- Email: [email protected]
Some questions in fuzzy topology
FUZZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 105 (1999) 277-285
Some questions in fuzzy topology 1 T. K u b i a k a'*, M . A . d e P r a d a V i c e n t e b a Wydzial Matematyki i Informatyki, Uniwersytet ira. Adama Mickiewicza. Matejki 48/49, 60-769 Poznah, Poland b Departamento de Matem(tticas, Universidad del Pals Vasco-Euskal Herriko Unibertsitatea, Apdo. 644, 48080 Bilbao, Spain
Received July 1998
Abstract
The purpose of this paper is to discuss some basic questions related (mainly) to Hutton's L-fuzzy unit interval. Some of those questions are stated here for the first time, while some of them are long-standing. Related results are surveyed in some cases. © 1999 Elsevier Science B.V. All rights reserved. Keywords: L-line; Unit L-interval; Complete L-regularity; L-normality; L-Hausdorffness; Stratification; Fuzzification
schemes; Lowen functors; Topological modification; Topologically generated; Normal and separating families of closed L-sets; L-compactness; Extension of functions; L-zero sets; Fixed points
In this paper we discuss some questions in fuzzy topology which should have been answered many years ago yet still remain open. Many such questions could be raised, but emphasis will be placed on those that centre around Hutton's L-interval. This canonical L-topological space is of interest for various reasons: it satisfies Urysohn's Lemma, the Tietze-Urysohn Extension Theorem, the Tychonoff Embedding Theorem and some others (all of them for L a complete lattice with an order-reversing involution - the minimal assumption about L), it has an L-uniform structure (when L is completely distributive), it has an L-topological lattice structure (when L is meetcontinuous), and also has certain algebraic operations (when L is a complete chain). When L is the twopoint lattice, the results mentioned above rank among the fundamental theorems in general topology. This Corresponding author. 1Partially supported by U.P.V. 127.310-EA052/96.
*
is evidence that the L-interval is a right generalization of the real unit interval and as such is an appropriate subject of investigation• It should be noted that it is not our intention to give a historical account of development with respect to assumptions about the underlying lattice L. Thus, in most cases, we shall make reference only to results involving the weakest assumptions about L. Finally, one personal comment. Although there has recently been some progress towards giving fuzzy topology the shape of a reasonably well established theory, there are still more than enough papers which do not do the subject any good• Too many workers are interested, for example, in to what extent a "semi-strongly open fuzzy set" can be made less open. We believe that answers to the kind of questions we pose here are far more likely to stimulate interest in fuzzy topology. This is an appropriate place to recall that many interesting questions of [29] still remain unanswered.
0165-0114/99/$ - see front matter (~ 1999 Elsevier Science B.V. All rights reserved. PII: S0165-0114(98)00326-1
278
T. Kubiak, M.A. de Prada Vicente/Fuzzy Sets and Systems 105 (1999) 277-285
1. Preliminaries We recall a bit of standard L-topological terminology and notation [6,8,37]. L-topological spaces. Let (L, ') be a complete lattice with an order-reversing involution t. A subfamily J of Lx ( : the complete lattice of all maps of a set X into L, with pointwisely defined ~ and ') is an L-topology and (X, 9-) is an L-topological space, if 3- is closed under arbitrary sups and finite infs, both formed in L x. The space will often be denoted just by X. Members of ~'- are called open L-sets, k E L x is a closed L-set if k r is open. We denote by ~ and Int a the closure and the interior of a E Lx. If A c X, the set of all restrictions {u IA: u E Y') is the subspace L-topology on A. A family 5e C Lx generates J - if ~Y-= (']{~ D 6e: ~ is an L-topology on X}. We do not assume in general that an L-topology contains all the constant members of L x. If the latter is the case, the L-topology is called stratified. Also, we write X c for the set X endowed with the L-topology generated by the original L-topology of X and all the constant members of Lx. A function f :X ~ Y is continuous if u o f (the composition of f and u) is open in X whenever u is open in Y. The product I-[jEjXj of a family {X~ : j E J} of L-topological spaces is the usual Cartesian product with the L-topology generated by {u o roy: u is open inXj, j E J } where nj is thejth projection. L-real line and L-intervals. We recall the concept of the L-line and L-intervals [8,3]. For (L, ~) a complete lattice, let ~L be the set of all order-reversing members 2 E L R such that VA(fl~) = 1 and AA(R) = 0. For every t E ~, let 2+(0 = V2(t, c~) and A-(t) = A 2 ( - o o , t). Define 2 ,,~/~ iff2 + -- #+ (this is equivalent to 2 - =/~[20]). The quotient set R ( L ) = ~L/,-~ is called the Lreal line (L-line). It is partially ordered by [2] ~<[/t] iff 2+~
2+ ~<~ - [2o]). For every t E ~ , Lt,RtEL R(L) are defined by Rt[2]=2+(t) and Lt[2]=2-(t)'. The natural Ltopology on R(L) is generated by all the Rt and Lt (t E ~). The subspace I(L) = {[2] E 0~(L): 2 - ( 0 f = 2+(1) = 0} is called the (unit) L-interval and (0, 1)(L) = {[2] E I(L): 2+(0) ~ = 2 - ( 1 ) = 0} is the "open"
L-interval. We note that ~(L) and (0,1)(L) are homeomorphic. We write C(X) (resp., C*(X)) for the collection of all continuous functions from X to ~(L) (resp., to I(L)). More about lattices. Our reference for lattices is [4], to which we refer for lattice concepts not defined in what follows. We note that terminologically we do not go beyond what is used in [12] and [20]. The Lowen fimctors o~L and zL. Let L be a continuous lattice with its Scott topology a(L). Let ~,L=(L,a(L)). Given a topological space X, let ~os/)( be the L-topological space obtained by providing the set X with the L-topology consisting of all continuous functions from the space X to XL (this is indeed an L-topology, see [4, the proof of II-4.17], [12] or [23]). Also, for Y an L-topological space, let zxLY be the topological space with Y as the underlying set and with the smallest topology making all the open L-sets of Y continuous to SL. Then both co~L:TOP--~ T O P ( L ) (the category of all L-topological spaces) and zsL : T O P ( L ) - ~ T O P are functors under preservation of functions. I f L is hypercontinuous, or completely distributive, then SL= TL=(L,v(L)) where ~(L) is the upper topology on L. (Note that always o(L) C a(L) and that complete distributivity =~ hypercontinuity ~ continuity.) We refer to [20] for more properties of ~o~L and zsL. We write ogL and ~L if no specific reference to the topology of L is made. Members of col ( T O P ) are called topologically generated and zz)( is called the topological modification of the L-topological space X. Finally, i f X is a topological space, xX denotes the L-topological space (in fact, a 2-topological space) with X as a set and the L-topology (in fact, the 2topology) consisting of all the characteristic functions of open subsets of X.
2. Open questions Separation axioms and compactness. We first recall the two most important separation axioms given by Hutton [8,9]. Let X be an L-topological space with (L, ') a complete lattice. Then X is called:
T. Kubiak, M.A. de Prada VicentelFuzzy Sets and Systems 105 (1999) 277-285
279
L-normal if, given a closed k and an open u with k<<.u, there exists an open v such that k<<.v<<.~<~u; completely L-regular if for every open u there exist a family ~1 CL x and a family {fa: a E s~[} C C*(X) such that u = V ~ ¢ and a<<.LIxfa<~Rof~<<.u for all aE~'. Besides Rodabaugh's [31] four criteria for Lcompactness ( = a compactness-type invariant for TOP(L)), any L-compactness notion must necessarily be related to what already exists in fuzzy topology and is well established. Thus, any L-compactness notion must be related to the two separation axioms just defined as well as to I(L) which must necessarily be L-compact. These are natural requirements towards the coherence of fuzzy topology. Clearly enough, one has to have an L-Hausdorffness which behaves properly with respect to L-compactness (e.g. Lcompact + L-Hausdorff ~ L-normal). Although I(L) is Hausdorff-like in two incomparable senses (see [17, 7.7] and [20, 9.5]), is L-normal i f L is infinitely distributive (see [20, 9.12]), and satisfies a number of various compactness-type properties (see [1,3,17]), the following question is in our opinion the most fundamental open question in fuzzy topology.
(8) L-Hausdorffness is productive too. (9) (cf. Rodabaugh's criterion III) L-compactness and L-Hausdorffness imply the existence of an LHausdorff L-compactification functor under some reasonable definition of density of crisp subsets. (See [17, 3.15(3)] for the reason we have included condition (2). In fact, a property which is not preserved under weakening (strengthening) of an L-topology cannot be called L-compactness (LHausdorffness).)
Question 1. Let (L, ~) be a complete lattice with ILl > 2 (more conditions on L are allowed). 2 Are there L-topological invariants, call them L-compactness and L-Hausdorffness, such that: (1) Every L-compact and L-Hausdorff space is Lnormal and completely L-regular. (2) L-compactness is preserved under weakening of an L-topology and L-Hausdorffness is preserved under strengthening of an L-topology. (3) Any L-topological space which is both Lcompact and L-Hausdorff is minimal L-Hausdorff and maximal L-compact. (4) I(L) (or I(L) c) is both L-compact and LHausdorff. (5) (cf. Rodabaugh's criterion I) I f X is a compact topological space, then xX is L-compact. (6) I f X is a Hausdorff topological space, then xX is L-Hausdorff. (7) (cf. Rodabaugh's criterion II) L-compactness satisfies the Product Tychonoff Theorem.
Fact 1. Let (L,') be a completely distributive lattice (hypercontinuity suffices; cf [20, 8.1]) with
2 W e agree that it is L = 2 which should satisfy all the assumptions imposed on L, but not necessarily L = [0,1].
It is always of interest to know if a topological invariant P and its L-topological version L-P are related in the following way: (Q)
~oL(P) = L-P f3 o)t (TOP).
Clearly, (f~) is nothing else but the statement, in the traditional terminology of Lowen, that L-P is a "good extension" of P, i.e. for any topological X, X has P iff coL)( has L-P. We alert the reader that a positive answer to Question 1 under (Q) is impossible (and to its weaker version as well). The following observation has been made in [17, 7.16]:
ILl>2. There do not exist L-compactness and L-Hausdorffness satisfyin9 (f~) and (2)-(4) of Question 1. Lowen [25] was the first to produce L-compactness and L-Hausdorffness (with L = I = [0, 1]) satisfying (2), (3) and (f~) (and clearly (5) and (6)). He proved that his invariants satisfy the following: (~)
L-Compact n L-Hausdorff --- ~L (Compact-Hausdorff).
We believe that condition (+) must now be regarded as a criterion for how not to define L-compactness and L-Hausdorffness. Thus, producing further properties for the sole purpose of showing that they satisfy (+) is of little interest, because it merely provides equivalent formulations of the conjunction of the two original Lowen's notions. Also, unlike the referee, we do not think one might use Fact 1 as an argument that an adjustment is needed in the definition of Hutton's
280
T. Kubiak, M.A. de Prada Vicente/Fuzzy Sets and Systems 105 (1999) 277-285
L-interval. If such an adjustment is needed, then certainly it is for different reasons. In general, any property (or a collection of properties) L-P cannot be regarded as a right generalization of P if L-P does satisfy (11)
L-P = ooL(P).
Otherwise, why not save effort and just use (11) as the definition of L-P? For further remarks on the issue see [17, 7.17]. This brings us to the following: Question 2 [18]. Let (L,') be a continuous (or hypercontinuous or completely distributive) lattice with ILl >2. Let L-compactness and L-Hausdorffness be defined so as to satisfy (~2), (2) and (3). Prove or disprove the following statement: Any L-compact and L-Hausdorff space is an object of coL(Compact-
Hausdorff). In view of [17, 3.17(2)] an atfurnative answer to the following question implies an affirmative answer to Question 2 (see also [17, 7.17] for more specific questions related to the invariants of Zhao [39]). Question 3. Do (f~), (2) and (3) imply that Lcompactness is equivalent to ultracompactness? IX is ultracompact iff zz/tt" is compact]. We also note that L-compactness and LHausdorffness satisfying (1) cannot satisfy (4) of Question 1, for the following holds:
Fact 2 (Zhang and Liu [38]). If(L, ') is completely
Fact 3. Let (L,') be a frame. The following statements hold for J with IJI ~<~0: (i) I(L ) J is perfectly L-normal. (ii) IflL I <<,~o, then (I(L)C) J is perfectly L-normal. [Perfect L-normality = L-normality plus closed Lsets are G6.] Proof. (i) If L is a frame, then I(L) J is completely L-regular (this holds for any J ) by [20, 2.4 and 2.5], hence L-regular. 3 If IJI ~<~0, then I(L) J is secondcountable, hence perfectly L-normal by [20, 9.11 ]. (ii) Since (I(L) c)J = (I(L) J)c, the same argument applies using [20, 3.15] (i.e. the fact that i f X is completely L-regular, so is X c (with no restrictions on the cardinality of L)). [] Regarding the higher-order separation axioms of Hutton and Reilly [10] (i.e. L-regularity, complete Lregularity, L-normality and perfect L-normality), we note that ~(L) (hence (0, 1)(L) and I(L)) do satisfy all of them provided L is a flame (see [20, 9.12]). Question 5. Which higher-order Hutton-Reilly separation axioms do R(L) and I(L) satisfy when (L, ') is merely supposed to be an arbitrary complete lattice? Fact 3 suggests the following questions: Question 6. Let (L, ') be a flame (or a complete lattice provided Question 5 has a reasonably affirmative answer). Are R(L) c and and I(L) c L-normal, completely L-normal or perfectly L-normal without any restriction on the cardinality of L?
is not topologically generated (and neither is I(L ) ).
Question 7. Let X be L-normal (completely Lnormal, perfectly L-normal). When is X c L-normal (completely L-normal, perfectly L-normal)?
An affirmative answer to (1), (7) and (8) of Question 1 would yield an affirmative answer to the following simple and long-standing question.
Question 8. Is there an (L,') with ILl>2 and such that ~ ( L ) = R(L) c and I(L)=l(L)C?
distributive, ILl > 2, and 1 E L is a coprime, then I(L )~
Question 4 (Rodabaugh [29,30]). Let (L, ') be a complete lattice with ILl > 2 (as always, extra assumptions about L are allowed). Is an arbitrary product I(L) J, (I(L )c )J L-normal?
As already mentioned (within the proof of Fact 3), if (L, ') is a flame and X is completely L-regular, then so is X c. The same holds for L-regularity by [23, 3.3],
It is really shocking that we still do not know the answer to this question and that we have only:
3An L-topological space (X,T) is L-regular [10] if u= v{vE Y: ~
T. Kubiak, M.A. de Prada VicentelFuzzy Sets and Systems 105 (1999) 277-285
in both cases with no restriction on the cardinality of L. Regarding Question 6, we have the following:
Fact 4. If(L,') is a frame with IL[ ~
zL(L-P) = P,
in which C is, of course, the most important inclusion. It is known that (I) holds for P = regular, with 2~L a continuous lattice [23] and for P = completely regular with TL a hypercontinuous lattice [20].
281
L-topological space M[L] such that M[2] becomes M with its interval topology. Both (b) and (c) are generalizations of Hutton's construction of I(L) from I. Klein's construction involves the strictly less-than relation < of L as well as certain subsets of the complete subchain Lc (--- all those elements of L which are comparable with each element of L). This is not far removed from the assumption that L is itself a chain. For further arguments against using L c and its subsets see the introduction to [17]. Thus, Klein's construction begs for a new reasonable lattice setting, e.g. a continuous lattice setting, perhaps with an additional assumption that primes are order-generating (such assumptions were used in [12] for a generalization of Klein's considerations with respect to the hypergraph functor). Anyway, solving Question 12, which follows, will provide another step towards delivering some important aspects of fuzzy topology from the strictly less-than relation (cf. [12,17]).
Question 9. For which L does (I) hold when P = normal, completely normal, perfectly normal? Question 10. Is (I) true for P = completely regular and L a continuous lattice? Question 11. Is ZZLI(L) completely regular for XL a continuous lattice? Note that this is known for L a hypercontinuous lattice [20], as a consequence of ZTLI(L) being compact and Hausdorff. A positive answer to Question 11 would provide a positive answer to Question 10.
Fuzzification schemes. Besides the object part of Lowen's functor ~OL,there are known various fuzzification procedures [7,11,26,27,34,35]. This rather informal notion refers to a reasonable way of producing an L-topological space X(L) from a topological space X so that (at least) X(2) ----xX. Here, we restrict ourselves to the following three fuzzification schemes: (a) The object part of ~Oc: T O P ~ TOP(L). (b) Klein's [11] construction X---~X(L) where X is a connected topological space (see below for lattice requirements). (c) The object part of a functor of Zhang and Liu [38] for (L, ') a completely distributive lattice: to each completely distributive lattice M there corresponds an
Question 12 (actually, a task). Redefine Klein's fuzzification scheme so that it works in a natural lattice setting (e.g. for L a continuous lattice in which, only if necessary, primes are order-generating ( = each element of L is an inf of prime elements). Klein has only a construction for objects. This suggests the following question which - we believe should be addressed only after having a reasonable solution to Question 12 at hand. In actual fact, this suggestion applies to Question 14 and 15 too.
Question 13. Is the Klein fuzzification scheme an object part of a reasonable functor? It has been shown by Sostak [35] that this is the case of linearly ordered connected spaces and L = [0, 1]. Many other questions could be raised. We note the following: Question 14. (i) Does Klein fuzzification preserve products and subspaces? (ii) (Klein [11]) What relationships exist between L-topological properties o f X (L) and topological properties of X? (iii) How is Klein fuzzification related to tOc, ZL or to the hypergraph functor SL (see [12])?
282
T. Kubiak, M.A. de Prada VicentelFuzzy Sets and Systems 105 (1999) 277-285
Question 14(i) was asked, in part, by Rodabaugh [28] some time ago: Is R2(L) homeomorphic to the usual L-topological product R(L) × R(L)? We note that Zhang-Liu fuzzification preserves products [38, 2.6]. Thus, since I(L)~-I[L], hence IJ[L]'~I(L) J for any J. (Recall that I J is a completely distributive lattice with componentwise ordering.) The following is a special case of Question 14(i):
Question 15. Under L for which Question 12 has a reasonably positive solution (we hope it will not go beyond complete distributivity), what is the relation between 1J(L) (Klein) and I(L)J? There is another natural way of producing new spaces from I(L). Namely, if (L,/) is a complete lattice, then (I(L), °) is a complete lattice with the order-reversing involution ° defined by [2]°= [2 °] where 2"(t)=2(1 - t ) t, tE R (see [13-15]). Thus, following [13] and [14], we can inductively define: Il(L) =I(L),
The first question that arises is this:
Question 16. What relation holds between In(L), I(L) n and In(L)? The above discussion then brings us to the following:
Question 17. Is the process of constructing In(L) (n E ~ ) somehow limited, i.e.; is there a limit, say I~(L), such that I ( I ~ ( L ) ) = I ~ ( L ) ? How is I ~ ( L ) related to I ~ ( L ) (Klein) and to I ( L ) ~ ? Continuous L-real functions. Let X be an Ltopological space with (L,') a complete lattice. We say that a, b E L x are completely separated if there exists f E C*(X) such that a <~L~f <.Rof <~b'. An a E L x is called an L-zero-set if a = R~of for some f E C ( X ) . This is equivalent [22,2.3] to the statement that a = R~g for some 0 E C*(X) and a = L~h for some h E C*(X). Let ZL(X) denote the set of all L-zero-sets of X.
In(L) =I(In-l(L))
for n ~> 2. Consequently, (In(L), °) is a complete lattice with an order-reversing involution (which is clearly an In-l(L)-topological space). Thus, one can have In(L)-fuzzy sets, a special case of nth order fuzzy sets. Fact 5 [14]. Let (L,') be a complete lattice. The followin9 statements hold: (1) In(L)~-ln+l(2) (both in a lattice and an Ltopological sense). (2) I f L is meet-continuous, then In+l(L) is an In(L )-topological lattice. (3) There is a complete and order-reversin9 involution preservin9 embeddin 9 of ln(L ) into 1~+1(L ). Also note that the assignment II : (L, I) ~ (I(L), °) is functorial in the category of all complete lattices with an order-reversing involution, morphisms being complete lattice homomorphism preserving orderreversing involutions. Namely, if ~o: (L1, i) ~ (L2, i) is a morphism, then so is II(~o) : (l(L1), °) ~ (I(L2), °) defined by II(~p)([2]) = [¢p o 2] (see [14] for details).
In dealing with topological complete regularity, the central property (cf. [5]) asserts that in any topological space every two disjoint zero-sets are completely separated. Unfortunately, this important feature has not yet been shown to hold in TOP(L) with ILl >2. For X a topological space, the proof is easy and involves the ring structure of C(X, •). [For, if f - l { 0 } N g - l { 0 } = ~ , then h = f / ( f + g ) completely separates these disjoint zero-sets]. In an L-topological setting, C(X) is merely a poset (or a lattice ifL is meet-continuous [20]), and the algebraic operations on ~(L) when L is a complete chain [32] are not strong enough. We thus have another important question (next to Question 1), already posed in [20,22].
Question 18. For which complete lattices (L, ') with ILl>2, are every two a, bEZL(X) with a<~b' completely separated? An affirmative answer to Question 18 would have numerous applications, particularly to the theory of completely L-regular spaces; one of these (not necessarily the most important) will be discussed in the next subsection.
T. Kubiak, M.A. de Prada VicentelFuzzy Sets and Systems 105 (1999) 277-285 We have used the symbol C*(X) to denote the family of all continuous functions from X to I(L), whereas in general topology it stands for the collection of all bounded continuous functions from X to ~. The issue is delicate, because a constant function in C(X) need not be "bounded" in the sense that it takes values in some [a, b](L) = {[2] E ~(L): 2 - ( a ) ' = 2+(b) = 0} (a
We do not know how much of what is known about ~(L) goes over to ~*(L); note that an attempt to define bounded ~(L)-valued functions can be found in [24].
Question 19. When does a continuous function f :X --~ R(L) take values in some [a,b](L)?
Question 20. For which continuous lattices (L,/) does the following hold: I f X is an arbitrary L-topological space and f : X - - + I ( L ) is continuous, there exists a continuous function 9 : ZZLX~ [0, 1] such that L~f 4 9 <~Rof? (This holds true for L = [0, 1] [19].) The following is one of the long-standing questions of Rodabaugh [29,30] conceming the ~(L)-valued case of the Tietze-Urysohn Extension Theorem for L-normal spaces. Question 21 (Rodabaugh [30]). For which L with ILl > 2 does every continuous f :A ~ •(L), where A is a subset of an L-normal space X with 1A being closed in X, have a continuous extension to the whole of X? We recall that this holds true for (L,') a meetcontinuous lattice provided X is perfectly L-normal (see [15] and cf. [20, the proof of 5.5]). In fact, from the discussion within the proof of Proposition 5.5 of [20], it suffices to assume that la C ZL(X) with X an L-normal space. One related question is this:
Question 22. Let X be L-normal with ILL>2. Do we have: a c L x is an L-zero-set ¢*, a is Gr? (Clearly, the point is to show ~ . ) A subset A of an L-topological space X is said to be C*-embedded in X if every function in C*(A) can
283
be extended to a function in C*(X). Similarly, A is called C-embedded if every function in C(A) can be extended to a function in C(X).
Question 23. Let ILl >2. When and only when is a C*-embedded subset also C-embedded ? (cf. [5, Theorem 1.18].) Let us go back to the case of L-normal spaces. Unlike the topological case, the Tietze-Urysohn Extension Theorem [16] does not characterize L-normality (just because an L-topological space need not have non-trivial crisp closed subsets). In fact, there are non L-normal spaces in which every crisp closed subset is C*-embedded [14].
Question 24 [14]. Under what conditions is Lnormality equivalent to C*-embedding of crisp closed subsets? We finish this subsection with two questions concerning fixed points. There are a number of contractive type fixed point theorems for various "fuzzy metric spaces". However, most of the latter carry a usual topology, unlike, e.g., the fuzzy metric spaces o f Erceg [2]. Question 25. Formulate and prove a Banach Contraction Theorem for Erceg L-fuzzy metric spaces. (This will require a definition of completeness for these spaces.) Question 26. Does every continuous function from I(L) J into I(L) J have a fixed point? 4 This is the case when IJl~<~t0 and (L,') is completely distributive with a countable base [21 ].
Completely L-regular spaces. Here we restate a few questions posed in [20,22].
Question 27. Let (L, ') be complete. Let ~ C L x generate the L-topology of X. Is X completely L-regular if each u E 6 ¢ has the separation property indicated in the definition of complete L-regularity? (This is the case when L is a frame.) 4 Added in proof: A positive answer to this question is given in: T. Kubiak and D. Zhang, On the L-fuzzyBrouwerfixedpoint theorem (this issue).
284
T. Kubiak, M.A. de Prada Vicente/Fuzzy Sets and Systems 105 (1999) 277-285
The questions which follow concern an intemal characterization o f complete L-regularity in terms o f normal and separating families o f closed L-sets. A family J d o f closed L-sets o f an L-topological space is called s e p a r a t i n g if for each open L-set u there exist two families {aj: j E J } , {bj: j E J } C OF such that u = V j e j aj and aj <~b~ <<,u for all j E J . It is called n o r m a l i f for every a, b E OF with a < b' there exist c, d E OF such that a <~c' <<.d <<.b'. W e note that with (L, ' ) a complete lattice, any X which admits a normal and separating family o f closed L-sets must necessarily be completely L-regular [22]. Question 28. Does every completely L-regular space have a normal and separating family o f closed L-sets? A positive answer to Question 18 yields a positive answer to the following which is actually equivalent to Question 18 (see [22,5.6]). Question 29. Is z L ( x ) a normal family i f (Note that it is always a separating family.)
[LI > 2 ?
Question 30. Does every completely L-regular space have a normal and separating family o f closed L-sets for some L # 2? For L-topological spaces in which open L-sets separate points (L-T0-spaces), using the Tychonoff embedding theorem, a positive answer to the following question yields a positive answer to Question 31 in the class o f L-T0-spaces with L a frame. Question 31. Is the property o f having a normal and separating family o f closed L-sets hereditary and productive if ]LI > 27 I f it is hereditary and (L, ' ) is a Boolean algebra, a positive answer to the following question will also yield a positive answer to Question 31 in the class o f L-T0-spaces. Question 32. Let ( L , ' ) be a complete Boolean algebra. Is there a bijection from the L-topology o f I ( L ) J onto the topology o f [0, 1] J which preserves arbitrary sups and finite infs? ( W e note that this is the case when IJI = 1 [8].)
References [1] G. Artico, R. Moresco, ~*-compactness of the fuzzy unit interval, Fuzzy Sets and Systems 69 0988) 243-249. [2] M.A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69 (1979) 205-230. [3] T.E. Gantner, R.C. Steinlage, R.H. Warren, Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62 (1978) 547-562. [4] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, A Compendium of Continuous Lattices, Springer, Berlin, 1980. [5] L. Gillman, M. Jerison, Rings of Continuous Functions, Springer, New York, 1976. [6] J.A. Goguen, The fuzzy Tychonoff theorem, J. Math. Anal. Appl. 43 (1973) 734-742. [7] U. Hthle, A.P. Sostak, Axiomatic foundation of fixed-basis fuzzy topology, Handbook of Fuzzy Set Theory, Kluwer Academic Publishers, Dordrecht, to appear. [8] B. Hutton, Normality in fuzzy topological spaces, J. Math. Anal. Appl. 50 (1975) 74-79. [9] B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal. Appl. 58 (1977) 559-571. [10] B. HuRon, I.L. Reilly, Separation axioms in fuzzy topological spaces, Fuzzy Sets and Systems 3 (1980) 93-104. [11] A.J. Klein, Generalizing the L-fuzzy unit interval, Fuzzy Sets and Systems 12 (1984) 271-279. [12] W. Kotzt, T. Kubiak, Fuzzy topologies of Scott continuous functions and their relation to the hypergraph functor, Quaest. Math. 15 (1992) 175-187. [13] T. Kubiak, A class of second order fuzzy sets, Proc. 10th National School for Scientists, Appl. of Math. in Techn., Vama, 1984. [14] T. Kubiak, On fuzzy topologies, Ph.D. Thesis, A. Mickiewicz University, Poznafi, 1985. [15] T. Kubiak, Extending continuous L-real functions, Math. Japon. 31 (1986) 875-887. [16] T. Kubiak, L-fuzzy normal spaces and Tietze extension theorem, J. Math. Anal. Appl. 125 (1987) 141-153. [17] T. Kubiak, The topological modification of the L-fuzzy unit interval, in Ref. [33], pp. 276-305. [18] T. Kubiak, Fourth open question, in Ref. [33], p. 350. [19] T. Kubiak, Are higher fuzzy separation axioms good extensions?, Quaest. Math. 16 (1993) 443-451. [20] T. Kubiak, On L-Tychonoff spaces, Fuzzy Sets and Systems 73 (1995) 25-53. [21] T. Kubiak, The fuzzy Brouwer fixed-point theorem, J. Math. Anal. Appl. 223 (1998) 62-66. [22] T. Kubiak, M.A. de Prada Vicente, On internal characterizations of completely L-regular spaces, J. Math. Anal. Appl. 216 (1997) 581-592. An extended abstract in: IFSA'97 Prague, Proc., Vol. 2. Academia, Prague, 1997, pp. 54-57. [23] T. Kubiak, M.A. de Prada Vicente, L-regular topological spaces and their topological modifications, Int. J. Math. & Math. Sci., to appear. [24] Liu Qijin, Fuzzy pseudocompactness, Fuzzy Math. 7 (1987) 69 -76.
72 Kubiak, M.A. de Prada VicentelFuzzy Sets and Systems 105 (1999) 277-285 [25] R. Lowen, Compact Hausdorff fuzzy topological spaces are topological, Topology Appl. 12 (1981) 65-74. [26] R. Lowen, Hyperspaces of fuzzy sets, Fuzzy Sets and Systems 9 (1983) 287-311. [27] R. Lowen, On the existence of natural fuzzy topologies on spaces of probability measures, Math. Nachr. 115 (1984) 33 -57. [28] S.E. Rodabaugh, Open questions: The L-fuzzy plane, in: E.P. Klement (Ed.), Proc. 3rd Int. Seminar on Fuzzy Set Theory, Johannes Kepler Universi~t, 7-12 September 1981, Linz, 1981, pp. 243-245. [29] S.E. Rodabaugh, The L-fuzzy real line and its subspaces, in: R.R. Yager (Ed.), Fuzzy Sets and Possibility Theory: Recent Developments, Pergamon Press, Oxford, 1982, pp. 402-418. [30] S.E. Rodabaugh, Separation axioms and the fuzzy real lines, Fuzzy Sets and Systems 11 (1983) 163-183. [31] S.E. Rodabaugh, Point-set lattice theoretic topology, Fuzzy Sets and Systems 40 (1991) 297-345. [32] S.E. Rodabaugh, Complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real line, Fuzzy Sets and Systems 15 (1985) 285-310.
285
[33] S.E. Rodabaugh, E.P. Klement, U. H6hle (Eds.), Applications of Category Theory to Fuzzy Subsets, Kluwer, Dordrecht, 1992. [34] A.P. Sostak, A fuzzy modification of a linearly ordered space, Colloq~ Math. Soc. Janos Bolyai 41, Topology and Applications, Eger (Hungary) (1983) pp. 581-604. [35] A. Sostak, A fuzzy modification of the category of linearly ordered spaces, Comment. Math. Univ. Carolinae 26 (1985) 421-442. [36] A. Sostak, Fuzzy topologies on spaces of probability measures, Topological Spaces and Mappings, Riga, 1989, pp. 421-442. [37] R.H. Warren, Neighborhoods, bases and continuity in fuzzy topological spaces, Rocky Motmtain J. Math. 8 (1978) 459-470. [38] Zhang De-xue, Liu Ying-ming, L-fuzzy modification of completely distributive lattices, Math. Nachr. 168 (1994) 79 -95. [39] Zhao Dongsheng, The N-compactness in L-fuzzy topological spaces, J. Math. Anal. Appl. 128 (1987) 64-79.
sets and systems ELSEVIER
Fuzzy Sets and Systems 105 (1999) 277-285
Some questions in fuzzy topology 1 T. K u b i a k a'*, M . A . d e P r a d a V i c e n t e b a Wydzial Matematyki i Informatyki, Uniwersytet ira. Adama Mickiewicza. Matejki 48/49, 60-769 Poznah, Poland b Departamento de Matem(tticas, Universidad del Pals Vasco-Euskal Herriko Unibertsitatea, Apdo. 644, 48080 Bilbao, Spain
Received July 1998
Abstract
The purpose of this paper is to discuss some basic questions related (mainly) to Hutton's L-fuzzy unit interval. Some of those questions are stated here for the first time, while some of them are long-standing. Related results are surveyed in some cases. © 1999 Elsevier Science B.V. All rights reserved. Keywords: L-line; Unit L-interval; Complete L-regularity; L-normality; L-Hausdorffness; Stratification; Fuzzification
schemes; Lowen functors; Topological modification; Topologically generated; Normal and separating families of closed L-sets; L-compactness; Extension of functions; L-zero sets; Fixed points
In this paper we discuss some questions in fuzzy topology which should have been answered many years ago yet still remain open. Many such questions could be raised, but emphasis will be placed on those that centre around Hutton's L-interval. This canonical L-topological space is of interest for various reasons: it satisfies Urysohn's Lemma, the Tietze-Urysohn Extension Theorem, the Tychonoff Embedding Theorem and some others (all of them for L a complete lattice with an order-reversing involution - the minimal assumption about L), it has an L-uniform structure (when L is completely distributive), it has an L-topological lattice structure (when L is meetcontinuous), and also has certain algebraic operations (when L is a complete chain). When L is the twopoint lattice, the results mentioned above rank among the fundamental theorems in general topology. This Corresponding author. 1Partially supported by U.P.V. 127.310-EA052/96.
*
is evidence that the L-interval is a right generalization of the real unit interval and as such is an appropriate subject of investigation• It should be noted that it is not our intention to give a historical account of development with respect to assumptions about the underlying lattice L. Thus, in most cases, we shall make reference only to results involving the weakest assumptions about L. Finally, one personal comment. Although there has recently been some progress towards giving fuzzy topology the shape of a reasonably well established theory, there are still more than enough papers which do not do the subject any good• Too many workers are interested, for example, in to what extent a "semi-strongly open fuzzy set" can be made less open. We believe that answers to the kind of questions we pose here are far more likely to stimulate interest in fuzzy topology. This is an appropriate place to recall that many interesting questions of [29] still remain unanswered.
0165-0114/99/$ - see front matter (~ 1999 Elsevier Science B.V. All rights reserved. PII: S0165-0114(98)00326-1
278
T. Kubiak, M.A. de Prada Vicente/Fuzzy Sets and Systems 105 (1999) 277-285
1. Preliminaries We recall a bit of standard L-topological terminology and notation [6,8,37]. L-topological spaces. Let (L, ') be a complete lattice with an order-reversing involution t. A subfamily J of Lx ( : the complete lattice of all maps of a set X into L, with pointwisely defined ~ and ') is an L-topology and (X, 9-) is an L-topological space, if 3- is closed under arbitrary sups and finite infs, both formed in L x. The space will often be denoted just by X. Members of ~'- are called open L-sets, k E L x is a closed L-set if k r is open. We denote by ~ and Int a the closure and the interior of a E Lx. If A c X, the set of all restrictions {u IA: u E Y') is the subspace L-topology on A. A family 5e C Lx generates J - if ~Y-= (']{~ D 6e: ~ is an L-topology on X}. We do not assume in general that an L-topology contains all the constant members of L x. If the latter is the case, the L-topology is called stratified. Also, we write X c for the set X endowed with the L-topology generated by the original L-topology of X and all the constant members of Lx. A function f :X ~ Y is continuous if u o f (the composition of f and u) is open in X whenever u is open in Y. The product I-[jEjXj of a family {X~ : j E J} of L-topological spaces is the usual Cartesian product with the L-topology generated by {u o roy: u is open inXj, j E J } where nj is thejth projection. L-real line and L-intervals. We recall the concept of the L-line and L-intervals [8,3]. For (L, ~) a complete lattice, let ~L be the set of all order-reversing members 2 E L R such that VA(fl~) = 1 and AA(R) = 0. For every t E ~, let 2+(0 = V2(t, c~) and A-(t) = A 2 ( - o o , t). Define 2 ,,~/~ iff2 + -- #+ (this is equivalent to 2 - =/~[20]). The quotient set R ( L ) = ~L/,-~ is called the Lreal line (L-line). It is partially ordered by [2] ~<[/t] iff 2+~
2+ ~<~ - [2o]). For every t E ~ , Lt,RtEL R(L) are defined by Rt[2]=2+(t) and Lt[2]=2-(t)'. The natural Ltopology on R(L) is generated by all the Rt and Lt (t E ~). The subspace I(L) = {[2] E 0~(L): 2 - ( 0 f = 2+(1) = 0} is called the (unit) L-interval and (0, 1)(L) = {[2] E I(L): 2+(0) ~ = 2 - ( 1 ) = 0} is the "open"
L-interval. We note that ~(L) and (0,1)(L) are homeomorphic. We write C(X) (resp., C*(X)) for the collection of all continuous functions from X to ~(L) (resp., to I(L)). More about lattices. Our reference for lattices is [4], to which we refer for lattice concepts not defined in what follows. We note that terminologically we do not go beyond what is used in [12] and [20]. The Lowen fimctors o~L and zL. Let L be a continuous lattice with its Scott topology a(L). Let ~,L=(L,a(L)). Given a topological space X, let ~os/)( be the L-topological space obtained by providing the set X with the L-topology consisting of all continuous functions from the space X to XL (this is indeed an L-topology, see [4, the proof of II-4.17], [12] or [23]). Also, for Y an L-topological space, let zxLY be the topological space with Y as the underlying set and with the smallest topology making all the open L-sets of Y continuous to SL. Then both co~L:TOP--~ T O P ( L ) (the category of all L-topological spaces) and zsL : T O P ( L ) - ~ T O P are functors under preservation of functions. I f L is hypercontinuous, or completely distributive, then SL= TL=(L,v(L)) where ~(L) is the upper topology on L. (Note that always o(L) C a(L) and that complete distributivity =~ hypercontinuity ~ continuity.) We refer to [20] for more properties of ~o~L and zsL. We write ogL and ~L if no specific reference to the topology of L is made. Members of col ( T O P ) are called topologically generated and zz)( is called the topological modification of the L-topological space X. Finally, i f X is a topological space, xX denotes the L-topological space (in fact, a 2-topological space) with X as a set and the L-topology (in fact, the 2topology) consisting of all the characteristic functions of open subsets of X.
2. Open questions Separation axioms and compactness. We first recall the two most important separation axioms given by Hutton [8,9]. Let X be an L-topological space with (L, ') a complete lattice. Then X is called:
T. Kubiak, M.A. de Prada VicentelFuzzy Sets and Systems 105 (1999) 277-285
279
L-normal if, given a closed k and an open u with k<<.u, there exists an open v such that k<<.v<<.~<~u; completely L-regular if for every open u there exist a family ~1 CL x and a family {fa: a E s~[} C C*(X) such that u = V ~ ¢ and a<<.LIxfa<~Rof~<<.u for all aE~'. Besides Rodabaugh's [31] four criteria for Lcompactness ( = a compactness-type invariant for TOP(L)), any L-compactness notion must necessarily be related to what already exists in fuzzy topology and is well established. Thus, any L-compactness notion must be related to the two separation axioms just defined as well as to I(L) which must necessarily be L-compact. These are natural requirements towards the coherence of fuzzy topology. Clearly enough, one has to have an L-Hausdorffness which behaves properly with respect to L-compactness (e.g. Lcompact + L-Hausdorff ~ L-normal). Although I(L) is Hausdorff-like in two incomparable senses (see [17, 7.7] and [20, 9.5]), is L-normal i f L is infinitely distributive (see [20, 9.12]), and satisfies a number of various compactness-type properties (see [1,3,17]), the following question is in our opinion the most fundamental open question in fuzzy topology.
(8) L-Hausdorffness is productive too. (9) (cf. Rodabaugh's criterion III) L-compactness and L-Hausdorffness imply the existence of an LHausdorff L-compactification functor under some reasonable definition of density of crisp subsets. (See [17, 3.15(3)] for the reason we have included condition (2). In fact, a property which is not preserved under weakening (strengthening) of an L-topology cannot be called L-compactness (LHausdorffness).)
Question 1. Let (L, ~) be a complete lattice with ILl > 2 (more conditions on L are allowed). 2 Are there L-topological invariants, call them L-compactness and L-Hausdorffness, such that: (1) Every L-compact and L-Hausdorff space is Lnormal and completely L-regular. (2) L-compactness is preserved under weakening of an L-topology and L-Hausdorffness is preserved under strengthening of an L-topology. (3) Any L-topological space which is both Lcompact and L-Hausdorff is minimal L-Hausdorff and maximal L-compact. (4) I(L) (or I(L) c) is both L-compact and LHausdorff. (5) (cf. Rodabaugh's criterion I) I f X is a compact topological space, then xX is L-compact. (6) I f X is a Hausdorff topological space, then xX is L-Hausdorff. (7) (cf. Rodabaugh's criterion II) L-compactness satisfies the Product Tychonoff Theorem.
Fact 1. Let (L,') be a completely distributive lattice (hypercontinuity suffices; cf [20, 8.1]) with
2 W e agree that it is L = 2 which should satisfy all the assumptions imposed on L, but not necessarily L = [0,1].
It is always of interest to know if a topological invariant P and its L-topological version L-P are related in the following way: (Q)
~oL(P) = L-P f3 o)t (TOP).
Clearly, (f~) is nothing else but the statement, in the traditional terminology of Lowen, that L-P is a "good extension" of P, i.e. for any topological X, X has P iff coL)( has L-P. We alert the reader that a positive answer to Question 1 under (Q) is impossible (and to its weaker version as well). The following observation has been made in [17, 7.16]:
ILl>2. There do not exist L-compactness and L-Hausdorffness satisfyin9 (f~) and (2)-(4) of Question 1. Lowen [25] was the first to produce L-compactness and L-Hausdorffness (with L = I = [0, 1]) satisfying (2), (3) and (f~) (and clearly (5) and (6)). He proved that his invariants satisfy the following: (~)
L-Compact n L-Hausdorff --- ~L (Compact-Hausdorff).
We believe that condition (+) must now be regarded as a criterion for how not to define L-compactness and L-Hausdorffness. Thus, producing further properties for the sole purpose of showing that they satisfy (+) is of little interest, because it merely provides equivalent formulations of the conjunction of the two original Lowen's notions. Also, unlike the referee, we do not think one might use Fact 1 as an argument that an adjustment is needed in the definition of Hutton's
280
T. Kubiak, M.A. de Prada Vicente/Fuzzy Sets and Systems 105 (1999) 277-285
L-interval. If such an adjustment is needed, then certainly it is for different reasons. In general, any property (or a collection of properties) L-P cannot be regarded as a right generalization of P if L-P does satisfy (11)
L-P = ooL(P).
Otherwise, why not save effort and just use (11) as the definition of L-P? For further remarks on the issue see [17, 7.17]. This brings us to the following: Question 2 [18]. Let (L,') be a continuous (or hypercontinuous or completely distributive) lattice with ILl >2. Let L-compactness and L-Hausdorffness be defined so as to satisfy (~2), (2) and (3). Prove or disprove the following statement: Any L-compact and L-Hausdorff space is an object of coL(Compact-
Hausdorff). In view of [17, 3.17(2)] an atfurnative answer to the following question implies an affirmative answer to Question 2 (see also [17, 7.17] for more specific questions related to the invariants of Zhao [39]). Question 3. Do (f~), (2) and (3) imply that Lcompactness is equivalent to ultracompactness? IX is ultracompact iff zz/tt" is compact]. We also note that L-compactness and LHausdorffness satisfying (1) cannot satisfy (4) of Question 1, for the following holds:
Fact 2 (Zhang and Liu [38]). If(L, ') is completely
Fact 3. Let (L,') be a frame. The following statements hold for J with IJI ~<~0: (i) I(L ) J is perfectly L-normal. (ii) IflL I <<,~o, then (I(L)C) J is perfectly L-normal. [Perfect L-normality = L-normality plus closed Lsets are G6.] Proof. (i) If L is a frame, then I(L) J is completely L-regular (this holds for any J ) by [20, 2.4 and 2.5], hence L-regular. 3 If IJI ~<~0, then I(L) J is secondcountable, hence perfectly L-normal by [20, 9.11 ]. (ii) Since (I(L) c)J = (I(L) J)c, the same argument applies using [20, 3.15] (i.e. the fact that i f X is completely L-regular, so is X c (with no restrictions on the cardinality of L)). [] Regarding the higher-order separation axioms of Hutton and Reilly [10] (i.e. L-regularity, complete Lregularity, L-normality and perfect L-normality), we note that ~(L) (hence (0, 1)(L) and I(L)) do satisfy all of them provided L is a flame (see [20, 9.12]). Question 5. Which higher-order Hutton-Reilly separation axioms do R(L) and I(L) satisfy when (L, ') is merely supposed to be an arbitrary complete lattice? Fact 3 suggests the following questions: Question 6. Let (L, ') be a flame (or a complete lattice provided Question 5 has a reasonably affirmative answer). Are R(L) c and and I(L) c L-normal, completely L-normal or perfectly L-normal without any restriction on the cardinality of L?
is not topologically generated (and neither is I(L ) ).
Question 7. Let X be L-normal (completely Lnormal, perfectly L-normal). When is X c L-normal (completely L-normal, perfectly L-normal)?
An affirmative answer to (1), (7) and (8) of Question 1 would yield an affirmative answer to the following simple and long-standing question.
Question 8. Is there an (L,') with ILl>2 and such that ~ ( L ) = R(L) c and I(L)=l(L)C?
distributive, ILl > 2, and 1 E L is a coprime, then I(L )~
Question 4 (Rodabaugh [29,30]). Let (L, ') be a complete lattice with ILl > 2 (as always, extra assumptions about L are allowed). Is an arbitrary product I(L) J, (I(L )c )J L-normal?
As already mentioned (within the proof of Fact 3), if (L, ') is a flame and X is completely L-regular, then so is X c. The same holds for L-regularity by [23, 3.3],
It is really shocking that we still do not know the answer to this question and that we have only:
3An L-topological space (X,T) is L-regular [10] if u= v{vE Y: ~
T. Kubiak, M.A. de Prada VicentelFuzzy Sets and Systems 105 (1999) 277-285
in both cases with no restriction on the cardinality of L. Regarding Question 6, we have the following:
Fact 4. If(L,') is a frame with IL[ ~
zL(L-P) = P,
in which C is, of course, the most important inclusion. It is known that (I) holds for P = regular, with 2~L a continuous lattice [23] and for P = completely regular with TL a hypercontinuous lattice [20].
281
L-topological space M[L] such that M[2] becomes M with its interval topology. Both (b) and (c) are generalizations of Hutton's construction of I(L) from I. Klein's construction involves the strictly less-than relation < of L as well as certain subsets of the complete subchain Lc (--- all those elements of L which are comparable with each element of L). This is not far removed from the assumption that L is itself a chain. For further arguments against using L c and its subsets see the introduction to [17]. Thus, Klein's construction begs for a new reasonable lattice setting, e.g. a continuous lattice setting, perhaps with an additional assumption that primes are order-generating (such assumptions were used in [12] for a generalization of Klein's considerations with respect to the hypergraph functor). Anyway, solving Question 12, which follows, will provide another step towards delivering some important aspects of fuzzy topology from the strictly less-than relation (cf. [12,17]).
Question 9. For which L does (I) hold when P = normal, completely normal, perfectly normal? Question 10. Is (I) true for P = completely regular and L a continuous lattice? Question 11. Is ZZLI(L) completely regular for XL a continuous lattice? Note that this is known for L a hypercontinuous lattice [20], as a consequence of ZTLI(L) being compact and Hausdorff. A positive answer to Question 11 would provide a positive answer to Question 10.
Fuzzification schemes. Besides the object part of Lowen's functor ~OL,there are known various fuzzification procedures [7,11,26,27,34,35]. This rather informal notion refers to a reasonable way of producing an L-topological space X(L) from a topological space X so that (at least) X(2) ----xX. Here, we restrict ourselves to the following three fuzzification schemes: (a) The object part of ~Oc: T O P ~ TOP(L). (b) Klein's [11] construction X---~X(L) where X is a connected topological space (see below for lattice requirements). (c) The object part of a functor of Zhang and Liu [38] for (L, ') a completely distributive lattice: to each completely distributive lattice M there corresponds an
Question 12 (actually, a task). Redefine Klein's fuzzification scheme so that it works in a natural lattice setting (e.g. for L a continuous lattice in which, only if necessary, primes are order-generating ( = each element of L is an inf of prime elements). Klein has only a construction for objects. This suggests the following question which - we believe should be addressed only after having a reasonable solution to Question 12 at hand. In actual fact, this suggestion applies to Question 14 and 15 too.
Question 13. Is the Klein fuzzification scheme an object part of a reasonable functor? It has been shown by Sostak [35] that this is the case of linearly ordered connected spaces and L = [0, 1]. Many other questions could be raised. We note the following: Question 14. (i) Does Klein fuzzification preserve products and subspaces? (ii) (Klein [11]) What relationships exist between L-topological properties o f X (L) and topological properties of X? (iii) How is Klein fuzzification related to tOc, ZL or to the hypergraph functor SL (see [12])?
282
T. Kubiak, M.A. de Prada VicentelFuzzy Sets and Systems 105 (1999) 277-285
Question 14(i) was asked, in part, by Rodabaugh [28] some time ago: Is R2(L) homeomorphic to the usual L-topological product R(L) × R(L)? We note that Zhang-Liu fuzzification preserves products [38, 2.6]. Thus, since I(L)~-I[L], hence IJ[L]'~I(L) J for any J. (Recall that I J is a completely distributive lattice with componentwise ordering.) The following is a special case of Question 14(i):
Question 15. Under L for which Question 12 has a reasonably positive solution (we hope it will not go beyond complete distributivity), what is the relation between 1J(L) (Klein) and I(L)J? There is another natural way of producing new spaces from I(L). Namely, if (L,/) is a complete lattice, then (I(L), °) is a complete lattice with the order-reversing involution ° defined by [2]°= [2 °] where 2"(t)=2(1 - t ) t, tE R (see [13-15]). Thus, following [13] and [14], we can inductively define: Il(L) =I(L),
The first question that arises is this:
Question 16. What relation holds between In(L), I(L) n and In(L)? The above discussion then brings us to the following:
Question 17. Is the process of constructing In(L) (n E ~ ) somehow limited, i.e.; is there a limit, say I~(L), such that I ( I ~ ( L ) ) = I ~ ( L ) ? How is I ~ ( L ) related to I ~ ( L ) (Klein) and to I ( L ) ~ ? Continuous L-real functions. Let X be an Ltopological space with (L,') a complete lattice. We say that a, b E L x are completely separated if there exists f E C*(X) such that a <~L~f <.Rof <~b'. An a E L x is called an L-zero-set if a = R~of for some f E C ( X ) . This is equivalent [22,2.3] to the statement that a = R~g for some 0 E C*(X) and a = L~h for some h E C*(X). Let ZL(X) denote the set of all L-zero-sets of X.
In(L) =I(In-l(L))
for n ~> 2. Consequently, (In(L), °) is a complete lattice with an order-reversing involution (which is clearly an In-l(L)-topological space). Thus, one can have In(L)-fuzzy sets, a special case of nth order fuzzy sets. Fact 5 [14]. Let (L,') be a complete lattice. The followin9 statements hold: (1) In(L)~-ln+l(2) (both in a lattice and an Ltopological sense). (2) I f L is meet-continuous, then In+l(L) is an In(L )-topological lattice. (3) There is a complete and order-reversin9 involution preservin9 embeddin 9 of ln(L ) into 1~+1(L ). Also note that the assignment II : (L, I) ~ (I(L), °) is functorial in the category of all complete lattices with an order-reversing involution, morphisms being complete lattice homomorphism preserving orderreversing involutions. Namely, if ~o: (L1, i) ~ (L2, i) is a morphism, then so is II(~o) : (l(L1), °) ~ (I(L2), °) defined by II(~p)([2]) = [¢p o 2] (see [14] for details).
In dealing with topological complete regularity, the central property (cf. [5]) asserts that in any topological space every two disjoint zero-sets are completely separated. Unfortunately, this important feature has not yet been shown to hold in TOP(L) with ILl >2. For X a topological space, the proof is easy and involves the ring structure of C(X, •). [For, if f - l { 0 } N g - l { 0 } = ~ , then h = f / ( f + g ) completely separates these disjoint zero-sets]. In an L-topological setting, C(X) is merely a poset (or a lattice ifL is meet-continuous [20]), and the algebraic operations on ~(L) when L is a complete chain [32] are not strong enough. We thus have another important question (next to Question 1), already posed in [20,22].
Question 18. For which complete lattices (L, ') with ILl>2, are every two a, bEZL(X) with a<~b' completely separated? An affirmative answer to Question 18 would have numerous applications, particularly to the theory of completely L-regular spaces; one of these (not necessarily the most important) will be discussed in the next subsection.
T. Kubiak, M.A. de Prada VicentelFuzzy Sets and Systems 105 (1999) 277-285 We have used the symbol C*(X) to denote the family of all continuous functions from X to I(L), whereas in general topology it stands for the collection of all bounded continuous functions from X to ~. The issue is delicate, because a constant function in C(X) need not be "bounded" in the sense that it takes values in some [a, b](L) = {[2] E ~(L): 2 - ( a ) ' = 2+(b) = 0} (a
We do not know how much of what is known about ~(L) goes over to ~*(L); note that an attempt to define bounded ~(L)-valued functions can be found in [24].
Question 19. When does a continuous function f :X --~ R(L) take values in some [a,b](L)?
Question 20. For which continuous lattices (L,/) does the following hold: I f X is an arbitrary L-topological space and f : X - - + I ( L ) is continuous, there exists a continuous function 9 : ZZLX~ [0, 1] such that L~f 4 9 <~Rof? (This holds true for L = [0, 1] [19].) The following is one of the long-standing questions of Rodabaugh [29,30] conceming the ~(L)-valued case of the Tietze-Urysohn Extension Theorem for L-normal spaces. Question 21 (Rodabaugh [30]). For which L with ILl > 2 does every continuous f :A ~ •(L), where A is a subset of an L-normal space X with 1A being closed in X, have a continuous extension to the whole of X? We recall that this holds true for (L,') a meetcontinuous lattice provided X is perfectly L-normal (see [15] and cf. [20, the proof of 5.5]). In fact, from the discussion within the proof of Proposition 5.5 of [20], it suffices to assume that la C ZL(X) with X an L-normal space. One related question is this:
Question 22. Let X be L-normal with ILL>2. Do we have: a c L x is an L-zero-set ¢*, a is Gr? (Clearly, the point is to show ~ . ) A subset A of an L-topological space X is said to be C*-embedded in X if every function in C*(A) can
283
be extended to a function in C*(X). Similarly, A is called C-embedded if every function in C(A) can be extended to a function in C(X).
Question 23. Let ILl >2. When and only when is a C*-embedded subset also C-embedded ? (cf. [5, Theorem 1.18].) Let us go back to the case of L-normal spaces. Unlike the topological case, the Tietze-Urysohn Extension Theorem [16] does not characterize L-normality (just because an L-topological space need not have non-trivial crisp closed subsets). In fact, there are non L-normal spaces in which every crisp closed subset is C*-embedded [14].
Question 24 [14]. Under what conditions is Lnormality equivalent to C*-embedding of crisp closed subsets? We finish this subsection with two questions concerning fixed points. There are a number of contractive type fixed point theorems for various "fuzzy metric spaces". However, most of the latter carry a usual topology, unlike, e.g., the fuzzy metric spaces o f Erceg [2]. Question 25. Formulate and prove a Banach Contraction Theorem for Erceg L-fuzzy metric spaces. (This will require a definition of completeness for these spaces.) Question 26. Does every continuous function from I(L) J into I(L) J have a fixed point? 4 This is the case when IJl~<~t0 and (L,') is completely distributive with a countable base [21 ].
Completely L-regular spaces. Here we restate a few questions posed in [20,22].
Question 27. Let (L, ') be complete. Let ~ C L x generate the L-topology of X. Is X completely L-regular if each u E 6 ¢ has the separation property indicated in the definition of complete L-regularity? (This is the case when L is a frame.) 4 Added in proof: A positive answer to this question is given in: T. Kubiak and D. Zhang, On the L-fuzzyBrouwerfixedpoint theorem (this issue).
284
T. Kubiak, M.A. de Prada Vicente/Fuzzy Sets and Systems 105 (1999) 277-285
The questions which follow concern an intemal characterization o f complete L-regularity in terms o f normal and separating families o f closed L-sets. A family J d o f closed L-sets o f an L-topological space is called s e p a r a t i n g if for each open L-set u there exist two families {aj: j E J } , {bj: j E J } C OF such that u = V j e j aj and aj <~b~ <<,u for all j E J . It is called n o r m a l i f for every a, b E OF with a < b' there exist c, d E OF such that a <~c' <<.d <<.b'. W e note that with (L, ' ) a complete lattice, any X which admits a normal and separating family o f closed L-sets must necessarily be completely L-regular [22]. Question 28. Does every completely L-regular space have a normal and separating family o f closed L-sets? A positive answer to Question 18 yields a positive answer to the following which is actually equivalent to Question 18 (see [22,5.6]). Question 29. Is z L ( x ) a normal family i f (Note that it is always a separating family.)
[LI > 2 ?
Question 30. Does every completely L-regular space have a normal and separating family o f closed L-sets for some L # 2? For L-topological spaces in which open L-sets separate points (L-T0-spaces), using the Tychonoff embedding theorem, a positive answer to the following question yields a positive answer to Question 31 in the class o f L-T0-spaces with L a frame. Question 31. Is the property o f having a normal and separating family o f closed L-sets hereditary and productive if ]LI > 27 I f it is hereditary and (L, ' ) is a Boolean algebra, a positive answer to the following question will also yield a positive answer to Question 31 in the class o f L-T0-spaces. Question 32. Let ( L , ' ) be a complete Boolean algebra. Is there a bijection from the L-topology o f I ( L ) J onto the topology o f [0, 1] J which preserves arbitrary sups and finite infs? ( W e note that this is the case when IJI = 1 [8].)
References [1] G. Artico, R. Moresco, ~*-compactness of the fuzzy unit interval, Fuzzy Sets and Systems 69 0988) 243-249. [2] M.A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69 (1979) 205-230. [3] T.E. Gantner, R.C. Steinlage, R.H. Warren, Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62 (1978) 547-562. [4] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, A Compendium of Continuous Lattices, Springer, Berlin, 1980. [5] L. Gillman, M. Jerison, Rings of Continuous Functions, Springer, New York, 1976. [6] J.A. Goguen, The fuzzy Tychonoff theorem, J. Math. Anal. Appl. 43 (1973) 734-742. [7] U. Hthle, A.P. Sostak, Axiomatic foundation of fixed-basis fuzzy topology, Handbook of Fuzzy Set Theory, Kluwer Academic Publishers, Dordrecht, to appear. [8] B. Hutton, Normality in fuzzy topological spaces, J. Math. Anal. Appl. 50 (1975) 74-79. [9] B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal. Appl. 58 (1977) 559-571. [10] B. HuRon, I.L. Reilly, Separation axioms in fuzzy topological spaces, Fuzzy Sets and Systems 3 (1980) 93-104. [11] A.J. Klein, Generalizing the L-fuzzy unit interval, Fuzzy Sets and Systems 12 (1984) 271-279. [12] W. Kotzt, T. Kubiak, Fuzzy topologies of Scott continuous functions and their relation to the hypergraph functor, Quaest. Math. 15 (1992) 175-187. [13] T. Kubiak, A class of second order fuzzy sets, Proc. 10th National School for Scientists, Appl. of Math. in Techn., Vama, 1984. [14] T. Kubiak, On fuzzy topologies, Ph.D. Thesis, A. Mickiewicz University, Poznafi, 1985. [15] T. Kubiak, Extending continuous L-real functions, Math. Japon. 31 (1986) 875-887. [16] T. Kubiak, L-fuzzy normal spaces and Tietze extension theorem, J. Math. Anal. Appl. 125 (1987) 141-153. [17] T. Kubiak, The topological modification of the L-fuzzy unit interval, in Ref. [33], pp. 276-305. [18] T. Kubiak, Fourth open question, in Ref. [33], p. 350. [19] T. Kubiak, Are higher fuzzy separation axioms good extensions?, Quaest. Math. 16 (1993) 443-451. [20] T. Kubiak, On L-Tychonoff spaces, Fuzzy Sets and Systems 73 (1995) 25-53. [21] T. Kubiak, The fuzzy Brouwer fixed-point theorem, J. Math. Anal. Appl. 223 (1998) 62-66. [22] T. Kubiak, M.A. de Prada Vicente, On internal characterizations of completely L-regular spaces, J. Math. Anal. Appl. 216 (1997) 581-592. An extended abstract in: IFSA'97 Prague, Proc., Vol. 2. Academia, Prague, 1997, pp. 54-57. [23] T. Kubiak, M.A. de Prada Vicente, L-regular topological spaces and their topological modifications, Int. J. Math. & Math. Sci., to appear. [24] Liu Qijin, Fuzzy pseudocompactness, Fuzzy Math. 7 (1987) 69 -76.
72 Kubiak, M.A. de Prada VicentelFuzzy Sets and Systems 105 (1999) 277-285 [25] R. Lowen, Compact Hausdorff fuzzy topological spaces are topological, Topology Appl. 12 (1981) 65-74. [26] R. Lowen, Hyperspaces of fuzzy sets, Fuzzy Sets and Systems 9 (1983) 287-311. [27] R. Lowen, On the existence of natural fuzzy topologies on spaces of probability measures, Math. Nachr. 115 (1984) 33 -57. [28] S.E. Rodabaugh, Open questions: The L-fuzzy plane, in: E.P. Klement (Ed.), Proc. 3rd Int. Seminar on Fuzzy Set Theory, Johannes Kepler Universi~t, 7-12 September 1981, Linz, 1981, pp. 243-245. [29] S.E. Rodabaugh, The L-fuzzy real line and its subspaces, in: R.R. Yager (Ed.), Fuzzy Sets and Possibility Theory: Recent Developments, Pergamon Press, Oxford, 1982, pp. 402-418. [30] S.E. Rodabaugh, Separation axioms and the fuzzy real lines, Fuzzy Sets and Systems 11 (1983) 163-183. [31] S.E. Rodabaugh, Point-set lattice theoretic topology, Fuzzy Sets and Systems 40 (1991) 297-345. [32] S.E. Rodabaugh, Complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real line, Fuzzy Sets and Systems 15 (1985) 285-310.
285
[33] S.E. Rodabaugh, E.P. Klement, U. H6hle (Eds.), Applications of Category Theory to Fuzzy Subsets, Kluwer, Dordrecht, 1992. [34] A.P. Sostak, A fuzzy modification of a linearly ordered space, Colloq~ Math. Soc. Janos Bolyai 41, Topology and Applications, Eger (Hungary) (1983) pp. 581-604. [35] A. Sostak, A fuzzy modification of the category of linearly ordered spaces, Comment. Math. Univ. Carolinae 26 (1985) 421-442. [36] A. Sostak, Fuzzy topologies on spaces of probability measures, Topological Spaces and Mappings, Riga, 1989, pp. 421-442. [37] R.H. Warren, Neighborhoods, bases and continuity in fuzzy topological spaces, Rocky Motmtain J. Math. 8 (1978) 459-470. [38] Zhang De-xue, Liu Ying-ming, L-fuzzy modification of completely distributive lattices, Math. Nachr. 168 (1994) 79 -95. [39] Zhao Dongsheng, The N-compactness in L-fuzzy topological spaces, J. Math. Anal. Appl. 128 (1987) 64-79.