) > ~ we have
Definition 2. (X, 3) is strong fuzzy compact iff for each e ~ I \ { 1 } = [0,1) every a-shading family in 3 has a finite a-shading subfamily.
Definition 3. (X, 3) is ultra fuzzy compact iff(X, z(6)) is compact, where z(b) is the initial topology on X generated by subsets of X of the form f - 1 ( 2 , 1] for a l l f e 3 and 2 ~ [0, 1). The following implications hold and are not reversible [15]: ultra ~ strong ~ Lowen. All three compactnesses have the Tychonoff product property, and Lowen later proved [12] that a (Lowen-) compact Hausdorff fuzzy topological
Nice compactness was originally defined in terms of closed fuzzy sets, a-nets, and remote neighbourhoods [16]. Starting with/-fuzzy sets we have the following definitions.
Definitions. (1) In a fuzzy topological space (fls) (X, 6), a fuzzy point e = x~ has a remote neighbourhood P if P is a closed fuzzy set such that 2 ~ P(x) (e¢P). Remote neighbourhoods will be written as R-nhbds. Here a fuzzy point x~ has value ,~ at x and zero elsewhere, and 0 < 2 ~ < 1 . Also x ~ 6 P iff ~ P(x). (2) A fuzzy net S = {S(n),n ~ D} is a function S:D ~ P where D is a directed set with order relation >~ and P is the collection of all fuzzy points of X. If the crisp net of values of S(n) converges in (0, 1] to e e (0, 1], then S is called an a-net. And if all the values of S(n) are e,S is a constant a-net.
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m.W. Warner / Fuzzy Sets and Systems 80 (1996) 15 22
(3) A fuzzy point e is a cluster point of a fuzzy net S if for each R-nhbd P of e we have frequently (see e.g. [-1]) S(n)¢P. (4) Let (X, 6) be an fts and A a fuzzy set in X. Then A is N-compact if each s-net (a ~ (0, 1]) contained in A has in A at least one cluster point with value ~. And when A = 1 is N-compact, (X,3) is called an N-compacttis. Wang proves that a space is strong fuzzy compact iffevery constant s-net has a cluster point with value ~ e (0, 1], and is thus able to deduce immediately that N-compactness implies strong compactness. He also shows that ultra fuzzy compactness implies N-compactness, and produces a version of Lowen-compactness in terms of cluster points. In /-fuzzy, then, N-compactness is inserted in the table of implications between strong and ultra fuzzy compactness, i.e.
Denoting by fl(e), the greatest minimal set of e, let fl*(e) = fl(e)nM(L). Then if 7 6 fl*(e) we write 7<1e. Definition. A fuzzy set A of an fts (X, 6) is N-compact if for any e e M(L) and every family q~ of closed e-remote nbhds of A, there exists a finite subfamily 4o - ~b of v-remote nbhds of A for some 7
Theorem. In antis (X, 6), a fuzzy set A is N-compact
This is easily seen to be an extension of the /-version since for 1 7 <~ e iff 0 < 7 < e. Zhao also proves that it is exactly equivalent to N-compactness defined in terms of cluster points of fuzzy nets - the statement being word for word the same as Wang's for L = I, but with the definition of s-net adjusted in terms of the relation
iff each e-remote nhbd family of A has a finite subset which is an r-remote nhbd family of A for some r e (0, o0.
6. S-compactness
Ultra ~ nice ~ strong ~ Lowen. It was left to Zhao [20] to give a geometric characterisation of Wang's N-compactness. Defining a family cb of closed fuzzy sets to be an e-remote nhbd family of a fuzzy set A (e ~ (0, 1]) if each fuzzy point x~ E A has an R-nhbd in 4, he proved
Turning to L-fuzzy topological spaces, L a fuzzy lattice, Zhao needed to use minimal sets. Definition. A subset B of L is called a minimal set of an element e of L if VB = e and for each b e B and every subset ~ of L with V~o >~ e, there exists 7 ~ ~o such that 7 ~> b where VB is the height of B. Definition. An element c~ of L is called (union-) irreducible if whenever p V q/> e then p/> e or
q>~e,p,q~L. The set of all such non-zero irreducible elements of L is denoted by M(L). Putting M(X, L) = {x,: x ~ X , c ~ M ( L ) } , the elements of M ( X , L ) are called points.
The (geometric) formulations of N-compactness are based on closed sets as remote neighbourhoods and the relation ~ representing the negation of set membership. The dual of this relation was actually suggested by Warner [17] to describe fuzzy set membership itself. Identification of the points of the frame of L-fuzzy subsets of a set X with its prime elements led to the fuzzy points of X being given as pairs (x, 2), x ~ X, 2 prime in L. They are written xz and for a fuzzy set A, x~ ~ A iff A(x) ~ 2. Recall that an element p of L is called prime if p 4:1 and wherever x A y ~< p then x ~< p or y ~< p. Primes are thus dual to the (union)-irreducible elements of N-compactness. This concept led to the following definition of compactness [18].
M.W. Warner / Fuzzy Sets and Systems 80 (1996) 15-22
Definition. An L-fuzzy topological space (X,6) is S-compact if for every prime p of L and every collection (fj)j~j of open L-fuzzy sets with ( V ~ j fj)(x) ~; p for all x e X, there is a finite subset F of J with (Vj~vfj)(x) 4 p for all x e X.
19
Reference has been made by Wang [16] and others to strong Q-compactness. It is probably the same as S-compactness, but translation from the Chinese is awaited to verify this.
When L = I this is simply strong compactness since in I all elements apart from 1 are prime. Warner and McLean [18] prove it to be a good extension and also prove that an S-compact Hausdorff space is topological. Wang [16] has shown that a fuzzy space is strong fuzzy compact iff every constant e-net has a cluster point with height e. It follows that N-compactness implies Scompactness. In any case a duality between S-compactness and a weakened form of the geometric version of Ncompactness is easy to detect since a family of eremote neighbourhoods gi of X satisfies ( / ~ i gi)(x) 2~ e for all x ~ X. Thus we have an equivalent version of S-compactness.
7. L-fuzzy subsets
Definition. (X, 6) is S-compact ifffor every e ~ M(L)
Theorem. If g is a compact fuzzy subset, then for
and every collection (gj)j~ of closed fuzzy sets with (/~j~s gj)(x) ~ e for all x ~ X there is a finite subset F of J with (/~j~F gfl(x) ~ e for all x e X. The difficulties encountered by Gantner et al. [3] in the Tychonoff theorem now vanish. The permitted values of e are precisely the primes, and the substitution of 2~ for < disposes of the comparability requirement. S-compactness of fuzzy subsets follows naturally, either by adapting the above definition or by using constant a-nets. It is inherited by closed fuzzy subsets, continuous images and finite suprema. Finally, since an S-compact Hausdorff space is topological and both N- and S-compactness are good extensions, it follows that on a Hausdroff L-fuzzy space these two compactnesses are equivalent. The difference in general between the two theories is only in the lack of necessity for an S-compact set to have a maximum; this, as we noted above, is a doubtful advantage for niceness. We note that L-fuzzy ultra-compactness can be formally defined as for/-fuzzy, and again implies L-fuzzy N-compactness.
Definition. For an L-fts (X, z) a fuzzy subset g is compact iff for every prime p e L and family (f)g~j of open L-fuzzy sets with (Vi~jfi)(x) ~ P for all fuzzy points xp with xp¢g' (i.e. for all x with g(x) >~p'), there exists a finite subset F of J such that (Vi~F f~)(x) ~ p for all x with g(x) >1p'.
Note: For g = X this is S-compactness. For L = I this is strong Q-compactness. We give an example of a proof of a typical theorem on compact L-fuzzy subsets.
every closed fuzzy subset h, h c~g is compact. Proof. Using the notation of the definition, we have {j~}i~jw{h'} a family of open sets with Vk~a k(x)g£ p for all x with g(x)>1 p'. Since if h(x) >>,p' then (hc~g)(x) >~p', so (~/k~a k)(x) ~ p. And if h(x) ~ p' then h'(x) ~ p, so (~/k~p k)(x) ~ p. Thus, if h(x) >>,p', no new point is added to hc~g, while if h(x) ~ p' then point p may be 'let in' but will be covered by h' (the open set). Hence h (via h') compensates for any points it lets in. Then by the compactness of g, there is a finite subfamily Y = {fbf2, ... ,f,,h'} with (VkE?, k)(x) ~ p ~TtX with g(x) >~p'. Then (Vi~ll ...... I f~)(x) 4 p for all x with (h~g)(x)>~p'. In fact, if (hc~g)(x)>~ p' then g(x) >~p' so (Vk~r k)(x) ~ p. So there exists 3k • 7 with k(x) 4p. But h'(x)<~p, so ~/i~11...... If~(x) ~;p. []
Note: This example shows that we need a 'Q'relation x~¢g' rather than xp ~ g. A 'tidier' formulation would occur in terms of closed sets and remote neighbourhoods, leading to every constant a-net in g having a cluster point of height e.
M.W. Warner / Fuzzy Sets and Systems 80 (1996) 15-22
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8. Some weakening of fuzzy compactness It is straightforward to obtain fuzzy versions of the classical weaker forms of compactness. We illustrate this briefly with a definition and goodness proof for almost compactness [7].
Definition. An fts (X,z) is almost compact iff Vp ~ pr(L) (where pr(L) is the set of primes of L) and every family (f~)i~1 of open sets with (V~lJ~) (x) 4; p Vx E X, there exists a finite family F __ I with (Vi~F cl(fi))(x) 4; P Vx ~ X. Lemma 4.1. Let (X,z) be a topological space and A ~_ X. Considering the L-fts (X,(co(z))), with e)(z) the set of all l.s.c, functions from x to L, e ~ L and f ( x ) -:
if x ~ A,
we have {0 cl(f)(x) =
if x ~ c l ( A ) , otherwise.
Proof. Let g(x) = { ;
if x E el(A), otherwise.
We shall prove that c l ( f ) = g. Since, for every b6L, g-~({t ~ L;t >~b})
=
X
if b = O ,
cl(A)
if e~>b
and b ~ O ,
0
ife~b
and b # O
is closed in (X, r), g is closed in (X, a)(z)). We also have g ~>f Then, f~< c l ( f ) ~< cl(g) = g. Thus, we have cl(f)(x) = 0 for all x~cl(A) and cl(f)(x) = e for all x ~ A. From c l ( f ) ~ < g we obtain ( c l ( f ) ) - l ( { t E L ; t ~ e}) ~_ g - l ( { t ~ L ; t ~ e } ) = (cl(A))'. Hence cl(f)(x) = e for all x ~ cl(A) and cl(f)(x) = 0 for all x¢cl(A) and e l ( f ) = g. []
Corollary 4.2. Let (X, z) be a topological space and A ~_ X. Considering the L-fts (X,(o~(z))), we have
~(cI(A) cl(ZA) where ZA is the characteristic function of A. ---~
Proof. This follows immediately from Lemma 4.1. [] Theorem 4.3. Fuzzy almost compactness is a good extension of almost compactness. Proof. (i) Let (Ai)i~s be an open cover of (X, 0. Then ()~Ai)iEJ is a family of open L-fuzzy sets in (X,o)(z)) with (Vi~j Za,)(x)= 1 4; p for all x ~ X and for all p ~ pr(L). From the almost compactness of (X,e)(z)) there exists a finite subset F of J with (Vi~rcl(zA,))(x) = 1 4; p for all x ~ X and for all p s pr(L). Since from Corollary 4.2 cl(zA,) = ,~cl(Ai) we have (Vi~F Zcl¢A,))(X)= 1 for all x s X. Hence Ui~F cl(Ai) = X and (X,r) is almost compact. (ii) Conversely, assume (X,z) almost compact. We use basic open sets in (X,¢o(z)) for this proof, namely a family fl = (fi)i~s with f~(x) = {;i
ifx~Ui~z, otherwise.
We sometimes write such an J~ as f ia,U~. So, taking p ~ pr(L) and such a family/3 with (Viii fi)(x) 4; p for all x e X, it is enough to prove that there is a finite subfamily whose closures have this property. Now for each x ~ X there is an i ~ J with f~'V'(x) 4; p, i.e. with a i 4; p. Let ~ = {Ui: 3i ~ J with ai 4; p and f.~,v,s/3}. Then cg is a family of open sets of (X,3) which covers the space. Take a finite subfamily ~¢ of off,say {U1.... , U, } such that Ui~,~l...... } cl(Ui) = X. Using Lemma 4.1, c l ( f i ) ( x ) = { ; i ifx~cl(U~),otherwise, and the result follows, since (Vi~fl ...... I cl(f/))(x) <~p for all x e X. []
9. Local compactness To model local compactness in L-fuzzy topology we should require goodness of extension and the fuzzy version of the 'Tychonoff theorem' for local
M.W. Warner / Fuzzy Sets and Systems 80 (1996) 15 22
compactness defined in Section 2. However, local compactness has another interesting property, namely that it implies the continuity of the locale of open sets of the topology. It would be interesting to look for an L-fuzzy local compactness which also implies the continuity of the fuzzy topological locale. For a discussion of local compactness and continuous locales see Johnstone [5]. Fuzzy local compactness immediately presents difficulties, however. Defining an fts to be locally compact iff for all fuzzy points Xp every open neighbourhood of Xp contains a compact neigbourhood of Xp, we have failed to verify even the goodness of extension criterion. A partial solution has been suggested as follows [6]:
21
expect a unique fuzzy version to emerge. But the task is not complete, as is instanced by the difficulties encountered in modelling L-fuzzy local compactness. However, it would be reasonable to expect satisfactory progress in this area in the not too distant future. Some other problems such as that of RS-compactness, of which a good version has not yet emerged, will also probably be solved. It is then possible that activity in positive modelling in this field will decline in favour of a study of the pathologies mentioned in our first paragraph. The extent to which fuzzy topologies can diverge from the classical could be as interesting and instructive as the modelling of General Topology discussed in this paper.
Definition. A fuzzy set C is very compact if it is given by C(x) = {f0, x ~ D , otherwise, where { ~ L and D is fuzzy compact. Then, replacing 'compact neighbourhood' by 'very compact neighbourhood', in the above suggested definition of L-fuzzy locally compact, Kudri and Warner [6] have proved that (i) L-fuzzy local compactness is a good extension of local compactness. (ii) A stratified compact Hausdorff fts is locally compact. (iii) The Tychonoff theorem for local compactness has an L-fuzzy version. (iv) For a stratified locally compact /-fuzzy space (X, ~) the locale ~ is continuous. We have not yet succeeded in extending property (iv) to a general lattice L and the question of whether our version of fuzzy local compactness does indeed produce a continuous locale remains open.
10. Conclusion
It appears, then, that the modelling of compactness in fuzzy topology is well advanced, although it is clearly unrealistic and probably undesirable to
References [1] J.J. Chadwick, A generalised form of compactness in fuzzy topological spaces, J. Math. Anal. Appl. 162 (1991) 92-110. [2] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182-190. 1-3] T.E. Gantner, R.C. Steinlage and R.H. Warren, Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62 (1978) 547-562. [4] J.A. Goguen, The fuzzy Tychonofftheorem, J. Math. Anal. Appl. 43 (1973) 734 742. [5] P.T. Johnstone, Stone Spaces (Cambridge University Press, Cambridge, 1982). [6] S.R.T. Kudri and M.W. Warner, L-fuzzy local compactness, Fuzzy Sets and Systems 67 (1994) 337 345. [7] S.R.T. Kudri and M.W. Warner, Some good L-fuzzy compactness-related concepts and their properties I, Fuzzy Sets and Systems 76 (1995) 141-155. I-8] Y.M. Liu and M.K. Luo, Induced spaces and fuzzy Stone-Cech compactifications, Scientia Sinica (A) 30 (1987) 1034-1044. 1-9] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976) 621-633. [10] R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446 454. [11] R. Lowen, Compact Hausdorff fuzzy topological spaces are topological, Topology Appl. 12 (1981) 65-74. [12] R. Lowen, On the Existence of Natural Non-Topological Fuzzy Topological Spaces (Heldermann Verlag, Berlin, 1985). [13] P.M. Pu and Y.M. Liu, Fuzzy topology l, neighbourhood structure of a fuzzy point and Moore Smith convergence, J. Math. Anal. Appl. 76 (1980) 571 599.
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[14] S.E. Rodabaugh, The Hausdorff separation axiom for fuzzy topological spaces, Topology Appl. 11 (1980) 319-331. [15] A.P. Shostak, Two decades of fuzzy topology: basic ideas, notions and results, Russian Math. Surveys 44 (1989) 125-189. [16] G.J. Wang, A new fuzzy compactness defined by fuzzy nets, J. Math. Anal. Appl. 94 (1983) 1 23.
1-17] M.W. Warner, Frame-fuzzy points and membership, Fuzzy Sets and Systems 42 (1991) 335-344. [18] M.W. Warner and R.G. McLean, On compact Hausdorff L-fuzzy spaces, Fuzzy Sets and Systems 55 (1993) 103-110. [19] M.W. Warner, Fuzzy topology with respect to continuous lattices, Fuzzy Sets and Systems 35 (1990) 85 91. [20] D.S. Zhao, The N-compactness in L-fuzzy topological spaces, J. Math. Anal. Appl. 128 (1987) 64-79.