Two decades of fuzzy topology: basic ideas, notions, and results

Transcription

1 Uspekhi Mat. Nauk 44:6 (1989), Russian Math. Surveys 44:6(1989), Two decades of fuzzy topology: basic ideas, notions, and results A.P. Shostak CONTENTS Introduction 0. Preliminaries: fuzzy sets Fuzzy topological spaces: the basic categories of fuzzy topology Fundamental interrelations between the category Top of topological 135 spaces and the categories of fuzzy topology 3. Local structure of fuzzy topological spaces Convergence structures in fuzzy spaces Separation in fuzzy spaces Normality and complete regularity type properties in fuzzy topology Compactness in fuzzy topology Connectedness in fuzzy spaces Fuzzy metric spaces and metrization of fuzzy spaces The fuzzy real line 3~ (R) and its subspaces Fuzzy modification of a linearly ordered space Fuzzy probabilistic modification of a topological space The interval fuzzy real line On hyperspaces of fuzzy sets Another view of the subject of fuzzy topology and certain categorical 170 aspects of it Conclusion: some reflections on the role and significance of fuzzy topology 176 References 177 Introduction The notion of a fuzzy set, introduced by Zadeh [169] in 1965, has caused great interest among both 'pure' and applied mathematicians. It has also raised enthusiasm among some engineers, biologists, psychologists, economists, and experts in other areas, who use (or at least try to use) mathematical ideas and methods in their research. We shall neither dwell upon the clarification of the reasons for such a considerable and diversified interest in this notion nor discuss its place and role in mathematics as a whole the reader will possibly find an answer to these and other similar questions after consulting the monographs [1], [69], [114], and others.

2 126 A.P. Shostak We are much more modest and concrete in our purpose, which is to present the basic concepts of fuzzy topology, the branch of mathematics which has resulted from a synthesis of the subject of general topology with ideas, notions, and methods of fuzzy set theory. General topology was one of the first branches of pure mathematics to which fuzzy sets have been applied systematically. It was in 1968, that is, three years after Zadeh's paper had appeared, that Chang [16] made the first "grafting" of the notion of a fuzzy set onto general topology. He introduced the notion that we call a Chang fuzzy space (1.1) and made an attempt to develop basic topological notions for such spaces. This paper was followed by others in which Chang fuzzy spaces and other topological type structures for fuzzy set systems were considered. Since the early eighties, the intensity of research in the area of fuzzy topology has increased sharply, and at present there are some six hundred publications in this area. In the present work we shall try to make the reader familiar with the basic ideas and categories of fuzzy topology, to present more or less systematically the basic notions, constructions, and results in this area, and to discuss the directions in which it is developing. We must state explicitly that our survey does not pretend to completeness. In particular, we shall only very briefly touch on such topics as fuzzy uniform structures [55], [6], [128], [173], [95], [8], fuzzy proximity structures [64], [65], [6], [8], [178], cardinal invariants of fuzzy spaces and their fuzzy subsets [155], [179], fuzzy topologies on groups and other algebraic objects [33], [113], [68], [66], and topics in fuzzy topological dynamics [81], [129]. Let us outline briefly the contents of our work. In 0 we present the minimal amount of information on fuzzy sets needed for reading the main body of the work. In 1 various approaches to the definition of a fuzzy space (and, accordingly, to the subject of fuzzy topology) are discussed and compared, the principal categories of fuzzy topology are considered, and a unified terminology is established. 2 is devoted to clarifying the fundamental interrelations between the categories of fuzzy topology and the category of topological spaces. Let us stress that the clarification of interrelations of such kind has both a technical interest and a fundamental (and even philosophical) importance for fuzzy topology. In 3 the notion of a fuzzy point is considered and the local structure of a fuzzy space is discussed. Let us note that the problem of finding an adequate analogue of a point in a fuzzy situation and the related problem of local study of fuzzy spaces have become a stumbling block for a number of authors. In 4 convergence structures in fuzzy spaces are studied. We draw the reader's attention to subsection 4.5, where the so-called fuzzy neighbourhood spaces are considered an essentially fuzzy phenomenon having no analogue in general topology and defined by means of filters. In 5-9 the most important topological properties for fuzzy spaces are considered. In 5 various approaches to the definition of the Hausdorff

3 Two decades of fuzzy topology: basic ideas, notions, and results 127 property for fuzzy spaces are discussed (see also 15.6). We think that this discussion is important not so much because it surveys various definitions of the Hausdorff property in fuzzy topology as because this simplest example demonstrates an inevitable branching process of ordinary topological notions under their extension to the categories of fuzzy topology. In 6 properties similar to normality and complete regularity are considered for fuzzy spaces. Assertions on maps from such spaces to the fuzzy interval, the fuzzy real line ( 10), and other 'standard' fuzzy spaces deserve the most attention. 7 is devoted to a rather detailed discussion of a most important topological property, that of compactness, as well as to the compactification problem for fuzzy spaces. (We shall return to the problem of compactness and compactifications in subsection 15.7, where we shall look at it from another point of view.) Properties similar to connectedness for fuzzy spaces are considered in 8. Finally, in 9 we shall discuss various approaches to the notions of a metric and metrizability in a fuzzy situation. Fuzzy stratifiable spaces a property similar to generalized metizability are also considered there. In constructions belonging, in the opinion of a number of authors, to the "gold reserve" of fuzzy topology are discussed these include the fuzzy interval and the fuzzy real line ( 10), fuzzy modification of a linearly ordered space ( 11), Klein modification of a connected space ( 11), fuzzy probability modification of a topological space ( 12), the interval real line ( 13), and the construction of hyperspaces of fuzzy subsets of a uniform space ( 14). In 15, which occupies a particular place in the survey, another (at a stretch, more general) view of the subject and objectives of fuzzy topology is presented. A wide use of category topology helps us very much in presenting this point of view. In this section we shall take a fresh look at a number of questions considered earlier. At the end of the work there is a conclusion, whose purpose is explained by its title. 0. Preliminaries: fuzzy sets (0.1) Fuzzy sets. Let X be a set. Following Zadeh [169] we define a fuzzy (sub)set of X as a map Μ : X -* I: = [0, 1 ]. In this connection M(x) is interpreted as the degree of membership of a point χ Ε Χ in a fuzzy set M, while an ordinary subset A C X is identified with its characteristic function A % A : X_> {0, 1} = : 2. (0.2) Ζ,-fuzzy sets. Having noticed that many properties of an interval prove to be inessential if not burdensome for one working with fuzzy sets, Goguen [41] introduced the notion of an L-fuzzy set, where L is an arbitrary lattice with both a minimal and a maximal element, 0 and 1 respectively. An L-fuzzy (sub)set

4 128 A.P. Shostak of X is a map M.X^-L. In particular, an I-fuzzy set in X is a "classical" fuzzy set in X (0.1) and a 2-fuzzy set in X is an ordinary subset of it. We denote by L x the totality of all Z-fuzzy subsets of a set X. (0.3) Fuzzy lattices. Although the definition of an Z-fuzzy set makes sense for an arbitrary lattice L, fuzzy topologists restrict themselves, as a rule, to the use of the so-called fuzzy lattices. Following Hutton [56], we define a fuzzy lattice as a complete completely distributive lactice with a minimal element 0 and a maximal element 1, on which an order-reversing involution a -* a c is fixed (that is, a < b, a, b G L =» b c < a c ). (For those lattice theory notions not defined here see, for example, [12]). In particular, having introduced an involution on the lattices / and 2 by the rule a -» a c : = 1 a and having endowed them with the natural order, one may consider them as fuzzy lattices. In what follows L always stands for a fuzzy lattice. (0.4) Orthocomplemented lattices. In some respects there is most resemblance between the lattice 2 and the so-called orthocomplemented lattices: an involution a -* a c on Ζ is called an orthocomplementation if a \J a c = 1 and a /\ a c 0 for every a G Z. An important example of an orthocomplemented lattice is the lattice 2 Z of all subsets of a set Ζ naturally ordered by inclusion and endowed with an involution A -* A c : = Z\A. (0.5) Operations on fuzzy sets. Let Λ' {A t : i Ξ ")} d L x be a family of L-fuzzy sets in X. By the union and the intersection of this family we mean respectively its supremum \J J: = V {Ar- i e.7} and infimum /\ Jr. = /\ {.4,-: i <Ξ 2}. The complement A c of an Ζ,-fuzzy set A is defined by the rule A c {x): = G4(x)) c, x I If Λ/ 7 is an Ζ,-fuzzy subset of a set X y for each γ G Γ, then we define the product of these fuzzy sets as the Ζ,-fuzzy subset Μ of the set X = EUv defined by Μ (χ) = Λ Λ/ ν (*ν) (see [ 169], [41 ]). It is easy to ν ν verify that for L = 2 these operations reduce to the ordinary set-theoretic operations of union, intersection, complement, and product, and that the behaviour of the operations just introduced is completely analogous to that of the corresponding set-theoretic operations. For example, the de Morgan law (\/ Aif = /\ A C j can be stated easily, and so on [ 169], [41 ]. If Ζ is an i i orthocomplemented lattice, then clearly the complementation in L x is an orthocomplementation as well: A \' A c = 1, A /\ A c 0. (Along with the definitions of basic operations on fuzzy sets presented above, there are other definitions of these operations occuring in "fuzzy mathematics" and especially in its applications. For example, the intersection and the union of fuzzy sets A, B G I x are given respectively by Α Β and min{^4 + β, 1}, and so on. The reader may become acquainted with various approaches to the definition of operations on Z-fuzzy sets, including

5 Two decades of fuzzy topology: basic ideas, notions, and results 129 axiomatic ones, by means of the papers [18], [78], [79]; see also [1], [69], [114]. In the present survey, as usual in fuzzy topology, operations on Z-fuzzy sets are always understood in the sense of the definition at the beginning of this subsection.) (0.6) Images and inverse images of fuzzy sets. Let Χ, Υ be sets and let /: X -* Υ be a map. The image f(a) G L Y of a fuzzy set Α <Ξ L x is defined by f (A )(y) = sup {Α (χ): χ e Γ 1 (y)} if/ -1 ( }')ΦΦ and f(a)(y) = 0 otherwise; the inverse image f' l (B) G L x of a fuzzy set Β L Y is defined by / -1 (S)(x) = Bf(x). Properties of images and inverse images of fuzzy sets are completely analogous to properties of images and inverse images of ordinary sets. For example, f 1 (V#i) = V 7 /" 1 ( #;), and i i so on (see [ 169], [41], [ 161], and others). If /: X -» Υ is a map, then by letting f(a): f(a) for each A GL^we obtain a map f:l x ->-L Y (this is the so-called Zadeh extension principle [169]). (0.7) Inclusion relation for -fuzzy sets. For Z-fuzzy subsets A and Β the inequality A < Β is treated as the statement "A is a subset of an L-fuzzy set B" [169], [41]. (0.8) Fuzzy inclusion relation for L-fuzzy sets. Along with the relation < on L x χ L x we shall need the fuzzy inclusion relation cz: L K χ L x L,defined by A CZ B: = inf (A c \J B){x) (see [22], [24]). It is evident that for ordinary subsets A and Β of a set X, ii A C Β then 4 c S = 1, and A CZ Β = 0 otherwise. There is a definite parallel between the properties of the ordinary inclusion CZ and the corresponding properties of the fuzzy inclusion C. This parallel manifests itself in the correspondence of the implication "if... then" to the inequality "<". For instance, to the statement "if A C 5 and A C C then A C Β η C" {A, B, C C X) there corresponds the inequality "(-4 T ) Λ 0 4 (Z C)< Λ C S Λ C" (Λ, 5, C G L x ); to the statement "if A CB then /G4) C /( )" (A 5 C X, /: X -» 7) there corresponds the inequality "(.4 c B) < (/ (^4)C / (β.))" W, 5 C I 1 ; /: X -> 7), and so forth. However, there are some deviations from this rule. For example, the "transitivity inequality" (A CZ B) /\ (B cz C) ^. Α ζΐ C in general is not valid. For more on the properties of fuzzy inclusion see [22], [141], [150]. (0.9) Fuzzy cardinals. We define a fuzzy cardinal [155] as a non-increasing map of the form κ: Κ > I, where Κ is the class of all (ordinary) cardinals, such that y. (0) = 1 and κ (α) = 0 for some a G I In [155] fuzzy cardinal arithmetic is developed, which is in a sense similar to the ordinary cardinal arithmetic. The cardinality of a fuzzy set Μ ε I x is a fuzzy cardinal κ Λί defined by ν-μ (α) = sup {t: ΛΓ 1 (t, i] > a}, where a G i X

6 130 A.P. Shostak 1. Fuzzy topological spaces: the basic categories of fuzzy topology 1.1. Fuzzy topological spaces: Chang's approach. As we noted earlier, the first definition of a fuzzy topological space is due to Chang [16]. According to Chang, a fuzzy topological space is a pair (Χ, τ), where X is a set and τ is a fuzzy topology on it, that is, a family of fuzzy subsets (τ C I x ) satisfying the following three axioms: (1)0,1 ex; (2) if U, FG T, then U Λ V GZ τ; (3) if ί/j Ε τ for every i G J, then \[U t e τ. 7 A map /: X -* Υ between fuzzy topological spaces (Χ, τ χ ) and (Υ, τ γ ) is said to be continuous if f~\v) G τ χ for each V G τ γ. Fuzzy topological spaces and continuous mappings form a category which we denote by CFT and call the category of [Chang] fuzzy [topological] spaces. (Hereafter we put in square brackets those words to be omitted in the sequel if no ambiguity arises.) A fuzzy subset A of a fuzzy space (X, r) is called closed if A c G r. We denote by T C the totality of all closed subsets of a space (Χ, r). One verifies easily that (l c ) 0, 1 G r c ; (2 C ) if A, Β G T C, then A\J Β Gr c ; (3 e ) if_ A t G T C for all i G Cf, then /\A t G T C. The smallest closed fuzzy set Μ i containing (0.7) a fuzzy set Μ G I x is called the closure of M, and the largest open (that is, belonging to r) fuzzy set Int Μ contained in Μ is called the interior of M. The basic properties of the closure and interior operations in fuzzy spaces bear a complete analogy to the corresponding properties of the closure and interior in topological spaces. For example, Μ = /\{Α:Α ΕΞχ ε, Α > Μ}; Μ \J Ν = Μ \J Ν; Μ/\Ν^Μ,\Ν, Μ = Μ for every Μ, Ν G I x (see, for example [ 161 ]; cf. [29]). Quite naturally one introduces also the notions of a base and a subbase of a fuzzy topology, an open and a closed map, a homeomorphism, and so on L-fuzzy topological spaces; Goguen's approach. Generalizing Chang's approach, Goguen [41] introduces the notion of an Ζ,-fuzzy topological space. Let L be an arbitrary (fixed) fuzzy lattice (0.3). We define a [Chang] L-fuzzy [topological] space as a pair (Χ, τ), where X is a set and τ is an L-fuzzy topology on it, that is, a family of Ζ,-fuzzy subsets (T C L X ) satisfying the axioms (l)-(3) from the preceding subsection. Ζ,-fuzzy spaces and their continuous maps (where the continuity is understood precisely as in 1.1) form a category CFT(L). Clearly, CFT{I) is just the category CFT, and CFT{2) is (up to an evident isomorphism) the category Top of topological spaces. All that we said on fuzzy spaces in 1.1 remains valid for Ζ,-fuzzy spaces.

7 Two decades of fuzzy topology: basic ideas, notions, and results Laminated fuzzy topological spaces: Lowen's approach. In 1976 Lowen [88] proposed a new notion of a fuzzy topological space which differs from Chang's definition (1.1) in that instead of the axiom (1) the following sharper axiom is used: (1 λ ) τ contains all the constants c Ε /. In what follows we shall refer to such spaces as laminated [Chang] fuzzy [topological] spaces. The category of laminated fuzzy spaces and their continuous maps (where Lowen continuity is understood precisely as in 1.1) is denoted by LCFT. An essential feature of laminated fuzzy spaces is that, as is easily verified, the constant maps of such spaces are a priori continuous (that is, they are morphisms in LCFT) [88]. We note also that there is a continuum of different Chang fuzzy topologies on a one-point set *, while there is only one laminated fuzzy topology (r = /*). From here Lowen and Wuyts [105] deduce that LCFT, unlike CFT, is a topological category in the sense of Herrlich [47]. Noting this and a number of other important advantages of laminated fuzzy spaces (see, for example, 1.6, 6.1, and so on), Lowen recommends that we persistently restrict ourselves to the study of such fuzzy spaces [88]-[105]. It is natural to call a Chang L-fuzzy space laminated if its Ζ-fuzzy topology contains all constants c S L. (Such spaces occur, for example, in [97], [125], [128], and others.) It is easily verified that LCFT{2) CFT{2) and up to an isomorphism it is the category Top of topological spaces Fuzzy topological spaces. A certain disadvantage of all the approaches considered above is some inconsistency in the use of the idea of fuzziness. In each of these approaches a fuzzy topology is an ordinary subset of the family of all fuzzy (or Z,-fuzzy) subsets of a given set X. In [146] another, more consistent approach to the use of ideas of "fuzzy mathematics" in general topology has been developed. According to [ 146] a fuzzy topological space is a pair (X, gf), where X is a set and %f: I x * I is a map (that is, a fuzzy subset of a set L x ) satisfying the following axioms: (1) $ (0) = g (1) = 1; (2) JT (U Λ V) > W (U) A f (V) for any U, V <= I x ; (3) W (V U t ) > f\s~ (U t ) for every family {U t : iej}c I x. i i In this connection the inequality f (U) > α, where a G (0, 1 ], is treated as the statement "the degree of openness of a fuzzy set U is not less than a", and the inequality ^ (U c ) >a as the statement "the degree of closedness of a fuzzy set U is not less than a". A map f:x^-y between fuzzy spaces (X, 3~ x ) and (Y, 3~ Y ) is called continuous if If χ (f~ l (V)) > ST Y (V) for each V / y. Loosely speaking,

8 132 A.P. Shostak one regards as continuous those maps that do not reduce the degree of openness of fuzzy sets under transition to the inverse image. We denote the category of fuzzy spaces and continuous maps by FT. A map f:x-+y between fuzzy spaces (X, Sf _ Y ) and (Y, Sf Y ) is said to be a homeomorphism if /is a bijection and both /and/" 1 are continuous. A map / is said to be closed (open) if Sf (U c ) < Sf (}U C ) (respectively & (U)< Sf (fu)) for every U G I x. We call a fuzzy space (X, Sf) laminated if (1' ) Sf (c) = 1 for every constant c G / (compare the definition in 1.3). Clearly, constant maps of laminated spaces are continuous. We denote by LFT the full subcategory in FT formed by laminated fuzzy spaces. By substituting an arbitrary fuzzy lattice L for an interval / in the definition of a (laminated) fuzzy space we arrive at the definition of a (laminated) L-fuzzy space. We denote the resulting category by FT(L) (respectively, LFT(L)) Certain interconnections between the categories CFT(L) and FT(L). By identifying, as usual, subsets of a given set with the corresponding characteristic functions one can consider CFT(L) as a full subcategory of FT(L). In this connection a fuzzy space (X, Si) (1.4) is evidently a Chang one (1.1), that is, it lies in CFT(L) if and only if a fuzzy topology Sf in addition to the axioms (l)-(3) satisfies the follo'wing: (4) Sf (L x ) C 2 = {0,1}. On the other hand, if (X, Sf) is an arbitrary L-fuzzy space and α e L + : = L\ {0}, then Sf a- = {U ΕΞ L x ': Sf (U) > a} is a Chang L-fuzzy topology on X (the so-called α-level Chang L-fuzzy topology of the given fuzzy topology Sf). In addition, Sf c a: = {U Ξ L x : Sf (U c ) > a} is precisely the family of all closed Ζ,-fuzzy subsets of the Chang space (X. Sf a ) These observations enable us to reduce the study of certain properties of a fuzzy topology Sf (1.4) to the study of much simpler objects, the corresponding α-level Chang fuzzy topologies Sf a - In particular, the continuity of a map /: (X, Sf x ) -* (Y, Sf Y ) is easily verified to be equivalent to the continuity of the maps /: (X, &i)-+ (Y, Sfl) for all a G L + [146], [152]. Clearly, LCFT(L) = CFT(L) Π LFT(L). The categories LCFT(L), CFT(L), and LFT(L) are coreflexive in FT(L) [138] Initial L-fuzzy topologies and the product of L-fuzzy spaces [146], [152]. Let : = {Sf y : γ Ξ Γ} be a family of L-fuzzy topologies on a set X (hereafter in this subsection we use the terminology from 1.4). The weakest among all L-fuzzy topologies on X majorizing (in the > sense) every $~ y we denote by sup Sfy and call it the supremum of that family. It is easy to see ν that every non-empty family of L-fuzzy topologies has a supremum and that

9 Two decades of fuzzy topology: basic ideas, notions, and results 133 if all JJVs are Chang (laminated) topologies, then sup if y is a Chang v (laminated) topology as well. Let Ibea set, (} r, ) an Z-fuzzy space, and /: X -* (Y, ) a map. We call the weakest Z-fuzzy topology $~: = f' 1 ($) on X making/ continuous the initial Z-fuzzy topology for /. One can show that for a map /: X -+ (Y, S) the initial Ζ,-fuzzy topology is determined by the formula 0- (U) = V [SO. Ψ) Λ «: α e L + }, where U e L*, f a : = {Γ 1 (Γ): V Ξ Π, 8 (V) > a}. The weakest Z-fuzzy topology on X making continuous all the maps f y : X > (Y y, y ), y r Γ, we call the initial one for that family of maps. It is easily shown that the initial topology for a family {/ Y : y ΕΞ Γ} is determined as sup Z" 1 ( Y ). If in addition all S Y 's are Chang (laminated) Z-fuzzy topologies, ν then the initial topology sup f' 1 ( y ) is Chang (laminated) as well. y Let {{X y, 3~y): γ ε Γ} be a family of Z-fuzzy spaces. The product of this family is the pair (X, if), where X = Π X y is a set product and 3~ is an ν Ζ,-fuzzy topology on X which is initial for the family of all projections /V X -» (Xy, c5" Y ), γ Γ. It is easy to verify that the operation so defined is indeed a product in FT. If in addition all (Xy, J" Y )'s are Chang (laminated) spaces, then so is their product. It should be stressed that in contrast to the situation in general topology the projection maps of products of fuzzy spaces (including Chang ones) need not be open. Another feature of the products of fuzzy spaces is that a "fibre" X Yo χ {χ Υ : γ Φ γ 0 } in a product 1 λ\ of fuzzy spaces is not, ν generally speaking, homeomorphic to the space X y<t. A necessary condition for projections to be open and, as a consequence, for fibres to be homeomorphic to the corresponding coordinate spaces, is that all the factors are laminated. (In this sense both LFT(L) and LCFT(L) are "more topological" categories than FT(L) and CFT(L)\ (cf. 1.3).) (For Chang Z-fuzzy spaces the product was first introduced by Goguen [42] and studied in detail in [89] and [118].) Dually, the final Z-fuzzy topology and the coproduct, or the direct sum of/--fuzzy spaces, are introduced ([146] ; for Chang spaces [89]) Subspaces of Ζ,-fuzzy spaces. The simplest and at the same time a very important operation of general topology is transition to a subspace. In considering a similar problem for /--fuzzy spaces, two different approaches are possible: an (L-fuzzy) subspace of an Z-fuzzy space (X, if) based on an ordinary crisp set A C X and an (L-fuzzy) subspace of an Z-fuzzy space (A", if) based on an L-fuzzy set Μ ε Ι χ. In the case of the first approach (which is by the way the only possible one from the category point of view) the problem causes no trouble: it is natural to mean by a subspace any pair (A, if A ), where

10 134 A.P.Shostak 3 A : LA L is a fuzzy topology on A defined by = sup {<r (y): v = /*, y Λ (It is clear that if gf is a Chang (laminated) topology, then so is S~ A.) For the second approach the problem has no reasonable solution within the categories considered; we shall return to it in subsection 15.5 after developing another point of view of the subject of fuzzy topology On the notion of continuity defect in fuzzy topology. In the vast majority of works on fuzzy topology the authors start from the fact that by analogy with general topology all maps between fuzzy spaces are divided into two classes: continuous maps, or morphisms in the corresponding category, and discontinuous ones. However, sucii a "crisp" classification of fuzzy spaces, based on two-valued logic, does not always seem natural in the context of fuzzy topology. In [174] another view of this problem is suggested, based on the notion of continuity defect of maps. We now present the basic ideas of that approach, restricting ourselves to the case of Chang fuzzy spaces (1.1). Let (Χ, τ χ ), (Υ, τ γ) be Chang fuzzy spaces. We define the continuity defect of a map f:x-* Υ as the number It is easy to show that cd(/) = sup sup (thv)- V x ΙηίΓ ι (ν))(χ). cd (/) *= sup sup (Γ 1 (B) - f 1 ( ))(*) = sup sup (f(m)-f (M))(y). X * ϊ Clearly, the condition cd(/) = 0 means precisely the continuity of/. If Χ, Υ are topological spaces and the map / is discontinuous, then cd(/) = 1. In the case of arbitrary fuzzy spaces Χ, Υ the continuity defect may be any number α G /; its value characterizes to what extent the given map differs from a continuous one. For illustrative purposes let us present the following assertions. The continuity defect of a composition of two maps does not exceed the sum of the defects of those maps. The continuity defect of the diagonal of a family of maps is equal to the supremum of the continuity defects of these maps [174]. By analogy with the continuity defect one may define also the defects of other properties of maps between fuzzy spaces, which enables us to view certain aspects of fuzzy topology as a whole in a different light Some features of the presentation of the material in the survey. The diversity of the original categories of fuzzy topology (see , and also 15) makes a unified presentation and perception of ideas, notions, and results of this field more difficult. Therefore let us agree to the following.

11 Two decades of fuzzy topology: basic ideas, notions, and results 135 Taking into account that most works in the area of fuzzy topology are based on Chang's definition (1.1) and that the presentation of the theory for Chang spaces is essentially easier than for fuzzy spaces in the sense of 1.4 and for L-fuzzy spaces (1.2), in the sequel we shall speak as a rule of Chang fuzzy spaces only and the term "fuzzy [topological] space" will be understood in the sense of 1.1. We refer to the category LCFT only if it is actually necessary. If constructions and results in a cited paper are presented for laminated spaces but are easily transferrable to the case of arbitrary Chang spaces, then we present them in such a situation without any reservations. We believe that under such a presentation, on one hand the role of the laminatedness condition, and on the other hand those "topological opportunities" afforded by the non-topological [47] category CFT, will be apparent. Let us remark that many results that are true for CFT can be extended in one form or another to CFT(L), sometimes for an arbitrary fuzzy lattice L, but more often for those fuzzy lattices meeting certain additional assumptions, of which the most common are separability, linearity of ordering on L, and the presence of orthocomplementation in L. In the present work we refer to CFT(L) occasionally and for illustrative purposes only. On the other hand, let us recall (1.5) that the study of the category FT can often be reduced to the study of Chang spaces by transition to α-levels, as(0, 1]. 2. Fundamental interrelations between the category Top of topological spaces and the categories of fuzzy topology Everyone working in the area of fuzzy topology has to answer (at least, for himself) the following two closely connected fundamental questions. (1) By virtue of which functors from the category Top to the categories of fuzzy topology are the most important and essential interrelations between them established? (2) What should be regarded as analogues of objects of the category Top in the categories of fuzzy topology? We shall consider here the simplest functors (correspondences) of this kind; certain specific more complicated constructions are described in Top as a subcategory of CFT: the natural inclusion functor e.top^ CFT. Identifying, as usual, subsets of a given set with the corresponding characteristic functions, we can treat a topological space (X, T) as an object of CFT. In this way an inclusion functor e : Top -* CFT arises, which identifies Top with the full subcategory of CFT whose objects are just those fuzzy spaces (X, r) satisfying the condition r C 2 X.

12 136 A.P. Shostak 2.2. Embedding Top into LCFT: the functor ω : Top -* LCFT. Following Lowen [88] we associate with any topological space (X, T) the laminated fuzzy space (Χ, ωτ), where ωτ is the totality of all lower semicontinuous maps of (X, T) to the interval /. It can easily be verified that the continuity of a map /: (X, T x ) -* (Y, T Y ) implies the continuity of the map /: (Χ, ωτ χ ) -> (Υ, ωτ γ ). Thus, ω can be viewed as a functor ω : Top -» LCFT mapping Top isomorphically onto a full subcategory of LCFT. Lowen restricts himself to laminated fuzzy spaces and suggests that the space ωχ: = (Χ, ωτ) should be considered as the fuzzy copy of a topological space X (see [88] -[105] and others). A fuzzy space (X, r) whose fuzzy topology is of the form τ = ωτ for an ordinary topology Τ on X is called topologically generated [88] or induced [164]. Let P be a topological property and $> the extension of that property to the category of fuzzy topological spaces. Lowen [90] calls an extension 3 ύ good if any topologically generated fuzzy space (Χ, ωτ) possesses the property Ρ if and only if the corresponding topological space (X, T) possesses the property 5 J - The significance of the notion of a good extension is that it enables us to distinguish among a large number of possible (and rather natural) extensions of a property p to CFT some of the (in a sense) most successful ones Laminated modification of a fuzzy space. In [152] the functor ω was extended to the so-called laminated modification functor λ : FT -* LFT. In particular, if (Χ, τ) is a Chang fuzzy space, then \x : = (Χ, λτ), where λτ is the weakest laminated fuzzy topology on X majorizing τ. It is easy to show that λτ results from closing the family π: = τ!j {c : c EE 1} with respect to finite intersections and arbitrary unions. (In other words, π is a subbase [35] of the fuzzy topology r.) It is not difficult to note that the restriction of the functor λ to Top coincides with the functor ω. The functor λ commutes with a product: λ (Π Χ γ) = Π (λ-χυ'ι \ γ ) ν for an arbitrary family of fuzzy spaces {X y : γεξ Γ}. The use of laminated modification often proves helpful in verifying various statements of fuzzy topology (see, for example, 8.1) The functor / : CFT -» Top and weakly induced fuzzy spaces. Let (Χ, τ) be a fuzzy space. As is easy to note, /(T) : = τ Π 2 Χ is an ordinary topology, moreover, it is the maximal among all topologies contained in r. Since evidently the continuity of a map /: (Χ, τ χ ) -* (Υ, τ Υ ) guarantees the continuity of the map /: (X, JT X ) -* {Υ, }τ Υ ), we can regard / as a functor / : CFT -* Top. Martin [111] distinguishes an important class of fuzzy spaces by means of the functor /; a fuzzy space (X, r) is called weakly induced if for all U Ε τ

13 Two decades of fuzzy topology: basic ideas, notions, and results 137 the maps U: (X, /τ) -> / are lower semicontinuous. A space (A', r) is weakly induced if and only if ιτ = /τ, where ι is the functor from 2.5. A fuzzy space is topologically generated (2.2) if and only if it is both laminated and weakly induced [111] Fuzzy extensions of a topology to a family of fuzzy spaces: the functor η : Top -* FT. Let us present here an example of a functor which, unlike those considered above, performs an embedding of the category Top into the category FT rather than the category CFT. A functor of this kind was first described by Diskin [22]. We present here the construction of a similar functor from [146]. Let (Χ, Γ) be a topological space; putting $ (M) = Μ C Into, Μ (0.8) for each Μ G I x, where Into, M: = V { u - u < M, U ΕΞ ωτ) (that is, \ntj\i is the interior of Μ in ωγ), we get a map ηγ: = if: I x > /, which is a fuzzy topology (1.4) on X. In addition, if a map /: (X, T x ) -> (Y, T Y ) is continuous, then the map/: (Χ, ητ χ ) -> (Υ, r\t Y ) is continuous as well. The functor η: Top -> FT arising in this way maps Top isomorphically onto a full subcategory η(τορ) of the category FT. In addition, η (Top) Π CFT = {0}, that is, to any non-empty topological space an essentially non-chang fuzzy space is assigned. For more details on the properties of this and other analogous functors, see [22], [152], [146] The functor 1: CFT -» Top. Following Lowen [88], let us associate with every fuzzy space (X, x) the ordinary topological space (Χ, ιτ), where ιτ is the weakest topology making all the U E τ into lower semicontinuous maps U: (Χ, ιτ) -* /. (Equivalently, ιτ = {U' 1 (a, 1]: α Ξ /, U (Ξ: τ}.) It is not hard to verify that if a map /: (Χ, τ.\) > (Υ, xr) is continuous, then the map /: (Χ, ιτ Α ) > (Υ, ixy) is continuous, hence we can regard ι as a functor 1: CFT * Top [88], [89]. The functor ι is the right inverse of the functor ω: obviously, in ω (Τ) Τ for every topology T. A fuzzy space (X, x) is topologically generated if and only if τ = ω ι (τ). For more on this and other properties of ι see [88], [89]. Let 3 J be a topological property. A fuzzy space (Χ, τ) is said [90], [110] to be an ultralfuzzyi.-p-space if the topological space (Χ, ιτ) has the property 3*. Clearly, the extension of a topological property 5 to the property of being an ultra-^-fuzzy space is a good one (2.2). In [152] the functor ι is extended to the category FT The functors ι α : CFT -> Top. The α-level functor i a, where a G [0, 1), associates with a fuzzy space (Χ, τ) the topological space (Χ, ι ατ ), where ι α τ: = {U' 1 (α, 1] : U ΕΞ τ} is the "cut" of the fuzzy topology τ at the level a, and with a continuous map /: (X, x x ) -> (Γ, Ty) the (continuous) map /: (Χ, ι α τχ)-^(υ, ι α τ Γ ).

14 138 A.P. Shostak The α-level functors were introduced in [90] and were later used by various authors ([121], [125], [73], and others) for solving some problems of fuzzy topology by reducing them to standard problems of general topology The functors i* a : CFT -> Top. Along with the functors ι α, the functors ι*: CFT -> Top, where a e (0, 1], are used not infrequently in fuzzy topology. The functor ι* associates with a fuzzy space (Χ, τ) the topological space (Χ, ι α τ), where ι*τ is the topology defined by the subbase n a : = {U' 1 [a, 1]: U Ξ τ}, while leaving morphisms unchanged [90], [121]. Let us note that the properties of the functors ι α and ι* and the information provided by them differ essentially from each other The hypergraph functor G : CFT -> Top. Let (Χ, τ) be a fuzzy space; we consider the topology GT on the product Χ χ [0, 1) defined by the subbase {{(x, a): U (χ) > α): χ X, a e [0, 1), U ΕΞ τ}. Now assigning to any fuzzy space (Χ, τ) the topological space G{X, τ) : = = (Χ χ [0, 1), GT) and to a continuous map /: (Χ, τ χ ) -* (Υ, τ ) the Y (continuous) map G(f): G{X, τ χ ) -» G(Y, r Y ) defined by G(f)(x, a): = - (fx, a), we get an embedding functor G : CFT -> Top. This and some other similar functors were first considered by Santos (in preprints) and Lowen [90]. Rodabaugh made use of them for studying separation in fuzzy spaces [121], [125]. Klein has shown that for each fuzzy topology τ the inclusion Gt d π: χ Τ, holds, where Τ ι {[0, α): α Ξ 1). In this connection the equality GT = ιτ χ T h is valid if and only if r is topologically generated [74]. 3. Local structure of fuzzy topological spaces 3.1. On the notion of a fuzzy point. In general topology, as in many other areas of pure mathematics, a fundamental role is played by the notion of a point. A point is a minimal object in the sense of the relation of belonging 6Ξ: it either does or does not belong to a set, while nothing can belong to a point. One of the fundamental peculiarities of fuzzy set mathematics and, in particular, fuzzy topology, is the absence of such 'minimal' objects. In order to fill the gap caused by this, many authors use, sometimes quite successfully, kinds of "substitutes" for points the so-called fuzzy points, appearing in the early 80s in works by Pu and Liu [117], [118], Sarkar [130], [131], Srivastava, Lai and Srivastava [133], de Mitri and Pascali [19]. (The definitions of a fuzzy point and of the corresponding relation <=? of a fuzzy point belonging to a fuzzy set given independently by these authors are very much alike, though they differ in some details. The definitions of a fuzzy point and the

15 Two decades of fuzzy topology: basic ideas, notions, and results 139 relation of belonging due to Wong [166] seem inadequate for a criticism of them see, for example, [44].) Following [ 117], [ 118], let us present here the basic definitions and facts connected with the notion of a fuzzy point. A fuzzy point of a set X is a map p: = pi,: X» /, where x 0 Ε Χ, t Ε (0, 1], defined by p(x 0 ) = t and p(x) = 0 for χ Φ χ 0 ; here x 0 is called the support of the fuzzy point and t its value. An ordinary point x 0 is treated as a fuzzy point pi,. A fuzzy point ρ belongs to a fuzzy set Μ (ρ&μ) if t <M(x 0 ). Along with the relation ΕΓ, PU and Liu consider the so-called quasicoincidence, or Q-coincidence relation: a fuzzy point p: = pi, is quasicoincident (Q-coincident) with a fuzzy set Μ (pqm) if M(x o )+t > 1. With respect to the operations of union and intersection of a family of fuzzy sets {U t : i ΕΞ 3} there are both analogies with the behaviour ofge and differences from it in the behaviour of ΕΞ and Q: pmu t ;pq/\ U t ^VipQur, ρεξ V U % A fuzzy point is a minimal object neither with respect to EE nor with respect to Q: having chosen s Ε (0, t) and rg(l-/, 1) for a given t Ε (0, 1 ], we have p*.ge />x. and pljqpl,- To conclude, we note that in [36] an axiomatic approach to the notion of a fuzzy point is developed, and in [ 159], [172] the notion of a molecular is introduced, which is a kind of analogue of a fuzzy point in Hutton spaces (15.4), and on the basis of it an attempt is made to develop a local theory of Hutton fuzzy spaces Neighbourhoods in fuzzy spaces. There are several approaches to the notion of a neighbourhood in a fuzzy space and, correspondingly, to the study of the local structure of fuzzy spaces: the approach by Warren [161], [162] and a (less successful) approach by Ludester and Roventa [106], using fuzzy neighbourhoods of ordinary points, the approaches by Pu and Liu [117], [118], Sarkar [130], [131], Srivastava, Lai and Srivastava [133], [134], and de Mitri and Pascali [19], based on fuzzy neighbourhoods of fuzzy points (a comparative analysis of the approaches from this group is carried out in [70]), and the approach by Rodabaugh [128], based on fuzzy neighbourhoods of fuzzy sets. Let us dwell on the approach by Pu and Liu. A fuzzy set Μ is called a neighbourhood (Q-neighbourhood) of a fuzzy point ρ in a fuzzy space {X, r) if ρ r Int Μ (respectively, pq Int M). Let ΛΛρ denote the family of all neighbourhoods (respectively, Q-neighbourhoods) of a fuzzy point ρ in (Χ, τ). Then: (1) U 6Ξ ^ v = ρξ= U (respectively pqu); (2) U, V EE ^ P =» U /\ V Ξ JV V \ (3) U (ΞΞ > p, U < V =» V ΕΞ <*%; (4) for every U ^JT V there is a V e ΛΤ ν such that V < U and Fe JT r for each fuzzy point r with r e F (respectively, rqv).

16 140 A. P. Shostak Conversely, for every fuzzy point ρ of a set X let a family of fuzzy sets JV P be fixed in such a way that the conditions (l)-(4) are satisfied. Then the family σ formed by all fuzzy sets U G I x such that U e JT V so long as ρ Ξ U (respectively, so long as pqu) forms a base for a fuzzy topology τ (a fuzzy topology σ = τ) on X. In this case Jf v is the base of a system of neighbourhoods (respectively, of the system of all Q-neighbourhoods) for the fuzzy point ρ in (Χ, τ). (Let us note that there are inaccuracies in the formulation of the converse statement in [117].) To characterize laminated fuzzy spaces in a similar way, it suffices to add to the axioms (l)-(4) the following one: (5) if p: = pi,, then t e -A r p (respectively, s ΕΞ Λ" ρ for all s > 1 - t) Local characterization of the closure and closedness of fuzzy sets. To illustrate some opportunities afforded by the study of the local structure of fuzzy spaces, we shall characterize in local terms the closure operation and the closedness property of a fuzzy set (cf. [29]). A fuzzy point ρ is a fuzzy space X is called an adherence point of a fuzzy set Μ e I x if for each Q-neighbourhood U there is a point χ such that U(x) + M(x) > 1. An adherence point ρ : = p{' of a fuzzy set Μ is called a limit point [117] if either ρ Ί± Μ or if ρ e= Μ, then for every Q-neighbourhood U of ρ there is an χ distinct from x 0 such that U(x) + M(x) > 1. The closure Μ of a fuzzy set Μ coincides with the totality of all its adherence points. A fuzzy set is closed if and only if it contains (ς=) all its limit points [117]. Also the following fuzzy version of the classical Yang theorem holds [71], the totality of all limit points for a given fuzzy point ρ is closed [117]. In [160] local properties of the boundary of a fuzzy set are studied. 4. Convergence structures in fuzzy spaces There are two different convergence theories used in general topology that lead to equivalent results. One of them is based on the notion of a net due to Moore and Smith (see [71 ]; another one, which goes back to the work of Cartan and Bourbaki (see [14]), is based on the notion of a filter. Each of these theories has a fuzzy analogue. We begin with an exposition of basic convergence theory of fuzzy nets developed (in the spirit of [71]) by Pu and Liu [117], [118] Fuzzy nets in fuzzy spaces [117], [118]. Let X be a fuzzy space and H the set of its fuzzy points. A map of the form φ: S -* X, where (S, >) is a directed set, is called a fuzzy net in X. We shall also write a fuzzy net in the form (p s ) sz s, or (p s ), where p s : = <^(i). Subnets are defined in an obvious way. A fuzzy net (p s ) is final in a fuzzy net Μ (finally Q-coincides with M) if there is an s 0 S such that p s E Μ (respectively, p s QM) for all s ^ s 0. A fuzzy net (p s ) is cofinal (cofinally

17 Two decades of fuzzy topology: basic ideas, notions, and results 141 Q-coincides) with Μ if for each s Ε S there is an s' > s such that p s / ξ= Μ (respectively p &.QM). A net (p s ) converges to a fuzzy point ρ if it is finally (2-coincident with each neighbourhood of it. A fuzzy point ρ is in the closure Μ if and only if some fuzzy net contained in Μ converges to p. A fuzzy set Μ is closed if and only if no fuzzy net contained in it converges to a p ^M. If a fuzzy net (p e ) s =s does not converge to p, then there is a subnet (p f ) e es> of which no subnet converges to p. A map f:x^ Υ is continuous if and only if the fact that a fuzzy net (p s ) converges to ρ in X implies that the fuzzy net (f(p s )) converges to /(p) in Y. These and many other properties of fuzzy nets are in complete analogy with properties of ordinary nets. However, there are significant differences. For example, there are universal nets that are cofinal but non-final in a fuzzy set. (It is natural to call a fuzzy net universal if from being final in U V V, where U, V S I x, it follows that it is final either in U or in V.) In [117] the notion of a convergence class of fuzzy nets is introduced (in the spirit of [71]). Convergence classes are used for describing all fuzzy topologies on a given set Fuzzy filters in fuzzy spaces. A theory of convergence of fuzzy filters was developed by Lowen [91] for laminated spaces and then extended to arbitrary fuzzy (Chang) spaces by Warren [163]. (We remark that this extension was in some respects nontrivial.) We now present its basic ideas in brief. A fuzzy filter in X is defined as a non-empty family of fuzzy subsets $ (that is, if (Z I x ) not containing 0 and such that: (1) if Μ, Ν e f, then Μ /\ Ν ef; (2) ifjlfs.f and Ν > Μ, then Ν <=.f. Unlike an ordinary filter, a fuzzy filter in general cannot be obtained as the intersection of all maximal fuzzy filters majorizing it. To surmount this difficulty, Lowen considers the family $ m (,<f) of all fuzzy filters that are minimal (with respect to inclusion) in the set φ (.IF) of all principal (in the spirit of [ 14]) fuzzy filters majorizing a given fuzzy filter &, and demonstrates that f = Λ {#: # <= sp m {s r)y The fuzzy adherence set and the fuzzy limit set of a fuzzy filter fr in a fuzzy space are defined by adh ζ = inf {/V: Ν (Ξ f) and lim = inf {adh f: $' <= $ m (f)} respectively. Loosely speaking, adh $ (x) (lim f (x)) is the degree with which χ is an adherence point (respectively, limit point) of the fuzzy filter f. If f, $' are fuzzy filters and f ZD.f, then adh f < adh f, but in contrast to the situation in topological spaces the inequality lim $ <ζ lim $' in general is not valid. A map /: (Χ, τ χ ) -* (Υ, τ γ ) is continuous if and only

18 142 A.P. Shostak if adh / (,f) > / (adh f) f every filter f in I or, equivalently, if t or lim / (&) Ξ> / (lim <f) for every principal fuzzy filter $ in X. Lowen successfully applies the theory of convergence of filters to the study of the compactness and separation properties of fuzzy spaces [91], [92], [168] The interrelation between the theories of convergence of fuzzy nets and fuzzy filters has been studied by Lowen (in preprints). As in the case of general topology, the two theories lead to essentially equivalent results. However, the transition from one theory to another is much more complicated both in conceptual and technical respects than that in general topology Other convergence theories in fuzzy spaces. The convergence theories considered above are the most advanced and seemingly the most successful ones in fuzzy topology. However, there are other (non-equivalent) convergence theories. For example, in [45] a certain convergence structure is introduced by means of fuzzy nets; fuzzy filters and other similar constructions underlie those convergence structures described in [45], [67], [ 17]. Convergence of fuzzy filters in Hutton fuzzy spaces (15.4) has been considered in [56], [57] Fuzzy neighbourhood spaces [93], [94]. With the help of fuzzy filters Lowen has distinguished an important subclass of the class of laminated fuzzy spaces the fuzzy neighbourhood spaces, characterized by the fact that their fuzzy topology may be restored from the so-called fuzzy neighbourhood systems. The class of fuzzy neighbourhood spaces is wide enough to contain, in particular, all topologically generated fuzzy spaces and all fuzzy uniform (in the sense of [95]) spaces. On the other hand, these spaces possess a number of important properties that do not hold for arbitrary fuzzy spaces. To define a fuzzy neighbourhood space, we consider a set X and for each χ G X fix a fuzzy filter f x. A family : = {f x : zge X} is called a [fuzzy] neighbourhood system on X if: (1) N(x) = 1 for each χ G X and each Ν Gr f x ; (2) if Ν ε ΕΞ f x for each ε > 0, then sup {Ν ε ε) <= f x ; ε (3) for arbitrary I G X, Ν GE f x, and ε > 0, there is a family {Nl: ζg= λ*}, where Nl <= f 2 is such that sup (Νϊ (ζ) Λ Ν', (y) - ε)< Ν (y) for every y G X. By putting Μ (χ) = inf {sup (M /\ N)(y): ΝGE f x ), where Μ G I x and ' x Gr, the fuzzy closure operator (1.1) is defined, and by the same token the fuzzy topology t ( ) on X. A fuzzy space (X, r) is called a neighbourhood space if τ = t ( )for a neighbourhood system. (In [93] the property of

19 Two decades of fuzzy topology: basic ideas, notions, and results 143 a fuzzy space of being a neighbourhood space is characterized internally as well, including a characterization by means of closure.) The closure operator in fuzzy neighbourhood spaces has a number of important features. For example, it is uniformly continuous in such spaces: if Μ, Ν I x, where X is a neighbourhood space, then Μ - Ν { < Μ - Ν ( Μ : = sup Μ (χ) ). χ For each constant α: / we have Μ /\ α = Μ f\ a (see also the example at the end of this subsection). The coincidence t ( ) = t ( ') where, ' are two neighbourhood systems on X, holds (if and) only if Ζ = Ζ'. A map {X, t (Z x ))~- (Υ, ί ( Y )) between neighbourhood spaces is continuous if and only if Z" 1 (!fi x,) CI &χ. for every χ 0 X If (X, T) is a topological space, then the neighbourhood system of the fuzzy space (Χ, ωτ) generated by it is of the form : = {sr x : = {N e / Λ : Ν (χ) = 1 and Λ Γ is lower semicontinuous at a point χ) : χ ΕΞ X} A fuzzy neighbourhood space X is Hausdorff (5.2) if and only if for every x, y e Χ, χ φ y, and ε > 0 there are N x e F«, N y S,f with Λ Γ /\ Λ' <ε. Χ ν In conclusion, we consider the following example. Let X be a set, \X\ > 2, τ = {α: α ΕΞ /} U {We / A ': Λ 7 <~ } Then for each xeiwe have % x Λ Φ %x = Xx Λ. an <3 so (X r) is a non-neighbourhood laminated space. 5. Separation in fuzzy spaces Here we shall discuss properties similar to r 0-7Yseparation in fuzzy spaces. The main attention will be paid to T 2 -separation, or Hausdorffness, as the most important property of this kind; the rest of the properties are mentioned only in passing. At present no less than ten approaches (or patterns) to the definition of Hausdorffness in fuzzy topology are known. Some of them differ negligibly from each other, but others do basically. We start with the spectral theory of Hausdorffness, which provides a framework for the description of many other theories of Hausdorffness as particular cases (for example, 5.2, 5.3, 5.4) Hausdorffness spectra of a fuzzy space [139]. For arbitrary a, b / and a fixed >0we put a <; b: ο a <z b for a < 1, 0 b > 0 and a = b for either a = 1 or b = 0; α < 5: ^ a < 6; a<z b: <=> α < b ε.

20 144 A. P. Shostak Let (Χ, r) be a fuzzy space. For each pair (i, j) e {0, 1, 2} 2 we define the Hausdorffness (i, j)-spectrum as the set Hj(X) formed by pairs (β, y) Ε I 2 such that for any distinct x, y G X (and an arbitrary ε > 0) there are U, V G r with U (χ) > β, V (y) > β, and i/ c V V c > γ. Loosely speaking, the fact that the pair (β, γ) belongs to the spectrum H((X) means that for any x, y G X there are neighbourhoods U and V that are "higher" than β at the corresponding point and that intersect "lower" than y c. In this connection "higher" and "lower" are understood either strictly, non-strictly, or "up to ε", depending on what values are taken by / and /. If (β, γ) G Hj(X) and β' < β, y' < y, then evidently (β', γ') Ε Η{(Χ). If i > /', / > /', then H\ (X) Z) H\'> (X); in this case H\(X) is closed and if (β, τ) e //!(*), 0' < 0, V < γ, then (j3', γ) G // (Z). Thus, (/, />spectra for various (/, /) may differ at the boundary only. (It is not difficult to construct examples showing that they can actually be different.) For any fuzzy space X and (i, j)?ξ {0, 1, 2} 2, the set (I X {0}) U ({0} X /): = FCZ Hi (X); in the case where X is laminated, {(β, γ): β + γ < 1} =: G CZ H\ (X). If X is a topological spiice, then HJ(X) = Η{(ωΧ) = I 2 if and only if X is Hausdorff; otherwise H{(X) = F and Η\(ωΧ) = Η\(ωΧ) = G. If Λ", X' are fuzzy spaces and there exists a continuous injection ψ : X -+ X', then H{(X) D HJ(X'); in particular, under transition to a subspace Hausdorffness spectra do not decrease. If X is a product of fuzzy spaces {X a : a(=a}, then H\ (Χ) Ζ) Π Η\ (X a ); if in addition all the X a 's are a laminated, then H\ (X) = Π H[ (X a ). (In the general case the equality need α not hold.) A very important characteristic property of a Hausdorff topological space is that the diagonal is closed in its square. To formulate a fuzzy analogue of this assertion, let us define the notion of closedness spectrum of a fuzzy set in a fuzzy space, which is necessary in what follows. We define the (/, j)-closedness spectrum of a fuzzy set Μ in a fuzzy space X as the set Clj(M, X) formed by pairs (β, y) G I 2 such that (for every ε > 0) there is a W G r with M c V W c > γ and in addition W (χ) ^> β whenever M c (χ) > β. (We stress that the closedness spectrum of a fuzzy set i generalizes the topological notion of closedness in an entirely different direction from that of the definition of a closed fuzzy set (1.1).) The Hausdorffness spectrum of a fuzzy space can now be characterized by HJ(X) = Cl((A, X 2 ). The closedness spectrum also enables us to present the following assertion (cf. [29], Russian ed., p. 171). Let Χ, Υ be fuzzy spaces and let /, g : X -* Υ be continuous maps. Then H\ (Y) CZ Cli ({x: f(x) = g (x)}, X).

21 Two decades of fuzzy topology: basic ideas, notions, and results 145 Properties of the boundary of the spectrum HJ(X) are investigated in [59]. In [139], [155] the spectra Τ{, (Χ) and ΤΪ (X) are also considered, corresponding to the topological properties of T t - and 7Vseparation. With the help of the closedness spectrum the ^-separation spectrum may be characterized as follows: T[i(X) = f)cl](x,x), which evidently corresponds χ- ίο the characterization of topological ΤΊ-spaces as those spaces whose onepoint subsets are closed. We note that H\ (X) f] (γ, i) 2 d T\i (X) for every fuzzy space but, generally speaking, Η] (Χ) φ Τ{ι (X) The Hausdorff property of a level: Rodabaugh's approach. In [121], [126] the α-hausdorff (a G [0, 1)) and a*-hausdorff (a G (0, 1]) properties are studied in detail. Making use of the terminology of 5.1, these properties can be characterized as follows. A fuzzy space is a-hausdorff {a*-hausdorff) if and only if (a, 1) G H&X) (respectively, (a, 1) G H\{X)). It is not difficult to note also that the α-hausdorff (a*-hausdorff) property of a fuzzy space (X, r) is equivalent to the Hausdorff property of the topological space (Χ. ι α τ) (respectively, (Χ, ι τ)), where ι α, tj are the α-level functors (2.6), (2.7). In [126] the a- and a*-hausdorff properties have been extended to the case of fuzzy subsets of fuzzy spaces Separation of disjoint fuzzy points. Pu and Liu [117] call a fuzzy space (Χ, r) Hausdorff if for any two of its fuzzy points p\ and p s y with distinct supports there are Q-neighbourhoods (3.2) U and V respectively such that U /\ V = 0. In [133] a fuzzy space (X, r) is called Hausdorff if for any two of its fuzzy points p* x and p s y such that χ Φ y and s, t < 1 there are U, V G r such that p' u e= F, p' x G? U and U Λ V = 0. It is not difficult to show that the two definitions of the Hausdorff property are equivalent and may be characterized by H\(X) = P. A fuzzy space is Hausdorff if and only if no fuzzy net in it converges to two fuzzy points with distinct supports [117] (see also 5.4). Every Hausdorff fuzzy space is ultra-hausdorff (2.5); on the other hand, there is an ultra- Hausdorff non-hausdorff fuzzy space Hausdorff property and filters. After characterizing Hausdorff (5.3) fuzzy spaces by the fact that the limit of every fuzzy filter in them does not vanish at one point at most, Lowen and Wuyts [168] introduce by means of fuzzy filters a series of axioms of TQ-T 2 type. In particular, they define T'^'-spaces as those fuzzy spaces in which lim ff, where if is an arbitrary fuzzy filter, reaches a positive maximum at one point at most; those spaces in which this requirement is met at least by all principal filters f are called Τ2-spaces. Clearly, T 2 =* T' 2 " => T' 2 '; the reverse implications do not hold in general. For fuzzy neighbourhood spaces (and for topological spaces as well) the conditions T 2, T'2', and Τ % are equivalent.

22 146 A.P. Shostak 5.5. Separation of fuzzy points: the approaches of Adnadjevic and Sarkar. In 5.3 (and implicitly in 5.1, 5.2, 5.4) the Hausdorff property of a fuzzy space was determined by the presence of disjoint ((^neighbourhoods of every two disjoint (that is, having distinct supports) fuzzy points. However, unlike ordinary points, fuzzy points may have a common support but differ in values. The possibility of separation of such points in one sense or another was by no means guaranteed in the preceding subsections. We shall consider here two approaches to the Hausdorff property problem, taking into account the possibility of separation of points with common support. One of them was developed by Adnadjevic [3] and his pupils [58], [72]; the other is due to Sarkar [ 130], [ 131 ]. (We note that the idea of separation of fuzzy points with a common support appears for the first time in [117], where a fuzzy space is called a quasi-t 0 -space if for any pair of its fuzzy points p s and p such that s < / we have p ςέ x x x ps.) x We call a fuzzy space {X, r) A-Hausdorff [3] (S-Hausdorff [130]) if for any pair of fuzzy points p x and p s : y (1A) if χ Φ y, then there are U, V <= τ such that pi e? U, p s ^V_, and y U J\^V = 0 (respectively, (15) if χ Φ y, then pi e U, pi ^ V, pi^v", and p% & U for some U, V r); (2) if χ = y and s < t, then pi e= U and pi ζ U for some U G r. Clearly, every yl-hausdorff space is both 5-Hausdorff and Hausdorff, but not vice versa. For a laminated space the Hausdorff and,4-hausdorff properties are obviously equivalent. Fuzzy points in an S-Hausdorff space are closed. (Adnadjevic calls a space in which every fuzzy point is closed a Τχ-space [3].) We stress that the closedness of fuzzy points (unlike the closedness of one-point sets) was in no way guaranteed by the approaches of Both A- and 5-Hausdorff properties are multiplicative and hereditary The reader may become familiar with some other definitions of the Hausdorff property, for example, via [37], [82], [134], [180], and others (see also 15.6). 6. Normality and complete regularity type properties in fuzzy topology 6.1. Normality. In contrast to the diversity of approaches to the notion of Hausdorffness ( 5), most authors are in agreement with each other when studying the notion of normality in fuzzy spaces, and take as a basis the definition by Hutton [54] or equivalent reformulations of it. To some extent this is explained by the fact that normality in general topology is defined in terms of open and closed sets only, without appealing to points, hence it can be carried over in a standard manner to the language of fuzzy topology.

23 Two decades of fuzzy topology: basic ideas, notions, and results 147 A fuzzy space (Χ, τ) is called normal [54] if for any pair of fuzzy sets A and U, where A ET C, U GT, and A < U, there is a V G τ satisfying the inequalities A < V < V < U. It is easy to verify [108] that a fuzzy space is normal if and only if for each pair of closed fuzzy sets A and Β such that A + B < 1 there are U, V G τ such that A < U, Β < V, and U+V < I. As in the case of topological spaces, the normality property is not preserved under taking products; in addition, contrary to the situation in Top, the normality of a product of fuzzy spaces does not guarantee the normality of the factors [126]. A sufficient condition for the normality of the factors to follow from the normality of a product is that all the factors are laminated. The normality property is inherited by closed subspaces. A closed continuous image of a normal fuzzy space is normal [108], [126]. Hutton [54] characterized the normality of a fuzzy space by means of maps from it to the fuzzy interval (/), having proved a kind of a fuzzy Urysohn lemma (for the definitions of (I), (R) and all the notation involved, see 10.1). A fuzzy space (X, r) is normal if and only if for each pair of fuzzy subsets A and U such that A G T C, U G r, and A < U there is a continuous map f : χ,+ f (/) with A(x) </(χ)(γ) </(x)(0 + ) < U(x) for all χ G X. Recently Kubiak [83] has obtained an analogue of the Tietze-Urysohn theorem: any continuous map /: A -»- (/), where A is a closed subspace of a normal fuzzy space X, has a continuous extension /: X -*- (/). We note that the problem of existence of such extensions has been raised repeatedly in the literature ([121], [123], and others). Kubiak's proof is based on a fuzzy analogue of Katetov's theorem (cf. [29], p. 88) due to him: a fuzzy space X is normal if and only if for each pair of maps g, h: X >- (R), where g is upper semicontinuous and h is lower semicontinuous and g < h, there is a continuous function/: X -> (R) withg <f<h Perfect normality. A normal fuzzy space of which every closed fuzzy set is the infimum of a countable family of open fuzzy sets is called perfectly normal [54]. A fuzzy space (Χ, τ) is perfectly normal if and only if for each pair of fuzzy subsets A and U such that A G T C, U G r, and A < U there is a continuous function /: X -> (I) with A(x) =/(χ)(γ) </(χ)(0 + ) = U(x) for all χ G X, see [54]. (Cf. Vedenissoff's theorem [29], p. 69.) Every continuous map /: A -*- Τ (R), where A is a closed subspace of a perfectly normal space X, has a continuous extension f: X -*- (R). (It is not known whether the condition of perfect normality may be weakened to that of normality (cf. 6.2) [84].) Important examples of perfectly normal fuzzy spaces are (7), (R), as well as their countable powers (f (/))«and ( (R))«. The spaces ( (R))* for k > S o are non-normal. It is not known whether the spaces ( (/))''" are normal for k > fc$ 0.

24 148 A.P. Shostak 6.3. Complete regularity. Functional characterizations of normal and perfectly normal spaces naturally suggest that the complete regularity of a fuzzy space should be defined by means of maps from it tof (/). This is what Hutton [55] and Katsaras [65] undertake, having first considered the properties of this type. Hutton calls a fuzzy space (Χ, r) completely regular if for each UET there are a family of fuzzy sets {M t : i e 3} and a family of maps {frx -*.f (/): i SE 3} such that V; M, = U and M t (x) </, (χ)(γ) </,(x)(0 + ) < U{x) for all i e Cf and χ Ε Χ. (Katsaras's definition [65] differs in form from the definition presented above, but it is easy to see that the two definitions are equivalent.) A completely regular topological space is completely regular in the category of fuzzy spaces as well. A normal fuzzy 7Vspace (5.5) is completely regular. A product of completely regular fuzzy spaces is completely regular; complete regularity is hereditary under transition to a subspace [65], [86]. The most important examples of completely regular fuzzy spaces are f(l), f (R), and products of them [65]. A fuzzy r r space (5.5) is a completely regular space of weight k, k > X o, if and only if it is homeomorphic to a subspace of (f (l)) k, see [65], [86]. A fuzzy space (Χ, r) is completely regular if and only if r is generated by a fuzzy uniformity [55], [86] (cf. [29], p. 523). A fuzzy space (X, r) is completely regular if and only if r is generated by a fuzzy proximity [65] (cf. [29], p. 554) ^-regularity [153]. We recall that a topological space X is called ^-regular [ 10], where Ε is a Hausdorff topological space, if X is homeomorphic to a subspace of E k for some cardinal k. In particular, /-regularity of a topological 7\ space is obviously equivalent to complete regularity. We shall now consider a fuzzy analogue of ^-regularity. We believe that in fuzzy topology the ^-regularity property should play a more important role than its particular case complete regularity (=,f (/)-regularity). The point is that in fuzzy topology along with the fuzzy interval f (1) there are other "canonical" objects: the laminated fuzzy interval $f l (1) (10.1), the fuzzy probabilistic interval Λ (7)(12.1), the interval real line 3 (R) ( 13), and so on, to say nothing of the versions of these constructions in the categories CFT(L). In the relevant situation each of these spaces becomes a "central object" pretending to the role of "interval". The second essential difference between the situation in CFT and the situation in Top is that while the spaces occurring in general topology "as a rule" satisfy at least the 7Yseparation axiom, in fuzzy topology even the axiom 7\ (for example, in the sense of 5.1 or 5.5) seems very restrictive (it is not satisfied by the spaces.f (/), f λ (1), Μ (1), and ^ λ (/)) Hence it is appropriate to distinguish between ^-regularity and -Tychonoff notions for fuzzy spaces.

25 Two decades of fuzzy topology: basic ideas, notions, and results 149 So, let Ε be a fixed fuzzy space. For a fuzzy space (X, r) we put C {X, ): = 0 c (Χ* -ε 1 ") (that is, C(X, E) is the set of continuous maps from the space X to all possible finite powers of E). A fuzzy space X is called Ε-regular if C{X, E) separates points and closed fuzzy sets in X, that is, for every A G T C, Χ 61, and ε > 0 there is an / G C(X, E) such that A (x) > / (.4)(/z) ε. We call a fuzzy space E-Tychonoff if it is homeomorphic to a subspace of fc for some cardinal k. A fuzzy space is.f (Z)-regular ( λ (Ι) -regular) if and only if it is completely regular (respectively, completely regular and laminated). A topological space is completely regular in Top if and only if it is /-regular in CFT. A product of "-regular fuzzy spaces is ^-regular. The ^-regularity property is preserved under transition to a subspace. A fuzzy space (X, r) is is-regular if and only if r is initial for the family of maps C((X, τ), Ε). A fuzzy space X is ^-regular if and only if for every divergent fuzzy net (p s ) in it, having a limit point, there is a continuous function /: X -> Ε sending it to a net divergent in E. A fuzzy space X is -Tychonoff if and only if it is ^'-regular and is-hausdorff (that is, for any x, y G Χ, χ φ y, there is an / G C(X, E) such that fx Φ fy). In the class of HVspaces the ^-regularity and -Tychonoff properties are equivalent. ((Χ, τ χ ) is called a W 0 -space if for any x, y G X, χ Φ y, there is a U G r x such that U(x) Φ U(y).) Hence it follows, in particular, that subspaces of the product f (/) R are precisely the completely regular fv 0 -spaces of weight k (cf. 6.3). 7. Compactness in fuzzy topology The compactness property of a topological space is one of the most important notions not only in topology but in the whole of pure mathematics. Therefore it is natural to pay particular interest to this notion in fuzzy topology, and as a consequence there are many publications devoted to it. We shall consider the main theories of compactness in the sections; on the way we shall touch on the Lindelof, countable compactness, and paracompactness properties. In subsections we consider the problem of fuzzy space compactifications (see also 15.7) Compactness of fuzzy spaces: the approach of Chang and Goguen. The first definition of compactness for fuzzy spaces was proposed in 1968 by Chang [16]; soon after, Goguen [42] extended it to the case of I-fuzzy spaces. It is convenient for us to present this definition here for Z-fuzzy spaces; for L = I it turns into Chang's definition. A family % α τ is called a cover of an Z-fuzzy space (Χ, τ) if \/ % = 1. An Z-fuzzy space is called compact if one can choose a finite subcover % 0 from an arbitrary cover % (that is, % 0 (Z %, \% ο I < Ko, and \J % 0 = 1) [16], [42].

26 150 A.P. Shostak For L = 2 this definition turns into the ordinary definition of compactness of a topological space. On the other hand (for L = Γ) it is clear that there do not exist any (non-empty) laminated compact fuzzy spaces, which is, in Lowen's opinion, a serious disadvantage of this definition. Tychonoff's theorem on the compactness of a product, which is one of the most important results of topological compactness theory, admits under this definition the following "coupe" fuzzy analogue only: the product X of a family of non-empty compact Ζ,-fuzzy spaces {X y : γge Γ}, where ΙΓΙ < k {k is a cardinal number), is compact if and only if the element 1 G L is fc-isolated in L (that is, sup A < 1 for each A C L with \A I < k and 1 A). On one hand, there follows the classical Tychonoff theorem (for L = 2), and on the other hand it follows that only finite products of compact fuzzy spaces are compact (for L I). The similarly defined Lindelof and countable compactness properties have been considered in [165] α-compactness and strong compactness of fuzzy spaces: the approach of Gantner, Steinlage, and Warren [35]. A fuzzy space {Χ, r) is called α-compact, where α G [0, 1), if for every % Z τ such that V % > «there is a finite % C % satisfying V %o > a. A fuzzy space that is α-compact for all α [0, 1) is called strongly compact [35]. A fuzzy space X is α-compact if and only if the topological space ι α Χ is compact (2.4). The notion of strong compactness (in contrast to the compactness of 7.1) is good (2.2). A continuous image of an α-compact space is α-compact. A product of non-empty fuzzy spaces is α-compact if and only if all factors are α-compact [35]. The spaces f (I), f>- (I) (10.1), Μ (I), and J/λ (I) (12.1) are strongly compact [35], [100]; on the other hand, the spaces ^(R),.<P (R) (10.1), M. (I), and Jt k (I) (12.1) are not α-compact for any α G [0, 1), see [35], [100]. The α-lindelof and α-countable compactness properties have been considered in [108]. In [147] the α-lin delof number of a fuzzy space (in the spirit of [29], p. 248, cf. [179]) has been used Compactness of fuzzy spaces: Lowen's approach. Lowen [88] calls a fuzzy space (Χ, τ) compact if for every a G I, every % α τ satisfying \J %^ a, and every ε > 0 there is a finite % 0 C % such that V "Wo > α ε. (If at least the constant α = 1 satisfies this condition, then the space is called weakly compact [90].) The properties of compactness and weak compactness are preserved by continuous maps. A product of non-empty fuzzy spaces is compact if and only if all the factors are compact; weak compactness is preserved under finite products only [88], [90].

27 Two decades of fuzzy topology: basic ideas, notions, and results 151 The compactness property is good. Every strongly compact space is compact. On the other hand, there is a compact fuzzy space (Χ, τ), I X I!> Xo, suc h that for every α [0, 1) the topological space (Χ, ι α τ) is discrete (hence, (Χ, T) is not α-compact for any α [0, 1)) [88], [90]. An interesting property of Hausdorff (5.2) compact fuzzy spaces has been established simultaneously and independently by Lowen and Martin: every compact Hausdorff fuzzy space is weakly induced (2.3) [111], [112]. Every compact Hausdorff laminated fuzzy space is topologically generated (2.2) [92]. Thus, the two properties of different nature, those of compactness and separation, imposed simultaneously on a fuzzy topology, guarantee a close connection of it with a certain ordinary topology on the same set. We believe that results of this kind bear additional witness to the specific role that compactness and Hausdorff properties play together, not only in general topology but in fuzzy topology as well. A closed continuous map /: X -* Υ between fuzzy spaces is called perfect if the inverse image f~ 1 (y) of each point y Ε Υ is compact [34]. Invoking the notion of a Q-neighbourhood, Friedler [34] has shown that a map / is perfect if and only if for every fuzzy space Ζ the product / χ id z is closed (cf. [14], English ed., p. 97; see also [10]). The inverse image of a compact space under a perfect map is compact [141] Spectral theory of compactness [141], [149]. This theory enables us to study very subtle properties of compactness type for fuzzy subsets of fuzzy spaces. It embraces the theories of subsections 7.1, 7.2, 7.3 as particular cases. It may be applied also to a description of compactness type properties of fuzzy subspaces of ordinary topological spaces. Without striving for maximal generality, we present here the basic ideas and results of the spectral theory of compactness. Let (Χ, τ) be a fuzzy space and (i,;) ΕΞ {0,1, 2} 2. We define the (i, j)-compactness spectrum of a fuzzy set Μ Ε Ι χ as the set Q(M) formed by numbers (3 / such that for each % d τ satisfying I c V ^ P ( an d i every ε > 0) there is a finite %* C % such that 71/ci V ^o > β (Here the j notation from 5.1 is used.) The set SC(M) formed by numbers β Ε / such that for each % a x satisfying M c \J (\/ %) > β there is a finite % B c % such that M c \J (\J % ) "^> β is called the strong compactness spectrum of the fuzzy set M. It is not difficult to note that a space X is compact (7.1) if and only if 1 C\(X); a space X is α-compact (strongly compact) if and only if α SC(X) (respectively, SC(X) = [0, 1)); the compactness of a space X in the sense of 7.3 is equivalent to Cl{X) = [0, 1]; finally, a space X is weakly compact if and only if 1 Cf(X).

28 152 A.P. Shostak The connection between various compactness spectra, as well as possible types of compactness spectra for various fuzzy spaces, has been studied by Steprans and Shostak. For example, they have established that a necessary and sufficient condition for a subset A C [0, 1) to be the strong compactness spectrum of some fuzzy space X of countable weight (or, equivalently, the strong compactness spectrum of a fuzzy subset of such a space) is \A\A <Kv Let us dwell on the properties of the spectrum C(M): = Cl(M). To begin with, we present the following simple characterization of it: β e C (M) ^ (V%CZ t {M CI V 11 > β =» sup {MCZ V %* %ocz%,\% 0 <N.}> β)). The number c(m): = inf(/\c(m)) (inf < > : = 1) is called the compactness degree of the fuzzy set M. Clearly, c{m) G C(M) for every M. The reader will easily recognize those facts from general topology that serve as prototypes for the following assertions. Let Χ, Υ be fuzzy spaces, let Μ I x, and let /: X -> Υ be a continuous map. Then C(M) C C(fM). Loosely speaking, continuous maps do not diminish the compactness spectrum. (The changes that the compactness spectrum may undergo under maps of a given defect (1.8) are described in [174].) If Μ, Ν, Κ(ΞΙ χ,μ<ξτ,μ<: Ν, then C (N)CZ C (M), C (M) CZ C (M f\ Κ) and C (N\/K) Z) C (N) f] C (K). The following analogue of Tychonoff s theorem holds. Let X = Π-ΧΊ- be a product of fuzzy spaces X h i e "J, and let Μ = Π ^ ε Ι χ be a product i of fuzzy sets M t e /**, iej. Then c (M) > inf c (M t ). (The inclusion i C (M) ZD f) C (Mj) is not valid in general.) If in addition all the A/,'s are normed, that is, sup M t (J ( ) = 1, then c (M) = inf c (Λ/,) (and, moreover, Π C {Mi) Z) C (M)). l i For a continuous map /: X -» Υ we define the compactness spectrum and compactness degree of it by C (/): = V\ C (f 1 ({/)) and c (/): = inf (7\ C (/)) V respectively. If /: X -* Y, g : Υ -+ Ζ are closed continuous maps and Ν G I Y, then c (/-1 (TV)) >c(j)/\c (N) and c (g «. /) > c (g) Λ c (/) (cf. [29], Russian ed., p. 278). Under the additional assumption f{x) = Υ is it true that c(g of) = c (g) Λ c (/)? (Cf. [29], Russian ed., p. 280.) In general topology statements on the interaction between compactness and separation type properties are of considerable interest (for example, statements on the normality of a Hausdorff compact space and on the equivalency of compactness and absolute closedness properties in the class of r 3 -spaces). The statements presented below testify that an interaction of this kind extends to the category of fuzzy spaces as well.

29 Two decades of fuzzy topology: basic ideas, notions, and results 153 Let (Χ, τ) be a fuzzy space, Μ, Ν G r c, β G Hf(X), β < c{m) Λ c(n), and Μ Α Ν < β. Then for each ε > 0 there are U, V G r such that Wci/ >p-i, JVCF>p-e, and 1/ Λ ^ < β + «Let {Χ, τ) be a fuzzy space and let Μ G /*. Then C(A/) Π Η(Χ) Π η (1/2, 1] C Cl{M) and Λ(7/(Λί) η R(X) Π Η(Χ) Π (1/2, 1] = C(M) η η R(X) Π #(*) Π (1/2, 1 ], where Cl(M) : = C/ftM); #(Λί): = HftM) (see 5.1); ACl(M) : = {/?: (Z is a fuzzy subspace, Ζ D I and J3 #(Z)) =* =* β G C7(M, Z)} is the absolute closedness spectrum of the fuzzy set M, and R(X) is the so-called regularity spectrum of the space X (see [ 141 ], [ 149], [139]). Limitations of space do not allow us to touch here on the problem of characterizing the compactness spectrum of a completely regular space by embedding it in a specific (so-called relatively closed) way into the cubes $ (I)" or the theory of ^-compactness in fuzzy topology closely connected with this problem [ 177]. In conclusion, let us present examples of compactness spectra in the simplest situation for fuzzy subsets of a topological space (X, T). If Μ C X, then C(M) = [0, 1] if and only ii Μ is compact; otherwise C (M) = {0}. Ιΐ Μ ei x is upper semicontinuous, then C(M) = [0, c(m)]; if in addition X is compact, then C(M) = [0, 1 ]. Let X = Υ U Z, where Υ Π Ζ = 0, let 0 < a < b < 1 and Μ : = ay+ bz. If X and Ζ are both compact, then C(M) = [0, 1]. If X is compact and Ζ is not, then C{M) = [0, b c ] U (a c, 1 ]. If X is non-compact and Ζ is compact, then C(M) = [0, a c ]. Finally, if both X and Ζ are non-compact, then C(M) - [0, b c }. Let X = R be the real line, let α G /, Λ/^χ) Ξ β> M 2 (x) = (2α/π) larctanxl, M 3 (x) = 1 -M 2 (x). Then C^) = C(M 2 ) = [0, a c ]; C(M 3 ) = [0, a]. Spectral theories of Lindelofness and countable compactness in fuzzy spaces have been developed [151], [142]. In [143] the so-called spectrum of hereditary Lindelofness of a fuzzy space is studied JT -compactness [158]. The notion of ^"-compactness (from the word "Net") seems to be a very successful one. It was introduced by Wang Guojun and used by a number of Chinese authors afterwards ([87] and others). A fuzzy net (Px^^s is called an α-net, where a G /, if the corresponding number net (a s )ses converges to a.. A fuzzy set Μ in a fuzzy space X is called J/'-compact if every α-net in Μ (α G (0, 1 ]) has a limiting fuzzy point in Μ with value a. A space X is called jr-compact if the constant Μ 1 is ^T-compact. The property of ^"-compactness is preserved under continuous maps and is inherited by closed fuzzy subsets. A product of fuzzy spaces is.^-compact if and only if all the factors are.y-compact. A topological space (X, T) is compact if and only if the fuzzy space (Χ, ω Τ) is ^Γ-compact.

30 154 A.P. Shostak ^"-compact sets possess interesting properties from the map-theoretic point of view. For example, every ^-compact fuzzy set reaches its supremum. In particular, closed (open) fuzzy sets reach their supremum (respectively, infimum) in an J^-compact space. Applying this assertion to (Χ, ωτ), where (X, T) is a topological space, we obtain a generalization of the Stone-Weierstrass theorem: Lower (upper) semicontinuous maps from a compact topological space to IR reach their infimum (respectively, supremum). A strongly compact fuzzy space is ^"-compact if and only if every closed fuzzy subset reaches its supremum Comparing various definitions of compactness. The basic relationships between compactness type properties of a fuzzy space can be represented in the form of the following diagram [35], [90], [104], [143]: ultracompactness (2.5) => JT-compactness =* strong compactness =* compactness (7.3) <* C(X) = [0, 1] => weak compactness *= «= compactness (7.1). The implications converse to those presented above are not valid in general. Nevertheless, within certain rather wide classes many of those properties become equivalent. For example, if a fuzzy space X is Hausdorff (5.2) or weakly induced (2.3), then the following are equivalent: (a) X is ultracompact; (b) X is ^-compact; (c) X is strongly compact; (d) X is α-compact for some a; (e) X is compact (7.3) ([110], [111], [176]; cf. [15]). Within the class of fuzzy neighbourhood spaces the following conditions are equivalent: (a) ultracompactness; (b) ^"-compactness; (c) strong compactness; (d) 0-compactness. A fuzzy neighbourhood space (4.5) is compact (7.3) if and only if it is α-compact for all α (0, 1); however, there are compact (7.3) neighbourhood spaces that are not 0-compact [94]. There are other definitions of compactness occuring in the fuzzy topology literature. For example, F /-compactness [67] defined by means of fuzzy filters; ^-compactness, based on the notion of a Q-cover [85], the so-called probabilistic compactness [51], and so on (see also 15.7). We now consider some approaches to the problem of fuzzy space compactifications. The first of them (7.7) reproduces in the fuzzy situation the idea of Aleksandroff one-point compactification. The idea of the second approach (7.8) worked out by Martin consists in a transition from the category CFT to the category Top via the functor ι with a subsequent return to the CFT. The third approach (7.9) is due to Cerutti; it is based on the ideas and techniques of categorical topology and is mainly applicable to weakly induced fuzzy spaces One-point α-compactification of a fuzzy space [35]. Let (X, r) be a fuzzy space, let α < 1, and let S?«be the family of all closed α-compact (crisp) subsets. We put X X \J {0}, where Ο is an arbitrary

31 Two decades of fuzzy topology: basic ideas, notions, and results 155 element not belonging to X, and for every ies a we define Κ : X -* 2 by R (O) = 1 and K(x) = K c (x) for χ Ε X Further, for each / Ε r we define 7: JT -> / by letting 0 (Q) = 0 and C/fx) = f/(x) for χ Ε Χ. Let Τ α be the fuzzy topology on X determined by the subbase {0: ί/ετ}υ {R- Κ e β α } Then the space (Χ, Ύ α ) is α-compact and (X, r) is a subspace of it; in addition, X is dense (7.8) in (Χ,Ύ α ) if (and only if) the space (X, r) is not α-compact. If in addition (Χ, τ) is locally α-compact (respectively, weakly locally α-compact), that is, for every χ Ε X there is a t/ G r such that U(x) = 1 (respectively, U(x) > a) and ί/~ηθ, 1 ] is α-compact and 1 *-Hausdorff (respectively, α-hausdorff), then the compactification (Χ, Ύ 01 ) is 1*-Hausdorff [35] (respectively, a-hausdorff [121]) Ultracompactifications of a fuzzy space [110], [112], [176]. A fuzzy subset Μ of a fuzzy space (X, r) is called dense if Μ = Χ; Μ is called ultradense if all the sets M~ l [a, 1], α < 1, are dense in the topological space (X, IT). It is not difficult to note that the ultradensity of Μ in the space (X, r) is equivalent to its density in the space (Χ, ωιτ). An ultracompact fuzzy space (bx, σ), containing a given fuzzy space (X, r) as an ultradense subspace, is called an ultracompactification of (Χ, τ). To construct ultracompactifications of a fuzzy space (X, r) we consider the compactification (bx, b(ir)) of the topological space (X, IT) and put T b := {Ui U Ξ ω (b (IT)) and U \x e? τ} It can be shown that T b is a fuzzy topology on bx and in addition (bx, r b ) is an ultracompactification of (X, T). If the space (Χ, ιτ) is completely regular and (βχ, βιτ) is its Stone-Cech compactification, then the corresponding ultracompactification (βχ, τ 0 ) of the initial space (X, r) has the following property, which enables us to call it the Stone-Cech ultracompactification. Every continuous map/: {Χ, τ) -> (Υ, σ), where (Υ, ο) is an ultracompact ultra-hausdorff fuzzy space, has a continuous extension/: (βχ, Τβ) -> (Υ, σ). The following description of all ultracompactifications of a given fuzzy space (Χ, τ) is due to Martin. Let (bx, r b ) be as above and let r b be a fuzzy topology on bx contained in r 6 and inducing τ on X (for example, τί = {0 = sup {F: F E T 6, F. Y = [/}}. Then (bx, r' b ) is an ultracompactification of (X, r). Conversely, an arbitrary ultracompactification of a space (Χ, τ) can be represented in the form (bx, r' b ) for a suitable choice of(bx, T b ) and r' b C r b. In conclusion we stress that though the scheme presented above enables us to describe ultracompactifications only, the scope of its applications is broader than may seem at first sight. The point is that for Hausdorff (5.2) and for weakly induced fuzzy spaces all the main definitions of compactness are equivalent to ultracompactness (7.6). We also remark that the existence of a Hausdorff compactification of a fuzzy space is equivalent to the soace itself being Hausdorff (5.2) and weakly induced [111].

32 156 A.P. Shostak 7.9. The categorical approach to compactification theory [15]. Let % be the category of 1*-Hausdorff (5.2) compact (7.3) fuzzy spaces, and let e : % -* CFT be the natural inclusion functor. The functor β: CFT -+'$, which is left adjoint to the functor e, has a number of properties making it similar to the functor β: Top -* Comp of the Stone-Cech compactification of a topological space. We point out the most important of these properties. The functor ρ is reflexive: for every fuzzy space X there is a morphism r x : X -* βχ such that each morphism /: X * Υ ~ % admits a unique factorization of the form / = φ r x. The equality (3ocj = cooj3is valid, hence the functor β acts on the category ω(τορ), which is an isomorphic copy of Top, precisely as the functor β acts on Top. The two functors β and ιβω are naturally isomorphic, hence β can be recovered from the functor β. Within the subcategory of weakly induced spaces the equality ιβ = βι holds, hence for weakly induced spaces the reflection r x : X -> β~χ coincides with the corresponding reflection r uy : ix ->- fnx. In particular, it follows that % is an epireflexive subcategory of the category of weakly induced 1 *-Hausdorff spaces. Nevertheless, in spite of the fact that the two functors β and jif are closely related to each other and in many respects seem alike, the situation with Stone-Cech compactification in CFT is much more complicated than in Top. One of the reasons is that the category 'β (unlike the category Comp of compact Hausdorff topological spaces) is not algebraic over Set or Top Other approaches to the compactification problem in fuzzy topology. Liu [85], [86] has constructed a Stone-Cech type compactification of a fuzzy completely regular (6.3) space by embedding it into the cube (5" (/))* of the corresponding weight and subsequently taking closure in it. Wang Goujun has investigated..-f-compactifications of a fuzzy space. An entirely different approach to the compactification problem in fuzzy topology is developed by Eklund [25] (see also 15.7). Compactifications of fuzzy subsets of fuzzy spaces have been considered in [177]. In conclusion we dwell on some unresolved problems in compactification theory. 1) To obtain an internal description of compactifications of fuzzy spaces and fuzzy subsets of them. 2) To construct a fuzzy Wallman type compactification [112]. 3) To describe compactifications of fuzzy spaces in terms of fuzzy proximities (in the spirit of Yu.M. Smirnov's theory) [112], involving either an already known [64], [65], [8], [178] or a new fuzzy proximity type structure. 4) To develop a general categorical theory of extensions of fuzzy spaces and fuzzy subsets Paracompactness of fuzzy spaces. The α-paracompactness property of a fuzzy space (in the spirit of 7.2) has been studied in [109]. A deep analysis of the Q-paracompactness property

33 Two decades of fuzzy topology: basic ideas, notions, and results 157 of fuzzy spaces and fuzzy subspaces (in the spirit of Q-compactness, 7.8) has been undertaken in [107]. Paracompactness type properties are considered in a number of other papers. 8. Connectedness in fuzzy spaces 8.1. Connectedness of fuzzy sets: the approach of Pu and Liu [117], [118]. We call a fuzzy set Μ in a fuzzy space (X, r) (C-)disconnected if there are Α, Β <Ξτ such that Μ /\ Α φ 0, Μ /\ Β φ 0, Μ < A V Β, and A /\ Β /\ Μ = 0. We call a fuzzy set (unconnected if it is not disconnected. A fuzzy space (X, r) is called connected if the constant Μ = 1 is connected. The closure of a connected fuzzy set is connected. The connectedness property is preserved by continuous maps. Any maximal connected fuzzy set Κ contained (in the sense of <) in a given fuzzy set Μ is called a component of M. Every connected fuzzy subset (<) of a fuzzy set Μ is contained in a component of M. In addition, if Ν λ and Λ' 2 are two components of a fuzzy set M, then A\ V N 2 is disconnected (both as a fuzzy subset in X and as a fuzzy subset of the subspace M~ l (0, 1 ]). If a fuzzy set is closed, then so is each component of it. A product of non-empty fuzzy spaces is connected if and only if every factor is connected. (Let us note that in the proof of this fact a transition to the laminated modification (2.2 ) is used, which does not influence the connectedness property.) A topological space (X, T) is connected if and only if the fuzzy space (Χ, ωγ) is connected connectedness of fuzzy sets. We call a fuzzy set Μ in a fuzzy space (X, r) O-disconnected if there are U, F r such that Μ /\ U Φ 0, Μ /\ V φ 0, Μ < U V V, and Μ /\ U /\ V = 0; otherwise we call Μ O-connected. Lowen [96] calls a fuzzy space connected if each strictly positive open-and-closed fuzzy subset is O-connected. In contrast to general topology, the use of closed fuzzy sets and open fuzzy sets in definitions of "connectedness" of fuzzy sets results in distinct concepts, C-connectedness and O-connectedness respectively. For example, let (X, 71 be a connected topological space represented as a union X = Υ U Z, Υ η Ζ = 0, Υ, Ζ Φ φ, and 0 < α < 1. We define fuzzy topologies τ,, i = 1, 2, on X by means of subbases m = {β: βge /} U {*/,, F, } U Γ respectively, where U x : = OLY, V X : = az, U 2 ' = U[, V 2 ' = V[. It is easy to verify that in (X, r t ) all the constants 3 G / are C-connected, but not O-connected for 3 < a; on the contrary, in (X, r 2 ) all the constants 3 7 are O-connected, but not C-connected for β < α Pathwise connectedness of fuzzy sets. Let σ be a fuzzy topology on / formed by all U: / -* I such that i7~ 1 (0, 1 ] is an open subset in /. Following Zheng Chong-you [171], we define a

34 158 A.P. Shostak fuzzy path in a fuzzy space (X, r) as a fuzzy subspace of the form φ(1), where φ: (/, σ) -+ (Χ, τ) is a continuous map and / is a C-connected fuzzy subset in (/, σ) such that 1(0) > 0, /(I) > 0. In this connection the fuzzy points ρ'φίο) and p 1^ are called, respectively, the beginning and the end of the fuzzy path φ(1). A fuzzy set Μ Ε Ι χ is called pathwise connected if for any two fuzzy points p, g g= Μ there is a fuzzy path ψ(1) in Μ (that is,?(/) < M) for which ρ is the beginning and q is the end [171]. A pathwise connected fuzzy set is both C-connected and 0-connected. If a family of pathwise connected fuzzy sets has a non-empty intersection, then the union is pathwise connected as well. Every pathwise connected fuzzy subset of a fuzzy set Μ is contained in some pathwise connected component of K. In addition, different pathwise connected components of a fuzzy set are disjoint [171]. (Certain mistakes in the proofs of these facts made in [171] have been corrected by Wuyts [167].) 8.4. A spectral theory of connectedness of fuzzy sets in fuzzy spaces (in the spirit of the spectral theories in 5.1, 7.4) is developed in [149], [150] Connectedness properties of the fuzzy real line and subspaces of it. Rodabaugh [124] has shown that there are no non-empty open fuzzy sets U and V in,<f (IR) such that U /\ V = 0 and (U V V) (x) > 0 for all χ G X (that is, making use of Rodabaugh's terminology, f (R) is 1-connected). Recently Kubiak [175], invoking the Helly space construction [29], has established that the fuzzy interval f (I), the fuzzy cube $ (I) k, and their laminated versions are ultraconnected (2.5), and therefore (because the properties of C- and O-connectedness are good ones) C-connected and O-connected as well. 9. Fuzzy metric spaces and metrization of fuzzy spaces There are several viewpoints of the notions of a metric and metrizability in fuzzy topology. They can be divided into two main groups The first group is formed by those papers in which a fuzzy (pseudo-)metric on a set X is treated as a map d: ϊ χ Χ >- R +, where X CZ I x (for example, -f = I x (Erceg [31]) or X = the totality of all fuzzy points of a set X (Deng [20], Hu Chang-ming [53])) satisfying some collection of axioms or other that are analogues of the ordinary (pseudo-) metric axioms. Thus, in such an approach numerical distances are set up between fuzzy objects. The main problems in which the authors of this approach are interested are: how a fuzzy metric induces a fuzzy (quasi)uniformity (in the sense of [59]) and a fuzzy topology [31], [20], [53]; critieria of (pseudo) metrizability (that in [31] is in the spirit of [29] p. 523, and that in [53] is a fuzzy version of the Nagata-Smirnov theorem); separation properties in (pseudo)metrie spaces [31], [7], [53], [21]; properties of completeness and total boundedness type [7], [21].

35 Two decades of fuzzy topology, basic ideas, notions, and results 159 It follows from the results of [31] and [7] that an approach to the (pseudo)metrizability problem in fuzzy topology from the viewpoint of fuzzy (quasi) uniformities due to Hutton [55], [57] is equivalent to Erceg's approach [31] We include in the second group those papers in which the distance between objects is fuzzy; the objects themselves may be either crisp, or (more seldom) fuzzy. In our opinion, the most interesting papers in this direction are those of Kaleva, Seikkala, and Eklund and Gahler [63], [61], [28]. We define a fuzzy metric on a set X as a map d: X X X -»- CJ (R), where J(R)is the interval real line ( 13), satisfying the axioms: (1) d(x, y) = 0 if and only if* = y; (2)d(x, y) = d( y, x); and (3) d(x, z) <d(x, y)+d(y, z), x, y, ζ Ε Χ. A number d(x, y)(t) is treated in this connection as the "possibility" [170] that the distance between χ and y is equal to t. The pair (X, d) is called a fuzzy metric space [63], [28]. (In [63] the authors proceed from a more general definition according to which a fuzzy metric space is a quadruple (X, d, L, R), where L, R :/ 2 -» / are symmetric decreasing functions with L(0, 0) = 0, R(l, 1) = 1. In the case where L = Min, R Max, this definition is equivalent to the one presented above. On the other hand, for an appropriate choice of L and R every Menger space [132] can be described as (X, d, L, R).) In [61] the notion of a Cauchy sequence in a fuzzy metric space is defined and on that basis the notion of completeness of a fuzzy metric space is introduced. If lim d (x, y) (t) = 0 for all x, y G X, then the space t-a (X, d) has a unique completion up to an isomorphism [61]. Eklund and Gahler [28] worked out a construction that enables us to assign a fuzzy (laminated) topology Ω (d, 'S) on a set X to an arbitrary fuzzy metric d on X and every family of maps φ: (0, 1] -*- 3 (R) satisfying certain conditions. (Unfortunately, we cannot present that interesting but rather cumbersome construction here.) A fuzzy space (X, r) is called metrizable if τ = Ω (d, ')for some d and <S. If % 0 ' = {ψι t ^ R + ), where φ ( Ξ t, then a fuzzy topology Ω (d, ^0) is topologically generated (2.2). In particular, if (X, T) is a topological space metrized with a metric m, then ωτ = Ω (ω (/»), 'S o ), where ω(νη)(χ, y) : = m(x, y). Applying the so-called method of levelwise representation of fuzzy numbers, Eklund and Gahler [28] define a fuzzy metric on the interval real line,7 (R) itself; for different ίί this metric generates different fuzzy topologies Ω (ρ, g) on J (R)(see also 13) Stratifiable spaces. The class of stratifiable topological spaces first distinguished by Ceder and studied in detail by Borges [13] is one of the most interesting representatives of the group of so-called generalized metrizable spaces. In [135] the notion

36 160 A.P. Shostak of a stratifiable space was extended to the fuzzy case. We remark that the property of a fuzzy space of being stratifiable may be used in some cases instead of fuzzy metrizability type properties, and at the same time it is comparatively easily amenable to study in the framework of fuzzy topology. A fuzzy space (X, r) is called stratifiable [135], [136] if it is possible to associate with every U G τ a sequence ( / ) C r in such a way that (l)d n <U for all η e Ν; (2) V U n = U, and (3) if U < V, VET, then U n < V n for all η e N. The definition introduced above is good in the sense of Lowen (2.2). Every fuzzy stratifiable space is perfectly normal (6.2). The class of fuzzy stratifiable spaces is invariant under transition to subspaces and closed images. A fuzzy space dominated by a family of fuzzy stratifiable spaces is stratifiable. A product of countably many laminated fuzzy spaces is stratifiable if and only if all the factors are stratifiable [135], [136]. In [137] maps of stratifiable spaces to the fuzzy interval f (I) are considered. In particular, it is shown that a fuzzy space (Χ, r) is stratifiable if and only if it is possible to associate with every U G τ a continuous map /tr: X -*- f (J) satisfying the condition fu{x){\~) - U c {x), χ G X, in such a way that U < V G τ implies f v <f v. If (Χ, τ) is a stratifiable space, then it is possible to associate with every pair (A, U), where A G r c, U G r and A < U, a function (f AU =) /: X ->- f (I) satisfying the inequalities A(x) </(*)(Γ) </Qc)(O + ) < U(x), χ G X, in such a way that if A < Β and U < V (where Β G r c ', V G τ, Β < V), then JBV <l 1AU- What should be regarded as the analogue of a given topological space (either arbitrary or at least among the most important ones, such as the real line R, the unit interval /, and so on) in fuzzy topology? A kind of answer to this question is contained in subsections 2.1, 2.2, 2.4, where the functors e, ω, and r? have been considered, enabling us to associate with a topological space (X, T) the fuzzy spaces (X, et), {Χ, ωτ), and (Χ, ητ), which can be viewed as fuzzy copies of the original space. An essential feature of these functors is that they change the topological structure only, leaving the underlying set of the space unchanged. However, being to a certain extent a copy of the topological space (X, T), a fuzzy space of type (Χ, μτ) cannot, as a rule, play in fuzzy topology the role taken by (X, T) in the category Top. We may say that objects of the type (Χ, μτ) are too "poor" to fulfil in fuzzy topology the functions of the object (X, T) in Top. In the next five sections we shall consider important constructions of fuzzy topology having an essentially different nature and enabling us to approach the problem in a different way. The most completely investigated construction among them is the so-called fuzzy real line f (R).

37 Two decades of fuzzy topology: basic ideas, notions, and results The fuzzy real line (R) and its subspaces In 1974 Hutton [54] constructed the Ζ,-fuzzy interval f L (I), and 4 years later Gantner, Steinlage, and Warren, by developing Hutton's idea, introduced the Ζ,-fuzzy real line f r, (R) (in the original papers the notations I(L) and R (L) are used, respectively). Laminated versions of the Ζ,-fuzzy real line ϊ (R) and the Ζ,-fuzzy interval \ (I) appear for the first time in [ 126]. The fundamental significance of the Z-fuzzy real line for fuzzy topology (and for the whole of "fuzzy set mathematics") stems from its topological and algebraic properties, which enable us to consider it as the (most important) fuzzy analogue of the ordinary real line, as well as the universal and categorical [125] nature of its construction. We recall also that the fuzzy real line underlies the definition of a fuzzy-valued measure [52], [79], and others. In the present work, following its main trend, we give most attention to the [/]-fuzzy real line (R), that is, to the case L = I; certain features of the general case will be noted in Construction of the fuzzy real lines (R), > (R). Let Ζ (R) be the set of all non-increasing maps z: R ->- / such that sup ζ (χ) = 1, inf ζ (χ) - 0. We introduce an equivalence relation ~ on Z(R) X X by putting z 1 ~ z 2 if and only if z 1 (x + ) = z 2 (x + ) and Zj(x~) = z 2 (x~) for all χ e R (here ζ (x + ): = sup ζ (ί), ζ (χ~): = inf ζ (t)), and let (R) be the t>x t<x set quotient of Ζ (R) by the relation ~. (It is useful to note that there is precisely one left semicontinuous function in each equivalence class [z] e (R), see [123], [97].) We introduce a partial order < on (R)by putting [zj < [z 2 ] if and only if z 1 (x + ) < z 2 (x + ) for every χ X. We define a fuzzy topology σ on f (R)by taking as a subbase the family of fuzzy sets {l b, r a : b, α e R) d / y^) ; where l b, r a are determined by l b [z] = z(b~) c (= 1 z(b~)), r a [z] = z(a + ) respectively. The fuzzy real line and the laminated fuzzy real line can now be introduced as follows: f (R): = (f (R), σ) and f λ modification functor (2.2 ). The subspaces of the fuzzy real line of the form (R) : = (f (R), λσ), where λ is the laminated f(a,b) :={[z]:2gz(r), 2 (» + ) = l,.(ii>0} and f la, b] := {[zh zez(r), ζ (<r) = 1, ζ (b + ) = 0}, where a, b 6Ξ R, a < b, are called open and closed fuzzy intervals, respectively. In particular, f 10, 1] = $ (I) in the Hutton closed fuzzy interval [54]. It is not difficult to show that the spaces jf (a, b) and $ (R) are homeomorphic. We expect the reader will have no difficulty now in defining the spaces.f la, b), (a, b], f [a, + oo), f ( e», b], (a, +oo), f ( oo, b) and their laminated versions, and in establishing which of these spaces are

38 162 A.P. Shostak homeomorphic to each other. In the sequel we shall also need the following notation: f (R + ) := (0, +00), f (R-) : =f (-00,0), f (R ) : = = U ^(-οο,-ε). ε>0 By identifying a number αε R withlx(_ cxi, a ]l, that is, with the equivalence class containing the characteristic function of the set (, a], we are able to consider [R as a subset of f (R). Moreover, the fuzzy topology σ induces on R the ordinary (order) topology T< (and λσ induces the fuzzy topology ω7 τ <), which makes it possible to speak about the real line as a (canonical) subspace of the fuzzy real line f (R). We remark that a construction of Μ (R)analogous to the construction of f (R) was proposed independently by Hohle [50]. Up to an isomorphism 3f, (R)may be characterized as the set f (R) endowed with the weakest fuzzy topology σ* containing σ such that (P6 σ*, a ^ I) -+ {{U + a) /\ ί ^ σ*, (C7 a) V 0 <Ξ σ*) (For properties of Μ (R) see also [ 100]-[102].) Algebraic properties of the fuzzy real line. In [123], [127] Rodabaugh defined the sum Θ and product Ο operations on the set $ (R). The definitions given by Rodabaugh are very cumbersome and require additional constructions. In [97] Lowen showed that the sum of elements [zj, [zj e f (R) can be characterized by [z t ] θ h 2 ] (x) = = sap (z x (t) /\ z 2 (χ ί))> where on the right hand side there appear left semicontinuous representatives of the corresponding equivalence classes. Without dwelling on the definition of the product O, let us present, following Rodabaugh [127], the basic algebraic properties of the system (f (R), θ-θ,ο- The operations and Ο induce on R, viewed as a subspace of & (R), the ordinary sum and product, respectively, (f (R), φ) is an additive Abelian semigroup with zero OeR, and (& (R), Q) is a multiplicative Abelian semigroup with unity 1 Ξ R; in addition, is distributive Let a,i f (R). Then a Q b = 0 <<=> either a = 0 or b = 0; a Q b <Ξ E R \ {0} =* a, b <Ξ R \ {0}. If a < b, then a Q < b 0 e f or e e f c (R+ ) and 6 0 c < α G c for c e " (R~) Both sets f (R + ) and f (R") are invariant under the operation φ. The elements of the set f (R ) are called fuzzy zeros. It is not difficult to show that f (R ) Π R = {0}; a Q b <=: f (R ) ^ either a e f (R ) or i 6 i (R ); if a, b ΕΞ f (R ) then a 0 6 e.f (R ); f(r ) I =! ^(R) = c. It is possible to cancel elements lying outside if (R ): ifaq^ = a O α φ Of (R ), then b = c.

39 Two decades of fuzzy topology: basic ideas, notions, and results 163 For every s f (R)weput-a: = (-l)aand A a := {b: a - - b ΕΞ f (R )} The elements of A a are called fuzzy additive inverses to a. It is not difficult to verify that -a e A a, A a f) R = {a} for ο Ε R and 4 0 Π R I = c for α φ R. The elements of the set f (1R 1 ) := {a + 1: fl6f(r 0 )} are called /wzzj units. Rodabaugh shows that f (R 1 ) Π f (R ) I = c; ifo,6ef (ir 1 ), then a Q J E f (R 1 ); ifa.l f (R 1 ) Π f (R ), then a + δ ΕΕ,f (R 1 ); ^ (R 1 ) Π R = {1} The elements of the set M a := {ίιε,ί (Κ): aqtef (R 1 )}, where α Φ Ο, α Ε ίγ (R), are called fuzzy multiplicative inverses to a. If α = R, thenm 0 Π R = $, otherwise Λ/ β Π R! = c; M a \ = M a f] A a = c. (Rodabaugh [127] calls any algebraic system (H, +,, <) satisfying the properties similar to those listed above a fuzzy hyperfield. The prefix "hyper" indicates that such a system is obtained as a fuzzy extension of an ordinary field, in our case R.) Continuity of algebraic operations on fuzzy real lines. The continuity of the sum.f (R) x.f (R) -+ f (R) was established [123] by using hypergraph functors (2.8); soon after, a simpler proof of the same fact was obtained [97]. In [127] the continuity of the product operation Of( R) xi( R)-^f (R) was proved. It is important to note, however, that for fixed a e= f (R) and 6 ε ί (R ++ ) U & (R") the maps h a, g b : $ (R) -»- & (R), defined by h a (s) :=-- s φ α and i t (s) := i 0 s, s ε f (R), in general are not homeomorphisms but only weak homeomorphisms, because the inverse maps Ία 1! gb 1 f (R) -* f (R)> being weakly continuous, need not be continuous [123], [127]. (The property of weak continuity for a map/: (Χ, τ χ ) -» -* (Υ, τγ) distinguished by Rodabaugh [122], which is intermediate between the property of continuity of/itself and the property of continuity of the map/: (X, LT X ) -> (Y, LT Y ), is an interesting and specifically fuzzy property of maps between fuzzy spaces.) If a, b e R and b Φ 0, then both h a and g b are homeomorphisms [123], [127]. In connection with what we have said above it is interesting to note that while f (R) is not α-compact for any a E [0, 1), the space f (R) φ α is 0-compact for α G f (R) \ R. Thus, a "shift" of the fuzzy real line by a "fuzzy vector" results in an essentially different object! It is not known [127] whether the space <f (R) Ο b is 0-compact (or at least α-compact for some α e [0, 1)) when l e f (R ++ ) U f (R"~). All we have said above remains valid for the laminated real line % (R) as well. It is not known if there exists a continuous extension of the algebraic operations + and from the complex plane C = R x R to the fuzzy space f (R) χ f (R) containing it in a canonical way [123].

40 164 A.P. Shostak Topological properties of fuzzy real lines and fuzzy intervals have been discussed in the corresponding subsections (6.2, 6.3, 7.2, 8.5, and so on) The role of the fuzzy real line in fuzzy topology. We have tried to clarify this role when considering the corresponding problems. In this connection we recall the "fuzzy Urysohn lemma" (6.1), the "fuzzy Vedenissoff theorem", the "fuzzy Tietze-Urysohn theorems" (6.1, 6.2), the functional characterization of stratifiable spaces (9.3), and so on. The fuzzy real line is successfully and "purposely" (that is, as an analogue of the ordinary real line) used also in the the theory of fuzzy uniformities [55], [128], fuzzy proximity theory [65], in certain constructions of compactifications ([85], [86], [177], and others). We note also that every statement in the category CFT involving the fuzzy real line remains true in the category LCFT with ;F(R) replaced by the laminated fuzzy real line f x (R) (by the way, this is a manifestation of the universality of the fuzzy real line construction) On the I-fuzzy real lines f L (R) and fl (R). Substituting an arbitrary fuzzy lattice L for the interval / in the definitions of 10.1, we arrive at the constructions of the Z-fuzzy real lines f L (R), fl (R) and the Ζ,-fuzzy intervals f L (7) and f\ (I), see [54], [35], [ 126]. In particular, for L = 2 the 2-fuzzy real lines \ (R)and ^" 2 (R) are isomorphic to the ordinary real line R, and the 2-fuzzy intervals fr* (7)and \ (/) are isomorphic to the ordinary interval /. Properties of Ζ,-fuzzy real lines and their subspaces may depend heavily on the choice of Z. Let us illustrate this by just one example. Artico and Moresco [9], while studying the property of a*-compactness (a G Z + ) of the space.ft (/)(= the compactness of the space ι α. (f L (7))(2.7)), established that if Ζ is a chain (for example, Ζ = Γ), then ^L (/) is a*-compact for all a; but if L = 2 Z for a set Z, then f L (I) is a*-compact if and only if either α < tf 0 or α > c. 11. Fuzzy modification of a linearly ordered space The main idea of constructing the fuzzy real line was used in [147] to construct the fuzzy modification (fuzzification) of an arbitrary linearly ordered space. Let X be a linearly ordered space. We define the set Z{X) and an equivalence relation ~ on it by analogy with 10.1, and let Ζ (Χ) be the set quotient of Z(X) by the ~ relation. Then by analogy with 10.1 we define a fuzzy topology σ on X (X) and put X (X) := (Χ (Χ), σ), X>- (X) := (Χ (Χ), λσ).

41 Two decades of fuzzy topology: basic ideas, notions, and results 165 The fuzzy spaces X (X) and Χ λ (Χ) are called respectively the fuzzy modification {fuzzification) and laminated fuzzy modification {laminated fuzzification) of the linearly ordered space X. Clearly, X (R) is precisely the fuzzy real line f (R), and Χ λ (R) is the laminated fuzzy real line $~? " (R). Although the spaces X (I) and f (/) are distinct, an isomorphism between them is easily constructed. By identifying an element a G X with the function class iy.(_, O ]] = X (X), we canonically identify X with a subspace of X (X), and ωχ with a subspace of Χ λ (Χ). The weight of a space X is equal to the weight of the fuzzy space X (X), and if Ζ > $ 0, it is equal to the weight of Χ λ (X) as well. (We define the weight of a fuzzy space as the minimal cardinality of a base for a fuzzy topology of it.) The following conditions are equivalent: (1) the space X is bounded (that is, there are maximal and minimal elements); (2) the space X (X) is strongly compact; (3) the space X (X) is α-compact for some a [0, 1). If X is not bounded, then for every a [0, 1) the α-lindelof number (7.2) of X (X) is equal to the cofinal character of X, see [147]. A fuzzy space X (X) is stratifiable if and only if X is metrizable [ 147]. Similar statements are also valid for λ (Χ). Let (Χ, <^χ), (Υ, <ζγ) be linearly ordered spaces and let /: X -* Υ be a nondecreasing continuous map. For every ζ = Z{X) we define u z Z{Y) by putting u z (y) : = inf {z (x): f (x) < y} if (<-, ν] ΓΊ f{x) Φ φ and u z {y): = 1 otherwise. The map / χ (Χ) -κ j (5 r ) defined by f[z] = [u z ] is continuous. If/: Χ-»- Υ is an increasing homeomorphism, then the map /: X {X) -*- X (Y) is a homeomorphism as well. Thus, the fuzzy modification X can be viewed as an embedding functor from the category Ord of linearly ordered spaces and continuous non-decreasing maps into the category CFT; this functor associates with a linearly ordered space X the fuzzy space X (X) and with a continuous non-decreasing map/: X-* Υ the continuous map f:x(x) -+ X(Y). Similarly, a laminated fuzzy modification can be treated as a functor &; Ord-*- LCFT [148]. In the case when the linearly ordered space X is connected, the fuzzy modification X (X) is isomorphic to the construction of tt (X) due to Klein [75], see [148]. 12. Fuzzy probabilistic modification of a topolbgical space It is not difficult to note [40] that elements of the fuzzy real line f (R) can be treated as distribution functions on R and by the same token one can arrive at a probabilistic interpretation of the fuzzy real line. The first to distinguish explicitly the probabilistic aspect of the fuzzy real line was Lowen [100] -[102]. (Similar ideas are traced in Hohle's works [49], [50], and others.) Developing the probabilistic treatment of the fuzzy real line,

42 166 A.P. Shostak Lowen extended this construction to the class of all separable metric spaces: he associates [100] with every separable metric space X a fuzzy space on the set JI (X) of probability measures on X; for X = R this construction leads to the fuzzy real line f (R)as a particular case. In [140], [44] this construction is extended to the case of an arbitrary topological space X (and even an arbitrary fuzzy (1.4) space X) Construction of the fuzzy probabilistic modification of a topological space [140], [144]. (For separable metric spaces see [100], [102].) Let (X, T) be a topological space, SB (X) the σ-algebra of all Borel subspaces, and Ji (X) the set of probability measures on X (that is, σ-additive maps p: 3d (X)-*-I). For every family ξ C ωτ (2.2) we define a fuzzy topology Tj on JI (X) by taking as a subbase the family of fuzzy sets where 6 V (p) : = U dp. e ξ} C We denote the resulting fuzzy space by j/ s (X) ;= {Ji (Χ), τξ). In the cases ξ = Τ and ξ = ωτ the construction of the fuzzy probabilistic modification Jl% (X) can be viewed as an embedding functor Ji%: Top ->- LCFT, associating with a topological space X the fuzzy space -Mi (X) and with a continuous map /: X -> Υ the (continuous) map /: M\x (X) -** *& (X) defined by f(p) {E):=p (Γ 1 (Ε)) {p<=m (X), E^SB (X)). Let Ζ <Ξ SB {X); we denote by J&\ (X) the subspace of JCi (X) formed by those measures ρ for which p(z) = 1. Then for ξ = Τ and ξ = ωτ the map φ: Μ % (Ζ) -> Jif (Ζ) defined by φ (ρ) (Ε):= ρ(ε [} Ζ){ρ(Ξ Ji (Ζ), Ε <Ξ SB (X)) is a homeomorphism. Let X be a r o -space. By assigning to each point χ X the corresponding Dirac measure p x (that is, p x {E) = l*x )we identify X with the subset 25 (X) of all Dirac measures of the space JI (X). Under this identification the topological space X (respectively, the fuzzy space ωχ) is homeomorphic to the subspace 3) (X) of the space Jl\ (X) if and only if ξ is a subbase for Τ (respectively, ξ is a subbase for the fuzzy topology ω Τ). Thus, by a "successful" choice of the system ξ the ΤΌ-space Χ "expands" to the fuzzy space J s (X), which contains the initial space as a (crisp) nucleus, or a basis; the measures ρ Ξ Μ\ {Χ) \ 3) (X) can be treated as kinds of fuzzy points of the space X (cf. 3.1). By means of the functor ι the fuzzy topologies considered above are closely related to the so-called Aleksandrov topology [ 5 ] W on the probability measure space JC (X). (In a number of cases, in particular, if Ζ is a separable metric space, W coincides with the so-called weak topology used in measure theory (see, for example [11], [157]). In particular,

43 Two decades of fuzzy topology: basic ideas, notions, and results 167 it T = ιτ,,,τ = W. However, as the "forgetful" functor should do, under transition from CFT to Top the functor ι loses part of the useful information, which makes it impossible to investigate the constructions considered here in the classical measure-theoretic framework. We shall illustrate this by the following two examples. For a separable metric space X, the set S> (A) is known to be closed in the weak topology ([11], and others). The following equality, which is valid for an arbitrary topological space X, is more subtle: 3(X)(p) = sup {/>{*}: xtex), where 3) (A) is the closure of 3J (X) in either.mr (A) or Μ ωτ (Χ), and ρ is an arbitrary smooth measure. The meaning of this equality is that the degree to which a smooth measure ρ is an adherence point for H> (X) in the spaces JKi (A') and Μ ωτ (Χ) is equal to the maximal value of its atoms. We present also the following equality characterizing the closure of the set CK (X) of all two-valued measures in the spaces Ji T (X) and Jt (dt {X) (X is a topological space): X (X) (p) = sup {/: if /, G Τ and p(t/, ) > 1 - t, i = 1,..., n, then ί/j f]... f] U n Φ 0} (compare with the well-known statement [157] on the weak closedness of X (X)) The connection between the topological properties of a space X and its fuzzy probabilistic modification we shall illustrate by the following assertions [144]. (For a separable metric space see [100].) The weight of X is equal to the weight of J! T (X), and for ω (A") > No is also equal to the weight of Jl at (X). The density of X is equal to both the density (that is, the least cardinality of dense subspaces) of Ji T (X) and the density of ~# ωτ (λ'). The following conditions are equivalent: (a) the topological space X is compact; (b) Jl T (A) is compact (7.3); (c) J( T (X) is countably compact (in the spirit of 7.3); (d) Μ ωτ (Χ) is compact (7.3); (e) J( at (A) is countably compact (7.3). Moreover, in every item b, c, d, e the condition of (countable) compactness can be replaced either by the condition of ultra (countable) compactness (7.6) or by the condition of strong (countable) compactness (7.2). The separation properties of the fuzzy modification Μ ωτ are very delicate. For example, if X is perfectly normal and \X\ > 2, then #ϊ(^ωτ(χ)) = {(β, γ): βίξ/, β + Τ<1}\{(0, 1)} (the notation follows 5.1). The separation properties in jf T significantly worse. (A) are Construction of the fuzzy probabilistic modification as a generalization of the fuzzy real line. Assigning to every ρ <Ξ Jl (R)the distribution function [40] - P :IR ->- / defined by z p {t): = p(-, t), t e R, and putting φ(ρ) := [ 1 z p ], we arrive

44 168 A.P. Shostak at a map ψ: Ji (R) -»- f (R). Endowing the set JK- (R) with the fuzzy topology r ff, where π := {( oo, a), (fc. -f co): a, fc lr}isa standard subbase of the real line, we make the map ψ: J! n (R) -» f (R) into a homeomorphism [100]. This enables us to consider the construction of the fuzzy probabilistic modification as an essential generalization and at the same time a standardization of the fuzzy real line construction. We also note the following [ 148]. If X is a linearly ordered space of countable character without isolated points, then ψ: Jl^ (X) -+ χ (X) is a homeomorphism. (π and φ are defined here by analogy with the preceding paragraph.) As Lowen [100] showed, in spite of the fact that the fuzzy topology τ π on Λ (R) is weaker than τ τ, and its laminated version τ is weaker than τ ωτ (in particular, their separation properties are considerably worse), the equality vr n = ιτ^' = W nevertheless remains valid (cf. 12.1), which, in Lowen's opinion, is of fundamental value. We also point out the identity of the compactness properties for all spaces of the type J! n (X), M T (X), Λ ωτ (Χ), where X is a linear space of countable character without isolated points [100], [148]. 13. The interval fuzzy real line Along with the fuzzy real line f (R)and its variations.f λ (Κ), Jl (R), Μ (R), and so on, in the literature on fuzzy topology another construction based on the real line R is used, which we call here the interval real line The construction and algebraic properties of the interval real line J(R) [63]. We define a regular fuzzy number as a map z: R >- / that is convex (that is, r < s < / implies min {z (r), ζ (<)} <; ζ (s) for all s, r, t 6Ξ R), normed, upper semicontinuous, and such that every level of it z a : = z~ x [a, 1 ], a G (0, 1 ], is a bounded subset of R (that is, taking into account the preceding conditions, z a = [z al, s a2 ] for some z al, z a2 e= P.) The set of all regular numbers forms the interval fuzzy real line J (R). Identifying a number α Ξ R with the characteristic function χ α, we may consider R as a subset of the interval real line. The sum and product of regular fuzzy numbers are defined respectively by (u + v) (t) := sup u(s) Λ V (r); (u-v) (t) := sup u (s)/\v(r). Clearly, these operations induce on R the ordinary sum and product operations respectively. We put u~v : = u + ( \)v and (-I)M = : u. Then {-u)(t) = = u( t) for all ier; (u + v) = ( «) + ( u); both equations a + z 0 and az = 1 (α, ζ ΕΞ % (R) and α Φ 0 in the second case) have a solution (which is unique) if and only if α Ξ R ([63], see also [115]). By putting u <; ν (u, ν ΕΞ 3 (R)) if and only if u ai < u ai, ζ = 1,2, for all α G (0, 1 ], we obtain a partial order on Cf (R). In addition, if u, ν, ζ > 0.

45 Two decades of fuzzy topology: basic ideas, notions, and results 169 then z(u + v) - zu + zv. If u < v, then υ < u and u + ζ < ν + ζ for every 2ΕΞ 3 (R). Each ζ ΕΞ J (R) can be represented as ζ = ζ + + ζ~, where z~, z + are regular fuzzy numbers determined by their α-levels zz% : min {z ai, 0} and Zai := max {z ai, 0}. We have u~ + v~ < (M + U)~, (t/ + u) + < u + + v +, (u-v)~ = inf {u"y +, u*v~), (u-v) + = sup {M~I>~, u + v + } [28] Fuzzy metric on y ((R). As we have mentioned, the interval real line plays a key role in the definition of a fuzzy metric (9.2). Following [28], we now show how a fuzzy metric is introduced on,*f (R) itself. For u, ν j(ir) and α Ε (0, 1 ] we put δ (u. υ) (a) : = \ u al Vai \ -r \ u a2 i' o2 I and let I u, ν α := [0. sup δ (u. ν) (β)]. It can be verified that the intervals \u, v\ a, a. E (0, 1] uniquely determine a regular fuzzy number lu, ι; I of which they are α-levels, and that the map d: J{R) χ J(R)-v J (R)defined by rf(«, u): = \u, v\ is a fuzzy metric (9.2) on J (R). It is also shown in [28] that \u + z, v + z\ = \u, v\ and \uz, i;zl< < 10, ζ I IM, υ I for all u, ν, ζ ΕΞ Cf ((R), but in general the equality 10, u v\ = \u, v\ fails (and therefore a fuzzy metric cannot be defined by means of a "fuzzy norm") 14. On hyperspaces of fuzzy sets Since we cannot consider the construction of hyperspaces of fuzzy sets in detail in this survey (this would require both a considerable amount of space and the introduction of new notions), we shall try to give the reader a certain intuitive idea of hyperspaces of fuzzy sets, constructions which aside from a theoretical interest for fuzzy topology itself may be used in applications The hyperspace of fuzzy sets of a uniform space: Lowen's approach [98], [99]. Let X be a uniform (topological) space; Lowen defines [95] two fuzzy uniformities on the hyperspace I x of fuzzy subsets of X, the so-called global fuzzy uniformity % s and the horizontal fuzzy uniformity % h ; we have % g C %,, We put I* := (/*, % g ), / ' := (/* \ {0}, %,). On the hyperspace 2 X of non-empty subsets of X, considered as a subspace of I x, the fuzzy uniformities % e and %h induce the classical Hausdorff-Bourbaki uniformity ([14], English ed., p. 172). Convergence in the space I x is determined by the topology of X, characterizing by the same token a "horizontal" convergence of fuzzy sets; convergence in the hyperspace I x is determined by both the topology of X and the topology of /, characterizing by the same token the "global" convergence of fuzzy sets.

46 170 A.P. Shostak Both in I x and in If the lower parts of the graphs of fuzzy sets influence the convergence less than the upper parts do. Therefore the two uniformities forms a sufficiently fine device for describing the proximity and convergence relations between fuzzy sets whose lower parts are either inaccurately defined or not defined at all, which may prove to be useful in certain applications (see, for example, [23], [81]). If /: X -* Υ is a uniformly continuous map between uniform spaces, then the maps /: I x -* 1%, / : -^/ ~* Ij (0-6) are uniformly continuous as well (in the sense of [95]) and therefore continuous. Let Φ χ (Χ) be the subspace of If formed by all U G I x that are upper semicontinuous. A uniform space X is compact (precompact) if and only if the fuzzy uniform space Φ χ (Χ) is compact (7.3) (respectively precompact [104]). The uniform space ι ν (Φ β (Χ)), where ι υ is the uniform analogue of the functor ι (2.4) (see [95]), is isomorphic to a closed subspace of the hyperspace 2 Xxl endowed with the Hausdorff-Bourbaki uniformity (the corresponding isomorphism sends each Μ G I x to its endograph {(*, t): t < Μ (χ)} e 2*x') [80], [81]. The fuzzy space If is not topologically generated, whatever the initial space X may be. Also, the subspace ΐζ\ of I x formed by those Μ G I x for which sup M(x) = a is not topologically generated for any a G (0, 1 ]. Lowen stresses that the existence of natural and important (laminated) fuzzy spaces such as I x and I* h that are not topologically generated (and therefore cannot be studied by means of ordinary topological methods) is evidence of the need for a general theory of fuzzy topological spaces Other structures on the hyperspace of fuzzy sets of a metric space X. Kloeden [80] defines a pseudometric on I x as the Hausdorff distance between endographs of the corresponding fuzzy sets. A similar pseudometric on I x was considered by Goetschel and Voxman [38], [39]. Heilpern [46] made use of a pseudometric on I x in which the distance between fuzzy sets is defined as the supremum of the Hausdorff distances between their α-levels over all α G (0, 1]. Kaleva and Seikhala [63] define convergence of a sequence of fuzzy sets (M n ) C I x to Μ G I x as convergence for each a G (0, 1] of the corresponding α-level sequences (M~^l[a., 1]) to M~ 1 [OL, 1] in the Hausdorff pseudometric on the space 2 X. A comparative analysis of these approaches in the case λ' = Κ" is performed in [62]. 15. Another view of the subject of fuzzy topology and certain categorical aspects of it In the preceding sections, as a rule, we have considered fuzzy topological spaces in the classical [169], [16] sense, that is, fuzziness has been treated as the possibility for characteristic functions to take values in the interval /.

47 Two decades of fuzzy topology: basic ideas, notions, and results 171 The general case of L-fuzzy topological spaces, where L is a fuzzy lattice, has been considered occasionally and for illustrative purposes only. We stress, however, that although properties of Z-fuzzy spaces may depend heavily on concrete properties of the lattice L, and their study sometimes meets additional serious technical difficulties, the theory of Ζ,-fuzzy spaces for an arbitrary (fixed) fuzzy lattice is based as a whole on the same ideas as the "classical" theory presented above, and in this sense it can be viewed as a direct generalization of the latter. An entirely different view of the subject and problematics of fuzzy topology arises if we consider Z-fuzzy spaces for various fuzzy lattices L simultaneously. This view of the subject of fuzzy topology, the idea of which goes back to Hutton's papers [56], [57], has been developed successfully by Rodabaugh [125], [128], Eklund and Gahler [25], [28], and the author [145], [154]. (Similar ideas are contained in T. Kubiak's dissertation (unpublished).) In this section, following the papers [145], [154] (based on the fundamental ideas of Hutton [56]), we shall define the category GFT (the so-called general category of fuzzy topological spaces), discuss certain specific properties of it, and show how all the categories of fuzzy topology described above may be identified with relevant subcategories of it. In we shall dwell on the category φ of Hutton fuzzy spaces and try to illustrate by an example of it the specific features of the topological theory that arises The category GFT [145], [154]. Let Lat be the category whose objects are fuzzy lattices (0.3) and whose morphisms are maps ψ: 1^ -* L 2 preserving the supremum, infimum, 0, 1, and involution. (The category Lat p opposite to Lat [43] is exactly the category FuzLat defined by Hutton [56], see also [27].) The objects of the category GFT are triples (X, L. if) where X is a set, L is a fuzzy lattice, and 0Γ: L x -* L is an I-fuzzy topology (1.4) on X. We declare the morphisms of GFT to be those pairs (/, φ): (Χ Ύ, L x, 3Ί) -> (Λ' 2, L 2, Γ 2 ) that are morphisms in the category Set X Lat 01 ' (that is, /: Χ γ -> Χ 2 is a settheoretic map and φ' 1 : L 2 -* L 1 is a morphism in Lat) and, moreover, such that #"j (φ' 1 ο Ν = /) > φ" 1 (<f (Ν)) for each A 7 G 7 y a kind of continuity condition! (Following the terminology of Eklund and Gahler [28], we may say that Set χ Lat p is declared to be the ground category for GFT. In contrast, for the categories Top, CFT, FT, CFT(L), LCFT(L), and so on, the ground category is the category Set\) If (/,, φ χ ):(Χ-,, L x, 3Ί) -»- (X 2, L 2, <f 2) and (/ s, φ 2 ): (Ζ 2, L 2, it. 2 )->- (X 3, L^f 3 ) are morphisms in GFT, then their composition is defined as (/ 2 f t, φ 2 cp^: (X,, L x, Γ,) -*- (Z, Z, $"? ). As identity morphisms we take pairs of the form (idx, id L ): (X. L, f) ->-

48 172 A.P. Shostak Product in GFT [145], [154]. We shall illustrate the features of the viewpoint presented above of the subject of fuzzy topology, as well as the specific character of the problems arising, by an example of the product in GFT. Let Χ, Υ be sets, L, Κ fuzzy lattices, 8: K Y ->- Κ a A^-fuzzy topology on Y, and (/ φ): (X, L) -* (Y, K) a morphism in Set χ Laf p. The weakest Z-fuzzy topology if on X making (/, φ): (X, L. f) -*-(Y. K, S) a morphism in GFT is called initial for the pair (/, φ). It is not difficult to verify that it can be defined by if (U) : = V i^a ~ & ( (/ ) Λ Φ" 1α : a GE K+), U G L x, where F a : = {<f l on i : ΝEE K Y, 8 (N) > a}. Now let {(Y y,l y, #\); γ6ξ Γ} be a family of fuzzy spaces and {(/ γ, ψγ): (λ 7, Ζ) ->- (Υ ν, L y ) : y Gr Γ} a family of morphisms in Set X Laf v. The weakest Z-fuzzy topology et on Ζ making all (f y, ψ Ί ) continuous is called initial for that family. It is easily verified that 3~ = sup SF y (1.6), where 3~ y is the Ζ,-fuzzy topology on X γ initial for (f y, ψ Ί ). Turning to the definition of the product in GFT, we first recall the operation of fuzzy lattice product introduced by Hutton [56]. Let {L y : γ ε Γ} be a family of fuzzy lattices. The elements of the lattice L = (g> L y are declared to be subsets a d II {L y : y Ξ Γ} (Π stands for the product in Set) such that 1) if t G a and s < t, then s G a (for j, / ΠΖ, + the inequality s < t means that s y < ί γ in L 7 for each γ G Γ) and 2) if 7 C Ζ,γ and b - Ub y C a, then /3 = (/? 7 ) G a, where 0 γ = sup b y. By means of the relation a < δ *> α C 6, where a, J ei, the set Z, is made into a lattice; finally, putting i c :== {x: (Vy (Ξ b)(3y e? T)(x y <,ϋ/γ)}, we can view Ζ as a fuzzy lattice. (For example, if L y = 2 z v, where Z y is a set, then <g) Z Y ss 2 nz v.) The equality πϋ, 1 (< Ye ) = {s e ΠΖ Υ : s Y, < ^J determines a map π Υ^: L Yi ->- Z. Hutton demonstrates that the operation Y Z Y defined in such a way together with the projections π 7 : Ζ -» Ζ γ is a product in Ζαί ''. We consider now a family {(^Y, Z Y, ^Υ):γ ΕΞ Γ} of fuzzy spaces and put X:= LI X y, Z:= g) L y. Let 5~ be an Z-fuzzy topology on X initial for the Υ family {(p y, π Υ ): (λ', Ζ) ->- (X Y, Ζ Υ, <^" ν ); γge Γ}, where p y : X -* X y are projection maps and ir y : L -* L y are defined as above. Then the fuzzy space (X. L, if) is the product of the family of fuzzy spaces {(X y, L y, ετ ν ):γ ΞΓ} in GFT. We call the reader's attention to the fact that every statement about the product in GFT contains information which is entirely different from that contained in a similar statement about the product either in FT{L) (1.4) or in any other category from 1. One of the reasons is that a lattice changes under the product operation in GFT (which seems natural if one remembers that the ground category for GFT is Set X Lat op rather than Set as, for example, in the case of FT(L)). Moreover, as Eklund [25] established, the lattice Z Y, where L y = Ζ for all γ G Γ, Irl > 2, is isomorphic to Ζ if and

49 Two decades of fuzzy topology: basic ideas, notions, and results 173 only if L = 2. Hence it follows, in particular, that for ordinary topological spaces and only for them the ordinary product coincides with that in GFT (and, therefore, with the product in any subcategory), which is one of the manifestations of the "invariance" of general topology in fuzzy topology! In conclusion we remark that the category GFT has equalizers (see, for example, [43]), which means, together with the presence of products in it, the completeness of GFT Certain subcategories of GFT. We denote by GCFT the full subcategory of GFT formed by all Chang Z-fuzzy spaces (1.2), by GLFT the full subcategory of GFT formed by all laminated Z-fuzzy spaces (1.3), by GOFT the full subcategory of GFT formed by all Z-fuzzy spaces, where Ζ runs over orthocomplemented lattices, and by GFT(L), where Ζ is a fixed fuzzy lattice, the full subcategory of GFT formed by Z-fuzzy spaces. An obvious meaning is assigned to the notation GLCFT, GOCFT, GCFT{L), GLFT(L), and so on. It is easily observed that GCFT is naturally isomorphic to the category Fuzz introduced by Rodabaugh [125]. The category GCFT is an epireflexive and epicoreflexive [154] subcategory of GFT. The category GLFT is epicoreflexive but not reflexive in GFT [154]. (To verify that GLFT is not reflexive in GFT, it suffices to note that the product of infinitely many copies of the space (X, L, if), where L Φ 2 and if (M) = 1 <H- Μ = const, is not in GLFT.) Similarly, GLCFT is epicoreflexive but not reflexive in GCFT. Eklund [25] has shown that GOCFT is epireflexive in GCFT (hence in GFT as well), while no subcategory of GCFT not containing Top can be reflexive in GCFT. We call the reader's attention to the fact that the category GFT(L) is not, in general, isomorphic to FT(L). (It follows from Rodabaugh's results that the two categories are isomorphic only if the only endomorphism φ: L -* L in Laf v is the identity map. This condition is satisfied by L 2, and one can show that both categories are isomorphic to Top.) We note also that the product is absent in GFT(L) for L Φ 2, unlike in FT{L). To identify categories of the form FT(L) with the corresponding subcategories in GFT, we denote by GFT{L, ψ), where L is a fuzzy lattice and φ : L -* L is a morphism in Laf 1 ', the subcategory of GFT(L) having the same objects as GFT(L) and morphisms of the form (/, ψ), with ψ = ιρ" for a given φ and some η IN. It is easy to check that the category FT(L) can be identified with the category GFT(L, d), the category CFT(L) of Chang Ζ,-fuzzy spaces (1.1) with GCFT(L, id), the category CLFT(L) of laminated Chang Z-fuzzy spaces (1.2) with GCLFT(L, id), and so forth Hutton fuzzy spaces. Let Ζ be a fuzzy lattice and (X, r) a Chang Z-fuzzy space (1.1). It is not difficult to note that in this case X:=L K is a fuzzy lattice as well, and r forms a subset in! invariant under taking supremums and finite infimums,

50 174 A.P. Shostak and containing 0 and 1. This simple observation enabled Hutton [56] to distinguish a category ξ>, which was later called the category of Hutton, or pointless, fuzzy topological spaces. (Similar ideas can be traced in Erceg's paper [32].) The objects of are pairs (Χ, τ), where X is a fuzzy lattice and r is a subset in X invariant under taking supremums and finite infimums and containing 0 and 1; the role of morphisms /: (X 1, τ,) -> (X^, τ.,) in. ) is played by maps /"': X % -> X x preserving sup, inf, involution, 0, 1, and such that f~\v) e Tj for all V r 2. By assigning to a Hutton space {%. τ) the L-fuzzy Chang space (*, Χ, τ), where * is a one-point set, an isomorphism is established between the category S? and the full (epireflexive [26]) subcategory >' of GCFT whose objects are of the form (* : L. τ), where L is a fuzzy lattice and τ is the Chang Z-fuzzy topology on * ([125]). It is not difficult to understand that the category of Hutton spaces S contains (up to an isomorphism) the category Top as a full subcategory; it suffices to assign the Hutton space (2 X, T) to any topological space (X, T). We stress here that although both the notion of a Hutton space (the category ) and the notion of an Z-fuzzy Chang space (the category CFT(L)) generalize the notion of a topological space (the category Top), the purposes and the ideological basis of these generalizations are diametrically opposite from the point of view of "classical" topology. The main goal of Hutton is to study "that part of topology which relates to the lattice theory" [56], and in order to realize it Hutton ignores points and considers sets as elements of a lattice Ζ and replaces the set-theoretic operations by the lattice supremum, infimum, and involution operations. On the other hand, in the theory of Z-fuzzy spaces the notion of a point extends to the notion of an Z-fuzzy point (in the spirit of 3.1) (and, either explicitly or implicitly, influences the corresponding theory heavily). We note in this connection Eklund's paper [26], in which the possibility of "extracting the maximal point part" of a Hutton space (Χ, τ) is studied, that is, representing it in the form (X, L, r), where X zz L x and in addition L is not representable as L ^ K z for any fuzzy lattice Κ and a set Z, \Z\ > 2. In conclusion we remark that the use of structure relations only in lattices X and τ d X, without appealing to points, makes the theory of Hutton spaces closer to the theory of locales ([60], [120]), and others). The connection between the two theories has been studied by Rodabaugh (in preprints) On subspaces of Hutton spaces and fuzzy subspaces of Ζ,-fuzzy spaces. Let (Χ, τ) be a Hutton space and let asi, For every ue^we put u a := (u /\ a)\/(a /\ a c ) and let X a : = {u a : u e X}. Restricting the V and Λ operations from Ζ to a and assigning to each u a E X a an element

51 Two decades of fuzzy topology: basic ideas, notions, and results 175 u*: = ul /\ a (=(i/ c ) 0 )e ϋ β, we get a fuzzy lattice X a with an involution u n ->- w*. In this case T O : = {U U : u e τ} is a fuzzy topology on X a, and hence (52 Q, τ α ) is a Hutton fuzzy space. (X a, τ α ) is called a subspace of the Hutton space (Χ, τ) [32]. In the case S = L A ' and a C X the subspace (,2! α, τ α ) of the Hutton space (Χ, τ) is isomorphic to the subspace (a, r a ) (1.7) of the Chang Ζ,-fuzzy space (X, r); on the other hand, the notion of a subspace of a category in φ makes it possible to speak about a fuzzy subspace ( a, τ α ) of the Chang Ζ,-fuzzy space (X, r) based on an Z-fuzzy subset a L x (cf. 1.7). We now consider the basic topological properties in the categories in question. Pretending neither to completeness nor to representativeness, we restrict ourselves here to the Hausdorff, compactness, and connectedness properties in the category ξ) of Hutton spaces. The choice of the category is not accidental: on the one hand, by replacing the space (L x, τ) e Ob (φ) by the space (X, L, r) Ob(GCFT) the notions we have introduced are extended to GCFT and, furthermore, to GFT, and on the other hand, the category enables us to understand both the specific character due to the use of different lattices and the ideology of the "pointless" approach Separation in ξ). Clearly, when defining the Hausdorff property and other lower separation axioms for Hutton spaces the ideas of the approaches described in 5 are inapplicable, for in spite of their diversity each of them appeals to points. In [57] a "pointless" scheme for separation axioms has been worked out; we shall present part of it here. A Hutton space (Χ, τ) is called a T 0 -space if X is the closure of the subset τ U T C (where T C : = {U C : U e= τ}) with respect to arbitrary supremums and infimums; (Χ, τ) is called an R 0 -space if r is contained in the closure of r c with respect to arbitrary supremums; (Χ, τ) is called an R r space if every u r can be represented in the form " = V {Λ u ab : b e= B a ) = ν {Λ "at,-- b e B a }, a b a b where u ah Ξ τ; (Χ, τ) is called a T^-space (T 2 -space) if it is simultaneously an R o - and 7O-space (respectively an /? r and r o -space). We remark that these definitions induce the corresponding standard definitions of separation axioms on Top considered as a subcategory of (15.4). All the properties considered above are multiplicative (naturally, in the sense of the definition of a product in or, equivalently, in GFT). A space (Χ, τ) is Hausdorff if and only if the diagonal Δ is closed in Χ χ Χ. Nevertheless, the authors have failed to characterize the Hausdorff property by means of the uniqueness of the limit (in one sense or other) of fuzzy filters [57].

52 176 A.P. Shostak Compactness in ξ> Let {Χ, τ) be a Hutton space. An element ο e X is said to be compact if for each % d τ such that a <; \/ C W there is a finite subset % 0 Cl % such that a <; \y #, 0. A Hutton space (.!, τ) is called compact [56] if all a r c are compact. It is easy to note that if (X, r) is a Chang L-fuzzy space and the Hutton space (L x, r) is compact, then (Χ, τ) is compact as well (7.1); the converse is not valid in general. A product ((χ, 3\., τ) of a family of Hutton spaces {(i v, τ γ ): γ e Γ} is compact if and only if all {Z y, T Y ) are compact [56]. However, the category Comp ξ) of compact Hutton spaces is reflexive neither in φ nor in GCFT (a similarly defined category Comp GCFT (and even the category Comp GOCFT) is not reflexive in GCFT either) [25], which rules out the possibility of constructing an adeuqate compactification theory in these categories (cf. 7.9) Connectedness in ;p. A Hutton space {Χ, τ) is called connected if τ Π τ = {0, 1} It is shown in [56] that a product ( 5 y, τ) of a family of Hutton spaces {(X y, τ Υ ): γ Ξ Γ} is connected if and only if all the factors are. Conclusion: some reflections on the role and significance of fuzzy topology Although fuzzy topology has already existed for two decades and there are no fewer than six hundred publications in this area, arguments about the "legality" of this branch of pure mathematics, and on its role and significance, are still raging. We make an attempt to express our main reflections in this connection. 1. Throughout the whole of our, survey we have tried to convince the reader that fuzzy topology, like any branch of pure mathematics is supposed to, has a quite definite subject of investigation, enjoys its own development dynamics and a certain inner harmony. 2. There are interesting and rather important mathematical constructions due to fuzzy topology in essence (that is, they have arisen in the framework of fuzzy topology and their study requires the involvement of the apparatus of fuzzy topology). Examples of such constructions have been considered in Throughout the whole of our work we have intended to trace at all levels a connection between fuzzy topology on the one hand and general topology and some other branches of mathematics on the other. Certain aspects of such a connection are made explicit by those functors and constructions considered in 2, 11, 12, Fuzzy topology has a certain philosophical significance. In particular, it provides us with a fresh look at numerous facts of general topology,

53 Two decades of fuzzy topology: basic ideas, notions, and results 177 and at the role of classical two-valued logic in general topology. Since we cannot dwell on this question (of which an investigation was undertaken by Rodabaugh in preprints, see also [129]), we suggest that the reader should look, for example, either at the classical Tychonoff theorem on a product of compacta from the viewpoint of "fuzzy Tychonoff theorems" ( ; 15.7) or at the compactification problem for topological spaces from the viewpoint of compactifications in fuzzy topology (7.9, 7.10, 15.7). It is also useful to look at the subject and problematics of general topology as a whole from the viewpoint of the theory presented in The notions and methods of fuzzy topology prove useful in some cases in posing questions and solving problems of classical mathematics. For example, in [103] Lowen consideres a certain family of fuzzy topologies on a metric space [X, d) and demonstrates how the study of this family by the methods and in the framework of fuzzy topology enables us to obtain useful (particularly for approximation theory) information on the space (X, d) itself. Examples of using fuzzy topology in the ordinary "crisp" mathematics can also be found in [129], [80], [102], and others. 6. When speaking about the use of fuzzy topology ideas, methods, and results in applied problems, one should note that as yet it has an occasional and rather superficial character. Among the works in which fuzzy topology is used to some extent, we mention a note by Ponsard [116] (a discussion of the application of fuzzy metrics to economics problems), papers by TopenCarov and Stoeva ([156] and others) (fuzzy topological automata) and the investigations of Averkin and Tarasov [2] and Logvinenko (attempts to model the perception process by means of fuzzy topology). We hope that our survey will promote further popularization of fuzzy topology and, possibly, an expansion of the so far very meagre list of its applications to applied problems. References [1] A.N. Averkin, I.Z. Batyrshin, A.F. Blishun, V.B. Silov, and V.B. Tarasov, Necketkie mnozhestva ν modelyakh upravleniya i iskusstvennogo intellekta (Fuzzy sets in models of control and artificial intelligence), Mir, Moscow [2] and V.B. Tarasov, Nechetkoe otnoshenie modelirovaniya i ego primenenie ν psikhologii i iskusstvennom intellekte (The fuzzy modelling relation and its application to psychology and artificial intelligence), Computing Centre of the USSR Academy of Sciences, Moscow [3] D. Adnadjevic, Separation properties of F-spaces, Mat. Vesnik 6 (1982), 1-8. MR 85d: 54003a; Zbl. 541 # [4], Dimension F-Ind of F-spaces, Baku Internat. Topology Conf., Abstracts, Part II, Baku 1987, p. 4. [5] A.D. Aleksandrov, Additive set-functions in abstract spaces, Mat. Sb. 13 (1943), MR 6-275; Zbl. 60 # 135.

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