Fuzzy metric neighbourhood spaces

Fuzzy Sets and Systems 45 (1992) 367-388 North-Holland

367

Fuzzy metric neighbourhood spaces Ali S. Mashhour Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

Nehad N. Morsi Department of Mathematics, Military Technical College, Kobry EI-Qubba, Cairo, Egypt Received October 1988 Revised July 1989

Abstract: H6hle founded the first coherent topological study of Menger's probabilistic metric spaces by endowing them with fuzzy T-uniformities. We continue H6hle's work in the special case when the triangular norm T used is Min, that is when the resulting probabilistic metric uniformities are Lowen fuzzy uniformities. In this case, we call the induced fuzzy topological spaces fuzzy pseudo-metric neighbourhood spaces. We introduce axioms of N-regularity and N-normality, which are satisfied in fuzzy pseudo-metric neighbourhood spaces. We introduce a Lowen fuzzy uniform topology on the fuzzy real line R(1), called the N-Euclidean topology, by means of a fuzzy (=probabilistic) metric on R(1). Using this N-Euclidean space, we introduce an axiom of N-complete regularity which characterizes the Lowen fuzzy uniform topologies.

Keywords: Fuzzy neighbourhood spaces; fuzzy uniform spaces; fuzzy metric; N-separation axioms; the N-Euclidean space; N-complete regularity.

Introduction In 1942, Menger introduced probabilistic metric spaces, where distances between points need not be sharply defined real numbers, but can be probability distribution functions on the nonnegative reals, and where addition of distances can be performed by employing any triangular norm T. Although attempts have been made at formulating some topological notions in probabilistic metric spaces, none could result in an integrated theory until the advent of fuzzy topology [5]. Although Kramosil and Michalek were the first to use the term fuzzy in connnection with probabilistic metric spaces [19], it was left to H6hle to set their study properly as an interesting special type of fuzzy topological spaces (fts for brevity). In [10], H6hle defined fuzzy T-uniformities (where T is any continuous t-norm) which subsume both the Lowen fuzzy uniformities [26] (when T = Min) and a particular type of the Hutton fuzzy uniformities [13] (when T = tm). He canonically associated fuzzy T-uniformities with probabilistic metric spaces, and he proved that a fuzzy T-uniformity is probabilistic pseudo-metrizable iff it has a countable basis. This paper is a continuation of [10], but only for the case of Lowen fuzzy uniformities (when T = Min). Since the fts's induced by those fuzzy uniformities are fuzzy neighbourhood spaces [27], it seems proper to call the fts's induced by fuzzy (=probabilistic) metric uniformities fuzzy metric

neighbourhood spaces. We call a (descending) probability distribution function on the real line ~ a fuzzy real number, and we call the set of those distributions the fuzzy real line, denoted ~ (I). This follows a custom established in some papers since Hutton introduced the fuzzy unit interval, in [12], subsequently expanded to the fuzzy real line, in [7]. Properly speaking, a fuzzy real line is this set endowed with a fuzzy topology. But, in the first four sections of this paper we shall have to consider it as just a partially ordered set with compatible operations of addition and nonnegative scalar multiplication, whereas, in Section 5, we endow this set I~(I) with a new fuzzy topology t(a); this time a Lowen fuzzy uniform topology. In this paper we proceed as follows. In Section 1, we list definitions and results needed in this paper: 0165-0114/92/$05.00 (~ 1992--Elsevier Science Publishers B.V. All rights reserved

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on Lowen's fuzzy neighbourhood spaces, on probabilistic metric spaces (called here fuzzy metric spaces) (X, d), on their fuzzy open balls and on their fuzzy metric topologies rd (introduced in [35]). In Sections 2 and 3, we outline H6hle's theory for fuzzy (=probabilistic) metric uniformities q/(d), but only in the special context of Lowen fuzzy uniformities. We discuss their induced fuzzy metric neighbourhood spaces (X, t(d)). We provide a number of new results: we introduce fuzzy metric proximities using Artico-Moresco fuzzy proximities [3]. We discuss the relationship between the (above mentioned) two fuzzy topologies rd and t(d) on the same fuzzy metric space (X, d). In Section 4, we introduce an axiom called N-regularity, which is satisfied by all fuzzy uniform spaces, and an axiom called N-normality, which is satisfied by all fuzzy psuedo-metric neighbourhood spaces. These are consistent with the Wuyts-Lowen lower separation axioms NT1 and NT2 of [40]. The set which deserves most to have nonnegative fuzzy real numbers for distances between its points is the set E* (I) of nonnegative fuzzy real numbers. In Section 5, we introduce a naturally defined fuzzy metric a on E*(1). The importance of the fts (R*(I), t(a)), which we call the N-Euclidean space, is demonstrated in Section 6. There, we use it to formulate an axiom of N-complete regularity which characterizes the Lowen fuzzy uniformizable fts's. This answers a long standing question in the theory of Lowen fuzzy uniform spaces. We use the customary notations I = [0, 1], I0 = ]0, 1] and 11 = [0, 1[ for unit intervals of real numbers. We consider the ordinary power set 2x, of a universe X, as a subfamily of its fuzzy power set I x. We denote the constant fuzzy subset of X with value tr • I by the symbol a (~ IX). We denote the (fuzzy) closure operator associated with a (fuzzy) topology r alternatively by - or cir. We end this introduction by a comment on examples. As to interesting examples of fuzzy metric neighbourhood spaces, the N-Euclidean space of Sections 5 and 6 should do for the time being. It has been applied and further studied in [42, 44, 47, 48], and higher dimensional analogues are proposed in [47]. (The articles [42-48] have been written after this article). As to the need for counterexamples to delimit the set of valid implications between Wuyts-Lowen's and our N-separation axioms, this need is adequately provided for by means of long established counterexamples in general topology; because all those N-axioms satisfy Lowen's criterion for goodness of extension [23]. Indeed, they satisfy a more demanding level-topologies criterion, which may also prove more useful [43, 45, 46, 48].

1. Fuzzy neighbourhood spaces and fuzzy pseudo-metric spaces In [24], Lowen introduced prefilters and prefilterbases. In [26], he introduced the operator on prefilterbases. We call it the presaturation operator. It is defined on a prefilterbase F in X by P = {~/o~t,, (°U - 0): °U ~ F} ~ I x. Evidently, F ~_ if'. Lowen showed that P is also a prefilterbase unless it contains 0, and that while F need not coincide with if', we have f' ~_ [F] = [~'F]. The saturation operator - is defined [26] on prefilterbases b y / ~ = [/~]. It follows that F ~ P c P =/~. A prefilterbase F is called presaturated when [" = F (in which case, [F] =/~). It is called saturated when /" = F (in which case, F is a prefilter). Given U ~ I x, we use the notation sup U = supx~x U(x). Definition 1.1 [27]. A fuzzy neighbourhood system on a universe X is a family v = (v(x))xEx of prefilters in X which satisfies: (N1) V c v(x) ~ V(x) = 1, for all x eX. (N2) v is presaturated; that is v = ~, = (¢'(x))x~x (hence, v is saturated), (N3) given x e X and 0 e Io, every V • v(x) has a 0-kernel in v; this consists of a family (°Vz)~ x such that for all y, z ~ X, °Vz e v(z) and °V~(z) ^ °Vz(y) <~V(y) + O. Lowen, (27), proved that the following operator - is a fuzzy closure operator on X. For U E I x and

x eX, U-(x)=

inf sup(U ^ V). V~v(x)

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The associated fuzzy topology on X is denoted by t(v). The fts (X, t(v)) is called a fuzzy neighbourhood space (abbreviated fns), and v is called its fuzzy neighbourhood system. Lowen showed that the association v ~ t(v) is a one-to-one correspondence between fuzzy neighbourhood systems and fuzzy neighbourhood spaces. A fuzzy neighbourhood basis for a fuzzy neighbourhood system v on X is a family/3 = (fl(x))x~x of prefilterbases in X which satisfies v =/~ = (fl(X))x,X [27]. In [37, Sections 2-4], the following feature of fns's was examined in some detail. Theorem 1.2 [37]. A fuzzy neighbourhood space is completely determined by the fuzzy closures of its

crisp subsets. Besides applying this theorem directly in two proofs (in Section 6), it will have a strong heuristic influence on our formulation of the axioms of N-regularity and N-normality (Section 4), and on formulating an Urysohn-type characterization of N-complete regularity (Theorem 6.14). Wuyts and Lowen introduced the following lower separation axioms on fns's. Definition 1.3 [40]. A fuzzy neighbourhood space (X, t(v)) is said to be (i) NT1 if all crisp singletons are closed, (ii) NT2 if for all x vey in X and all 0 el0, ther exist U c v(x) and V ~ v(y) with U ^ V ~<0, (iii) WNT1 if for all x :/:y in X, there exists U ~ v(x) with U(y) < 1, (iv) WNT2 if for all x :/:y in X, there exists U ~ v(x), V ~ v(y) and o~~ 11 with U ^ V ~
These four axioms are related as follows [40]: NT2

;, NT1

WNT2 ~

WNT1

The definition of the fuzzy real line has several equivalent phrasings (cf. [7, 12, 18, 28, 38]). The following one is adequate for our treatment. Definition 1.4. A fuzzy real number is a descending, left continuous real function It~~ 1, with supremum 1 and infimum 0. The set E (I) of all fuzzy real numbers is called the fuzzy real line. The real line ~ is canonically embedded in E(1) by sending every r e ~ onto the fuzzy real number defined by f(s)=

1 ifs<~r, 0 ifs>r.

(I) is partially ordered by the natural ordering of its elements as real functions. This extends the usual order on ~ , which shows the advantage of defning the fuzzy real numbers as descending, rather than ascending, functions. We use the notations ~ + = { r ~ : r > O } , [ ~ * = { r c ~ : r ~ > 0 } and ~*(I)={~7~R(I):o>~O}. It follows that I~* = ~*(I) fq I~. A subset G of ~ ( I ) which is bounded below (above) has its infimum (supremum) in I~(1). This is simply the infimum function of G (of the set of upper bounds of G in It~(1)) (cf. [28, 38]). Elements of I~(I) can also be considered as fuzzy subsets in 1B. This is how their addition and nonnegative scalar multiplication are defined.

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Definition 1.5 [18, 28, 38]. Addition is defined on R(1) by; for r/, ~ e R(1) and r e R, (r/~) ~)(r) = sup{r/(a) ^ ~(b): a + b = r}. This addition is commutative, associative, and has identity 0. It is also order-preserving. Definition 1.6 [28]. Let U be a fuzzy subset of a set X, and let a~ e I. (i) The re-cut (also called the strong q-cut) of U is the crisp subset U" = {x e X: U(x) > o~} of X. (ii) The tr*-cut (also called the weak tr-cut, or the re-level set) of U is the crisp subset U~. = {x ~ X: U(x)/> o~} of X. It is easily seen that if U, V are fuzzy subsets of some semigroup, then for all t r e l , (U ~) V) ~ = U ~ @ V ~ (cf. [28, 35]), and (U (~ V)o,. _~ U~. ~) V~. (cf. [35]). Applying these facts to the addition in ~ (I), we can conclude that the sum of two fuzzy real numbers is indeed a fuzzy real number [28], and (R(I), (~) is an Abelian cancellation monoid.

Proposition 1.7. Let ~, ~, rb ~ R (I), where j runs on some indexing set J. Then (i) for all s • R, (~ ~ ¢)(s) = inf{~(b) v ¢(s - b): b e ~ ) . (ii) ~ ~ infj~ rb = infj~j [~ ~ rb], if the left hand side exists.

Proof. (i) This identity is proved in [28, Lemma 2.2.1], and independently in [35, Theorem 1.2]. (ii) Assuming infj ~j exists, we get for all s • N by applying the identity in (i),

[ ~ ~ inf Tb](s) = inf [ ~(b ) v inf Tb(s - b ) ] j

her

j

= inf inf [~(b) v rb(s - b)] = inf inf [~(b) v rb(s - b)] b~R

j

j

be:~

= inf [~ ~) rb](s ) = (inf [~ ~) rb])(s), j

\J

which demonstrates the claimed identity. Definition 1.8 [12]. For every r • R, the fuzzy subset

Lr(Tl) = 1 - o(r),

Lr

of ~ (I) is defined by

rl • ~(I).

Evidently, Lr(rl) is ascending in r and descending in r/. Part (i) of Proposition 1.7 entails:

Theorem 1.9. Given rl, ~ • ~ (I) and r • ~, we find Lr(r ! ~ ~) --sup{Lb(r/) ^ L,-b(¢): b c R} = inf{Lb(r/) v Lr-b(~): b ~ ff~}. This theorem confirms the equivalence of using either descending or ascending functions to define the fuzzy real numbers. The original versions of the next definition are due to Menger, HOhle and others. Definition 1.10 (cf. [35]). A fuzzy pseudo-metric (abbreviated f.p.metric) d on a set X is a function d : X x X---> R *(I) which satisfies: for x, y, z • X, (i) d(x, x) = O, (ii) d(x, y) = d(y, x) (symmetry), (iii) d(x, y)(~ d(y, z)>-d(x, z) (triangle inequality). Such d is called a fuzzy metric (abbreviated f.metric) if in addition it satisfies:

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(iv) x ~ y ~ d ( x , y ) > O . It is called a f u z z y strict metric (s.metric) if it satisfies the stronger axiom: (iv)' x ~ y ~ d(x, y ) ( 0 + ) = 1. The cases of f.metric and s.metric correspond to the axioms WNT1 and NT1, respectively, on the induced fuzzy neighbourhood spaces (see Theorem 3.3). Fuzzy open balls were originally introduced in [35] with radii in ~ * ( 1 ) - {0}. However, it would suffice to consider only radii from ~+, as can be seen from [35]. In this special case, we get: Definition 1.11 [35]. A f u z z y open ball B(x; ~), in a f.p.metric space (X, d) with centre x • X and radius r • R +, is the fuzzy subset of X given, for all y • X, by

B(x; ?)(y) = tr[d(x, Y)I. Definition 1.12 [35]. The f u z z y (pseudo-metric) topology 3d associated with a f.p.metric space (X, d) is the coarsest fully stratified fuzzy topology on X which contains all the fuzzy open balls of (X, d).

The above definition is equivalent to, though different from, the original one of [35]. This is due to the next theorem. Theorem 1.13 [35]. Let (X, d) be a f.p.metric space. Then: (i) if U(x) > it for U • 7:d, X • X and tr • I, then there is r • R ÷ such that B(x; f) ^ a <~ U (notice that B(x; f)(x) = Lr(O) = 1); (ii) the collection {B(x; f) ^ a: x • X and r, o: • Io.1} is a basis for 3d. Definition 1.14 ([9], see also [16, 35]). Nonnegative real scalar multiplication is defined on ~ ( I ) by: for 77• R(I) and s • ~*, so • ~ ( I ) is given for all r • ~ by

fO(r) if s = 0, (srl)(r) = ~ " "[~(r/s) if s:/:0. Definition 1.15 ([9], see also [16,35]). A f u z z y pseudo-norm (abbreviated f.p.norm) on a real or complex vector space X is a function II II : x ~ ~*(I) which satisfies: for x, y • X and s in the field,

(i) Ilsxll = Isl Ilxll, (ii) IIx +yll ~< Ilxll @ IlYll (triangle inequality). Such II II is called a f u z z y norm (abbreviated f.norm) if in addition it satisfies: (iii) x:/:0 ~ Ilxll>0. It is called a f u z z y strict norm (s.norm) if it satisfies the stronger axiom: (iii)' x :/:0 ~ Ilxll (0+) = 1. The f.p.metric d associated with a f.p.normed space (X, II II) is given by: for x, y • X,

d(x, y) = IlY -xll. This d will be a s.metric (f.metric) iff II II is a s.norm (f.norm). The associated 3d will be called the f.p.normed topology associated with II IIDefinition 1.16 [14]. Let r be the usual topology on the real or complex field ~. Then t0(r), the fuzzy topology topologically generated by x, is called the f u z z y usual topology on ~. Definition 1.17 [14]. A fully stratified fuzzy topology 3 on a real or complex vector space X is called linear if both vector addition, (X, 3) × (X, 3)---> (X, 3), and scalar multiplication, (~, to(g)) × (X, 3)-+ (X, 3), are continuous functions. Theorem 1.18 [35]. Every f u z z y pseudo-normed topology is linear.

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2. Fuzzy metric uniformities and fuzzy metric proximities

A fuzzy relation in a set X is a fuzzy subset ~p e I x×x. In [26], Lowen defines for ~p, ~/,' e I x×x and U • I x, the section of ~p over U by ~p(U)(x)=sup U(y) ^ ~O(y,x), y~X

x •X,

the composition of lp and ~p' by

~/' o~p(x, y ) = s u p ~p(x, z) ^ ~p'(z, y), zEX

x, y e X ,

the symmetric of ~p by ~p(x, y) = ~p(y, x),

x, y e X.

Definition 2.1 [26]. A fuzzy uniformity on X is a subset ¢ / c I x~x which fulfills the following properties: (FU1) 0-//is a prefilter, (FU2) 0//is saturated, (FU3) for all ~p ~ atl and for all x • X, ~p(x, x) = 1, (FU4) for all ~p • 0//, s~p • o//, (FU5) for all ~p • a//and for all 0 ~ Io, there exists ~Vo• 0//such that ~Poo ~/'o - 0 ~< W. The pair (X, a//) will be called a fuzzy uniform space. The members of a//are called fuzzy entourages. A function f between fuzzy uniform spaces is called uniformly continuous if ( f x f ) -1 takes fuzzy entourages onto fuzzy entourages. In [27], Lowen showed that the following is indeed a fns.

Definition 2.2 [26, 27]. If (X, 0//) is a fuzzy uniform space, then the fuzzy neighbourhood space induced by (X, °l0 and denoted by (X, t(°-/t)) is given by the fuzzy closure operator

u-= y.'e~ A r(u), Equivalently, (X, t(°//)) is given by the fuzzy neighbourhood system v(x) = (~p(x): ~p ~ 0-//}, x ~ X. The above association is functorial. A fts (X, r) will be called fuzzy uniformizable when r = t(q/) for some fuzzy uniformity q / o n X.

Definition (FUB1) (FUB2) (FUB3)

2.3 [26]. A subset 7/" c I x×x is called a fuzzy uniform base in X if it satisfies: of is a prefilterbase, for all ~ • 7/" and all x • X, ~p(x, x) = 1, for all ~v • of and for all 0 ~ Io, there exists ~Oo • of such that ~Po ° ~Po - 0 ~
It is easily verified that of is a fuzzy uniform base in X iff ~ is a fuzzy uniformity on X. Also, if in Definition 2.2 the fuzzy uniformity ~ is replaced by any of its bases, then we still obtain the same fuzzy closure operator. For product fuzzy uniformities (required in the sequel), the reader can consult [26]. Definition 2.4 [3]. A fuzzy proximity on a set X is a function 6 : I x x I x ~ I which satisfies, for any U, V, W • I x, the following conditions: (FP1) 6(0, 1) = 0, (FP2) 6(U, V ) = 6(V, U), (FP3) 6(U, V) v 6(W, V) = 6(U v W, V),

A.S. Mashhour, N.N. Morsi / Fuzzy metric neighbourhood spaces

(FP4) if 6 ( U , V ) < o l

for some t r • l o ,

then there exists C • 2 x such that

373

6(U,C)
and

6 ( x - C, V) < o~,

(FP5) 6(U, V) i> sup(U ^ V), (FP6) if I V - WI ~<0 for 0 • I, then for every U • I x , 16(U, V ) - 6(U, W)I ~< 0. The pair (X, 6) is said to be a fuzzy proximity space. The fts (X, t(6)) induced by 6 is given by the fuzzy closure operator

U-(x)=6(U,x),

U•I x

and

xeX.

This definition is equivalent to the original one of [3], albeit somewhat different in axiom (FP4) (cf. [37, Remark 5.2]). Theorem 2.5. [3, 4]. Every fuzzy uniform space (X, 91) induces a fuzzy proximity space (X, 6(91)),

where 6 = 6(all) is given by 6(U, V ) = inf sup(ap(U) ^ ~p(V)),

U, V • I x.

(In this formula, all can be replaced by any of its bases.) This fuzzy proximity is compatible with °ll; that is, t(6( U)) = t(91). Moreover, every fuzzy proximity is induced by at least one fuzzy uniformity. In [37], it is shown that a fuzzy proximity on X is essentially a fuzzy binary relation of nearness between the crisp subsets of X. This led to the following notion, which we shall use in Theorem 4.6 in proving the N-regularity of fuzzy proximity spaces. Definition 2.6 [37]. A fuzzy proximal neighbourhood system on X is a family P = (P(M))Me2 x of prefilters in X (with the exception that 0 • P(0)) which satisfy for all M, S • 2 x and o: • 11: (PN1) if U • P(M), then U >>-M,

(PN2) if (X - S) v a • P(M), then (X - M) v a • P(S), (PN3) if S v a • P ( M ) , then for every 7•]off 1[ there is C • 2 x such that C v T • P ( M ) S v y e P(C), (PN4) P(M) is saturated. A fuzzy proximal neighbourhood basis for P is a family Q of prefilterbases in X with Q = P.

and

Theorem 2.7 [37]. (i) The fuzzy proximal neighbourhood systems are in a canonical one-to-one

correspondence with the fuzzy proximity spaces. (ii) If 91 is a fuzzy uniformity (a fuzzy uniform base) on X, then the fuzzy proximal neighbourhood system of 6(91) (a fuzzy proximal neighbourhood basis for 6(°~)) is P = (P(M))M~2~ given by P(M) = {~p(M): ~p • 91}, M 6 2 x. We now proceed to outline H6hle's construction of a fuzzy pseudo-metric uniformity 91(d) on each f.p.metric space (X, d), in the special context of Lowen fuzzy uniformity. Throughout, H will denote a subset of R + with infimum O. Theorem 2.8 [10]. Let (X, d) be a tip.metric space, and define for every r • H a fuzzy subset ~r • I x×x

by ~pr(X, y) = Lr[d(x, y)],

X, y • X.

Then the family ~V= {~Pr: r 6 H} is a fuzzy uniform base in X. The fuzzy uniformity 91(d) = ~ is independent of the choice of H, and it has a countable basis. Proof. (FUB1) Direct, from

~r A ~b : ~rAb"

(2.1)

374

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(FUB2) lp,(x, x) = Lr[d(x, x)] = L,(O) = 1 for all r e H and x ~ X. (FUB3) Since d is symmetric, then % = sip, for all r. Also for all r, b e H and all x, y e X, lPb ° ~,(x, y) = sup ~r(X, Z) A ~.lb(Z,y) zEX

= sup Lr[d(x, Z)] ^ Lb[d(z, y)] zEX

~< sup Lr+b[d(x, z) t~) d(z, y)] zqg

<~L,+b[d(x, y)]

(from Theorem 1.9)

(from the triangle inequality)

= ~2,+b(X, Y). Hence, ~Pb° ~Pr<~~2r+b, which completes the proof of (FUB3). The above shows that T" is a fuzzy uniform base. That q/(d) = ~ is independent of the choice of H follows from (2.1). Finally, by taking a countable H we obtain a countable basis for °//(d). D e f i n i t i o n 2.9 [10]. The fuzzy pseudo-metric uniformity (abbreviated f.p.metric uniformity) associated with a f.p.metric space (X, d) is the fuzzy uniformity q/(d) of the above theorem. The above family = {~p,: r e H} is called a canonical fuzzy uniform basis for a//(d).

fuzzy pseudo-metric proximity (abbreviated f.p.metric proximity) associated with a f.p.metric space (X, d) is the fuzzy proximity 6(d) = 6(q/(d)) induced by the fuzzy uniformity °//(d).

D e f i n i t i o n 2.10. The

T h e o r e m 2.11.

Let (X, d) be a f.p.metric space. Then its associated f.p.metric proximity 6 = 6(d) is

given by 6(U, V) = inf

sup

U(x) ^ Lr[d(x, y)] A V(y),

sup

L,[d(x, y)],

r ~ (x, y ) ~ X x X

U, V ~ I x.

In particular, 6(M, S) = inf

M, S e 2 x.

r~H (x, y)EMxS

Also, this 6 has a fuzzy proximal neighbourhood basis Q(M)={UrelX:reH},

M ~ 2 x,

where U , = W B(Y;F) elx. y~M

Proof. Let ~ = {~pr: r e H} be a canonical fuzzy uniform basis for 9/(d). Then for all

6(u, v)=

inf supOp(V) ^

qJ(V))

.q., ~ T"

inf sup(~0(U) A V)

(from (FUB 3))

inf

sup

U(x) A ~pr(X, y) A V ( y )

inf

sup

U(x) ^ Lr[d(x, y)] ^ V(y).

rEH (x,y)~X×X

r¢H (x,y)~X×X

U, V ~ I x,

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Also from Theorem 2.7, 6 has a fuzzy proximal neighbourhood basis

O(M)={Ur=apr(M):r•H},

M e 2 x,

where for all x • X,

Ur(x) = sup ~p~(y, x) = sup Lr[d(x, y)] y~M

y~M

Definition 2.12. Let (X, d) be a f.p.metric space. For all M, S • 2 x, define

d(M, S) --

d(x, y) • R *(I).

inf (x,y)eM×S

This definition extends d : X x X --~ • *( I ) to a symmetric function d : 2x x 2x --* R *( I ). Theorem 2.13. Let 6 = 6(d) be the f.p.metric proximity associated with a f.p.metric space (X, d). Then for all crisp subsets M, S of X,

6(M, S) = 1 - d(M, S)(O+). Proof. From Theorem 2.11 and [35, Theorem 1.1(viii)],

6(M, S ) = inf

Lr[d(x,y)]

sup

r~_~ ( x , y ) ~ M × S

= inf Lr[ reH

inf

d(x,y)J

L(x,y)~M×S

= inf L,[d(M, S)] = 1 - d(M, S)(O+). rE~

Proposition 2.14. Let x, y e X and M e 2 x be in a f.p.metric space (X, d). Then d(x, y) ~) d(y, M) >I

d(x, M). Proof.

d(x, y) ~ d(y, M) = d(x, y) ~ inf d(y, z) zEM

= inf [d(x, y) ~) d(y, z)]

(by Proposition 1.7)

z~M

/> inf d(x, z)

(by the triangle inequality)

zeM

= d(x, M).

3. Fuzzy pseudo-metric neighbourhood spaces Definition 3.1. The fuzzy pseudo-metric neighbourhood space (abbreviated N-p.metric space) induced

by a f.p.metric space (X, d) is the fuzzy neighbourhood space (X, t(d)) -- (X, t(all(d))) induced by the fuzzy uniformity a//(d). If d is a s.metric (f.metric), then (X, t(d)) will be called a N-s.metric space (N-metric space).

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Theorem 3.2. Let (X, d) be a f.p.metric space. (i) The N-p.metric space (X, t(d)) has a f u z z y neighbourhood basis fid = (fid(X))x~X given for all x ~ X by fid(X) = {(B(x, f): r e H}. (ii) The f u z z y closure operator - of (X, t(d)) is given for all U ~ I x and x ~ X by U - ( x ) = inf sup U(y) ^ L,[d(x, y)]. r~H yc=X

Proof. (i) Let {~p, elX×X: r e l l } be a canonical fuzzy uniform basis for ~ ( d ) (as given in Theorem 2.8). Then (cf. Definition 2.2), (X, t(d)) has a fuzzy neighbourhood basis { ~Pr(x):r e H} at each x E X. But for all y e X,

~Vr(X ) ( y ) = ~Pr(X, y) = Lr[d(x, y)] = B(x; f)(y). This proves that ~Or(X) = B(X; f). (ii) This follows directly from (i).

Theorem 3.3. Let (X, d) be a f.p.metric space. (i) (X, t(d)) is NT2 iff it is a N-s.metric space [17], (ii) (X, t(d)) is W N T 2 iff it is a N-metric space [10]. Proof. (i) From [40, Theorem 5.1], (X, t(d)) is NT2 iff A a//(d) = the diagonal of X x X iff for all x 4=y in X, 1 - d(x, y)(0+) = inf Lr[d(x, y)] = inf ~pr(X, y) re~

r~C-O

= [A ~(d)](x, y) = 0 iff for all x 4:y in X, d(x, y)(O+) = 1 iff (X, d) is a s.metric space. (ii) From [40, Theorem 5.1], (X, t(d)) is WNT2 iff for all x=Py in X, 1 - d ( x , y ) ( O + ) = [A q/(d)](x, y) < 1 iff for all x :#y in X, d(x, y ) ( 0 + ) > 0 iff (X, d) is a f.metric space.

Theorem 3.4. The f u z z y closure of a crisp subset M of a N-p.metric space (X, t(d)) is given by M - ( x ) = 1 - d(x, M)(O+),

x e X.

Proof. This follows directly from Theorem 2.13, since t(d) = t(6(d)), and hence M - ( x ) = 6(x, M) = 1 - d(x, M)(0+). We now turn our attention to an alternative, interesting method of inducing the fns (X, t(d)); from the fts (X, rd) by means of the functor t- of Definition 3.7 below.

Definition 3.5 [36]. Let (X, r) be a fts. (i) fir - (fit(x)) is the family of prefilterbases in X given by fir(x) = {U ~ r: U(x) = 1}. (ii) A family F = (F(x))x~x of prefilterbases in X is called a family of local bases for r if (LB1) for all x e X and V ~ F(x), V ~ r and V ( x ) = 1, (LB2) for all U ~ r, x e X and tr~/1 such that cr < U(x), there is V e F(x) such that V ^ a ~< U. (iii) (X, z) is said to be a f u z z y neighbourhood-base space if for all U ~ r and t r e 11, U v U ~ e l= (U ~ e 2x is the c~-cut of U, cf. Defnition 1.6). In the category FTOP of all fuzzy topological spaces (according to Chang's definition), the full subcategory of all fuzzy neighbourhood-base spaces is denoted by FNBS.

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Theorem 3.6 [36]. Let (X, r) be a fuzzy neighbourhood-base space. Then: (i) f~ = ( f , ( x ) ) ~ x is the family of the largest local bases for r. (ii) If (F(x))x~X is a family of local bases for r, then F is a fuzzy neighbourhood base in X, and

In [36], it is shown that the following is indeed a functor. Definition 3.7 [36]. Let (X, r) be a fuzzy neighbourhood-base space. Then (X, t(r)) denotes the fns with the fuzzy neighbourhood basis fl~. The functor t-:FNBS---, FNS is identity on functions, and is defined on objects (X, r) • FNBS by t-(X, r) = (X, t(r)).

Theorem 3.8. If a fts (X, r) has a family F = (F(x))x~X of local bases, then (X, r) is a fuzzy neighbourhood-base space. Proof. Let U • ~, tr • I~ and V = U v U °'. We need to prove that V • z. We realize this by constructing for every x • X a fuzzy subset Vx such that V~ • r,

Vx ~< V

and

Vx(x) = V(x).

(3.1)

If x ~ U ~, we take V, = U, which obviously satisfies (3.1). If x • U ~, that is U(x) > tr, then there are V, • F(x) c r and 0 • ]or, U(x)[ such that V~ ^ 0 ~< U.

(3.2)

Since (0) ~ = 1, then Vx ~< [Vx ^ 0] v [V~ ^ 0]" ~< U v U ~ (from (3.2)) = V. Also, V~(x) = 1 = V(x). So, in this case too Vx satisfies (3.1). In conclusion, V = Vx~x Vx • r. This proves that (X, r) • FNBS.

Theorem 3.9. Let (X, d) be a f.p.metric space. (i) The family fd = (fld(X))x~X given for all x • X by fd(X) = {B(x; f): r • H} is a family of local bases for rd. (ii) (X, rd) is a fuzzy neighbourhood-base space. (ii) t-(X, Td) = (X, t(d)). Proofl (i) follows immediately from Theorem 1.13. (ii) follows immediately from (i) and Theorem 3.8. (iii) Since, by Theorem 3.2(i), fld is a fuzzy neighbourhood basis for (X, t(d)), then from Definition 3.7 and Theorem 3.6(ii), (X, t(rd)) and (X, t(d)) have the common fuzzy neighbourhood basis fla. Hence, they coincide. By [36, Theorem 5.7], unless (X, Td) is topologically generated, it will differ from (X, t(d)). We also have:

Theorem 3.10. Let (X, d) be a classical pseudo-metric space, and let A be the associated pseudo-metric topology. Then, if we consider (X, d) as a f.p.metric space, we get

t(d) = r d = to(A). Proof. Observe that in both the classical and the fuzzy cases, B(x; ~) denotes the same crisp member of I x. From Theorem 1.13, Td has subbase A U collection of all constant fuzzy subsets of X. Hence from [32, Theorem 5.1], rd = tO(A). Consequently from Theorem 3.9 and [36, Theorem 5.7], t(d) = t(td) = t(to()t)) to(A). =

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Theorem 3.11 [36]. Let F = (F(x))x~X be a family of local bases for a fully stratified fts (X, r). Then the collection {U ^ ¢t: U • [._Jx~XF(x) and ol • 10,a} is a basis for 3. Theorem 3.12. Let Y be a nonempty crisp subset of a f.p.metric space (X, d), and let d' = dlr be the

restriction o l d to Y x Y. Then obviously (Y, d') is a f.p.metric space. Also, (i) 3d, = (Td)lr (=Td relativized to Y), (ii) t(d') = t(d) I,. (=t(d) relativized to Y). Proof. (i) Since the family fld = (fld(X))x,X, of Theorem 3.9, is a family of local bases for (X, 3d), then, evidently, the family (fld(X)lY)x~r, given for all x • Y by

fla(x)lv = { U ^ Y: U • t~d(X) C1 x} c l r, is a family of local bases for (Y, (3d)]r). But, applying Theorem 3.9 to the f.p.metric space (Y, d') we find that (fld(x)lv)x~r is also a family of local bases for (Y, 3d,). Hence from Theorem 3.11, 3d, = (3d)ly(ii) From Theorem 3.2, (fld(x)lr)x~v is a fuzzy neighbourhood basis for (Y, t(d')). On the other hand, the fuzzy neighbourhood system of the fns (Y, t(d)lr ) is the initial fuzzy neighbourhood system for the inclusion Y---)(X, t(d)) [1]. So, it too has the fuzzy neighbourhood basis (fld(X)lr)x~r. Consequently, t(d') = t(d) ly. Definition 3.13 [35]. Let (X, d) be a f.p.metric space, x • X and b • ~+. We define the fuzzy closed ball with centre x and radius 6 by C(x; b) = Ar~lb.-t B(x; ~). Theorem 3.14. For all x in a N-p.metric space (X, t(d)) and all b • ~+, (i) B(x; 6)- <~C(x; 6) (as in the classical case, equality need not hold), (ii) C(x; 6) is closed.

Proof. (i) For all z E X, we have from Theorem 3.2,

B(x; b)-(z) = inf sup(B(x; b) ^ B(z; ~)) r~H

= inf sup Lb[d(x, y)] ^ Lr[d(z, y)] rEH yEX

<~ inf sup Lb+r[d(x, y) • d(y, z)] r~H

(from Theorem 1.9)

y~X

~< inf Lb+r[d(x, z)]

(from the triangle inequality)

r~H

=

This proves that B(x; 6)- <~C(x; 6). (ii) Since for all b < r < a in R +, C(x; 6) ~ B(x; ~) <~C(x; f) ~ B(x; 6), then

C(x; 6) = A

r~]b,~[

C(x; O.

Hence from (i),

C(x;g)-=[,~. B(x;~)]A

r~]b,~[

A C(x;O=C(x;6).

re]b,~[

This proves that C(x; 6) is closed.

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4. N-Regularity and N.Normality

We provide axioms of N-regularity and N-normality, which are meaningful only for fuzzy neighbourhood spaces, and which are satisfied by all N-p.metric spaces. It is possible, but out of the scope of this paper, to extend them to axioms of regularity and normality which are meaningful for all fts's (cf. [40] for lower separation axioms). Notation 4.1. Let ~6 =

(~(x))x~x be

a fuzzy neighbourhood base in X. For every M • 2x, we put

fl(M) = { yM Uy: Uy • fl(Y) for all y • M }. Y

Definition 4.2. A fuzzy neighbourhood space (X, t(v)) is said to be: (i) N-regular if for all M • 2x, x • X and 0 • Io, there exist Uo • v(M) and Vo • v(x) such that M-(x) + 0 >I sup(Uo ^ Vo); (ii) NT3 if it is N-regular and NT1; (iii) N-normal if for all M , S • 2 x and 0 • I o , there exist U o • v ( M ) and V o • v ( S ) such that s u p ( m - ^ s - ) + 0/> sup(Uo ^ 11o); (iv) NT4 if it is N-normal and NT1. Theorem 4.3. For a fns (X, t(v)), the following properties are equivalent: (i) N-regularity; (ii) for all M e 2x and x e X,

M-(x) >!

inf

(U,V)Ev(M)×v(x)

sup(U ^ V);

(iii) equality holds in (ii); (iv) for all M • 2x, M - = A {U-: U • v(M)}; (v) for all M • 2x, x • X and 0 • Io, there is U • v(M) such that M-(x) >! U-(x) - O. Proof. (i) ::> (ii) Direct from definition. (ii) ~ (iii) Suppose (ii) holds. Then

M-(x) >i

inf

(U, V)~v(M)× v(x)

sup(U ^ V)

/> inf sup(M A V) = M - ( x ) . V~v(x)

Hence, equality holds. (iii) ~ (iv) Suppose (iii) holds. Then M - ( x ) = u,v(M,infI s u~nfx p(U^v =

inf

Uev(M)

U-(x)

V)]

for a l l x e X .

This proves (iv). (iv) ~ (v) Direct. (v) ::> (i) Suppose (v) holds. Then for all M • 2x, x e X and 0 • I0, there is U • v(M) such that

M-(x) + ½0 >i U-(x) = inf sup(U ^ V) Vev(x)

~>sup(UAV)-½0 This proves (i).

for some V • v ( x ) .

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It is easily seen that in Definition 4.2 and Theorem 4.3, the fuzzy neighbourhood system v can be replaced by any of its bases.

Theorem 4.4. N-regularity and N-normality are good extensions (cf. [23]). Proof. Observe that topologically generated fls's are fns's [27]. We prove the goodness of N-normality. That of N-regularity will have a similar proof. Let (X, Z) be a topological space. For x • X, denote by Z(x) the (X, Z)-neighbourhood basis at x, {C • Z: x • C}. Then (k(X))x~X is also a fuzzy neighbourhood basis for (X, to(Z)) [27]. Also for all M • 2x, cl,o(x)(M) = clx(M) • 2x [32]. Hence, (X, to(A)) is N-normal iff for all M, S • 2x, sup(M- ^ S-) I>

inf

(U,V)eX(M)×X(S)

sup(U ^ V);

iff (X, Z) is normal. This proves that N-normality is a good extension.

Theorem 4.5. (i) NT4 ~ NT3 ~ NT2. (ii) N-regularity and WNT1 ~ WNT2. Proof. (i) it follows directly from definitions. (ii) Suppose a fns (X, t(v)) is N-regular and WNT1. Then for any x ~ y in X, we get from the WNT1 property and Theorem 4.3, 1>

inf V(y) =y-(x) =

Vev(x)

inf

(U,V)ev(y)xv(x)

sup(U A V).

This proves that (X, t(v)) is WNT2. That N-normality and WNT1 ¢:> N-regularity is because we did not attempt to formulate weak versions of N-normality and N-regularity. However, such an implication and such weak axioms would have been of no consequence for N-p.metric spaces; because we now show that those spaces are always N-normal and N-regular, and because in fuzzy uniform spaces, WNT1 is equivalent to WNT2 [40].

Theorem 4.6. Every fuzzy proximity space is N-regular. Consequently, every fuzzy uniform space is

N-regular. Proof. Let P = (P(M))m~2x be the fuzzy proximal neighbourhood system of a fuzzy proximity space (X, 6). Since by [37, Theorem 7.1], v = (P(x))x~x is the fuzzy neighbourhood system of (X, t(6)), then for all M • 2x, P(M) ~_(-'ly~MP(Y) (by [37, Theorem 6.6]) = v(m). So from [37, Theorem 6.9], we have for all M • 2x and x • X,

M-(x) = 6(M, x) --i>

inf

inf

(U,V)¢P(M)×P(x)

(U, V)e:v(M)x v(x)

sup(U ^ V)

sup(U ^ V).

But by Theorem 4.3, this implies that (X, t(6)) is N-regular.

Theorem 4.7. Every N-p.metric space is N-normal. Proof. Let (X, d) be a f.p.metric space, and let Va be the fuzzy neighbourhood system of (X, t(d)). Choose any M, S • 2x and 0 • Io, and denote y = sup(M- ^ S - ) and a~ = y + 0. If tr t> 1, there will be nothing to prove. So, suppose that (r < 1. Given x • M,

d(x, S)(0+) = 1 - S-(x) ~l-y.

(from Theorem 3.4)

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So, there is r x e ~ + such that d ( x , S ) ( 2 r x ) ~ l - m Define for every x e M , U~=B(x;fx), and let U = Vx~M Ux ~ Vd(M). For every y e S, define ry~ R ÷ as above with M and S interchanged, and put Vy = B(y; fy). Let V = Vy~s Vy ~ Vd(S). Now for every z e X, x e M and y e S,

(Ux ^ Vy)(z) = B(x; fx)(z) ^ B(y; Py)(Z) = L~Jd(x, z)] ^ Lrjd(y, z)] <~L~ +r~[d(x, z) 0 d(y, z)] <~L,~+,jd(x, y)]

(from Theorem 1.9)

(from the triangle inequality).

Without loss of generality, we assume that rx >! ry. Hence, (Ux A V y ) ( Z ) ~< t2r~[d(x, y)] ~< LerJd(x, S ) ] = 1 - d(x, S)(2r~) ~< o~.

This gives s u p ( f ^ V) =

sup (Ux ^ Vy)(z) (x,y,z)eMxS×X

~< o~ = sup(M- ^ S - ) + 0, which proves that (X, t(d)) is N-normal.

5. The N-Euclidean topology on ~ ( I ) We introduce a f.metric a on the fuzzy real line R(1). We call this a the f.Euclidean metric, and t(O) the N-Euclidean topology, on ~(I). If +oo is not acceptable as a possible value for a, then we have to narrow R(I) down to include only those fuzzy real numbers r/which are normalized; i.e. satisfy r/(r) = 1 for some r e E.

(5.1)

This additional condition is not stringent, since it is satisfied in E*(1) and in the fuzzy unit interval

I(I), which are the subspaces of ~ (1) of interest in the sequel. The f.metric a will be the restriction (to E(I)) of a f.norm on the smallest real vector space ~ ( I ) which includes E (I). So, we begin by discussing this vector space. ~ ( I ) is constructed by means of a standard algebraic procedure as follows: (R (I), ~ ) is an Abelian cancellation monoid, and scalar multiplication in ~ (I) by nonnegative reals is associative, distributive over + and ~ , and has unit 1 (cf. [35, 38]). Then the equivalence relation - given on R(I) x R(1) by

bestows its substitution property on addition and nonnegative scalar multiplication. Hence, the coordinatewise operations on E ( 1 ) x E(1) induce vector space operations on the collection of equivalence classes. We denote the resulting real vector space by ~ ( I ) . If we denote a member of ~ ( I ) simply by any of its representative ordered pairs of fuzzy real numbers, then the vector space operations on ~ ( 1 ) will be given explicitly by (,~, ¢) • (7', ¢') = ( 7 / ~ O', ¢ • ¢'), and for every real t, t(r/,

f (tr/, t¢) ¢)--t(Itl ¢, Ill rl)

if t/> 0, if t < 0 .

The zero vector is 0 = (0, 0), and the opposite of (r/, ~) is (~, r/). ~(1) is embedded in ~ ( I ) by the canonical monomorphism r/~-~ (r/, 0). It is easily seen that ~ ( I ) is the smallest (in the usual algebraic sense) real vector space which thus includes ~ (I), and that ~ ~ I~(I) is a vector subspace. The order of

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382

•(I) can be unambiguously extended over M(1) as follows:

(77, ~)/> (r/', ~') in ~ ( I ) ¢:> r/(3 ~' I> ~ (3 r/' in ~(I). This partial order on I~(1) is a vector order; that is, it is preserved under translations and nonnegative scalar multiplication, and is reversed by taking opposites. Theorem 5.1. Define the mapping [[ ~ : ~ ( I) ---~~ *( I) by [[(r/, ~)] = inf{~ • R*(I): ~ i> (r/, ~) and ~ >>-(~, r/)} = inf{~ • I~*(I): ~ (~ ~ i> r/and ~ (~ r1 >i ~}. Then: (i) [[ ]] is a finorm on ~ ( I ) , (ii) [[(r/, ~)]]/> (r/, ~), and equality holds if and only if (rl, ~) • ~ *(I), (iii) [[r~= ]rl- for all r • R (that is, [[ ~ extends the absolute value function on ~ ). Proof. That [[ ]] is well defined is ensured by the stipulation (5.1) and the fact that I> is well defined on ~ ( I ) . We first show that [l [[ is a f.p.norm. Given (r/, ~), (r/', ~') • ~ ( I ) and t • I~, we have [[(r/, ~)~ (3 [[(r/', ~')]] = inf{~ • R *(I): ~ (3 r/I> ~ and ~ (3 ~ i> r/} (3 inf{~' • ~*(1): ~ ' ( 3 r / ' ~> ~' and ~ ' ( 3 ~'~> r/'} = i n f { ~ ~' • I~*(I): ~(3 r/~> ~, ~(3 ~ ~> r/, ~' (3 r/' ~> ~ ' and ~' (~ ~'~> r/'} (by applying Proposition 1.7 twice) >~inf{x • ~*(1): X (3 Tl ~3 0' >>-~ ~ ~ ' a n d x ( ~ ( ~ ' ~ > r / ~ r / ' } = [[(~, ~) • (,7', ~')[[, and if t t> 0, lit(r/, ¢)]] = [[(tO, t~)]] = inf{~ • R *(I): ~ (~ tr/i> t¢ and ~ ~3 t¢/> tr/} = t inf{~ • ~*(1): ~ • r/~> ¢ and ~ ~ ¢ t> r/} = t[[(r/, ~)]], and hence for t < 0, [[t(r/, ¢)1]= Illtl (¢, r/)ll = Itl [[(¢, r/)ll = Ill I[(r/, ¢)ll, the last equality being a direct consequence of the definition of [[ ]], which also entails (r/, ¢):/: 0 ~(r/, ¢)] > 0. This proves that I[ ] is a f.norm. (ii) and (iii) are direct consequences of the definition. Definition 5.2. The f.Euclidean norm on M(1) is the f.norm I[ [[ of the above theorem. The associated f.metric is denoted by a. It is called the f.Euclidean metric on ~ ( I ) , while t(a) is called the N-Euclidean topology on ~ ( I ) . Same terminology applies to the restrictions of these notions to the fuzzy real line ~ (I). Observe that by Theorem 1.18, the fuzzy topology za on ~ ( I ) is linear. While, by [16, Proposition 8.2], the WNT1 fuzzy neighbourhood space ( ~ ( I ) , t(c~)) is also linear. The f.Euclidean metric O is given on II~(1)c l~(I) explicitly by c~(r/; r/') = [[(r/, r/')]] = inf{~ • ~*(I): ~ ~ r//> r/' and ~ ~ 71' I> r/}. By considering the elements of ~(1) as fuzzy subsets of I~ (cf. Section 1), it is direct to verify that for

77, r/' in I~ (I), a(n; 77') = V [Ir~.. - ,~'.1 ^ a] • R*(t) c I ~,

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where I I is the absolute value function from ~ ( c ~(I) c I a) to R* ( c ~(I) c IR). However, this last formula will not be used in this paper.

Proposition 5.3. The f.Euclidean metric 3 on ~(I) restricts to the usual metric (which we denote by 3) on ~ ~ ~(I), and the N-Euclidean topology t( 3) on ~ ( I ) relativizes to the fuzzy usual topology on ~. Proof. The first assertion can be deduced directly from the definition of 3. Consequently from Theorems 3.12 and 3.10, t(a)l. - t(al~) = t(~) = the fuzzy usual topology on ~. The usual metric ~ on R* satisfies three universal properties with respect to all pseudo-metric spaces. We show that the f.Euclidean metric 3 on ~*(I) extends those universal properteis over all f.p.metric spaces.

Proposition 5.4. Let x, y • X and M • 2x be in a f.p.metric space (X, d). Then O(d(x, M); d(y, M)) <- d(x, y).

Proof. By Proposition 2.14, d(x, M)<~ d(y, M ) ~ d(x, y) and d(y, M)<-d(x, M ) • d(x, y). Hence,

a(d(x, M); d(y, M)) = inf{~ • ~*(I): d(x, M) <~d(y, M) ~ ~ and d(y, M) <~d(x, M) ~9 ~ } <- d(x, y). We say that two f.p.metrics dl and d2 on a set X are N-equivalent if t(dl)= t(d2), and are U-equivalent if a//(d 0 = °//(d2).

Theorem 5.5. The f. Euclidean metric 3 on ~*(I) satisfies: (UP1) 3(r/; 0) = Tl for all rI • ~*(I). Moreover, in the presence of (UP1), 3 is the unique, up to U-equivalence, f.p.metric on ~*(I) which satisfies: (UP2) given any f.p.metric space (X, d), the f.p.metric d is a uniformly continuous function

d:(X, ~t(d)) × (X, ~t(d))--, (~ *(I), ~z(a)), where the domain is given the product fuzzy uniformity. Proof. By Theorem 5.1(ii), 3 satisfies (UP1). Let {~Pr• l~'(n×~*(+): r • ~+} and {~Pr • IX×X: r • ~+} be canonical fuzzy uniform bases for o//(3) and °//(d), respectively. Then for all r•lI~ + and (Xl, x2, Yl, Y2) • X2 x X 2,

(d × d)-l( ~Ar)(x~, x2, y,, y2) = W~(d(x,, x2), d(y,, Y2)) = Lz~[a(d(x~, x2); d(y,, Y2))] Lz~[ O(d(x~, x2); d(x2, YO) ~ O(d(x2, Y0; d(yl, Y2))] (from the triangle inequality for O) L2,[d(xl, Yl) ~) d(x2, Y2)] (from Proposition 5.4) L~[d(xl, Yl)] ^ L~[d(x2, Y2)] (from Theorem 1.9) = ~p~(x~,Yl) ^ ~P~(x2, Y2) ~- ~)r :~ ~¢?r(Xl, X2,

Y~, Y2)"

This shows that (d x d)-l(~,~r)~ > ~0~* ~p~ for all r e ~+, which proves that d is uniformly continuous, and demonstrates (UP2). For uniqueness, suppose a f.p.metric co' on ~*(I) satisfies (UP1) and (UP2). Then the identity function (~*(/), °-t/(co'))---~(~*(I), o-//(co))is uniformly continuous, because it is the restriction of co' to

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384 the domain • * ( I ) x U-equivalent to a.

{0}. Similarly, the inverse function is uniformly continuous. Hence, a' is

It follows directly from the above theorem that d:(X,t(d))x(X,t(d))-->($~*(l),t(O)) d: (X, Zd) × (X, rd)"-* (R*(1), ra) are both continuous. In Section 6, we also need:

and

Notation 5.6. Let M be a crisp subset of a f.p.metric space (X, d). We denote by dM :X---> ~*(I) the function given by dM(X) = d(x, M), x e X. Theorem 5.7. In the above notations, dM is uniformly continuous (X, all(d))---> (~*(I), °//(0)). Proof. Using the canonical fuzzy uniform bases {Wr': r e ~+} of (R*(1), q/(a)) and {~Pr: r e ~+} of (X, °//(d)), we get for all r e ~ + and (x, y) e X × X,

(dM x dM)-'(~O))(x, y) = W(dM(X), dM(y)) = Lr[a(d(x,

M); d(y, M))]

>t Lr[d(x , y)]

(from Proposition 5.4)

= ~.lr(X , y).

This shows that (dM× dM)-~(~Pr') I> ~Pr for all r 6 ~+, which proves that dM is uniformly continuous.

6. N-Complete regularity We introduce an axiom of N-complete regularity, on fts's, which directly implies fuzzy uniformizability. To prove the equivalence of these two properties, we establish their equivalence to embeddability in products of N-p. metric spaces. Definition 6.1. A gauge for a fuzzy uniformity o//.on a set X is a f.p.metric d on X such that °//(d) ~_ ~. Proposition 6.2 [17]. Let ~p be a fuzzy entourage in a fuzzy uniform space (X, oF). Then there is a gauge d for ~ such that ~ ~ all(d). Proof (cf. [17]). Given ~p e °F, a subfamily °//"v = {~Pi,,.... , , e ~: all indices are positive integers} can be constructed, by induction, such that ~Pl = ~P, and for any lp' = ~Pin,..... , e ~'v and any positive integer ni, the fuzzy entourage ~P"= ~(i+1),,...,, ,,, ~ ~o is symmetric, and it satisfies lp"° lp" -- 1/ni <~ lp '. The family of all finite intersections of members of ~ , is a countable fuzzy uniform base °/4/"which contains ~p and is included in oF'. By [10, Theorem 3.2], there is a f.p. metric d on X such that 0//(d ) = o/~.. This d is a gauge for ~ , and ~p ~ q/(d). (The 1-Hausdorff-separatedness requirement in [10, Theorem 3.2] is easily seen to be needed only if d is to be a f.metric.) Proposition 6.3 [17]. A fuzzy topological space is fuzzy uniformizable if and only if it is embeddable in a product of N-p.metric spaces. Proof. Let D be the family of all gauges for a fuzzy uniform space (X, ~). Let I]a~o (X, °It(d)) be the product fuzzy uniform space. Then, it follows from Proposition 6.2 that the initial fuzzy uniformity (cf.

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385

[26]) on X for the diagonal inclusion X---~IJd,o(X, °ll(d)) is ~ . This proves that this diagonal inclusion is an embedding (X, t(~F))--->IJd~O (X, t(d)). The inverse implication is direct; because fuzzy uniformizability is productive and hereditary [26]. The fuzzy unit interval I(I) is the subset of E (I) {r/e E(I): r/(0) = 1 and ~/(1+) =0} = {r/e E(1): 0 ~< ~/~< i}. For the notion of initial fuzzy topologies, we refer the reader to [22].

Definition 6.4. A fully stratified fts (X, ~) is said to be N-completely regular if T is the initial fuzzy topology for the family of all continuous functions (X, ~)--~ (I(I), t(a)). Proposition 6.5. N-complete regularity ~ fuzzy uniformizability. Proof. By [26, Theorem 4.3], an initial fuzzy topology for fuzzy uniformizable fuzzy topologies is fuzzy uniformizable. Corollary 6.6. N-complete regularity ~ N-regularity.

Proposition 6.7. N-complete regularity is productive and hereditary. Proof. Let {(Xj, rj): ] e J} be a family of N-completely regular fts's, and let Pjo:IIj,j(X/, r j ) ~ (Xjo, rio) be the coordinate projection, Jo e J. It follows that for every continuous function f:(Xjo, rio)---> (I(I), t(a)), the function FI = f OPjo:I-Ij,j (Xj, rj)---~(I(I), t(3)) is continuous, and for every U e t(a), F;I(U) = e ~ ' ( f - ' ( U ) ) . Since all the fts's (Xj, Tj) are N-completely regular, then the family of all fuzzy subsets of the type

Fi~(U) is a subbasis for the product fts Ilj,j (Xj, Tj). Hence, this product fts is N-completely regular. That N-complete regularity is hereditary is a direct conclusion from its definition.

Proposition 6.8. Every N-p.metric space is N-completely regular. Proof. Let (X, d) be a f.p.metric space, and truncate d at i e E*(I). This truncation does not affect 9/(d). Let {~p): r e E*} and {~p,: r e ~*} be the canonical fuzzy uniform bases of (I(I), °-t/(a)) and (X, 9/(d)), respectively. Given M e 2 x, let f = dM:(X, ~(d))--~ (I(I), ~ ( a ) ) be the function f(x) = d(x, M), x e X. From Theorem 5.7, f is uniformly continuous. For all x e X,

f-l([f(M)l-)(x) = [f(M)l-(f(x)) = O-(f(x)) = 1 - a(f(x); 0)(0+) = 1 -f(x)(0+)

(from Theorem 3.4)

(by (UP1) of Theorem 5.5)

= 1 - d(x, M ) ( 0 + ) = M-(x)

(from Theorem 3.4 again).

This proves that M - = d~t~([dM(M)] -) for all M e 2 x. Hence, t(d) and the initial fuzzy topology on X for all continuous functions (X, t(d))---> (I(I), t(O)) have the same fuzzy closures for members of 2 x. Since both are fuzzy neighbourhood topologies, then, by Theorem 1.2, they must coincide. This proves that (X, t(d)) is N-completely regular.

Proposition 6.9. If afts is embeddable in a product of N-p. metric spaces, then it is N-completely regular.

386

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Proof. This follows from Propositions 6.7 and 6.8. We arrive at: Theorem 6.10. A fuzzy topological space is fuzzy uniformizable if and only if its is N-completely

regular; if and only if it is embeddable in a product of N-p.metric spaces. Proof. A compendium of Propositions 6.3, 6.5 and 6.9. Corollary 6.11. N-complete regularity is a good extension. Proof. Let (X, ,~) be a topological space. Then, (X, to().)) is N-completely regular ¢:> (X, to(A)) is fuzzy uniformizable ¢:> (X,).) is uniformizable (by [26, Corollary 3.2]) ¢:> (X, A) is completely regular. Corollary 6.12. Neither (I(I), t( a)) nor (I(I), ra) is topologically generated. Proof. By [22, Theorem 1.4], an initial fuzzy topology of topologically generated fts's is topologically generated. Consequently, if (I(I), t(a)) was topologically generated, then all N-completely regular fts's, and hence all fuzzy uniformizable fts's, would have been topologically generated too. But this conclusion is false; see [27, Example 5.D]. Hence, (I(1), t(a)) is not topologically generated. Consequently, by [36, Theorem 5.7] and Theorem 3.9, (l(I), To) is not topologically generated either. It follows from the above corollary that neither (~(I), t(a)) nor (•(I), To) is topologically generated. Proposition 6.13. Let M be a crisp subset of X, and let a function f :X---~(I(I), t(a)) be such that f ( M ) = O. Then for all x • X, 1 - f ( x ) ( 0 + ) = [f(M)]-(f(x)). Proof.

1 - f ( x ) ( 0 + ) = 1 - O(f(x); 0)(0+) =O-(f(x))

(by (UP1) of Theorem 5.5)

(from Theorem 3.4)

= [f(M)]-(f(x)). We end by a characterization of N-complete regularity within the category of fuzzy neighbourhood spaces. Observe that this category has, in turn, several characterizations within the category of fuzzy topological spaces (see [34, 36, 37, 39]). Theorem 6.14. Let (X, ~) be a fuzzy neighbourhood space. The following two statements are equivalent:

(i) (X, 3) is N-completely regular. (ii) For all M • 2x, x • X and 0 • lo, there is a continuous function f :(X, ~)---~ (I(I), t(a)) such that f ( M ) = 6 and M-(x) + 0/> 1 - f ( x ) ( 0 + ) . Proof. (i) f f (ii) Suppose that (X, 1-) is N-completely regular. Then there is a fuzzy uniformity T" on X such that ~-= t(T') (by Theorem 6.10). By the definition of t(T') (Definition 2.2), given M • 2x, x • X and 0 • I0, there is a fuzzy entourage ~p • T" such that M-(x) + 0 >t ~p(M)(x).

(6.1)

By Proposition 6.2, there is a gauge d for °V such that ~p • 9/(d) and d is truncated at i. In the N-p.metric space (X, t(d)), let W • I x be the fuzzy closure of M. Then applying Definition 2.2 this time

A.S. Mashhour, N.N. Morsi / Fuzzy metric neighbourhood spaces to

(x, t(d)),

387

w e get

~p( M ) (x) >>-W(x) = 1 - d(x, M)(0+)

(from Theorem 3.4)

= 1 - dM(x)(O+). H e n c e f r o m (6.1), M - ( x ) + 0 >i 1 - dM(x)(O+ ).

(6.2)

But since 9 / ( d ) _ _ ~ , t h e n t ( d ) ~ _ t ( ~ ) = v . H e n c e , by T h e o r e m 5.7, c o n t i n u o u s . This t o g e t h e r with (6.2) a n d t h e o b v i o u s fact t h a t d M ( M ) = 0 satisfies the r e q u i r e m e n t s o f c o n d i t i o n (ii). (ii) ~ (i) S u p p o s e (ii) h o l d s , a n d let v n b e t h e initial fuzzy t o p o l o g y o n X ( X , t)--->(l(1), t(a)). L e t M b e a crisp s u b s e t o f X. T h e n for e v e r y x c o n t i n u o u s function f : ( X , v)---~ ( I ( I ) , t( O)) such t h a t

d ~ t : ( X , t ) - - - , ( l ( 1 ) , t ( a ) ) is p r o v e t h a t the f u n c t i o n dM for all c o n t i n u o u s functions e X a n d 0 e I0, t h e r e is a

M-(x) + 0/> 1 -f(x)(0+) = [f(M)]-(f(x))

( f r o m P r o p o s i t i o n 6.13)

=f-~([f(M)]-)(x). Hence, M - >i A {g-l([g(M)]-): g is continuous from (X, v) to (I(I), t(3))}/> the closure of M in

vn> M- (because v is finer than vn). Hence, equality holds. Consequently, by Theorem 1.2, v = vn; that is (X, v) is N-completely regular.

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