Two decades of fuzzy topology: basic ideas, notions, and ......§15. Another view of the subject of...

Uspekhi Mat. Nauk 44:6 (1989), 99-147 Russian Math. Surveys 44:6(1989), 125-186

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

A.P. Shostak

CONTENTS

Introduction§ 0. Preliminaries: fuzzy sets 125§ 1. Fuzzy topological spaces: the basic categories of fuzzy topology 127§ 2. Fundamental interrelations between the category Top of topological 135

spaces and the categories of fuzzy topology§ 3. Local structure of fuzzy topological spaces 138§ 4. Convergence structures in fuzzy spaces 140§ 5. Separation in fuzzy spaces 143§ 6. Normality and complete regularity type properties in fuzzy topology 146§ 7. Compactness in fuzzy topology 149§ 8. Connectedness in fuzzy spaces 157§ 9. Fuzzy metric spaces and metrization of fuzzy spaces 158§10. The fuzzy real line 3~ (R) and its subspaces 161§11. Fuzzy modification of a linearly ordered space 164§12. Fuzzy probabilistic modification of a topological space 165§13. The interval fuzzy real line 168§14. On hyperspaces of fuzzy sets 169§15. Another view of the subject of fuzzy topology and certain categorical 170

aspects of itConclusion: some reflections on the role and significance of fuzzy topology 176References 177

Introduction

The notion of a fuzzy set, introduced by Zadeh [169] in 1965, hascaused great interest among both 'pure' and applied mathematicians. It hasalso 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 dwellupon the clarification of the reasons for such a considerable and diversifiedinterest in this notion nor discuss its place and role in mathematics as awhole—the reader will possibly find an answer to these and other similarquestions after consulting the monographs [1], [69], [114], and others.

126 A.P. Shostak

We are much more modest and concrete in our purpose, which is to presentthe basic concepts of fuzzy topology, the branch of mathematics which hasresulted 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 towhich fuzzy sets have been applied systematically. It was in 1968, that is,three years after Zadeh's paper had appeared, that Chang [16] made thefirst "grafting" of the notion of a fuzzy set onto general topology. Heintroduced the notion that we call a Chang fuzzy space (1.1) and made anattempt to develop basic topological notions for such spaces. This paper wasfollowed by others in which Chang fuzzy spaces and other topological typestructures 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 thebasic ideas and categories of fuzzy topology, to present more or lesssystematically the basic notions, constructions, and results in this area, andto discuss the directions in which it is developing. We must state explicitlythat our survey does not pretend to completeness. In particular, we shallonly 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 theminimal amount of information on fuzzy sets needed for reading the mainbody of the work. In § 1 various approaches to the definition of a fuzzyspace (and, accordingly, to the subject of fuzzy topology) are discussed andcompared, the principal categories of fuzzy topology are considered, and aunified terminology is established. § 2 is devoted to clarifying the fundamentalinterrelations between the categories of fuzzy topology and the category oftopological spaces. Let us stress that the clarification of interrelations ofsuch kind has both a technical interest and a fundamental (and evenphilosophical) importance for fuzzy topology. In § 3 the notion of a fuzzypoint is considered and the local structure of a fuzzy space is discussed. Letus note that the problem of finding an adequate analogue of a point in afuzzy situation and the related problem of local study of fuzzy spaces havebecome a stumbling block for a number of authors. In §4 convergencestructures in fuzzy spaces are studied. We draw the reader's attention tosubsection 4.5, where the so-called fuzzy neighbourhood spaces areconsidered—an essentially fuzzy phenomenon having no analogue in generaltopology and defined by means of filters.

In §5-9 the most important topological properties for fuzzy spaces areconsidered. In § 5 various approaches to the definition of the Hausdorff

Two decades of fuzzy topology: basic ideas, notions, and results 127

property for fuzzy spaces are discussed (see also 15.6). We think that thisdiscussion is important not so much because it surveys various definitions ofthe Hausdorff property in fuzzy topology as because this simplest exampledemonstrates an inevitable branching process of ordinary topological notionsunder their extension to the categories of fuzzy topology. In § 6 propertiessimilar to normality and complete regularity are considered for fuzzy spaces.Assertions on maps from such spaces to the fuzzy interval, the fuzzy realline (§10), and other 'standard' fuzzy spaces deserve the most attention.§ 7 is devoted to a rather detailed discussion of a most important topologicalproperty, that of compactness, as well as to the compactification problemfor fuzzy spaces. (We shall return to the problem of compactness andcompactifications in subsection 15.7, where we shall look at it from anotherpoint of view.) Properties similar to connectedness for fuzzy spaces areconsidered in §8. Finally, in §9 we shall discuss various approaches to thenotions of a metric and metrizability in a fuzzy situation. Fuzzy stratifiablespaces—a property similar to generalized metizability—are also consideredthere.

In § 10-14 constructions belonging, in the opinion of a number ofauthors, to the "gold reserve" of fuzzy topology are discussed—these includethe fuzzy interval and the fuzzy real line (§10), fuzzy modification of alinearly ordered space (§11), Klein modification of a connected space (§11),fuzzy probability modification of a topological space (§12), the interval realline (§13), and the construction of hyperspaces of fuzzy subsets of auniform space (§14).

In § 15, which occupies a particular place in the survey, another (at astretch, more general) view of the subject and objectives of fuzzy topologyis presented. A wide use of category topology helps us very much inpresenting this point of view. In this section we shall take a fresh look at anumber of questions considered earlier. At the end of the work there is aconclusion, 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 asa map Μ : X -*• I: = [0, 1 ] . In this connection M(x) is interpreted as thedegree of membership of a point χ Ε Χ in a fuzzy set M, while an ordinarysubset 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 ifnot burdensome for one working with fuzzy sets, Goguen [41] introducedthe notion of an L-fuzzy set, where L is an arbitrary lattice with both aminimal and a maximal element, 0 and 1 respectively. An L-fuzzy (sub)set

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. Wedenote by Lx 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 arbitrarylattice L, fuzzy topologists restrict themselves, as a rule, to the use of theso-called fuzzy lattices. Following Hutton [56], we define a fuzzy lattice asa complete completely distributive lactice with a minimal element 0 and amaximal element 1, on which an order-reversing involution a -*• ac is fixed(that is, a < b, a, b G L =»• bc < ac). (For those lattice theory notions notdefined here see, for example, [12]). In particular, having introduced aninvolution on the lattices / and 2 by the rule a -»· ac : = 1 — a and havingendowed them with the natural order, one may consider them as fuzzylattices. 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 theso-called orthocomplemented lattices: an involution a -*• ac on Ζ is calledan orthocomplementation if a \J ac = 1 and a /\ ac — 0 for every a G Z.An important example of an orthocomplemented lattice is the lattice 2Z ofall subsets of a set Ζ naturally ordered by inclusion and endowed with aninvolution A -*• Ac: = Z\A.

(0.5) Operations on fuzzy sets.Let Λ' — {At: i €Ξ ·")} d Lx be a family of L-fuzzy sets in X. By the unionand the intersection of this family we mean respectively its supremum\J J: = V {Ar- i e .7} and infimum /\ Jr. = /\ {.4,-: i <Ξ 2}. Thecomplement Ac of an Ζ,-fuzzy set A is defined by the rule Ac{x): = G4(x))c,x £ I If Λ/7 is an Ζ,-fuzzy subset of a set Xy for each γ G Γ, then wedefine 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 toν ν

verify that for L = 2 these operations reduce to the ordinary set-theoreticoperations of union, intersection, complement, and product, and that thebehaviour of the operations just introduced is completely analogous to thatof the corresponding set-theoretic operations. For example, the de Morganlaw (\/ Aif = /\ ACj can be stated easily, and so on [ 169], [41 ]. If Ζ is an

i i

orthocomplemented lattice, then clearly the complementation in Lx is anorthocomplementation as well: A \' Ac = 1, A /\ Ac — 0.

(Along with the definitions of basic operations on fuzzy sets presentedabove, there are other definitions of these operations occuring in "fuzzymathematics" and especially in its applications. For example, the intersectionand the union of fuzzy sets A, B G Ix are given respectively by Α ·Β andmin{^4 + β, 1}, and so on. The reader may become acquainted with variousapproaches to the definition of operations on Z-fuzzy sets, including


axiomatic ones, by means of the papers [18], [78], [79]; see also [1],[69], [114]. In the present survey, as usual in fuzzy topology, operationson Z-fuzzy sets are always understood in the sense of the definition at thebeginning 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 LY of afuzzy set Α <Ξ Lx is defined by f (A )(y) = sup {Α (χ): χ e Γ 1 (y)} i f/ - 1 ( }')ΦΦand f(A)(y) = 0 otherwise; the inverse image f'l(B) G Lx of a fuzzy setΒ € LY is defined by / - 1(S)(x) = Bf(x). Properties of images and inverseimages of fuzzy sets are completely analogous to properties of images andinverse images of ordinary sets. For example, f1 (V#i) = V7/"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 G L ^ w e

obtain a map f:Lx->-LY (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 Lx χ Lx we shall need the fuzzy inclusionrelation cz: LK χ Lx — L,defined by A CZ B: = inf (Ac \J B){x) (see [22],

X

[24]). It is evident that for ordinary subsets A and Β of a set X, ii A C Βthen 4 c S = 1, and A CZ Β = 0 otherwise.

There is a definite parallel between the properties of the ordinaryinclusion 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 5and A C C then A C Β η C" {A, B, C C X) there corresponds the inequality"(-4 £T £) Λ 04 (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 theinequality "(.4 c B) < (/ (^4)C / (β.))" W, 5 C I 1 ; / : X -> 7), and soforth. 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 notvalid. 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 formκ: Κ —> I, where Κ is the class of all (ordinary) cardinals, such thaty. (0) = 1 and κ (α) = 0 for some a G I In [155] fuzzy cardinal arithmeticis developed, which is in a sense similar to the ordinary cardinal arithmetic.The cardinality of a fuzzy set Μ ε Ix is a fuzzy cardinal κΛί defined byν-Μ (α) = sup {t: | ΛΓ1 (t, i] | > a}, where a G i£

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 dueto 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 offuzzy subsets (τ C Ix) satisfying the following three axioms:

(1)0,1 e x ;(2) if U, F G T, then U Λ V GZ τ;(3) if ί/j Ε τ for every i G J, then \[Ut e τ.

7

A map / : X -*• Υ between fuzzy topological spaces (Χ, τχ) and (Υ, τγ) issaid to be continuous if f~\V) G τχ for each V G τγ.

Fuzzy topological spaces and continuous mappings form a category whichwe denote by CFT and call the category of [Chang] fuzzy [topological]spaces. (Hereafter we put in square brackets those words to be omitted inthe sequel if no ambiguity arises.)

A fuzzy subset A of a fuzzy space (X, r) is called closed if Ac G r. Wedenote by TC the totality of all closed subsets of a space (Χ, r). One verifieseasily that ( l c ) 0, 1 G rc; (2C) if A, Β G TC, then A\J Β Grc; (3e) if_At G TC for all i G Cf, then /\At G TC. The smallest closed fuzzy set Μ

i

containing (0.7) a fuzzy set Μ G Ix is called the closure of M, and thelargest open (that is, belonging to r) fuzzy set Int Μ contained in Μ is calledthe interior of M. The basic properties of the closure and interior operationsin fuzzy spaces bear a complete analogy to the corresponding properties ofthe closure and interior in topological spaces. For example,

Μ = /\{Α:Α ΕΞχε, Α > Μ}; Μ \J Ν = Μ \J Ν; Μ/\Ν^Μ,\Ν·,

Μ = Μ for every Μ, Ν G Ix (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.

1.2. L-fuzzy topological spaces; Goguen's approach.Generalizing Chang's approach, Goguen [41] introduces the notion of anΖ,-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 isa 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 understoodprecisely as in 1.1) form a category CFT(L). Clearly, CFT{I) is just thecategory CFT, and CFT{2) is (up to an evident isomorphism) the categoryTop of topological spaces.

All that we said on fuzzy spaces in 1.1 remains valid for Ζ,-fuzzy spaces.


1.3. Laminated fuzzy topological spaces: Lowen's approach.In 1976 Lowen [88] proposed a new notion of a fuzzy topological spacewhich 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 theircontinuous 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 aremorphisms in LCFT) [88]. We note also that there is a continuum ofdifferent Chang fuzzy topologies on a one-point set *, while there is onlyone laminated fuzzy topology (r = /*) . From here Lowen and Wuyts [105]deduce that LCFT, unlike CFT, is a topological category in the sense ofHerrlich [47].

Noting this and a number of other important advantages of laminatedfuzzy spaces (see, for example, 1.6, 6.1, and so on), Lowen recommendsthat 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-fuzzytopology 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 itis the category Top of topological spaces.

1.4. Fuzzy topological spaces.A certain disadvantage of all the approaches considered above is someinconsistency in the use of the idea of fuzziness. In each of these approachesa 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 theuse 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 aset and %f: Ix —*• I is a map (that is, a fuzzy subset of a set Lx) satisfyingthe following axioms:

(1) $ (0) = g (1) = 1;(2) JT (U Λ V) > W (U) A f (V) for any U, V <= Ix;(3) W (V Ut) > f\S~ (Ut) for every family {Ut: i e J } C Ix.

i i

In this connection the inequality f (U) > α, where a G (0, 1 ] , is treated asthe statement "the degree of openness of a fuzzy set U is not less than a",and the inequality ^ (Uc) > a as the statement "the degree of closedness ofa fuzzy set U is not less than a".

A map f:X^-Y between fuzzy spaces (X, 3~x) and (Y, 3~Y) is calledcontinuous if If χ (f~l (V)) > STY (V) for each V € / y . Loosely speaking,

132 A.P. Shostak

one regards as continuous those maps that do not reduce the degree ofopenness 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, SfY) is said to be

a homeomorphism if / i s a bijection and both / a n d / " 1 are continuous.A map / is said to be closed (open) if Sf (Uc) < Sf (}UC) (respectively& (U)< Sf (fU)) for every U G Ix.

We call a fuzzy space (X, Sf) laminated if (1'·) Sf (c) = 1 for everyconstant c G / (compare the definition in 1.3).

Clearly, constant maps of laminated spaces are continuous. We denote byLFT the full subcategory in FT formed by laminated fuzzy spaces.

By substituting an arbitrary fuzzy lattice L for an interval / in thedefinition 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)).

1.5. Certain interconnections between the categories CFT(L) and FT(L).By identifying, as usual, subsets of a given set with the correspondingcharacteristic functions one can consider CFT(L) as a full subcategory ofFT(L). In this connection a fuzzy space (X, Si) (1.4) is evidently a Changone (1.1), that is, it lies in CFT(L) if and only if a fuzzy topology Sf inaddition to the axioms (l)-(3) satisfies the follo'wing:

(4) Sf (Lx) C 2 = {0,1}.

On the other hand, if (X, Sf) is an arbitrary L-fuzzy space andα e L+: = L \ {0}, then Sf a- = {U ΕΞ Lx': Sf (U) > a} is a Chang L-fuzzytopology on X (the so-called α-level Chang L-fuzzy topology of the givenfuzzy topology Sf). In addition, Sfc

a: = {U €Ξ Lx: Sf (Uc) > a} is preciselythe family of all closed Ζ,-fuzzy subsets of the Chang space (X. Sfa)· Theseobservations enable us to reduce the study of certain properties of a fuzzytopology Sf (1.4) to the study of much simpler objects, the correspondingα-level Chang fuzzy topologies Sfa- In particular, the continuity of a map/: (X, Sfx) -* (Y, SfY) is easily verified to be equivalent to the continuity ofthe 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].

1.6. Initial L-fuzzy topologies and the product of L-fuzzy spaces [146],[152].Let £ : = {Sfy: 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 weakestamong all L-fuzzy topologies on X majorizing (in the > sense) every $~y wedenote by sup Sfy and call it the supremum of that family. It is easy to see

νthat every non-empty family of L-fuzzy topologies has a supremum and that


if all JJVs are Chang (laminated) topologies, then sup ify is a Chang(laminated) topology as well. v

Let I b e a set, (}r, §) an Z-fuzzy space, and /: X -* (Y, §) a map. Wecall the weakest Z-fuzzy topology $~: = f'1 ($) on X making/ continuousthe initial Z-fuzzy topology for /. One can show that for a map/: X -+ (Y, S) the initial Ζ,-fuzzy topology is determined by the formula0- (U) = V [SO. Ψ) Λ «: α e L+}, where

U e L*, fa: = {Γ1 (Γ): V €Ξ Π, 8 (V) > a}.The weakest Z-fuzzy topology on X making continuous all the mapsfy: X —> (Yy, §y), y £r Γ, we call the initial one for that family of maps. Itis easily shown that the initial topology for a family {/Y: y ΕΞ Γ} is determinedas sup Z"1 (§Y). If in addition all SY's are Chang (laminated) Z-fuzzy topologies,

νthen the initial topology sup f'1 (§y) is Chang (laminated) as well.

y

Let {{Xy, 3~y): γ ε Γ} be a family of Z-fuzzy spaces. The product of thisfamily is the pair (X, if), where X = Π Xy is a set product and 3~ is an

νΖ,-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 definedis 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 topologythe projection maps of products of fuzzy spaces (including Chang ones) neednot be open. Another feature of the products of fuzzy spaces is that a

"fibre" XYo χ {χΥ: γ Φ γ0} in a product 1|λ\ of fuzzy spaces is not,ν

generally speaking, homeomorphic to the space Xy<t. A necessary conditionfor projections to be open and, as a consequence, for fibres to behomeomorphic to the corresponding coordinate spaces, is that all the factorsare laminated. (In this sense both LFT(L) and LCFT(L) are "moretopological" 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 sumof/--fuzzy spaces, are introduced ([146] ; for Chang spaces [89]).

1.7. Subspaces of Ζ,-fuzzy spaces.The simplest and at the same time a very important operation of generaltopology is transition to a subspace. In considering a similar problem for/--fuzzy spaces, two different approaches are possible: an (L-fuzzy) subspaceof an Z-fuzzy space (X, if) based on an ordinary crisp set A C X andan (L-fuzzy) subspace of an Z-fuzzy space (A", if) based on an L-fuzzy setΜ ε Ιχ. In the case of the first approach (which is by the way the onlypossible one from the category point of view) the problem causes notrouble: it is natural to mean by a subspace any pair (A, if A), where

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.) Forthe second approach the problem has no reasonable solution within thecategories considered; we shall return to it in subsection 15.5 afterdeveloping another point of view of the subject of fuzzy topology.

1.8. On the notion of continuity defect in fuzzy topology.In the vast majority of works on fuzzy topology the authors start from thefact that by analogy with general topology all maps between fuzzy spacesare divided into two classes: continuous maps, or morphisms in thecorresponding category, and discontinuous ones. However, sucii a "crisp"classification of fuzzy spaces, based on two-valued logic, does not alwaysseem natural in the context of fuzzy topology. In [174] another view ofthis problem is suggested, based on the notion of continuity defect of maps.We now present the basic ideas of that approach, restricting ourselves to thecase of Chang fuzzy spaces (1.1).

Let (Χ, τχ), (Υ, τ γ) be Chang fuzzy spaces. We define the continuitydefect of a map f:X-* Υ as the number

cd(/) = sup sup (tHV)- ΙηίΓι(ν))(χ).V x

It is easy to show that

cd (/) *= sup sup (Γ 1 (B) - f1 (£))(*) = sup sup (f(M)-f (M))(y).X * ϊ

Clearly, the condition cd(/) = 0 means precisely the continuity of/. IfΧ, 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 anynumber α G /; its value characterizes to what extent the given map differsfrom 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 ofa family of maps is equal to the supremum of the continuity defects ofthese maps [174].

By analogy with the continuity defect one may define also the defects ofother properties of maps between fuzzy spaces, which enables us to viewcertain aspects of fuzzy topology as a whole in a different light.

1.9. Some features of the presentation of the material in the survey.The diversity of the original categories of fuzzy topology (see 1.1-1.4, andalso §15) makes a unified presentation and perception of ideas, notions, andresults of this field more difficult. Therefore let us agree to the following.


Taking into account that most works in the area of fuzzy topology arebased on Chang's definition (1.1) and that the presentation of the theory forChang spaces is essentially easier than for fuzzy spaces in the sense of 1.4and for L-fuzzy spaces (1.2), in the sequel we shall speak as a rule of Changfuzzy spaces only and the term "fuzzy [topological] space" will beunderstood in the sense of 1.1.

We refer to the category LCFT only if it is actually necessary. Ifconstructions and results in a cited paper are presented for laminated spacesbut are easily transferrable to the case of arbitrary Chang spaces, then wepresent them in such a situation without any reservations. We believe thatunder such a presentation, on one hand the role of the laminatednesscondition, and on the other hand those "topological opportunities" affordedby the non-topological [47] category CFT, will be apparent.

Let us remark that many results that are true for CFT can be extended inone 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, andthe presence of orthocomplementation in L. In the present work we referto CFT(L) occasionally and for illustrative purposes only.

On the other hand, let us recall (1.5) that the study of the category FTcan often be reduced to the study of Chang spaces by transition to α-levels,a S ( 0 , 1].

§2. Fundamental interrelations between the category Top of topologicalspaces 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 categoriesof fuzzy topology are the most important and essential interrelationsbetween them established?

(2) What should be regarded as analogues of objects of the category Topin the categories of fuzzy topology?

We shall consider here the simplest functors (correspondences) of thiskind; certain specific more complicated constructions are described in§10-14.

2.1. Top as a subcategory of CFT: the natural inclusion functore.Top^ CFT.Identifying, as usual, subsets of a given set with the correspondingcharacteristic functions, we can treat a topological space (X, T) as an objectof CFT. In this way an inclusion functor e : Top -*• CFT arises, whichidentifies Top with the full subcategory of CFT whose objects are just thosefuzzy spaces (X, r) satisfying the condition r C 2X.

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) thelaminated fuzzy space (Χ, ωΤ), where ωΤ is the totality of all lowersemicontinuous maps of (X, T) to the interval /. It can easily be verifiedthat the continuity of a map / : (X, Tx) -*• (Y, TY) implies the continuity ofthe map / : (Χ, ωΤχ) -> (Υ, ωΤγ). Thus, ω can be viewed as a functorω : Top -»• LCFT mapping Top isomorphically onto a full subcategory ofLCFT. Lowen restricts himself to laminated fuzzy spaces and suggests thatthe space ωΧ: = (Χ, ωΤ) should be considered as the fuzzy copy of atopological space X (see [88] -[105] and others).

A fuzzy space (X, r) whose fuzzy topology is of the form τ = ωΤ for anordinary topology Τ on X is called topologically generated [88] or induced[164].

Let £P be a topological property and $> the extension of that property tothe category of fuzzy topological spaces. Lowen [90] calls an extension 3ύ

good if any topologically generated fuzzy space (Χ, ωΤ) possesses theproperty Ρ if and only if the corresponding topological space (X, T)possesses the property 5J- The significance of the notion of a goodextension is that it enables us to distinguish among a large number ofpossible (and rather natural) extensions of a property £p to CFT some of the(in a sense) most successful ones.

2.2°. Laminated modification of a fuzzy space.In [152] the functor ω was extended to the so-called laminated modificationfunctor 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 Xmajorizing τ. It is easy to show that λτ results from closing the familyπ: = 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 notdifficult 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\ ν γ )

for an arbitrary family of fuzzy spaces {Xy: γΕΞ Γ}.The use of laminated modification often proves helpful in verifying

various statements of fuzzy topology (see, for example, 8.1).

2.3. The functor / : CFT -»• Top and weakly induced fuzzy spaces.Let (Χ, τ) be a fuzzy space. As is easy to note, /(T) : = τ Π 2Χ is anordinary topology, moreover, it is the maximal among all topologiescontained in r. Since evidently the continuity of a map / : (Χ, τχ) -*• (Υ, τΥ)guarantees the continuity of the map / : (X, JTX) -*• {Υ, }τΥ), we can regard /as a functor / : CFT -* Top.

Martin [111] distinguishes an important class of fuzzy spaces by means ofthe functor /; a fuzzy space (X, r) is called weakly induced if for all U Ε τ


the maps U: (X, /τ) -> / are lower semicontinuous. A space (A', r) is weaklyinduced 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 bothlaminated and weakly induced [111].

2.4. Fuzzy extensions of a topology to a family of fuzzy spaces: thefunctor η : Top -*• FT.Let us present here an example of a functor which, unlike those consideredabove, performs an embedding of the category Top into the categoryFT rather than the category CFT. A functor of this kind was first describedby 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 Ix, 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: Ix —>• /, which isa fuzzy topology (1.4) on X. In addition, if a map / : (X, Tx) -> (Y, TY) iscontinuous, then the m a p / : (Χ, ηΤχ) ->• (Υ, r\TY) is continuous as well.

The functor η: Top -> FT arising in this way maps Top isomorphicallyonto a full subcategory η(Τορ) of the category FT. In addition,η (Top) Π CFT = {0}, that is, to any non-empty topological space anessentially non-Chang fuzzy space is assigned. For more details on theproperties of this and other analogous functors, see [22], [152], [146].

2.5. The functor 1: CFT -» Top.Following Lowen [88], let us associate with every fuzzy space (X, x) theordinary topological space (Χ, ιτ), where ιτ is the weakest topology makingall 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, ιτΑ) —>· (Υ, ixy) iscontinuous, 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) — Τfor every topology T. A fuzzy space (X, x) is topologically generated if andonly if τ = ω°ι (τ). For more on this and other properties of ι see [88],[89].

Let 3J 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 theproperty 3*. Clearly, the extension of a topological property 5° to theproperty of being an ultra-^-fuzzy space is a good one (2.2).

In [152] the functor ι is extended to the category FT.

2.6. The functors ια: CFT -> Top.The α-level functor i a, where a G [0, 1), associates with a fuzzy space (Χ, τ)the topological space (Χ, ιαΤ) , where ιατ: = {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, xx) -> (Γ, Ty) the (continuous) map /: (Χ, ιατχ)-^(Υ, ι α τ Γ ).

138 A.P. Shostak

The α-level functors were introduced in [90] and were later used by variousauthors ([121], [125], [73], and others) for solving some problems offuzzy topology by reducing them to standard problems of general topology.

2.7. The functors i*a : CFT -> Top.Along with the functors ια, the functors ι*: CFT -> Top, where a e (0, 1],are used not infrequently in fuzzy topology. The functor ι* associates witha fuzzy space (Χ, τ) the topological space (Χ, ιατ), where ι*τ is the topologydefined by the subbase na: = {U'1 [a, 1]: U £Ξ τ}, while leaving morphismsunchanged [90], [121].

Let us note that the properties of the functors ια and ι* and the informationprovided by them differ essentially from each other.

2.8. The hypergraph functor G : CFT ->• Top.Let (Χ, τ) be a fuzzy space; we consider the topology GT on the productΧ 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, τχ) -*• (Υ, τ Y) the(continuous) map G(f): G{X, τχ) -»· G(Y, rY) 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 (inpreprints) and Lowen [90]. Rodabaugh made use of them for studyingseparation 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 equalityGT = ιτ χ Th 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, afundamental role is played by the notion of a point. A point is a minimalobject in the sense of the relation of belonging 6Ξ: it either does or doesnot belong to a set, while nothing can belong to a point. One of thefundamental peculiarities of fuzzy set mathematics and, in particular, fuzzytopology, is the absence of such 'minimal' objects. In order to fill the gapcaused by this, many authors use, sometimes quite successfully, kinds of"substitutes" for points—the so-called fuzzy points, appearing in the early80s in works by Pu and Liu [117], [118], Sarkar [130], [131], Srivastava,Lai and Srivastava [133], de Mitri and Pascali [19]. (The definitions of afuzzy point and of the corresponding relation <=? of a fuzzy point belongingto 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


relation of belonging due to Wong [166] seem inadequate—for a criticism ofthem see, for example, [44].) Following [ 117], [ 118], let us present herethe 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 x0 Ε Χ,t Ε (0, 1], defined by p(x0) = t and p(x) = 0 for χ Φ χ 0 ; here x0 is calledthe support of the fuzzy point and t its value. An ordinary point x 0 istreated as a fuzzy point pi,. A fuzzy point ρ belongs to a fuzzy setΜ (ρ&Μ) if t <M(x0).

Along with the relation ΕΓ, PU and Liu consider the so-called quasi-coincidence, or Q-coincidence relation: a fuzzy point p: = pi, is quasi-coincident (Q-coincident) with a fuzzy set Μ (pQM) if M(xo)+t > 1.

With respect to the operations of union and intersection of a family offuzzy sets {Ut: i ΕΞ 3} there are both analogies with the behaviour of GE anddifferences from it in the behaviour of ΕΞ and Q:

pmUt;pQ/\ Ut ̂ VipQur, ρ ΕΞ V U %

A fuzzy point is a minimal object neither with respect to EE nor withrespect to Q: having chosen s Ε (0, t) and r G ( l - / , 1) for a givent Ε (0, 1 ], we have p*. GE />x. and pljQpl,-

To conclude, we note that in [36] an axiomatic approach to the notionof a fuzzy point is developed, and in [ 159], [172] the notion of a molecularis 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 theoryof Hutton fuzzy spaces.

3.2. Neighbourhoods in fuzzy spaces.There are several approaches to the notion of a neighbourhood in a fuzzyspace and, correspondingly, to the study of the local structure of fuzzyspaces: the approach by Warren [161], [162] and a (less successful)approach by Ludester and Roventa [106], using fuzzy neighbourhoods ofordinary 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 analysisof the approaches from this group is carried out in [70]), and the approachby Rodabaugh [128], based on fuzzy neighbourhoods of fuzzy sets. Let usdwell on the approach by Pu and Liu.

A fuzzy set Μ is called a neighbourhood (Q-neighbourhood) of a fuzzypoint ρ in a fuzzy space {X, r) if ρ £r Int Μ (respectively, pQ Int M). LetΛΛρ denote the family of all neighbourhoods (respectively, Q-neighbourhoods)of a fuzzy point ρ in (Χ, τ). Then: (1) U 6Ξ ̂ v =£ ρξ= U (respectivelypQU); (2) U, V EE ̂ P =» U /\ V €Ξ JVV\ (3) U (ΞΞ > p , U < V =» V ΕΞ <*%;(4) for every U ̂ JTV there is a V e ΛΤν such that V < U and F e JTr foreach fuzzy point r with r e F (respectively, rQV).

140 A. P. Shostak

Conversely, for every fuzzy point ρ of a set X let a family of fuzzy sets•JVP be fixed in such a way that the conditions ( l)-(4) are satisfied. Thenthe family σ formed by all fuzzy sets U G Ix such that U e JTV so long asρ €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 Jfv is the base of a system ofneighbourhoods (respectively, of the system of all Q-neighbourhoods) for thefuzzy point ρ in (Χ, τ). (Let us note that there are inaccuracies in theformulation of the converse statement in [117].)

To characterize laminated fuzzy spaces in a similar way, it suffices to addto the axioms ( l)-(4) the following one: (5) if p: = pi,, then t e -Ar

p

(respectively, s ΕΞ Λ"ρ for all s > 1 - t).

3.3. Local characterization of the closure and closedness of fuzzy sets.To illustrate some opportunities afforded by the study of the local structureof fuzzy spaces, we shall characterize in local terms the closure operationand 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 fuzzyset Μ e Ix 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-neighbour-hood U of ρ there is an χ distinct from x0 such that U(x) + M(x) > 1.

The closure Μ of a fuzzy set Μ coincides with the totality of all itsadherence points. A fuzzy set is closed if and only if it contains (ς=) all itslimit points [117]. Also the following fuzzy version of the classical Yangtheorem holds [71], the totality of all limit points for a given fuzzy pointρ 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 thatlead to equivalent results. One of them is based on the notion of a net dueto Moore and Smith (see [71 ]; another one, which goes back to the workof Cartan and Bourbaki (see [14]), is based on the notion of a filter. Eachof these theories has a fuzzy analogue. We begin with an exposition of basicconvergence theory of fuzzy nets developed (in the spirit of [71]) by Puand Liu [117], [118].

4 . 1 . F u z z y nets in fuzzy spaces [ 1 1 7 ] , [ 1 1 8 ] .Let X be a fuzzy space and H the set of its fuzzy points. A map of theform 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 (ps)szs, or (ps), where ps: = < (̂i).Subnets are defined in an obvious way. A fuzzy net (ps) is final in a fuzzynet Μ (finally Q-coincides with M) if there is an s0 € S such that ps€E Μ(respectively, psQM) for all s ^ s0. A fuzzy net (ps) is cofinal (cofinally


Q-coincides) with Μ if for each s Ε S there is an s' > s such that ps/ ξ= Μ(respectively p&.QM). A net (ps) 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 netcontained 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 (pe)s=s doesnot converge to p, then there is a subnet (pf)ees> of which no subnetconverges to p. A map f:X^ Υ is continuous if and only if the fact that afuzzy net (ps) converges to ρ in X implies that the fuzzy net (f(ps))converges to /(p) in Y.

These and many other properties of fuzzy nets are in complete analogywith properties of ordinary nets. However, there are significant differences.For example, there are universal nets that are cofinal but non-final in afuzzy set. (It is natural to call a fuzzy net universal if from being final inU V V, where U, V S Ix, 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 fuzzytopologies on a given set.

4.2. Fuzzy filters in fuzzy spaces.A theory of convergence of fuzzy filters was developed by Lowen [91] forlaminated spaces and then extended to arbitrary fuzzy (Chang) spaces byWarren [163]. (We remark that this extension was in some respects non-trivial.) 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 Ix) not containing 0 and such that:

(1) if Μ, Ν e f, then Μ /\ Ν e f ;(2) i f J l f S . 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 thisdifficulty, Lowen considers the family $ m (,<f) of all fuzzy filters that areminimal (with respect to inclusion) in the set φ (.IF) of all principal (in thespirit of [ 14]) fuzzy filters majorizing a given fuzzy filter &, and demonstratesthat f = Λ {#: # <= spm {sr)y

The fuzzy adherence set and the fuzzy limit set of a fuzzy filter fr in afuzzy 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 withwhich 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 incontrast to the situation in topological spaces the inequality lim $ <ζ lim $'in general is not valid. A map / : (Χ, τχ) -*• (Υ, τγ) is continuous if and only

142 A.P. Shostak

if adh / (,f) > / (adh tf) for every filter f in I or, equivalently, iflim / (&) Ξ> / (lim <f) for every principal fuzzy filter $ in X.

Lowen successfully applies the theory of convergence of filters to thestudy of the compactness and separation properties of fuzzy spaces [91],[92], [168].

4.3. The interrelation between the theories of convergence of fuzzy netsand fuzzy filters has been studied by Lowen (in preprints). As in the caseof general topology, the two theories lead to essentially equivalent results.However, the transition from one theory to another is much more complicatedboth in conceptual and technical respects than that in general topology.

4.4. Other convergence theories in fuzzy spaces.The convergence theories considered above are the most advanced andseemingly the most successful ones in fuzzy topology. However, there areother (non-equivalent) convergence theories. For example, in [45] a certainconvergence structure is introduced by means of fuzzy nets; fuzzy filtersand other similar constructions underlie those convergence structuresdescribed in [45], [67], [ 17]. Convergence of fuzzy filters in Hutton fuzzyspaces (15.4) has been considered in [56], [57].

4.5. Fuzzy neighbourhood spaces [93], [94].With the help of fuzzy filters Lowen has distinguished an important subclassof the class of laminated fuzzy spaces—the fuzzy neighbourhood spaces,characterized by the fact that their fuzzy topology may be restored fromthe so-called fuzzy neighbourhood systems. The class of fuzzy neighbourhoodspaces is wide enough to contain, in particular, all topologically generatedfuzzy spaces and all fuzzy uniform (in the sense of [95]) spaces. On theother hand, these spaces possess a number of important properties that donot hold for arbitrary fuzzy spaces.

To define a fuzzy neighbourhood space, we consider a set X and for eachχ G X fix a fuzzy filter fx. A family £ : = {fx: zGE X} is called a [fuzzy]neighbourhood system on X if:

(1) N(x) = 1 for each χ G X and each Ν Gr fx;(2) if Νε ΕΞ fx for each ε > 0, then sup {Νε — ε) <= fx;

ε

(3) for arbitrary I G X , Ν GE fx, and ε > 0, there is a family{Nl: ζ G= 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): Ν GE f x ) , where Μ G Ix and

§'x Gr £ , the fuzzy closure operator (1.1) is defined, and by the same tokenthe fuzzy topology t (£) on X. A fuzzy space (X, r) is called a neighbourhoodspace if τ = t (£)for a neighbourhood system £. (In [93] the property of


a fuzzy space of being a neighbourhood space is characterized internally aswell, including a characterization by means of closure.)

The closure operator in fuzzy neighbourhood spaces has a number ofimportant features. For example, it is uniformly continuous in such spaces:if Μ, Ν £ Ix, 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 atthe end of this subsection).

The coincidence t (£) = t (£') where £, £ ' are two neighbourhoodsystems on X, holds (if and) only if Ζ = Ζ'. A map {X, t (Zx))~- (Υ, ί (£Y))

between neighbourhood spaces is continuous if and only if Z"1 (!fix,) CI &χ.for every χ 0 € X

If (X, T) is a topological space, then the neighbourhood system of thefuzzy space (Χ, ωΤ) generated by it is of the form

£ : = {srx: = {N e /Λ : Ν (χ) = 1

and ΛΓ is lower semicontinuous at a point χ) : χ ΕΞ X}·

A fuzzy neighbourhood space X is Hausdorff (5.2) if and only if for everyx, y e Χ, χ φ y, and ε > 0 there are Nx e F«, Ny S ,f „ with ΛΓ

Χ /\ Λ'ν < ε .In conclusion, we consider the following example. Let X be a set,

\X\ > 2, τ = {α: α ΕΞ /} U {We /A': Λ7 < ~ }· Then for each x e i w e

— 1 1 1 1

have %x Λ — — — Φ —£• %x = Xx Λ — . an<3 so (X r) is a non-neighbourhoodlaminated space.

§5. Separation in fuzzy spaces

Here we shall discuss properties similar to r0-7Yseparation in fuzzyspaces. The main attention will be paid to T2-separation, or Hausdorffness,as the most important property of this kind; the rest of the properties arementioned only in passing.

At present no less than ten approaches (or patterns) to the definition ofHausdorffness in fuzzy topology are known. Some of them differ negligiblyfrom each other, but others do basically. We start with the spectral theoryof Hausdorffness, which provides a framework for the description of manyother theories of Hausdorffness as particular cases (for example, 5.2, 5.3, 5.4).

5.1. Hausdorffness spectra of a fuzzy space [139].For arbitrary a, b £ / and a fixed £ > 0 w e 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.

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 I2

such that for any distinct x, y G X (and an arbitrary ε > 0) there areU, V G r with U (χ) > β, V (y) > β, and i/c V Vc > γ. Loosely speaking,

the fact that the pair (β, γ) belongs to the spectrum H((X) means that forany x, y G X there are neighbourhoods U and V that are "higher" than β atthe corresponding point and that intersect "lower" than yc. In thisconnection "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). Ifi > /', / > /', 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 forvarious (/, /) may differ at the boundary only. (It is not difficult toconstruct 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 Xis a topological spiice, then HJ(X) = Η{(ωΧ) = I2 if and only if X isHausdorff; 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 subspaceHausdorffness spectra do not decrease. If X is a product of fuzzy spaces{Xa: a(=A}, then H\ (Χ) Ζ) Π Η\ (Xa); if in addition all the Xa's are

a

laminated, then H\ (X) = Π H[ (Xa). (In the general case the equality needα

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 ofthis assertion, let us define the notion of closedness spectrum of a fuzzy setin a fuzzy space, which is necessary in what follows.

We define the (/, j)-closedness spectrum of a fuzzy set Μ in a fuzzy spaceX as the set Clj(M, X) formed by pairs (β, y) G I2 such that (for everyε > 0) there is a W G r with Mc V Wc > γ and in addition W (χ) ^> β

whenever Mc (χ) > β. (We stress that the closedness spectrum of a fuzzy seti

generalizes the topological notion of closedness in an entirely differentdirection from that of the definition of a closed fuzzy set (1.1).) TheHausdorffness spectrum of a fuzzy space can now be characterized byHJ(X) = Cl((A, X2). The closedness spectrum also enables us to present thefollowing assertion (cf. [29], Russian ed., p. 171). Let Χ, Υ be fuzzy spacesand let /, g : X -*• Υ be continuous maps. Then

H\ (Y) CZ Cli ({x: f(x) = g (x)}, X).


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 Tt- and 7Vseparation. Withthe help of the closedness spectrum the ^-separation spectrum may becharacterized as follows: T[i(X) = f)Cl](x,X), which evidently corresponds

χ-

ίο the characterization of topological ΤΊ-spaces as those spaces whose one-

point 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).5.2. 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, theseproperties 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) propertyof a fuzzy space (X, r) is equivalent to the Hausdorff property of thetopological space (Χ. ιατ) (respectively, (Χ, ι£τ)), where ια, tj are the α-levelfunctors (2.6), (2.7). In [126] the a- and a*-Hausdorff properties have beenextended to the case of fuzzy subsets of fuzzy spaces.

5.3. Separation of disjoint fuzzy points.Pu and Liu [117] call a fuzzy space (Χ, r) Hausdorff if for any two of itsfuzzy points p\ and ps

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 ps

y suchthat χ Φ y and s, t < 1 there are U, V G r such that p'u e= F, p'x G? U andU Λ V = 0. It is not difficult to show that the two definitions of theHausdorff 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 twofuzzy points with distinct supports [117] (see also 5.4). Every Hausdorfffuzzy space is ultra-Hausdorff (2.5); on the other hand, there is an ultra-Hausdorff non-Hausdorff fuzzy space.

5.4. Hausdorff property and filters.After characterizing Hausdorff (5.3) fuzzy spaces by the fact that the limitof every fuzzy filter in them does not vanish at one point at most, Lowenand Wuyts [168] introduce by means of fuzzy filters a series of axioms ofTQ-T2 type. In particular, they define T'^'-spaces as those fuzzy spaces inwhich lim ff, where if is an arbitrary fuzzy filter, reaches a positivemaximum at one point at most; those spaces in which this requirement ismet at least by all principal filters f are called Τ2-spaces. Clearly,T2 =*• T'2" => T'2'; the reverse implications do not hold in general. For fuzzyneighbourhood spaces (and for topological spaces as well) the conditions T2,T'2', and Τ % are equivalent.

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 fuzzyspace was determined by the presence of disjoint ((^neighbourhoods ofevery two disjoint (that is, having distinct supports) fuzzy points. However,unlike ordinary points, fuzzy points may have a common support but differin values. The possibility of separation of such points in one sense oranother was by no means guaranteed in the preceding subsections. We shallconsider here two approaches to the Hausdorff property problem, takinginto 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 ofseparation of fuzzy points with a common support appears for the first timein [117], where a fuzzy space is called a quasi-T0-space if for any pair of its

fuzzy points ps

x and px such that s < / we have px ςέ ps

x.)We call a fuzzy space {X, r) A-Hausdorff [3] (S-Hausdorff [130]) if for

any pair of fuzzy points px and ps

y:(1A) if χ Φ y, then there are U, V <= τ such that pi e? U, ps

y ^V_, andU J\^V = 0 (respectively, (15) if χ Φ y, then pi e U, pi ^ V, pi^V", andp% & 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-Hausdorffproperties are obviously equivalent. Fuzzy points in an S-Hausdorff spaceare closed. (Adnadjevic calls a space in which every fuzzy point is closed aΤχ-space [3].) We stress that the closedness of fuzzy points (unlike theclosedness of one-point sets) was in no way guaranteed by the approaches of5.1-5.4. Both A- and 5-Hausdorff properties are multiplicative andhereditary.

5.6. The reader may become familiar with some other definitions of theHausdorff property, for example, via [37], [82], [134], [180], and others(see also 15.6).

§6. Normality and complete regularity type propertiesin 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 thenotion of normality in fuzzy spaces, and take as a basis the definition byHutton [54] or equivalent reformulations of it. To some extent this isexplained by the fact that normality in general topology is defined in termsof open and closed sets only, without appealing to points, hence it can becarried over in a standard manner to the language of fuzzy topology.


A fuzzy space (Χ, τ) is called normal [54] if for any pair of fuzzy sets Aand U, where A ETC, U GT, and A < U, there is a V G τ satisfying theinequalities A < V < V < U. It is easy to verify [108] that a fuzzy spaceis normal if and only if for each pair of closed fuzzy sets A and Β such thatA + 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 notpreserved under taking products; in addition, contrary to the situation inTop, the normality of a product of fuzzy spaces does not guarantee thenormality of the factors [126]. A sufficient condition for the normality ofthe factors to follow from the normality of a product is that all the factorsare 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 ofmaps from it to the fuzzy interval £ (/), having proved a kind of a fuzzyUrysohn lemma (for the definitions of £(I), £ (R) and all the notationinvolved, see 10.1).

A fuzzy space (X, r) is normal if and only if for each pair of fuzzysubsets A and U such that A G TC, U G r, and A < U there is a continuousmap 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-Urysohntheorem: any continuous map /: A -»- £ (/), where A is a closed subspaceof a normal fuzzy space X, has a continuous extension /: X -*- £ (/).

We note that the problem of existence of such extensions has been raisedrepeatedly in the literature ([121], [123], and others). Kubiak's proof isbased 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.

6.2. Perfect normality.A normal fuzzy space of which every closed fuzzy set is the infimum of acountable 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 offuzzy subsets A and U such that A G TC, U G r, and A < U there is acontinuous 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 aperfectly normal space X, has a continuous extension f: X -*- £ (R). (It isnot known whether the condition of perfect normality may be weakened tothat of normality (cf. 6.2) [84].)

Important examples of perfectly normal fuzzy spaces are £ (7), £ (R), aswell as their countable powers ( f (/))«· and (£ (R))«·. The spaces (£ (R))*for k > S o are non-normal. It is not known whether the spaces (£ (/))''" arenormal for k > fc$0.

148 A.P. Shostak

6.3. Complete regularity.Functional characterizations of normal and perfectly normal spaces naturallysuggest that the complete regularity of a fuzzy space should be defined bymeans of maps from it t o f (/). This is what Hutton [55] and Katsaras [65]undertake, having first considered the properties of this type. Hutton calls afuzzy space (Χ, r) completely regular if for each UET there are a family offuzzy sets {Mt: i e 3} and a family of maps {frX -*· .f (/): i SE 3} such thatV; M, = U and Mt(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 presentedabove, but it is easy to see that the two definitions are equivalent.)

A completely regular topological space is completely regular in thecategory of fuzzy spaces as well. A normal fuzzy 7Vspace (5.5) is completelyregular. 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 aref(l), f (R), and products of them [65]. A fuzzy r rspace (5.5) is acompletely regular space of weight k, k > Xo, if and only if it ishomeomorphic 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 bya fuzzy uniformity [55], [86] (cf. [29], p. 523). A fuzzy space (X, r) iscompletely regular if and only if r is generated by a fuzzy proximity [65](cf. [29], p. 554).

6.4. ^-regularity [153].We recall that a topological space X is called ^-regular [ 10], where Ε is aHausdorff topological space, if X is homeomorphic to a subspace of Ek forsome cardinal k. In particular, /-regularity of a topological 7\ space isobviously equivalent to complete regularity.

We shall now consider a fuzzy analogue of ^-regularity. We believe thatin fuzzy topology the ^-regularity property should play a more importantrole than its particular case—complete regularity ( = ,f (/)-regularity). Thepoint is that in fuzzy topology along with the fuzzy interval f (1) there areother "canonical" objects: the laminated fuzzy interval $fl (1) (10.1), thefuzzy 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 thecategories 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 thesituation in Top is that while the spaces occurring in general topology "as arule" satisfy at least the 7Yseparation axiom, in fuzzy topology even theaxiom 7\ (for example, in the sense of 5.1 or 5.5) seems very restrictive (itis not satisfied by the spaces .f (/), fλ (1), Μ (1), and ^ λ (/))· Hence it isappropriate to distinguish between ^-regularity and £-Tychonoff notions forfuzzy spaces.


So, let Ε be a fixed fuzzy space. For a fuzzy space (X, r) we put

C {X, £): = 0 c (Χ* -ε1") (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 iscalled Ε-regular if C{X, E) separates points and closed fuzzy sets in X, thatis, for every A G TC, Χ 6 1 , 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 ishomeomorphic to a subspace of £ f c for some cardinal k.

A fuzzy space is .f (Z)-regular (£λ(Ι) -regular) if and only if it is completelyregular (respectively, completely regular and laminated). A topological spaceis completely regular in Top if and only if it is /-regular in CFT.

A product of £"-regular fuzzy spaces is ^-regular. The ^-regularityproperty is preserved under transition to a subspace. A fuzzy space (X, r) isis-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 (ps)in it, having a limit point, there is a continuous function / : X ->• Ε sending itto a net divergent in E.

A fuzzy space X is £-Tychonoff if and only if it is ^'-regular andis-Hausdorff (that is, for any x, y G Χ, χ φ y, there is an / G C(X, E) suchthat fx Φ fy). In the class of HVspaces the ^-regularity and £-Tychonoffproperties are equivalent. ((Χ, τχ) is called a W0-space if for any x, y G X,χ Φ y, there is a U G rx such that U(x) Φ U(y).) Hence it follows, inparticular, that subspaces of the product f (/)R are precisely the completelyregular fV0-spaces of weight k (cf. 6.3).

§7. Compactness in fuzzy topology

The compactness property of a topological space is one of the mostimportant notions not only in topology but in the whole of pure mathematics.Therefore it is natural to pay particular interest to this notion in fuzzytopology, and as a consequence there are many publications devoted to it.We shall consider the main theories of compactness in the sections; on theway we shall touch on the Lindelof, countable compactness, andparacompactness properties. In subsections 7.7-7.10 we consider theproblem of fuzzy space compactifications (see also 15.7).

7.1. Compactness of fuzzy spaces: the approach of Chang and Goguen.The first definition of compactness for fuzzy spaces was proposed in 1968by Chang [16]; soon after, Goguen [42] extended it to the case of I-fuzzyspaces. It is convenient for us to present this definition here for Z-fuzzyspaces; 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].

150 A.P. Shostak

For L = 2 this definition turns into the ordinary definition of compactnessof a topological space. On the other hand (for L = Γ) it is clear that theredo 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 ofthe most important results of topological compactness theory, admits underthis definition the following "coupe" fuzzy analogue only: the product Xof a family of non-empty compact Ζ,-fuzzy spaces {Xy: γ GE Γ}, whereΙΓΙ < k {k is a cardinal number), is compact if and only if the element1 G L is fc-isolated in L (that is, sup A < 1 for each A C L with \A I < kand 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 ofcompact fuzzy spaces are compact (for L — I).

The similarly defined Lindelof and countable compactness properties havebeen considered in [165].

7.2. α-Compactness and strong compactness of fuzzy spaces: the approachof 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 ιαΧ iscompact (2.4). The notion of strong compactness (in contrast to thecompactness of 7.1) is good (2.2). A continuous image of an α-compactspace is α-compact. A product of non-empty fuzzy spaces is α-compact ifand 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 stronglycompact [35], [100]; on the other hand, the spaces ^ ( R ) , .<P (R) (10.1),M. (I), and Jtk (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 beenconsidered in [108]. In [147] the α-Lin delof number of a fuzzy space (inthe spirit of [29], p. 248, cf. [179]) has been used.

7.3. 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 % suchthat 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 bycontinuous maps. A product of non-empty fuzzy spaces is compact if andonly if all the factors are compact; weak compactness is preserved underfinite products only [88], [90].


The compactness property is good. Every strongly compact space iscompact. On the other hand, there is a compact fuzzy space (Χ, τ),I X I !> Xo, s u c h that for every α € [0, 1) the topological space (Χ, ιατ) isdiscrete (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 beenestablished simultaneously and independently by Lowen and Martin: everycompact 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 andseparation, imposed simultaneously on a fuzzy topology, guarantee a closeconnection of it with a certain ordinary topology on the same set. Webelieve that results of this kind bear additional witness to the specific rolethat compactness and Hausdorff properties play together, not only in generaltopology but in fuzzy topology as well.

A closed continuous map / : X -*• Υ between fuzzy spaces is called perfectif the inverse image f~1(y) of each point y Ε Υ is compact [34]. Invokingthe notion of a Q-neighbourhood, Friedler [34] has shown that a map / isperfect 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 acompact space under a perfect map is compact [141].

7.4. Spectral t h e o r y o f c o m p a c t n e s s [ 1 4 1 ] , [ 1 4 9 ] .This theory enables us to study very subtle properties of compactness typefor fuzzy subsets of fuzzy spaces. It embraces the theories of subsections7.1, 7.2, 7.3 as particular cases. It may be applied also to a description ofcompactness type properties of fuzzy subspaces of ordinary topologicalspaces. Without striving for maximal generality, we present here the basicideas 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 Ιχ as the set Q(M) formedby numbers (3£/ such that for each % d τ satisfying I c V ^ P ( a nd

i

every ε > 0) there is a finite %* C % such that 71/ci V ^o > β· (Here thej

notation from 5.1 is used.) The set SC(M) formed by numbers β Ε / suchthat for each % a x satisfying Mc \J (\/ %) > β there is a finite %B c %such that Mc \J (\J %„) "̂ > β is called the strong compactness spectrum ofthe fuzzy set M.

It is not difficult to note that a space X is compact (7.1) if and only if1 £ C\(X); a space X is α-compact (strongly compact) if and only ifα £ SC(X) (respectively, SC(X) = [0, 1)); the compactness of a space X inthe sense of 7.3 is equivalent to Cl{X) = [0, 1]; finally, a space X isweakly compact if and only if 1 £ Cf(X).

152 A.P. Shostak

The connection between various compactness spectra, as well as possibletypes of compactness spectra for various fuzzy spaces, has been studied bySteprans and Shostak. For example, they have established that a necessaryand sufficient condition for a subset A C [0, 1) to be the strong compactnessspectrum of some fuzzy space X of countable weight (or, equivalently, thestrong compactness spectrum of a fuzzy subset of such a space) is\A\A | < K v

Let us dwell on the properties of the spectrum C(M): = Cl(M). To beginwith, we present the following simple characterization of it:

β e C (M) ̂ (V% CZ t {M CI V 11 > β =» sup {M CZ V %*•

%oCZ%, \%0 | < N . } > β)).The number c(M): = inf(/\C(M)) (inf <£> : = 1) is called the compactnessdegree of the fuzzy set M. Clearly, c{M) G C(M) for every M.

The reader will easily recognize those facts from general topology thatserve as prototypes for the following assertions.

Let Χ, Υ be fuzzy spaces, let Μ £ Ix, and let / : X -> Υ be a continuousmap. Then C(M) C C(fM). Loosely speaking, continuous maps do notdiminish the compactness spectrum. (The changes that the compactnessspectrum may undergo under maps of a given defect (1.8) are described in[174].)

If Μ, Ν, Κ(ΞΙχ,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 Xh i e "J, and let Μ = Π ^ ε Ιχ be a producti

of fuzzy sets Mt e /**, i e J . Then c (M) > inf c (Mt). (The inclusioni

C (M) ZD f) C (Mj) is not valid in general.) If in addition all the A/,'s are

normed, that is, sup Mt (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 IY,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 truethat c(g of) = c (g) Λ c (/)? (Cf. [29], Russian ed., p. 280.)

In general topology statements on the interaction between compactnessand separation type properties are of considerable interest (for example,statements on the normality of a Hausdorff compact space and on theequivalency of compactness and absolute closedness properties in the class ofr3-spaces). The statements presented below testify that an interaction ofthis kind extends to the category of fuzzy spaces as well.


Let (Χ, τ) be a fuzzy space, Μ, Ν G r c, β G Hf(X), β < c{M) Λ c(N), andΜ Α Ν < β€. Then for each ε > 0 there are U, V G r such thatW c i / > p - i , J V C F > p - e , a n d 1/ Λ ^ < β° + «·

Let {Χ, τ) be a fuzzy space and let Μ G /*. Then C(A/) Π Η(Χ) Πη (1/2, 1] C Cl{M) and Λ(7/(Λί) η R(X) Π Η(Χ) Π (1/2, 1] = C(M) ηη 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, andR(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 ofcharacterizing the compactness spectrum of a completely regular space byembedding it in a specific (so-called relatively closed) way into the cubes$ (I)" or the theory of ^-compactness in fuzzy topology closely connectedwith this problem [ 177].

In conclusion, let us present examples of compactness spectra in thesimplest 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; otherwiseC (M) = {0}. Ιΐ Μ e i 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 Ζis not, then C{M) = [0, bc] U (ac, 1 ]. If X is non-compact and Ζ iscompact, then C(M) = [0, ac]. Finally, if both X and Ζ are non-compact,then C(M) - [0, bc}.

Let X = R be the real line, let α G /, Λ/^χ) Ξ β > M2(x) = (2α/π) larctanxl,M3(x) = 1 -M2(x). Then C ^ ) = C(M2) = [0, a c ] ; C(M3) = [0, a ] .

Spectral theories of Lindelofness and countable compactness in fuzzyspaces have been developed [151], [142]. In [143] the so-called spectrumof hereditary Lindelofness of a fuzzy space is studied.

7.5. JT -compactness [158].The notion of ^"-compactness (from the word "Net") seems to be a verysuccessful 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 (as)ses convergesto a.. A fuzzy set Μ in a fuzzy space X is called J/'-compact if every α-netin Μ (α G (0, 1 ]) has a limiting fuzzy point in Μ with value a. A space Xis called jr-compact if the constant Μ — 1 is ^T-compact.

The property of ^"-compactness is preserved under continuous maps andis inherited by closed fuzzy subsets. A product of fuzzy spaces is .^-compactif and only if all the factors are .y-compact. A topological space (X, T) iscompact if and only if the fuzzy space (Χ, ω Τ) is ^Γ-compact.

154 A.P. Shostak

^"-compact sets possess interesting properties from the map-theoreticpoint of view. For example, every ^-compact fuzzy set reaches itssupremum. 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 ofthe Stone-Weierstrass theorem:

Lower (upper) semicontinuous maps from a compact topological space toIR reach their infimum (respectively, supremum).

A strongly compact fuzzy space is ^"-compact if and only if every closedfuzzy subset reaches its supremum.

7.6. Comparing various definitions of compactness.The basic relationships between compactness type properties of a fuzzyspace can be represented in the form of the following diagram [35], [90],[104], [143]: ultracompactness (2.5) => JT-compactness =* strongcompactness =*· compactness (7.3) <* C(X) = [0, 1] => weak compactness ·*=«= compactness (7.1).

The implications converse to those presented above are not valid ingeneral. Nevertheless, within certain rather wide classes many of thoseproperties 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 isultracompact; (b) X is ^-compact; (c) X is strongly compact; (d) X isα-compact for some a; (e) X is compact (7.3) ([110], [111], [176]; cf.[15]).

Within the class of fuzzy neighbourhood spaces the following conditionsare equivalent: (a) ultracompactness; (b) ^"-compactness; (c) strongcompactness; (d) 0-compactness. A fuzzy neighbourhood space (4.5) iscompact (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 topologyliterature. For example, F£/-compactness [67] defined by means of fuzzyfilters; ^-compactness, based on the notion of a Q-cover [85], the so-calledprobabilistic compactness [51], and so on (see also 15.7).

We now consider some approaches to the problem of fuzzy spacecompactifications. The first of them (7.7) reproduces in the fuzzy situationthe idea of Aleksandroff one-point compactification. The idea of the secondapproach (7.8) worked out by Martin consists in a transition from thecategory CFT to the category Top via the functor ι with a subsequentreturn to the CFT. The third approach (7.9) is due to Cerutti; it is basedon the ideas and techniques of categorical topology and is mainly applicableto weakly induced fuzzy spaces.

7.7. 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 closedα-compact (crisp) subsets. We put X — X \J {0}, where Ο is an arbitrary


element not belonging to X, and for every i e s a w e define Κ : X -*• 2 byR (O) = 1 and K(x) = Kc(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 Τα be thefuzzy topology on X determined by the subbase {0: ί / ε τ } υ {R- Κ e βα}·Then the space (Χ, Ύα) is α-compact and (X, r) is a subspace of it; inaddition, X is dense (7.8) in (Χ,Ύα) if (and only if) the space (X, r) is notα-compact. If in addition (Χ, τ) is locally α-compact (respectively, weaklylocally α-compact), that is, for every χ Ε X there is a t/ G r such thatU(x) = 1 (respectively, U(x) > a) and ί/~Ηθ, 1 ] is α-compact and 1 *-Hausdorff(respectively, α-Hausdorff), then the compactification (Χ, Ύ01) is 1*-Hausdorff[35] (respectively, a-Hausdorff [121]).

7 . 8 . U l t r a c o m p a c t i f i c a t i o n s o f a f u z z y s p a c e [ 1 1 0 ] , [ 1 1 2 ] , [ 1 7 6 ] .A fuzzy subset Μ of a fuzzy space (X, r) is called dense if Μ = Χ; Μ iscalled ultradense if all the sets M~l[a, 1], α < 1, are dense in the topologicalspace (X, IT). It is not difficult to note that the ultradensity of Μ in thespace (X, r) is equivalent to its density in the space (Χ, ωιτ). An ultracompactfuzzy space (bX, σ), containing a given fuzzy space (X, r) as an ultradensesubspace, is called an ultracompactification of (Χ, τ).

To construct ultracompactifications of a fuzzy space (X, r) we considerthe compactification (bX, b(ir)) of the topological space (X, IT) and putTb : = {Ui U €Ξ ω (b (IT)) and U \x e? τ} It can be shown that Tb is afuzzy topology on bX and in addition (bX, rb) is an ultracompactificationof (X, T).

If the space (Χ, ιτ) is completely regular and (βΧ, βιτ) is its Stone-Cechcompactification, then the corresponding ultracompactification (βΧ, τ0) ofthe initial space (X, r) has the following property, which enables us to callit the Stone-Cech ultracompactification.

Every continuous m a p / : {Χ, τ) -> (Υ, σ), where (Υ, ο) is an ultracompactultra-Hausdorff fuzzy space, has a continuous extension/: (βΧ, Τβ) ->· (Υ, σ).

The following description of all ultracompactifications of a given fuzzyspace (Χ, τ) is due to Martin. Let (bX, rb) be as above and let rb be afuzzy topology on bX contained in r 6 and inducing τ on X (for example,τί = {0 = sup {F: F E T6, F |.Y = [/}}. Then (bX, r'b) is an ultra-compactification of (X, r). Conversely, an arbitrary ultracompactification ofa space (Χ, τ) can be represented in the form (bX, r'b) for a suitable choiceof(bX, Tb) and r'b C rb.

In conclusion we stress that though the scheme presented above enables usto describe ultracompactifications only, the scope of its applications isbroader 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 compactnessare equivalent to ultracompactness (7.6). We also remark that the existenceof a Hausdorff compactification of a fuzzy space is equivalent to the soaceitself being Hausdorff (5.2) and weakly induced [111].

156 A.P. Shostak

7.9. The categorical approach to compactification theory [ 1 5 ] .

Let % be the category of 1*-Hausdorff (5.2) compact (7.3) fuzzy spaces, andlet 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 itsimilar to the functor β: Top -*• Comp of the Stone-Cech compactificationof 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 morphismrx: X -*• βΧ such that each morphism /: X —*• Υ ~ % admits a uniquefactorization of the form / = φ° rx. The equality (3ocj = cooj3is valid,hence the functor β acts on the category ω(Τορ), which is an isomorphiccopy of Top, precisely as the functor β acts on Top. The two functors βand 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 rx: X -> β~Χ coincides withthe corresponding reflection ru Y: iX ->- fnX. In particular, it follows that %is an epireflexive subcategory of the category of weakly induced 1 *-Hausdorffspaces.

Nevertheless, in spite of the fact that the two functors β and jif are closelyrelated to each other and in many respects seem alike, the situation withStone-Cech compactification in CFT is much more complicated than in Top.One of the reasons is that the category 'β (unlike the category Comp ofcompact Hausdorff topological spaces) is not algebraic over Set or Top.

7.10. Other approaches to the compactification problem in fuzzy topology.Liu [85], [86] has constructed a Stone-Cech type compactification of afuzzy completely regular (6.3) space by embedding it into the cube (5" (/))*of the corresponding weight and subsequently taking closure in it. WangGoujun has investigated ..-F-compactifications of a fuzzy space. An entirelydifferent approach to the compactification problem in fuzzy topology isdeveloped by Eklund [25] (see also 15.7). Compactifications of fuzzysubsets of fuzzy spaces have been considered in [177]. In conclusion wedwell on some unresolved problems in compactification theory.

1) To obtain an internal description of compactifications of fuzzy spacesand 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 eitheran already known [64], [65], [8], [178] or a new fuzzy proximity typestructure.

4) To develop a general categorical theory of extensions of fuzzy spacesand fuzzy subsets.

7.11. Paracompactness of fuzzy spaces.The α-paracompactness property of a fuzzy space (in the spirit of 7.2) hasbeen studied in [109]. A deep analysis of the Q-paracompactness property


of fuzzy spaces and fuzzy subspaces (in the spirit of Q-compactness, 7.8) hasbeen undertaken in [107]. Paracompactness type properties are consideredin a number of other papers.

§8. Connectedness in fuzzy spaces

8 . 1 . C o n n e c t e d n e s s o f f u z z y s e t s : the approach o f Pu and Liu [ 1 1 7 ] , [ 1 1 8 ] .We call a fuzzy set Μ in a fuzzy space (X, r) (C-)disconnected if there areΑ, Β <Ξτ° such that Μ /\ Α φ 0, Μ /\ Β φ 0, Μ < A V Β, andA /\ Β /\ Μ = 0. We call a fuzzy set (unconnected if it is notdisconnected. A fuzzy space (X, r) is called connected if the constantΜ = 1 is connected.

The closure of a connected fuzzy set is connected. The connectednessproperty is preserved by continuous maps.

Any maximal connected fuzzy set Κ contained (in the sense of < ) in agiven fuzzy set Μ is called a component of M. Every connected fuzzysubset ( < ) of a fuzzy set Μ is contained in a component of M. In addition,if Νλ and Λ'2 are two components of a fuzzy set M, then A\ V N2 isdisconnected (both as a fuzzy subset in X and as a fuzzy subset of thesubspace 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 everyfactor is connected. (Let us note that in the proof of this fact a transitionto the laminated modification (2.2°) is used, which does not influence theconnectedness property.) A topological space (X, T) is connected if andonly if the fuzzy space (Χ, ωΓ) is connected.

8.2. 0-connectedness of fuzzy sets.We call a fuzzy set Μ in a fuzzy space (X, r) O-disconnected if there areU, F £ r such that Μ /\ U Φ 0, Μ /\ V φ 0, Μ < U V V, andΜ /\ U /\ V = 0; otherwise we call Μ O-connected. Lowen [96] calls afuzzy space connected if each strictly positive open-and-closed fuzzy subsetis O-connected.

In contrast to general topology, the use of closed fuzzy sets and openfuzzy sets in definitions of "connectedness" of fuzzy sets results in distinctconcepts, 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: β GE /} U {*/,·, F,·} U Γrespectively, where Ux: = OLY, VX : = aZ, U2 '• = U[, V2 '• = V[. It is easy toverify that in (X, r t ) all the constants |3 G / are C-connected, but notO-connected for |3 < a; on the contrary, in (X, r 2) all the constants | 3 £ 7are O-connected, but not C-connected for β < α.

8.3. 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

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 fuzzysubset in (/, σ) such that 1(0) > 0, /(I) > 0. In this connection the fuzzy

points ρ'φίο) and p1^ are called, respectively, the beginning and the end ofthe fuzzy path φ(1). A fuzzy set Μ Ε Ιχ is called pathwise connected if forany 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. Ifa family of pathwise connected fuzzy sets has a non-empty intersection,then the union is pathwise connected as well. Every pathwise connectedfuzzy subset of a fuzzy set Μ is contained in some pathwise connectedcomponent of K. In addition, different pathwise connected components ofa fuzzy set are disjoint [171]. (Certain mistakes in the proofs of these factsmade in [171] have been corrected by Wuyts [167].)

8.4. A spectral theory of connectedness of fuzzy sets in fuzzy spaces (inthe spirit of the spectral theories in 5.1, 7.4) is developed in [ 1 4 9 ] , [ 1 5 0 ] .

8.5. Connectedness properties of the fuzzy real line and subspaces of it.Rodabaugh [124] has shown that there are no non-empty open fuzzy sets Uand 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], hasestablished that the fuzzy interval f (I), the fuzzy cube $ (I)k, and theirlaminated versions are ultraconnected (2.5), and therefore (because theproperties of C- and O-connectedness are good ones) C-connected andO-connected as well.

§9. Fuzzy metric spaces and metrization of fuzzy spacesThere are several viewpoints of the notions of a metric and metrizability in

fuzzy topology. They can be divided into two main groups.

9.1. 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+, whereX CZ Ix (for example, -f = Ix (Erceg [31]) or X = the totality of all fuzzypoints of a set X (Deng [20], Hu Chang-ming [53])) satisfying somecollection of axioms or other that are analogues of the ordinary (pseudo-)metric axioms. Thus, in such an approach numerical distances are set upbetween fuzzy objects.

The main problems in which the authors of this approach are interestedare: 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 completenessand total boundedness type [7], [21].

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 offuzzy (quasi)—uniformities due to Hutton [55], [57] is equivalent to Erceg'sapproach [31].

9.2. We include in the second group those papers in which the distancebetween objects is fuzzy; the objects themselves may be either crisp, or(more seldom) fuzzy. In our opinion, the most interesting papers in thisdirection 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), whereJ(R)is the interval real line (§13), satisfying the axioms: (1) d(x, y) = 0 ifand 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. Thepair (X, d) is called a fuzzy metric space [63], [28]. (In [63] the authorsproceed from a more general definition according to which a fuzzy metricspace is a quadruple (X, d, L, R), where L, R : / 2 -»• / are symmetricdecreasing functions with L(0, 0) = 0, R(l, 1) = 1. In the case whereL = 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 Mengerspace [132] can be described as (X, d, L, R).)

In [61] the notion of a Cauchy sequence in a fuzzy metric space isdefined and on that basis the notion of completeness of a fuzzy metricspace 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 fuzzymetric d on X and every family £ of maps φ: (0, 1] -*- 3 (R) satisfyingcertain conditions. (Unfortunately, we cannot present that interesting butrather cumbersome construction here.) A fuzzy space (X, r) is calledmetrizable if τ = Ω (d, £')for some d and <S. If %0 '•= {ψι· t ^ R+), whereφ ( Ξ t, then a fuzzy topology Ω (d, ^ 0 ) is topologically generated (2.2). Inparticular, if (X, T) is a topological space metrized with a metric m, thenωΤ = Ω (ω (/»), 'So), where ω(νη)(χ, y) : = m(x, y).

Applying the so-called method of levelwise representation of fuzzynumbers, Eklund and Gahler [28] define a fuzzy metric on the interval realline ,7 (R) itself; for different ίί this metric generates different fuzzytopologies Ω (ρ, g) on J (R)(see also §13).

9.3. Stratifiable spaces.The class of stratifiable topological spaces first distinguished by Ceder andstudied in detail by Borges [13] is one of the most interesting representativesof the group of so-called generalized metrizable spaces. In [135] the notion

160 A.P. Shostak

of a stratifiable space was extended to the fuzzy case. We remark that theproperty of a fuzzy space of being stratifiable may be used in some casesinstead of fuzzy metrizability type properties, and at the same time it iscomparatively easily amenable to study in the framework of fuzzy topology.

A fuzzy space (X, r) is called stratifiable [135], [136] if it is possible toassociate with every U G τ a sequence (£/„)„ C r in such a way that(l)Dn<U for all η e Ν; (2) V Un = U, and (3) if U < V, VET, then

Un < Vn 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 fuzzystratifiable spaces is invariant under transition to subspaces and closedimages. A fuzzy space dominated by a family of fuzzy stratifiable spaces isstratifiable. A product of countably many laminated fuzzy spaces isstratifiable if and only if all the factors are stratifiable [135], [136].

In [137] maps of stratifiable spaces to the fuzzy interval f (I) areconsidered. In particular, it is shown that a fuzzy space (Χ, r) is stratifiableif 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){\~) - Uc{x), χ G X, in such away that U < V G τ implies fv <fv.

If (Χ, τ) is a stratifiable space, then it is possible to associate with everypair (A, U), where A G rc, U G r and A < U, a function (fAU = ) /: X ->- f (I)satisfying the inequalities A(x) < / ( * ) ( Γ ) </Qc)(O+) < U(x), χ G X, in sucha way that if A < Β and U < V (where Β G rc', V G τ, Β < V), thenJBV <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 realline R, the unit interval /, and so on) in fuzzy topology? A kind of answerto this question is contained in subsections 2.1, 2.2, 2.4, where the functorse, ω, and r? have been considered, enabling us to associate with a topologicalspace (X, T) the fuzzy spaces (X, eT), {Χ, ωΤ), and (Χ, ηΤ), which can beviewed as fuzzy copies of the original space. An essential feature of thesefunctors is that they change the topological structure only, leaving theunderlying set of the space unchanged. However, being to a certain extenta 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 thecategory 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 offuzzy topology having an essentially different nature and enabling us toapproach the problem in a different way. The most completely investigatedconstruction among them is the so-called fuzzy real line f (R).


§10. The fuzzy real line £ (R) and its subspaces

In 1974 Hutton [54] constructed the Ζ,-fuzzy interval fL (I), and 4 yearslater Gantner, Steinlage, and Warren, by developing Hutton's idea, introducedthe Ζ,-fuzzy real line f r, (R) (in the original papers the notations I(L) andR (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 topologicaland algebraic properties, which enable us to consider it as the (mostimportant) fuzzy analogue of the ordinary real line, as well as the universaland categorical [125] nature of its construction. We recall also that thefuzzy 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 tothe [/]-fuzzy real line £ (R), that is, to the case L = I; certain features ofthe general case will be noted in 10.6.

10.1. Construction of the fuzzy real lines £ (R), £>• (R).Let Ζ (R) be the set of all non-increasing maps z: R ->- / such thatsup ζ (χ) = 1, inf ζ (χ) —- 0. We introduce an equivalence relation ~ on Z(R)

X X

by putting z1 ~ z2 if and only if z1(x+) = z2(x+) and Zj(x~) = z2(x~) for allχ 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 isprecisely one left semicontinuous function in each equivalence class[z] e £ (R), see [123], [97].) We introduce a partial order < on £ (R)byputting [zj < [z2] if and only if z1(x+) < z2(x+) for every χ £ X.

We define a fuzzy topology σ on f (R)by taking as a subbase the familyof fuzzy sets {lb, ra: b, α e R) d / y ^ ) ; where lb, ra are determined bylb[z] = z(b~)c(= 1 — z(b~)), ra[z] = z(a+) respectively. The fuzzy real lineand the laminated fuzzy real line can now be introduced as follows:f (R): = (f (R), σ) and fλ (R) : = (f (R), λσ), where λ is the laminatedmodification functor (2.2°).

The subspaces of the fuzzy real line of the form

f(a,b) :={[z] :2GZ(R), 2 (» + ) = l , . ( i i > 0 } andf la, b] := {[zh z e Z ( 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]. Itis 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) andtheir laminated versions, and in establishing which of these spaces are

162 A.P. Shostak

homeomorphic to each other. In the sequel we shall also need the followingnotation: f (R+) : = £ (0, +00), f (R-) : = f (-00,0), f (R°) : =

= U ^(-οο,-ε).ε>0

By identifying a number α ε R withlx(_cxi,a]l, that is, with the equivalenceclass containing the characteristic function of the set (—°°, a], we are able toconsider [R as a subset of f (R). Moreover, the fuzzy topology σ induces onR the ordinary (order) topology T< (and λσ induces the fuzzy topologyω7τ<), 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 off (R) was proposed independently by Hohle [50]. Up to an isomorphism3f, (R)may be characterized as the set f (R) endowed with the weakestfuzzy topology σ* containing σ such that

( P 6 σ*, a ^ I) -+ {{U + a) /\ ί ^ σ*, (C7 — a) V 0 <Ξ σ*)·

(For properties of Μ (R) see also [ 100]-[102].)

10.2. Algebraic properties of the fuzzy real line.In [123], [127] Rodabaugh defined the sum Θ and product Ο operationson the set $ (R). The definitions given by Rodabaugh are very cumbersomeand require additional constructions. In [97] Lowen showed that the sumof elements [zj, [zj e f (R) can be characterized by [zt] θ h2] (x) == sap (zx (t) /\ z2 (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 Abeliansemigroup with zero O e R , and (& (R), Q) is a multiplicative Abeliansemigroup with unity 1 £Ξ R; in addition, © is distributive over @. Leta , i £ f (R). Then a Q b = 0 <<=> either a = 0 or b = 0; a Q b <ΞE R \ {0} =* a, b <Ξ R \ {0}. If a < b, then a Q c < b 0 e f or e e f (R+)and 6 0 c < α G c for c e £" (R~)· Both sets f (R+) and f (R") areinvariant under the operation φ .

The elements of the set f (R°) are called fuzzy zeros. It is not difficultto show that f (R°) Π R = {0}; a Q b <=: f (R°) ^ either a e f (R°) ori 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°): i f a Q ^ = a Oα φ Of (R°), then b = c.


For every s £ f (R)weput-a : = (- l)aand Aa : = {b: a -|- b ΕΞ f (R°)}·The elements of Aa are called fuzzy additive inverses to a. It is not difficultto verify that -a e Aa, Aa f) R = — {a} for ο Ε R and | 4 0 Π R I = c forα φ R.

The elements of the set f (1R1) : = {a + 1: fl6f(R0)} are called /wzzjunits. Rodabaugh shows that | f (R1) Π f (R°) I = c; i f o , 6 e f (iR1),then a Q J E f (R1); i f a . l £ f (R1) Π f (R°), then a + δ ΕΕ ,f (R1);^ (R1) Π R = {1}·

The elements of the set Ma : = { ί ι ε , ί (Κ): a Q t e f (R1)}, whereα Φ Ο, α £Ε ίΓ (R), are called fuzzy multiplicative inverses to a. If α €= R,

thenM0 Π R = $ , otherwise | Λ/β Π R ! = c; | Ma \ = | Ma f] Aa | = c.

(Rodabaugh [127] calls any algebraic system (H, +, ·, < ) satisfying theproperties similar to those listed above a fuzzy hyperfield. The prefix"hyper" indicates that such a system is obtained as a fuzzy extension of anordinary field, in our case R.)

10.3. 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 samefact was obtained [97]. In [127] the continuity of the product operationO f ( R ) x i ( R ) - ^ f (R) was proved.

It is important to note, however, that for fixed a e= f (R) and6 ε ί (R++) U & ( R " ) the maps ha, gb: $ (R) -»- & (R), defined byha (s) :=-- s φ α and i t (s) : = i 0 s, s ε f (R), in general are nothomeomorphisms but only weak homeomorphisms, because the inverse mapsΊα1! gb1· f (R) -*• f (R)> being weakly continuous, need not be continuous[123], [127]. (The property of weak continuity for a m a p / : (Χ, τχ) -»·-* (Υ, τγ) distinguished by Rodabaugh [122], which is intermediate betweenthe property of continuity of/itself and the property of continuity of them a p / : (X, LTX) ->· (Y, LTY), is an interesting and specifically fuzzy propertyof maps between fuzzy spaces.) If a, b e R and b Φ 0, then both ha and gb

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 is0-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 forsome 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) aswell.

It is not known if there exists a continuous extension of the algebraicoperations + and · from the complex plane C = R x R to the fuzzy spacef (R) χ f (R) containing it in a canonical way [123].

164 A.P. Shostak

10.4. Topological properties of fuzzy real lines and fuzzy intervals havebeen discussed in the corresponding subsections (6.2, 6.3, 7.2, 8.5, and soon).

10.5. The role of the fuzzy real line in fuzzy topology.We have tried to clarify this role when considering the correspondingproblems. 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 soon. The fuzzy real line is successfully and "purposely" (that is, as ananalogue of the ordinary real line) used also in the the theory of fuzzyuniformities [55], [128], fuzzy proximity theory [65], in certainconstructions of compactifications ([85], [86], [177], and others).

We note also that every statement in the category CFT involving thefuzzy real line remains true in the category LCFT with ;F(R) replaced bythe laminated fuzzy real line fx (R) (by the way, this is a manifestation ofthe universality of the fuzzy real line construction).

10.6. On the I-fuzzy real lines fL (R) and fl (R).Substituting an arbitrary fuzzy lattice L for the interval / in the definitionsof 10.1, we arrive at the constructions of the Z-fuzzy real lines fL (R),fl (R) and the Ζ,-fuzzy intervals fL (7) and f\ (I), see [54], [35], [ 126].In particular, for L = 2 the 2-fuzzy real lines §\ (R)and ^"2 (R) areisomorphic 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 onthe choice of Z. Let us illustrate this by just one example. Artico andMoresco [9], while studying the property of a*-compactness (a G Z + ) ofthe space .ft (/)(= the compactness of the space ια. (fL (7))(2.7)),established that if Ζ is a chain (for example, Ζ = Γ), then ^ L (/) is a*-compactfor all a; but if L = 2Z for a set Z, then fL (I) is a*-compact if and only ifeither | α | < tf0 or | α | > c .

§11. Fuzzy modification of a linearly ordered space

The main idea of constructing the fuzzy real line was used in [147] toconstruct the fuzzy modification (fuzzification) of an arbitrary linearlyordered space.

Let X be a linearly ordered space. We define the set Z{X) and anequivalence relation ~ on it by analogy with 10.1, and let Ζ (Χ) be the setquotient of Z(X) by the ~ relation. Then by analogy with 10.1 we define afuzzy topology σ on X (X) and put

X (X) : = (Χ (Χ), σ), X>- (X) : = (Χ (Χ), λσ).


The fuzzy spaces X (X) and Χλ (Χ) are called respectively the fuzzymodification {fuzzification) and laminated fuzzy modification {laminatedfuzzification) of the linearly ordered space X. Clearly, X (R) is precisely thefuzzy real line f (R), and Χλ (R) is the laminated fuzzy real line $~?" (R).

Although the spaces X (I) and f (/) are distinct, an isomorphism betweenthem 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 subspaceof Χλ (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 Χλ (X) as well. (We definethe weight of a fuzzy space as the minimal cardinality of a base for a fuzzytopology of it.)

The following conditions are equivalent: (1) the space X is bounded (thatis, there are maximal and minimal elements); (2) the space X (X) is stronglycompact; (3) the space X (X) is α-compact for some a £ [0, 1). If X is notbounded, then for every a € [ 0 , 1) the α-Lindelof number (7.2) of X (X) isequal 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 non-

decreasing continuous map. For every ζ €= Z{X) we define uz £ Z{Y) byputting uz (y) : = inf {z (x): f (x) < y} if (<-, ν] ΓΊ f{X) Φ φ and uz{y): = 1otherwise. The map /· χ (Χ) -κ j£ (5r) defined by f[z] = [uz] 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 viewedas an embedding functor from the category Ord of linearly ordered spacesand continuous non-decreasing maps into the category CFT; this functorassociates with a linearly ordered space X the fuzzy space X (X) and with acontinuous non-decreasing m a p / : 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 fuzzymodification 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 canarrive at a probabilistic interpretation of the fuzzy real line. The first todistinguish explicitly the probabilistic aspect of the fuzzy real line wasLowen [100] -[102] . (Similar ideas are traced in Hohle's works [49], [50],and others.) Developing the probabilistic treatment of the fuzzy real line,

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 onthe set JI (X) of probability measures on X; for X = R this constructionleads to the fuzzy real line f (R)as a particular case. In [140], [44] thisconstruction is extended to the case of an arbitrary topological space X (andeven an arbitrary fuzzy (1.4) space X).

12.1. Construction of the fuzzy probabilistic modification of a topologicalspace [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 mapsp: 3d (X)-*-I). For every family ξ C ωΤ (2.2) we define a fuzzy topologyTj on JI (X) by taking as a subbase the family of fuzzy sets

e ξ} C

where 6V (p) : = § U dp. We denote the resulting fuzzy space by

j / s (X) ; = {Ji (Χ), τξ). In the cases ξ = Τ and ξ = ωΤ the construction ofthe fuzzy probabilistic modification Jl% (X) can be viewed as an embeddingfunctor Ji%: Top ->- LCFT, associating with a topological space X the fuzzyspace -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 bythose measures ρ for which p(Z) = 1. Then for ξ = Τ and ξ = ωΤ the mapφ: 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 ro-space. By assigning to each point χ £ X the corresponding

Dirac measure px (that is, px{E) = l * x € £ ) w e identify X with thesubset 25 (X) of all Dirac measures of the space JI (X). Under thisidentification the topological space X (respectively, the fuzzy space ωΧ) ishomeomorphic to the subspace 3) (X) of the space Jl\ (X) if and only if ξ isa 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" tothe 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 offuzzy points of the space X (cf. 3.1).

By means of the functor ι the fuzzy topologies considered above areclosely related to the so-called Aleksandrov topology [ 5 ] W on the probabilitymeasure space JC (X). (In a number of cases, in particular, if Ζ is aseparable metric space, W coincides with the so-called weak topology usedin measure theory (see, for example [11], [157]). In particular,


itT = ιτ,,,τ = W. However, as the "forgetful" functor should do, undertransition from CFT to Top the functor ι loses part of the useful information,which makes it impossible to investigate the constructions considered here inthe classical measure-theoretic framework. We shall illustrate this by thefollowing two examples.

For a separable metric space X, the set S> (A) is known to be closed inthe weak topology ([11], and others). The following equality, which is validfor 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 ΜωΤ (Χ), and ρ is anarbitrary smooth measure. The meaning of this equality is that the degreeto which a smooth measure ρ is an adherence point for H> (X) in the spacesJKi (A') and ΜωΤ (Χ) is equal to the maximal value of its atoms.

We present also the following equality characterizing the closure of theset CK (X) of all two-valued measures in the spaces JiT (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] Un Φ 0 } (compare with the well-knownstatement [157] on the weak closedness of X (X)).

12.2. The connection between the topological properties of a space X andits fuzzy probabilistic modification we shall illustrate by the followingassertions [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 isalso equal to the weight of JlaT (X). The density of X is equal to both thedensity (that is, the least cardinality of dense subspaces) of JiT (X) and thedensity of ~# ω Τ (λ').

The following conditions are equivalent: (a) the topological space X iscompact; (b) JlT (A) is compact (7.3); (c) J(T (X) is countably compact(in the spirit of 7.3); (d) ΜωΤ (Χ) is compact (7.3); (e) J(aT (A) iscountably compact (7.3). Moreover, in every item b, c, d, e the conditionof (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 ΜωΤ 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 jfT (A) aresignificantly worse.

12.3. Construction of the fuzzy probabilistic modification as a generalizationof the fuzzy real line.Assigning to every ρ <Ξ Jl (R)the distribution function [40] -P: IR ->- /defined by zp{t): = p(-°°, t), t e R, and putting φ(ρ) := [ 1 — z p ] , we arrive

168 A.P. Shostak

at a map ψ: Ji (R) -»- f (R). Endowing the set JK- (R) with the fuzzytopology rff, where π : = {(— oo, a), (fc. -f co): a, fc£lR}isa standardsubbase of the real line, we make the map ψ: J!n (R) -» f (R) into ahomeomorphism [100]. This enables us to consider the construction of thefuzzy probabilistic modification as an essential generalization and at thesame time a standardization of the fuzzy real line construction.

We also note the following [ 148]. If X is a linearly ordered space ofcountable character without isolated points, then ψ: Jl^ (X) -+• χ (X) is ahomeomorphism. (π and φ are defined here by analogy with the precedingparagraph.)

As Lowen [100] showed, in spite of the fact that the fuzzy topology τ π

on Λ (R) is weaker than ττ, and its laminated version τ£ is weaker than τ ω Τ

(in particular, their separation properties are considerably worse), theequality vrn = ιτ^' = W nevertheless remains valid (cf. 12.1), which, inLowen's opinion, is of fundamental value. We also point out the identity ofthe compactness properties for all spaces of the type J!n(X), MT(X), ΛωΤ(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 basedon the real line R is used, which we call here the interval real line.

13.1. The construction and algebraic properties of the interval real lineJ ( 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, uppersemicontinuous, and such that every level of it za : = z~x[a, 1 ], a G (0, 1 ],is a bounded subset of R (that is, taking into account the precedingconditions, za = [zal, sa2] for some z a l, za 2 e= P.)· The set of all regularnumbers forms the interval fuzzy real line J (R). Identifying a numberα €Ξ R with the characteristic function χα, we may consider R as a subset ofthe 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 operationsrespectively. We put u~v : = u + (—\)v and ( - I ) M = : —u. Then {-u)(t) == u(—t) for all i e R ; —(u + v) = (—«) + (—u); both equations a + z — 0 andaz = 1 (α, ζ ΕΞ % (R) and α Φ 0 in the second case) have a solution (whichis 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 uai < u a i, ζ = 1,2, for allα G (0, 1 ] , we obtain a partial order on Cf (R). In addition, if u, ν, ζ > 0.


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 {zai, 0} and

Zai : = max {zai, 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+} [ 2 8 ] .

13.2. Fuzzy metric on y ((R).As we have mentioned, the interval real line plays a key role in the definitionof a fuzzy metric (9.2). Following [28], we now show how a fuzzy metricis introduced on ,*f (R) itself. For u, ν £ j ( I R ) and α Ε (0, 1 ] we putδ (u. υ) (a) : = \ ual — Vai \ -r \ ua2 — i 'o 2 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 aregular fuzzy number lu, ι; I of which they are α-levels, and that the mapd: 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 equality10, u — v\ = \u, v\ fails (and therefore a fuzzy metric cannot be defined bymeans of a "fuzzy norm")·

§ 14. On hyperspaces of fuzzy sets

Since we cannot consider the construction of hyperspaces of fuzzy sets indetail in this survey (this would require both a considerable amount of spaceand the introduction of new notions), we shall try to give the reader acertain intuitive idea of hyperspaces of fuzzy sets, constructions which asidefrom a theoretical interest for fuzzy topology itself may be used inapplications.

14.1. 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 fuzzyuniformities on the hyperspace Ix of fuzzy subsets of X, the so-called globalfuzzy uniformity %s and the horizontal fuzzy uniformity %h; we have%g C %,, We put I* : = (/*, %g), /£' : = (/* \ {0}, %,).

On the hyperspace 2X of non-empty subsets of X, considered as asubspace of Ix, the fuzzy uniformities %e and %h induce the classicalHausdorff-Bourbaki uniformity ([14], English ed., p. 172). Convergence inthe space Ix is determined by the topology of X, characterizing by the sametoken a "horizontal" convergence of fuzzy sets; convergence in thehyperspace Ix is determined by both the topology of X and the topology of/, characterizing by the same token the "global" convergence of fuzzy sets.

170 A.P. Shostak

Both in Ix and in If the lower parts of the graphs of fuzzy sets influencethe convergence less than the upper parts do. Therefore the two uniformitiesforms a sufficiently fine device for describing the proximity and convergencerelations between fuzzy sets whose lower parts are either inaccuratelydefined or not defined at all, which may prove to be useful in certainapplications (see, for example, [23], [81]).

If / : X -*• Υ is a uniformly continuous map between uniform spaces, thenthe maps / : Ix -*• 1%, / : -̂ / ~* Ij (0-6) are uniformly continuous as well (inthe sense of [95]) and therefore continuous.

Let Φχ(Χ) be the subspace of If formed by all U G Ix that are uppersemicontinuous. A uniform space X is compact (precompact) if and only ifthe fuzzy uniform space Φχ(Χ) is compact (7.3) (respectively precompact[104]). The uniform space ιν (Φβ (Χ)), where ι υ is the uniform analogue ofthe functor ι (2.4) (see [95]), is isomorphic to a closed subspace of thehyperspace 2Xxl endowed with the Hausdorff-Bourbaki uniformity (thecorresponding isomorphism sends each Μ G Ix to its endograph{(*, t): t < Μ (χ)} e 2*x') [80], [81].

The fuzzy space If is not topologically generated, whatever the initialspace X may be. Also, the subspace ΐζ\ of Ix formed by those Μ G Ix forwhich sup M(x) = a is not topologically generated for any a G (0, 1 ].Lowen stresses that the existence of natural and important (laminated) fuzzyspaces such as Ix and I*h that are not topologically generated (and thereforecannot be studied by means of ordinary topological methods) is evidence ofthe need for a general theory of fuzzy topological spaces.

14.2. Other structures on the hyperspace of fuzzy sets of a metric space X.Kloeden [80] defines a pseudometric on Ix as the Hausdorff distancebetween endographs of the corresponding fuzzy sets. A similar pseudometricon Ix was considered by Goetschel and Voxman [38], [39]. Heilpern [46]made use of a pseudometric on Ix in which the distance between fuzzy setsis defined as the supremum of the Hausdorff distances between their α-levelsover all α G (0, 1]. Kaleva and Seikhala [63] define convergence of asequence of fuzzy sets (Mn) C Ix to Μ G Ix as convergence for eacha 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 2X.

A comparative analysis of these approaches in the case λ' = Κ" isperformed in [62].

§15. Another view of the subject of fuzzy topology and certaincategorical aspects of it

In the preceding sections, as a rule, we have considered fuzzy topologicalspaces in the classical [169], [16] sense, that is, fuzziness has been treatedas the possibility for characteristic functions to take values in the interval /.


The general case of L-fuzzy topological spaces, where L is a fuzzy lattice,has been considered occasionally and for illustrative purposes only. Westress, however, that although properties of Z-fuzzy spaces may dependheavily on concrete properties of the lattice L, and their study sometimesmeets additional serious technical difficulties, the theory of Ζ,-fuzzy spacesfor an arbitrary (fixed) fuzzy lattice is based as a whole on the same ideasas the "classical" theory presented above, and in this sense it can be viewedas a direct generalization of the latter.

An entirely different view of the subject and problematics of fuzzytopology arises if we consider Z-fuzzy spaces for various fuzzy lattices Lsimultaneously. This view of the subject of fuzzy topology, the idea ofwhich goes back to Hutton's papers [56], [57], has been developedsuccessfully by Rodabaugh [125], [128], Eklund and Gahler [25], [28],and the author [145], [154]. (Similar ideas are contained in T. Kubiak'sdissertation (unpublished).)

In this section, following the papers [145], [154] (based on thefundamental ideas of Hutton [56]), we shall define the category GFT (theso-called general category of fuzzy topological spaces), discuss certainspecific properties of it, and show how all the categories of fuzzy topologydescribed above may be identified with relevant subcategories of it. In15.5-15.8 we shall dwell on the category φ of Hutton fuzzy spaces andtry to illustrate by an example of it the specific features of the topologicaltheory that arises.

15.1. The category GFT [145], [154].Let Lat be the category whose objects are fuzzy lattices (0.3) and whosemorphisms are maps ψ: 1^ -*• L2 preserving the supremum, infimum, 0, 1,and involution. (The category Lat°p opposite to Lat [43] is exactly thecategory FuzLat defined by Hutton [56], see also [27].) The objects ofthe category GFT are triples (X, L. if) where X is a set, L is a fuzzy lattice,and 0Γ: Lx -*• L is an I-fuzzy topology (1.4) on X. We declare themorphisms of GFT to be those pairs (/, φ): (ΧΎ, Lx, 3Ί) ->• (Λ'2, L2, £Γ2) thatare morphisms in the category Set X Lat01' (that is, / : Χγ -> Χ2 is a set-theoretic map and φ'1 : L2 -*• L1 is a morphism in Lat) and, moreover, suchthat #"j (φ' 1 ο Ν = /) > φ"1 (<f „ (Ν)) for each A7 G 7y—a kind of continuitycondition! (Following the terminology of Eklund and Gahler [28], we maysay that Set χ Lat°p is declared to be the ground category for GFT. Incontrast, for the categories Top, CFT, FT, CFT(L), LCFT(L), and so on,the ground category is the category Set\) If (/,, φχ): (Χ-,, Lx, 3Ί) -»- (X2, L2, <f 2)and (/s, φ2): (Ζ 2 , L2, iT.2)->- (X3, L^f3) are morphisms in GFT, then theircomposition is defined as (/2 ° ft, φ2 ° cp^: (X,, Lx, ?°Γ,) -*- (Z 3 , Z 3, $"3). Asidentity morphisms we take pairs of the form (idx, idL): (X. L,£f) ->-

172 A.P. Shostak

15.2. Product in GFT [145], [154].We shall illustrate the features of the viewpoint presented above of thesubject of fuzzy topology, as well as the specific character of the problemsarising, by an example of the product in GFT.

Let Χ, Υ be sets, L, Κ fuzzy lattices, 8: KY ->- Κ a A -̂fuzzy topology onY, and (/ φ): (X, L) -*• (Y, K) a morphism in Set χ Lafp. The weakestZ-fuzzy topology if on X making (/, φ): (X, L. f) -*- (Y. K, S) a morphismin GFT is called initial for the pair (/, φ). It is not difficult to verify thatit can be defined by if (U) : = V i^a ~ & ((/) Λ Φ" 1 α : a GE K+), U G Lx,where Fa: = {<floN°i : Ν EE KY, 8 (N) > a}. Now let {(Yy,Ly, #\); γ6Ξ Γ}be a family of fuzzy spaces and {(/γ, ψγ): (λ7, Ζ) ->- (Υν, Ly) : y Gr Γ} afamily of morphisms in Set X Lafv. The weakest Z-fuzzy topology eT on Ζmaking all (fy, ψΊ) continuous is called initial for that family. It is easilyverified that 3~ = sup SFy (1.6), where 3~y is the Ζ,-fuzzy topology on X

γinitial for (fy, ψΊ).

Turning to the definition of the product in GFT, we first recall theoperation of fuzzy lattice product © introduced by Hutton [56]. Let{Ly: γ ε Γ} be a family of fuzzy lattices. The elements of the latticeL = (g> Ly are declared to be subsets a d II {Ly: y €Ξ Γ} (Π stands for theproduct 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 sy < ίγ in L 7 for each γ G Γ) and 2) if£ 7 C Ζ,γ and b - Uby C a, then /3 = (/?7) G a, where 0γ = sup by. Bymeans of the relation a < δ *> α C 6, where a, J e i , the set Z, is made intoa lattice; finally, putting i c :== {x: (Vy (Ξ b)(3y e? T)(xy < ,ϋ/γ)}, we can view

Ζ as a fuzzy lattice. (For example, if Ly = 2zv, where Zy is a set, then

<g) ZY ss 2nzv.) The equality πΫ,1 (<Ye) = {s e ΠΖΥ: sY, < ^ J determines amap πΥ^ : LYi ->- Z. Hutton demonstrates that the operation ®YZY defined insuch a way together with the projections π7 : Ζ -»· Ζ γ is a product in Ζαί°''.

We consider now a family {(^Y, ZY, ^ Υ ) : γ ΕΞ Γ} of fuzzy spaces and put

X:= LI Xy, Z : = g) Ly. Let 5~ be an Z-fuzzy topology on X initial for theΥ

family {(py, πΥ): (λ', Ζ) ->- (XY, ΖΥ, < "̂ν); γ GE Γ}, where py : X -*• Xy areprojection maps and iry : L -*• Ly are defined as above. Then the fuzzyspace (X. L, if) is the product of the family of fuzzy spaces{(Xy, Ly, ε Τ ν ) :γ€ΞΓ} in GFT.

We call the reader's attention to the fact that every statement about theproduct in GFT contains information which is entirely different from thatcontained in a similar statement about the product either in FT{L) (1.4) orin any other category from § 1. One of the reasons is that a lattice changesunder the product operation in GFT (which seems natural if one remembersthat the ground category for GFT is Set X Latop rather than Set as, forexample, in the case of FT(L)). Moreover, as Eklund [25] established, thelattice ® ZY, where Ly = Ζ for all γ G Γ, Irl > 2, is isomorphic to Ζ if and


only if L = 2. Hence it follows, in particular, that for ordinary topologicalspaces and only for them the ordinary product coincides with that in GFT(and, therefore, with the product in any subcategory), which is one of themanifestations of the "invariance" of general topology in fuzzy topology!

In conclusion we remark that the category GFT has equalizers (see, forexample, [43]), which means, together with the presence of products in it,the completeness of GFT.

15.3. Certain subcategories of GFT.We denote by GCFT the full subcategory of GFT formed by all ChangZ-fuzzy spaces (1.2), by GLFT the full subcategory of GFT formed by alllaminated Z-fuzzy spaces (1.3), by GOFT the full subcategory of GFTformed 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 ofGFT formed by Z-fuzzy spaces. An obvious meaning is assigned to thenotation GLCFT, GOCFT, GCFT{L), GLFT(L), and so on.

It is easily observed that GCFT is naturally isomorphic to the categoryFuzz introduced by Rodabaugh [125]. The category GCFT is an epireflexiveand epicoreflexive [154] subcategory of GFT. The category GLFT isepicoreflexive but not reflexive in GFT [154]. (To verify that GLFT is notreflexive in GFT, it suffices to note that the product of infinitely manycopies 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 inGCFT. Eklund [25] has shown that GOCFT is epireflexive in GCFT (hencein GFT as well), while no subcategory of GCFT not containing Top can bereflexive 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 thatthe two categories are isomorphic only if the only endomorphism φ: L -*• Lin Lafv is the identity map. This condition is satisfied by L — 2, and onecan show that both categories are isomorphic to Top.) We note also thatthe product is absent in GFT(L) for L Φ 2, unlike in FT{L).

To identify categories of the form FT(L) with the correspondingsubcategories in GFT, we denote by GFT{L, ψ), where L is a fuzzy latticeand φ : L -*• L is a morphism in Laf1', the subcategory of GFT(L) having thesame objects as GFT(L) and morphisms of the form (/, ψ), with ψ = ιρ" fora given φ and some η € IN. It is easy to check that the category FT(L) canbe identified with the category GFT(L, d), the category CFT(L) of ChangΖ,-fuzzy spaces (1.1) with GCFT(L, id), the category CLFT(L) of laminatedChang Z-fuzzy spaces (1.2) with GCLFT(L, id), and so forth.

15.4. Hutton fuzzy spaces.Let Ζ be a fuzzy lattice and (X, r) a Chang Z-fuzzy space (1.1). It is notdifficult to note that in this case X:=LK is a fuzzy lattice as well, and rforms a subset in !£• invariant under taking supremums and finite infimums,

174 A.P. Shostak

and containing 0 and 1. This simple observation enabled Hutton [56] todistinguish a category ξ>, which was later called the category of Hutton, orpointless, fuzzy topological spaces. (Similar ideas can be traced in Erceg'spaper [32].)

The objects of © are pairs (Χ, τ), where X is a fuzzy lattice and r is asubset in X invariant under taking supremums and finite infimums andcontaining 0 and 1; the role of morphisms /: (X1, τ,) ->• (X^, τ.,) in .£) isplayed by maps /"': X% -> Xx preserving sup, inf, involution, 0, 1, and suchthat 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 thecategory S? and the full (epireflexive [26]) subcategory £>' of GCFT whoseobjects are of the form (* : L. τ), where L is a fuzzy lattice and τ is theChang Z-fuzzy topology on * ([125]).

It is not difficult to understand that the category of Hutton spaces Scontains (up to an isomorphism) the category Top as a full subcategory; itsuffices to assign the Hutton space (2X, T) to any topological space (X, T).We stress here that although both the notion of a Hutton space (thecategory §) and the notion of an Z-fuzzy Chang space (the category CFT(L))generalize the notion of a topological space (the category Top), the purposesand the ideological basis of these generalizations are diametrically oppositefrom the point of view of "classical" topology. The main goal of Hutton isto study "that part of topology which relates to the lattice theory" [56],and in order to realize it Hutton ignores points and considers sets aselements of a lattice Ζ and replaces the set-theoretic operations by the latticesupremum, infimum, and involution operations. On the other hand, in thetheory of Z-fuzzy spaces the notion of a point extends to the notion of anZ-fuzzy point (in the spirit of 3.1) (and, either explicitly or implicitly,influences the corresponding theory heavily). We note in this connectionEklund's paper [26], in which the possibility of "extracting the maximalpoint part" of a Hutton space (Χ, τ) is studied, that is, representing it in theform (X, L, r), where X zz Lx and in addition L is not representable asL ^ Kz for any fuzzy lattice Κ and a set Z, \Z\ > 2.

In conclusion we remark that the use of structure relations only in latticesX and τ d X, without appealing to points, makes the theory of Huttonspaces closer to the theory of locales ([60], [120]), and others). Theconnection between the two theories has been studied by Rodabaugh (inpreprints).

15.5. On subspaces of Hutton spaces and fuzzy subspaces of Ζ,-fuzzy spaces.Let (Χ, τ) be a Hutton space and let a s i , For every u E ^ w e putua:= (u /\ a)\/(a /\ ac) and let Xa: = {ua: u e X}. Restricting the V and Λoperations from Ζ to £a and assigning to each ua £E Xa an element


u*: = ul /\ a (=(i/ c ) 0 )e ϋ£β, we get a fuzzy lattice Xa with an involutionun ->- w*. In this case TO: = {UU: u e τ} is a fuzzy topology on Xa, and hence(52Q, τα) is a Hutton fuzzy space. (Xa, τα) is called a subspace of the Huttonspace (Χ, τ) [32].

In the case S = LA' and a C X the subspace (,2!α, τα) of the Hutton space(Χ, τ) is isomorphic to the subspace (a, ra) (1.7) of the Chang Ζ,-fuzzy space(X, r); on the other hand, the notion of a subspace of a category in φmakes it possible to speak about a fuzzy subspace (£a, τα) of the ChangΖ,-fuzzy space (X, r) based on an Z-fuzzy subset a £ Lx (cf. 1.7).

We now consider the basic topological properties in the categories inquestion. Pretending neither to completeness nor to representativeness, werestrict ourselves here to the Hausdorff, compactness, and connectednessproperties in the category ξ) of Hutton spaces. The choice of the category§ is not accidental: on the one hand, by replacing the space (Lx, τ) e Ob (φ)by the space (X, L, r) £ Ob(GCFT) the notions we have introduced areextended to GCFT and, furthermore, to GFT, and on the other hand, thecategory § enables us to understand both the specific character due to theuse of different lattices and the ideology of the "pointless" approach.

15.6. Separation in ξ).Clearly, when defining the Hausdorff property and other lower separationaxioms for Hutton spaces the ideas of the approaches described in § 5 areinapplicable, 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 T0-space if X is the closure of the subsetτ U TC (where TC: = {UC: U e= τ}) with respect to arbitrary supremums andinfimums; (Χ, τ) is called an R0-space if r is contained in the closure of r c

with respect to arbitrary supremums; (Χ, τ) is called an Rrspace if everyu £ r can be represented in the form

" = V {Λ uab: b e= Ba) = ν {Λ "at,-- b e Ba},a b a b

where uah £Ξ τ; (Χ, τ) is called a T^-space (T2-space) if it is simultaneouslyan Ro- and 7O-space (respectively an / ? r and ro-space). We remark thatthese definitions induce the corresponding standard definitions of separationaxioms on Top considered as a subcategory of § (15.4).

All the properties considered above are multiplicative (naturally, in thesense 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 propertyby means of the uniqueness of the limit (in one sense or other) of fuzzyfilters [57].

176 A.P. Shostak

15.7. Compactness in ξ>·Let {Χ, τ) be a Hutton space. An element ο e X is said to be compact iffor each % d τ such that a <; \/ CW there is a finite subset %0 Cl % such thata <; \y °#,0. A Hutton space (.!£, τ) is called compact [56] if all a £ r c arecompact.

It is easy to note that if (X, r) is a Chang L-fuzzy space and the Huttonspace (Lx, r) is compact, then (Χ, τ) is compact as well (7.1); the converseis not valid in general.

A product ((χ, 3\., τ) of a family of Hutton spaces {(i£v, τ γ): γ e Γ} iscompact if and only if all {Zy, TY) are compact [56]. However, the categoryComp ξ) of compact Hutton spaces is reflexive neither in φ nor in GCFT(a similarly defined category Comp GCFT (and even the category CompGOCFT) is not reflexive in GCFT either) [25], which rules out the possibilityof constructing an adeuqate compactification theory in these categories(cf. 7.9).

15.8. 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 {(Xy, τΥ): 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 thereare 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 thisconnection.

1. Throughout the whole of our, survey we have tried to convince thereader that fuzzy topology, like any branch of pure mathematics is supposedto, has a quite definite subject of investigation, enjoys its own developmentdynamics and a certain inner harmony.

2. There are interesting and rather important mathematical constructionsdue to fuzzy topology in essence (that is, they have arisen in the frameworkof fuzzy topology and their study requires the involvement of the apparatusof fuzzy topology). Examples of such constructions have been consideredin §§10-14.

3. Throughout the whole of our work we have intended to trace at alllevels a connection between fuzzy topology on the one hand and generaltopology and some other branches of mathematics on the other. Certainaspects of such a connection are made explicit by those functors andconstructions considered in § § 2, 11, 12, 14.

4. Fuzzy topology has a certain philosophical significance. In particular,it provides us with a fresh look at numerous facts of general topology,


and at the role of classical two-valued logic in general topology. Since wecannot dwell on this question (of which an investigation was undertaken byRodabaugh—in preprints, see also [129]), we suggest that the reader shouldlook, for example, either at the classical Tychonoff theorem on a product ofcompacta from the viewpoint of "fuzzy Tychonoff theorems" (7.1-7.5;15.7) or at the compactification problem for topological spaces from theviewpoint of compactifications in fuzzy topology (7.9, 7.10, 15.7). It is alsouseful to look at the subject and problematics of general topology as awhole from the viewpoint of the theory presented in § 15.

5. The notions and methods of fuzzy topology prove useful in some casesin posing questions and solving problems of classical mathematics. Forexample, in [103] Lowen consideres a certain family of fuzzy topologies ona metric space [X, d) and demonstrates how the study of this family by themethods 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" mathematicscan also be found in [129], [80], [102], and others.

6. When speaking about the use of fuzzy topology ideas, methods, andresults in applied problems, one should note that as yet it has an occasionaland rather superficial character. Among the works in which fuzzy topologyis used to some extent, we mention a note by Ponsard [116] (a discussionof the application of fuzzy metrics to economics problems), papers byTopenCarov and Stoeva ([156] and others) (fuzzy topological automata)and the investigations of Averkin and Tarasov [2] and Logvinenko (attemptsto model the perception process by means of fuzzy topology). We hopethat our survey will promote further popularization of fuzzy topology and,possibly, an expansion of the so far very meagre list of its applications toapplied problems.

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Translated by V. Pestoff Latvian State University

Received by the Editors 28 February 1989

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