On inductive dimensions for fuzzy topological spaces

On inductive dimensions for fuzzy topological spaces

sets and systems ELSEVIER Fuzzy Sets and Systems 73 (1995) 5-12 On inductive dimensions for fuzzy topological spaces D. Adnadjevi6 a'*, A.P. Sostak ...

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sets and systems ELSEVIER

Fuzzy Sets and Systems 73 (1995) 5-12

On inductive dimensions for fuzzy topological spaces D. Adnadjevi6 a'*, A.P. Sostak b "PMF, University of Beograd, Studentski trg 16, Beograd 11000, Yugoslavia bDepartment of Mathematics, University of Latvia, Riga 226098, Latvia

Received April 1993; revised November 1994

Abstract

An approach to the dimension theory for fuzzy topological spaces is being developed. The appropriate context for this theory is not the category CFT of Chang fuzzy topological spaces or some of its modifications, but the category Hut introduced in the paper (this category is a slight extension of the category H of Hutton fuzzy topological spaces Hutton (1980). The frames of this category allow us to make exposition simple and uniform, and on the other hand to make it applicable in quite a general setting. Keywords: Chang fuzzy topological space; Hutton fuzzy topological space; Subspace; Inductive dimensions

O. Introduction

The dimension theory for fuzzy topological spaces is probably the least developed branch of Fuzzy Topology. The few works devoted to the problem of dimension in the fuzzy setting include Zougdani's paper [27] in which the definition of dimension dim for (Chang) fuzzy topological spaces was introduced and some elementary facts about it were established and Adnadjevic's papers [2, 3] in which an attempt to define and to study dimensions ind and Ind for (Chang) fuzzy topological spaces was undertaken. The aim of this paper is to develop an alternative approach to the theory of inductive dimensions ind and Ind for fuzzy spaces. Our starting point here is the idea borrowed from I-2, 3] according to which

*Corresponding author.

the border of a fuzzy set in a fuzzy space X is to be understood (for the purposes of dimension theory) as a fuzzy subspace (in some sense) of X, rather than as a fuzzy subset of X. However, as different from 1-2, 3] we assume that the appropriate context for this theory is not the category C F T of (Chang) fuzzy topological spaces [5] or its extensions defined in [2, 3], but the category Hut introduced below which is a slight modification of the category H of Hutton fuzzy topological spaces. The frames of the category Hut will allow us, from one side, to make the exposition simple and uniform, and from the other to make it applicable in a very general situation.

1. Preliminaries

1.1. Definition. By a H u t t o n f u z z y (topological) space we mean a triple ( f f , z, tr), where ~ is a

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complete completely distributive lattice with a minimal 0 and a maximal 1 elements, • is a subset of closed under taking arbitrary suprema and finite infima, a is a subset of L# closed under taking arbitrary infima and finite suprema and, besides, 0,1 • T n t r . If E • L?, then the closure of E in (&e, z, tr) is defined by the equality/~ = A{A: A • a, A/> E}. The elements of z are interpreted as open fuzzy sets of the space ( i f , z, tr) while the elements oftr are interpreted as its closed fuzzy sets. 1.2. D e f i n i t i o n . L e t ( f f 1, z 1, tr i ), ( A ° 2, "~2, o"2 ) be two Hutton fuzzy spaces a n d f - 1 : "~'2 ----I."~I be a mapping which preserves arbitrary suprema, finite infima, 0 and 1. T h e n f : ( ~ x , zl, trl) ~ (L#2, z2, a2) is called a morphism from ( ~ 1 , Zl, trl) to ( ~ 2 , z2, a2) (or a continuous mapping from (L#~,zl,al) to (~e2,z2,tr2)) i f f - l ( V ) • Z l for each V • z 2 and f - l ( A ) • a l for each A • a2.

Hutton fuzzy topological spaces and continuous mappings between them form a category which will be denoted Hut. 1.3. Example (cf. Hutton [10]). Let (X,T) be a topological space and S be the family of its closed subsets. Then the triple (2x, T, S) is a Hutton fuzzy space. Besides, if f : (X, Tx) ~ (Y, Tr) is a continuous mapping of topological spaces, then defining the mapping f - 1 : 2 ~ ' ~ 2 x by the equality f-l(M):=f-t(M) for each M • 2 r we obtain a morphism f :(2 x, Tx, Sx) ~ (2 Y, Tl,, St) of the category Hut. In such a way the category Top of topological spaces can be embedded as a subcategory of the category Hut. (Notice that the image of Top under this embedding is not a full subcategory of Hut: given a mapping f - 1 : 2 r _ _ . 2 x, generally there does not exist a mapping
L = I is a unit interval, then (X, z) is a Chang fuzzy topological space.) Let tr be the family of all closed L-fuzzy subsets of (X, z). Then the triple (Lx, ~, a) is a Hutton fuzzy space. Besides, i f f : (X, zx) ~ (Y, zr) is a continuous mapping of L-fuzzy spaces, then defining the mapping f - 1 : Lr ~ Lx by the equality f-l(M):=f-l(M) for each M • L r we obtain a m o r p h i s m f : (Lx, Zx, ax). ~ .(L r, Tr, trr) in the category Hut. Thus for each L the category CFT(L) of (Chang-Goguen) D_-fuzzytopological spaces can be considered as a (non-full) subcategory of the category Hut. 1.5. Remark. In the original Hutton's paper [10], see also [11], a fuzzy topological space is defined as a pair (L,a,z) where .L# is a complete completely distributive lattice containing 0, 1 and equipped with an order reversing involution c: ~ __, ~ , and z is the same as in Definition 1.1. The family tr of closed fuzzy sets in the space (L#,r) is defined as tr = {A: Ac e z} and hence it is not to be explicitly indicated. The one-to-one relation between open and closed (fuzzy) sets in a (fuzzy) topological space (both Chang and Hutton [10]) realized by involution is very useful for establishing a large number of different facts of (Fuzzy) Topology. However, we have to sacrifice here the involution and hence to sacrifice also the important connections between the properties of openness and closedness in order to enable an easy and uniform transition from a fuzzy space to a fuzzy subspace; this transition is very essential for our approach and, on the other hand, it completely destroys the involution of the lattice of the original fuzzy space (in case it was equipped with some involution). It is easy to notice also that the original category H of Hutton fuzzy spaces in a natural way can be considered as a full subcategory of the category Hut; the objects of H are triples (.LP,r, tr) e Ob(Hut) where .W is equipped with an order reversing involution ~: ~e ~ ~e and tr = {A: A~ • z}. 1.6. Definition. A morphism f:(.~,('l,'t'l,O'l)---~ ("~2, "C2,0"2) is called a homeomorphism if the following conditions are satisfied: (1) f - 1 : t~ 2 ~ ~ 1 is a bijection: (2) f - l ( V ) e z x iff Vez2; (3) f - l ( A ) e a x iffA e a r .

D. Adnadjevib, A.P. Sostak / Fuzzy Sets and Systems 73 (1995) 5-12

Two

Hutton

fuzzy

spaces

(so~,z~,ox)

and

(SO2, "g2,0"2) are called homeomorphic if there exists a homeomorphism f : (SO~, z~, 0.t) --* (SO2, z2,02). 1.7. Definition. Let (SO,z,0.) be a Hutton fuzzy space, ~t,fl ~ SO and ~ ~< ft. Let SO~ := {M ~ SO: ct ~< M ~ < f l } , z ~ : = { ( C A fl) V ~ : U 6 z } , a ~ : = { ( A A fl) V c t : A ~ a } . T h e t r i p l e ( >>SO~,,,,o~! t~ .p ~ is a Hutton fuzzy space; it will be called a fuzzy subspace of the space (SO, z, a). We shall sometimes write SO~ instead of SO~ and SOa instead of SOlo. A fuzzy subspace (L~'~,z~, a~) is called open if a e 0. and fl e z; it is called closed if ~ ~ z and f l e a. 1.8. Remark. Notice that the morphism i:(SO~, zPat, u~t -axJ ~ (SO, z, a) defined by the equality i- ~(A):= (A A 13) V at for each A e S° is a monomorphism in the category Hut and hence (SO~,z~,a~) a a a is a subobject of the object (SO, z, 0.). On the other hand, it is easy to notice that there are subobjects of an object (SO, z, a) in Hut which are not its subspaces. 1.9. Remark. Notice that even if(so, z,a) is an object of H (see Remark 1.5), its subspace (SO~, z~, 0.~) may fail to be an object of H (and "usually" it does not belong to H). 1.10. Example. Let (X, T) be a topological space and (M, T') be its subspace where M c X; besides let S' be the set of closed subsets of the space (M, T'). Then (M, T') can be identified with the object (2 u, T', S') of Hut (see Example 1.3). Taking in Definition 1.7 ~ = 0 and fl = M (we identify here and in the sequel a subset M of X with its characteristic function) we can identify the space (2M, T ', S') with the fuzzy subspace ((2x)~, Tff, SoM) of the space (2 x, T, S) which is evidently a copy of the space (X, T) in the category Hut (Example 1.3). Besides, a fuzzy subspace ((2X)ou , Tff, SoM) is open (closed) in (2x, T, S) iff the subspace (M, T') is open (resp. closed) in (X, T). 1.11. Example. If (X,T) is a C h a n g - G o g u e n ~_fuzzy space and (M, T') is its subspace where M c X, then (M, z') as a subspace of (X, z) can be identified with the fuzzy subspace (0_X)oM, Zo M, aoM) of the Hutton fuzzy space 0_x, z, a) which is a copy of

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the space (X, z) in the category Hut. (Here 0. := {A: A c ~ z}.) Besides, a fuzzy subspace ((n_)o, x u To, u aou ) is open (closed) iff the subspace (M, z') is open (resp. closed) in (X, z). 1.12. Definition. An element 7 E SO is called irreducible [4] if the inequality y ~< ~ V fl where ~, fl ~ SO implies that either 7 ~< ~t or 7 ~< ft. An irreducible non-zero element is called a molecule (see e.g. [23, 24, 26]). In the theory of L-fuzzy topological spaces [8] and in the theory of Hutton fuzzy spaces molecular elements play the role which is similar to that of fuzzy points (see e.g. [15]) in the theory of Chang fuzzy spaces. Besides, it is easy to notice that an element ~ of the lattice I x is molecular iff it is a fuzzy point, i.e. if y = x t for some x e X and t ~(0,1]. For a lattice SO let SO, denote the set of all its molecules.

2. The small inductive dimension for Hutton fuzzy topological spaces Let (SO, r, a) be a Hutton fuzzy space. If no confusion is possible, we write sometimes just SO instead of (so, z,a). Let n denote a non-negative integer. 2.1. Definition. We define the small inductive dimension ind for Hutton fuzzy topological spaces as follows: (i) ind SO = - 1 if I SO I = 1, i.e. if SO consists of a single element 0 = 1; (ii) indso ~< n if for each ~ E SO. and each W ~ satisfying 7 ~< W there exists U e z such that y ~< U ~< l_7 <~ W and indsoW ~ n - 1; (iii) ind SO = n if ind SO ~< n and the inequality ind SO ~< n - 1 does not hold. 2.2. Remark. From Definition 2.1 it follows that the equality ind SO = 0 just means that SO is not a trivial lattice (i.e. 0 # 1) and for each y e SO, and each W e z satisfying Y ~< W there exists U ~ z c~0. such that y ~< U ~< W.

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2.3. Remark. Obviously the definition of ind.w makes sense only in case when (.W, z, a) is regular. (The space (.W, z,a) is called regular if for each 2 • .W, and each W • z satisfying ~. ~< W there exists U • z s u c h t h a t 2 ~< U ~< [7 ~< W ; s e e [ l l ] , c f . also [1, 18].) In the sequel in this section we shall always assume that (.W, z, a) is regular. 2.4. Proposition. Let Hutton fuzzy spaces (.W, z, a) and ( .W', z', a') be homeomorphic. / f ind .W is defined, then ind.w' is defined, too, and besides ind . w ' =

ind .W. Proof. We prove this statement by induction. Assume first that ind.w = - 1 and let f :(.W, z, a) (.W', z', tr') be a h o m e o m o r p h i s m . Then obviously I.wI = 1 implies I.w'l = 1, i.e. ind.w' = - 1. Assume now that the statement is true in case ind.w~
( f - ' ) - I(U') = U. Consider the subspace (.W'OOI,zW:,ooo:) of (.W', r', a') and let f o ~ be the restriction o f f - ~ to .WOO. It is easy to notice that the corresponding morphism fo" (.WOO,TU, t7 GU)...+f v ,U' -' t ~~-,U' w,zv,,a'W,) is a h o m e o m o r p h i s m and hence according to the inductive assumption, i n d . w v~ = ind.w'v°: ~< n - 1, and therefore ind .W ~< n. Since h o m e o m o r p h i s m is an equivalence relation it follows that ind .W' ~< n implies ind .W ~< n and hence ind.w = ind.w'. [] 2.5. Theorem. I f ind .W <<.n, then ind .W~ <<.n for

each subspace .W~ of .W. 2.6. Lemma. I f ind .W ~< n, then ind .Wr ~ n for

every y • .W. Proof. We shall prove by induction a more general fact. Namely, it will be shown that for each sub-

space .W~ of a space .W and each 7 • .W the inequality ind .W~ ~< n implies the inequality ind(.w~a)r ~< n where (.w~)r = ~'~ v r The statement is obvious in case ind .W~ ~< - 1. Assume that it is established that the inequality ind.w~ ~< n - 1 implies the inequality ind(.w~)r <~ n - 1 for each 7 • .W and each subspace .W~ of .W and let ind .W~ ~< n. , # I f 2 • ((.w~)r), and W • (z~)~ satisfy the inequality 2 ~ < W ' , then obviously 2 • ( . W ~ ) , and W ' = W V 7 for some W • z ~ such that 2 ~< W. The inequality ind .W~ <~ n implies that there exists U • z~ satisfying 2 ~< U ~< [7 ~< W and such that ind(.w~)W ~< n - 1 (the closure [7 is taken in .W~). Since obviously ct ~< U ~
=

2.7. Lemma. I f ind .W <~n, then ind.W e <~ n for

each 6 • .W. Proof. We shall prove by induction a more general fact. Namely, it will be shown that for each subspace .W~ of a space .W and each ~ • .W the inequality ind .W~ ~< n implies the inequality ind(.w~) e ~< n where ( ~ ) e := cap ^ e The statement is obvious in case ind .W~ = - 1. Assume that the inequality ind(LP~) e ~< n - 1 holds for each g • .W and each subspace .W~ of a space .W such that ind .Lea ~< n - 1 and let ind .W~ ~< n. I f 2 • ((.w~)e), and W ' • (z~) e satisfy the inequality 2 ~< W', then obviously 2 •(.W~), and W ' = W A ~ for some W • z ~ such that 2 ~< W. F r o m the condition ind .W~ ~< n it follows that there exists U • z~ satisfying the inequality 2 ~< U ~< [7 ~< W and such that ind(.w~)W ~< n - 1. (The closure [7 is taken from .wa.) Since obviously • ~< U ~< [7 ~< fl, it follows that (.w~)ut7 = .WOO.Applying the inductive assumption we can conclude that ind(.woo)e,,< n - 1. T o complete the proof notice that (.wvu)6 = ^6 = .WOO:where U' = U A 6 •(z~) e and [7' is .WOO^e its closure in (.w~)e and hence, taking into account the inequality 2 ~< U' ~< [7' ~< W, conclude that ~.z~ ~ ^ e .

D. AdnadjeviO, A.P. Sostak / Fuzzy Sets and Systems 73 (1995) 5-12

indSegl ~< n - 1. However this just means that i n d ( ~ ) ~ ~< n. [] To prove Theorem 2.5 it is sufficient now to notice that (Ae,)a = ~ and to apply successively Theorem 2.5 and Lemma 2.6.

2.8. Small inductive dimension for D_-fuzzytopological spaces. To transfer the definition of the small inductive dimension to the case of (ChangGoguen) D_-fuzzytopological spaces we propose the following approach. Let (X, z) be an Q_-fuzzytopological space where 0_ is a complete completely distributive lattice with 0, 1 and an order reversing involution ~: D_--, 0_ and let r c := {A: A c ~ ~}. Then we define the small inductive dimension of (X, ~) by the equality ind(X, z) = ind(Q_x, r, ~). In [21] we have called two C h a n g - G o g u e n l_fuzzy topological spaces (X1, D-l,zl) and (X2, Q-2,z2) parahomeomorphic if the Hutton fuzzy spaces (0-x~, zl) and (Q_2 x~, z2) are homeomorphic. If ~-1 = Q-2 and D_-fuzzy topological spaces (X~,Q_I,z~) and (XE,~E,Z'2) are homeomorphic, then obviously Hutton fuzzy space (O_X',zl,Zl) ~ and (0-2X2,'152,l'2)c are homeomorphic and hence homeomorphic 1_fuzzy topological spaces are parahomeomorphic; the converse is not generally true. It is obvious that if 0_-fuzzy topological spaces (X~, D_~,rl) and (X2,D_2,Zz) are parahomeomorphic and ind(X1, D_~,z t) is defined, then ind(X2, 0_2,Zz) is defined, too, and ind(X~, D_I,z~) = ind(X2, ~_~,rE).

2.9. Proposition. If (X, T) is a regular topological space, then ind(X, T) = n / f f i n d ( 2 x, T, T c) = n. Proof. The statement is obviously true in case ind(X, T) = -- 1. Assume that the statement is true for each topological space such that ind(X, T) ~< n - 1 and let ind(X, T) ~< n. Take a fuzzy point x t ~ I x and let W ~ T satisfy the inequality x' ~< W; obviously in this situation it just means that x ~ W. Since ind(X, T) ~< n, there exists U ~ T such that x e U c O c W and ind(/.7\U, T v \ v ) < ~ n - 1 , where T o w is the restriction of the topology T to the subspace [7\U. According to the inductive assumption ind(2 v\v, To\v, T~7\v) ~< n -- 1. Noticing

-

C

that the Hutton fuzzy spaces (2 vw, To\v, To\v) and ((2x)vo, Tv°, (TC)vo) are homeomorphic we conclude that ind(2 x, T, T c) ~< n. Conversely, assume that for a topological space (X, T) the inequality ind(2 x, T, T c) ~< n holds. Let x e X and W be a neighborhood of x in (X, T). Then, identifying the point x with the fuzzy point x ~ e 2 x we can find U e T such that x ~ ~< U ~< /_7 ~< W and ind((2x)W,T~,(T~)W) <~ n - 1. Noticing that Hutton fuzzy spaces ((2x)~,TV,(TC)W) and (20\V, To\v,(T~)v\u) are homeomorphic, we can conclude by the inductive assumption that ind(/_7\U, Tear) ~< n - 1 and hence ind(X, T) ~< n. [] 2.10. Remark. Unfortunately, it follows from Proposition 2.4 that the definition of the small inductive dimension for fuzzy topological spaces is not a good extension (in the sense of Lowen 1-123) of the usual (topological) definition of the small inductive dimension for topological spaces.

3. The large inductive dimension for Hutton fuzzy topological spaces As above, let (LP, z,a) be a Hutton fuzzy topological space. 3.1. Definition. We define the large inductive dimension Ind L~a of a Hutton fuzzy topological space (5¢, z, a) as follows: (i) Ind ~ = - 1 if 5 ° consists of a single element 0 = 1; (ii) Ind Sa ~< n if for each A e a and each W e satisfying A ~< W there exists U ~ z such that A ~< U ~< G ~< W and IndL#~ ~< n - 1; (iii) Ind LP = n if Ind £~ ~< n and the inequality Ind 5a < n - 1 does not hold. 3.2. Remark. From Definition 3.1 it follows that the equality Ind L~ = 0 just means that 5 a is not a trivial lattice and that for each A E ~r and each W e T satisfying A ~< W there exists U E r c~a such thatA~
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D. Adnadjevik, A.P. Sostak / Fuzzy Sets and Systems 73 (1995) 5-12

each A • tr and each W • z satisfying the inequality A~
3.5. Theorem. If Ind A° ~< n, then Ind ~ each closed subspace ~ of ~ .

<. n for

3.6. Lemma. If Ind LP ~< n, then Ind ~ each ~ • z.

<~n for

Proof. We shall prove by induction a more general fact. Namely, it will be proved that for each subspace L~'~of a space Ae and each ~, • z the inequality Ind Aa~ ~< n implies the inequality Ind(Ae~)y ~< n where (Aa~)~ = <,a~ ~ V y " v~ The statement is obvious in case Ind Ae~ ~< - 1. Assume that it is established that the inequality Ind L,¢~ ~< n - 1 implies the inequality Ind(L~'~)~ ~< n - 1 for each ), • z and each subspace Ae~ of L~' and let Ind . ~ ~< n. If A' • (L~a~)~, W ' • (z~)~ ~ and A' ~< W' then obviously A ' = A VV and W ' = W VV for some A • t r ~ , W • z ~ . Since A¢ is normal, 7 • z and A ~< W V y it follows that there exists U • z~ such t h a t A ~ < U ~ < / . 7 ~ < W a n d Ind(LP~)v ~t7 ~< n - 1 (the closure /.7 is taken in Aa~). Since, obviously, #t7 ~< U ~< 0.7 ~< fl it follows that (Ae~)v = .LeW. Applying the inductive assumption we can conclude that Ind(L~v°)r ~< n - 1. Noticing that (LPW)r = L/~va~ = ~W; where U ' = U V ~ •(z~)r and LT' is its closure in (Ae~)r and that A' ~< U' ~< t.7' ~< W', we conclude that IndAa~/~< n - 1, and hence Ind(L~'~)~ ~< n. [] 3.7. Lemma. If Ind L~' ~< n, then Ind Lt"~ <. n for each 6 • tr. Proof. We shall show by induction a more general fact. Namely, it will be proved that for each sub-

space Ae~ and each 6 • a the inequality Ind Ae~ ~< n implies the inequality Ind(Ae~) 6 ~< n where (A¢~)6 c.p#

A6

The statement is obvious in case Ind Aa~ = _ 1. Assume that the inequality Ind(Aa~) ~< n - 1 implies the inequality Ind(Ae~) ~ ~< n - 1 for each 6 • a and each subspace Ae~ of a space Ae and assume that Ind Aa~ ~< n. Let A' • (try) ~ and W' • (z~) 6 satisfy A' ~< W'. Since 6 • a, we can assume without loss of generality that A'~< 6 and hence A ' = A •try. Choose W • z ~ such that W ' = W V 6; then obviously A ~< W. Reasoning in the same way as in the proof of Lemma 2.7 and substituting A for 2 we obtain that I n d ( ~ ) ~ ~< n. [] Applying Lemmas 3.6 and 3.7 to the subspace Ae~ = (Le~)P of the fuzzy space A" where ~ • z and fl • tr (see Definition 1.7) we obtain the statement of the theorem.

3.8. Large inductive dimension for 0_-fuzzy topological spaces. Patterned after 2.8, we can transfer the definition of the large inductive dimension to the case of (Chang-Goguen) B_-fuzzy topological spaces. Namely, let (X, z) be an D_-fuzzytopological space (Example 1.4). Then we define its large inductive dimension by the equality I n d ( X , z ) = Ind(k x, z, zc). Patterned after the proof of Proposition 2.9 one can easily establish th.,, "J~Iidity of the next statement:

3.9. Proposition. L e t (X, T ) be a normal topological space. Then Ind(X, T) = n / f f I n d ( 2 x, T , T c) = n. 3.10. Remark. The definition of the large inductive dimension Ind for Hutton fuzzy topological spaces is not a good extension of the usual (topological) definition of the large inductive dimension for topological spaces (see Proposition 4.5).

4. Zero-dimensionality in fuzzy topology 4.1. Definition. A Hutton fuzzy topological space (Ae, z, a) is called zero-dimensional if ind A¢ = 0;

D. Adnadjevik, A.P. Sostak / Fuzzy Sets and Systems 73 (1995) 5-12

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( ~ , z , a ) is called strongly zero-dimensional if Ind L~' = 0.

clopen fuzzy sets iff it is homeomorphic to a subspace of the space Z ~ where x is some cardinal. Besides, ~c can be chosen equal to the weight of the space

4.2. Remark. A similar approach to the concept of zero-dimensionality for Chang fuzzy topological spaces can be found also in some earlier on fuzzy topology. For example in [14] (see also [16]) a Chang fuzzy space (X, z) is called zero-dimensional if each U e ~ can be expressed as a union of clopen (i.e. closed and open) fuzzy sets (in other words if z has a base consisting of clopen sets). Obviously, a Chang fuzzy space is zero-dimensional in our sense (Definition 4.1) iff it is regular and each U e ~ is a union of clopen fuzzy sets. An example of a Hausdorff [22] fuzzy space every open fuzzy set of which is a union ofclopen fuzzy sets but which fails to be regular and hence is not zerodimensional in our sense is presented below.

(x,~).

4.3. Example. A Hausdorff Chang fuzzy topological space with a base consisting of clopen fuzzy sets which fails to be regular (and hence is not zerodimensional). Let (X, z) be a Hausdorff zero-dimensional topological space and let z be the fuzzy topology on X generated by the subbase P = T u U {c: c e [0, ½] u ('~, 1] }. An elementary verification shows that the space (X, z) thus obtained has the desired properties. In [16] a universal space for fuzzy topological spaces having a base consisting of clopen fuzzy sets was constructed. For completeness we shall exposit this construction below (3.3). Let Z denote the unit interval I endowed with the fuzzy topology z the subbase of which is {0, 1, e, e~}, where e: 1 ~ I is the identity mapping. It is easy to notice that Z is homeomorphic to the fuzzy modification J~(T) of the three point space T = {0,1,2} ([16], cf. also [13, 14,20]). Recall that a fuzzy space (X, ~) is called a Wospace if for every two points x,y E X there exists U e r such that U ( x ) = U(y). (This property first appeared in [25, 17]: spaces with this property were called by To-spaces there.) 4,4. Theorem (Pujate and Sostak [16]). A fuzzy space (X, ~) is a Wo-space with a base consisting of

In other words the space Z ~ is universal for the class of Wo-spaces which have a base of cardinality x consisting of clopen fuzzy sets. To each topological space (X, T) Lowen [12] assignes a Chang fuzzy topological space (X, toT) where toT is the family of all lower-semicontinuous functions from (X, T) into the usual unit interval I. Obviously, toT contains all constants. Lowen considers the space (X, toT) as the fuzzy copy of the topological space (X, T). 4.5. Proposition. If (X, T) is a completely regular topological Tl-space, then ind(X, toT) = 0. Proof. Let x' be a fuzzy point in X and W e toT be its neighborhood, i.e. W(x) >1 t. It is easy to notice that x t is a closed fuzzy set in (X, toT) and hence the function x t : X - ~ I is uppersemicontinuous. Since W is lower-semicontinuous and x'~< W we can apply Tong-Katetov insertion theorem according to which there exists a continuous function U : X ---,I such that x t <~ U <. W. To complete the proof it is sufficient to notice that a continuous function U:(X, T ) ~ I is both closed and open in coT. [] Quite similarly we can prove the next statement: 4.6. Proposition. If (X, T) is a normal topological Tl-space then Ind(X, toT) = 0.

References [1] D. Adnadjevic, Separation properties of F-spaces, Mat. Vesnik 6 (1982) 1-8. [2] D. Adnadjevic, Dimension F-Ind of F-spaces, in: Baku Internat. Topoi. Conference, Abstracts, Part II. Baku (1987) 4. [31 D. Adnadjevic, Dimension of fuzzy spaces, Fuzzy Sets and Systems 26 (1988) 85-92. ['4] G. Birkhoff, Lattice Theory (AMS, New York, 1948). [-5] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182-190. [.6] R. Engelking, General Topology (PWN, Warszawa, 1977).

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