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Invariant probabilistic metrizability of fuzzy neighbourhood groups
Fuzzy Sets and Systems 47 (1992) 233-245 North-Holland
233
Invariant probabilistic metrizability of fuzzy neighbourhood groups T.M.G. Ahsanullah* Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh-11451, Saudi Arabia
N e h a d N. M o r s i Department of Mathematics, Military Technical College, Kobry EI-Qubba, Cairo, Egypt Received March 1990 Revised November 1990
Abstract: We introduce the notions of N-local compactness and N-openness in fuzzy neighbourhood spaces and in fuzzy neighbourhood groups, together with some characterizations for them. In particular, we prove that a fuzzy neighbourhood space is N-locally compact if and only if its or-level spaces, for all 0 < ot < 1, are locally compact. A similar criterion is established for N-open functions. We also introduce the notion of fuzzy absolute value functions on groups. We show that there exists a 1-1 correspondence between invariant fuzzy (probabilistic) pseudo-metrics on groups and fuzzy absolute value functions. We establish theorems on fuzzy (probabilistic) pseudo-metrizability of fuzzy neighbourhood groups. Finally, fuzzy (probabilistic) metrizability of quotient fuzzy neighbourhood groups is also taken into account.
AMS Subject Classification: 54A40. Keywords: Fuzzy neighbourhood space; fuzzy uniform space; fuzzy neighbourhood group, N-local compactness; N-open function; fuzzy real numbers; fuzzy (probabilistic) metric; fuzzy absolute value function; fuzzy metric neighbourhood space; o~-Ievel space, WNT2-space.
Introduction
The compatibility of Lowen's well known fuzzy neighbourhood systems with the group structure has been studied in [1]. It was also shown there that a fuzzy neighbourhood group (fng) is fuzzy uniformizable in the sense of Lowen [19]. However, other open questions remain unsolved: First, how to provide some * Permanent Address: Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh.
kind of fuzzy metrizability of fng's which would be a good extension of its classical counterpart? Second, does there exist an accompanying notion of local compactness for fuzzy neighbourhood spaces (fns's) which would also be a good extension? In 1982, H6hle [11] established a famous metrization theorem, which states: "A fuzzy T-uniformity is probabilistic metrizable if and only if it is Hausdorff separated and has a countable basis of fuzzy vicinities". His Tuniformity coincides with the Lowen fuzzy uniformity when the triangular norm T is Min, which is the one adopted throughout this paper. A probabilistic metric induces a fuzzy uniformity. This in turn induces a fuzzy neighbourhood space, said in [26] to be a fuzzy metric neighbourhood space. As probabilistic metrics have managed to occupy a prominent place in the theory of fuzzy neighbourhood spaces [6, 11, 13, 26, 33, 34], it now seems reasonable to also term them fuzzy metrics, especially whenever their fuzzy topological aspects are being considered. Taking fuzzy metric neighbourhood spaces as our starting point, we provide answers to the two questions listed above: In Section 2, we study open functions between fns's. We show that openness of functions is equivalent to a new notion of N-openness (neighbourhood openness) of functions, and that a function between fns's is open iff it is open in the entering re-level topologies for all 0 < tr < 1. Section 3 is devoted to studying a notion of N-local compactness in fuzzy neighbourhood spaces. We prove that this notion enjoys a level-topologies criterion identical to the one above. It is therefore a good extension. In Section 4, we present a characterization theorem for N-local compactness of fuzzy neighbourhood groups. Invariant fuzzy metrizability of fng's is considered in Section 5, with the help of a notion of fuzzy absolute value functions. Employing a canonical transformation
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between those two notions, several characterizations for the fuzzy metrization of fng's are provided. Finally, in Section 6, we investigate the fuzzy metrizability of quotient fng's.
for all v e / x and x • X. The pair (X, t(2")) is called a fuzzy neighbourhood space (fns), and X is referred to as its fuzzy neighbourhood system. We denote by FNS the category of fns's and their continuous maps.
1. Preliminaries
Definition 1.2 [1]. Let (X, -) be a fuzzy neighbourhood system on triple (X, .,t(2")) is said to neighbourhood group (fng) if conditions are fulfilled: (FNG1) the mapping
Let X be a set, I0=10, 11, I1=[0, 1[, Io,1=]0,1[
and
I=[0,1].
A fuzzy subset of X is an element of I x (={#:X--->I}). We denote by a • I x the constant fuzzy subset of X having value tr • L The power set 2x. of X is considered as a subfamily of IX; by identifying the subsets of X with their characteristic functions. The height of a fuzzy set # e I x is the real number hgt(#) = sup #(x). xEX
We follow the definition of fuzzy topological spaces (fts's) due to Lowen [16]. (See also [25].) We say that (X, -) is a fts to indicate that - is the fuzzy closure operator of that fts. We refer to [19] and [20] for fuzzy uniformities and fuzzy neighbourhood systems, respectively. However, we recall here: Definition 1.1 [20]. A family 2' = (2"(X))xEx of prefilters in a set X is said to be a f u z z y neighbourhood system on X if the following
conditions are satisfied: (N1) for all x e X and for all # • 2 " ( x ) , #(x) = 1; (N2) for all x e X and for every family (#o)o~o of elements of 2"(x), sup (Uo - O) e X(x); 0~!o
(N3) for all x • X, for all # ~ X(x) and for all 0 ~ Io, there exists a family (#~zEx ~ [L~x2"(z) such that for all y, z • X, #O(z) ^ #O(y) _ 0 ~<#(y). The elements of 2"(x) are called the fuzzy neighbourhoods of x. A fuzzy topology t(X) on X is given by the fuzzy closure operator -: v - ( x ) = inf hgt(v ^ #), u~(x)
group and ~7 a X. Then the be a f u z z y the following
m : ( X x X, t(X) x t(2"))--~ (X, t(2")): (x, y) ~--~xy
is continuous, (FNG2) the mapping g : ( X , t(2"))--~ (X, t(2")):
x t'-"-)x - 1
is continuous. Theorem 1.3 [1]. Let (X, .) be a group and X a fuzzy neighbourhood system on X. Then ( X , . , t(X)) is a fuzzy neighbourhood group if and only if the following conditions are satisfied: (a) for all b ~ X,
X(b) = {~b(#): # ~ X(e)} (alternatively, X(b) = {9~b(#): # e Z(e)}), where ~b, ~tb :X--'~X are the left and right translations on X by the element b, respectively; (b) for all ~ • X(e) and for all 0 > O, there exists v e X ( e ) such that v - 0 ~ g ( # ) ; i.e. the inversion map g : x ~ x -t is continuous at the neutral element e; (c) for all # ~ X(e) and for all 0 > O, there exists v ~ X(e) such that v . v - 0 <- #; i.e. the multiplication m :(x, y) ~ x y is continuous at (e, e); (d) for all # ~ X(e), for all 0 > 0 and for all x ~ X, there exists v ~ X(e) such that x • v . x -1 0 <~#; i.e. the inner automorphism Intx : z xzx -~ is continuous at e. (Notice that, according to our notation, {x} and lx are the same thing.) Definition 1.4 [40]. Let X be a set, # e I x and oce L The oc-cut (respectively c~*-cut) of # is the
crisp subset of X u ~ = {x ~ x : u ( x ) > a }
T.M.G. AhsanuUah, N.N. Morsi / Metrizability of.fuzzy neighbourhood groups
(respectively
then
U~. = {x e X: U(x) >I a}).
v ~ inf #j = inf (v ~/uj).
Definition 1.5 [18]. Let (X, r) be a fts, and let oce I1. The m-level topology of (X, t) is the ordinary topology on X ~ ( r ) = {U~: U ~ r}.
The topological space (X, t~(t)) is called the re-level space of the fts (X, t). Theorem 1.6 [4, 15, 31]. Let - be the fuzzy closure operator of a fuzzy neighbourhood space (X, t(Z)), and let (M, o:) e 2x x I 1. Then (M-)(1-,~). = the to,(t(X))-closure of M. 1.7 [31]. Let o be the fuzzy interior operator of a fuzzy neighbourhood space (X, t(Z)), and let tz e I x. Then
Theorem
# ° = sup [a ^ (# ~)o]. a~¢h
We denote by R* the set of nonnegative real numbers. Definition 1.8 [9, 11, 22, 35, 36]. A nonnegative fuzzy real number ~7 is a descending, left continuous real function: R*-+I, with 7/(0)= 1 and with infimum 0. The set of all nonnegative fuzzy real numbers is denoted by ~*(I). Nonnegative fuzzy real numbers are ordered by their natural ordering as real functions on R * Under this order, arbitrary subsets of R*(I) have their infima in R*(1). Hence, subsets of R*(I) which are bounded above also have their suprema in R*(I). The bottom element 0 of R*(1) is given by 1 ifs =0, 0(s)=
0
j~J
235
jeJ
Definition 1.11 [11,36] (see also [26,30]). A fuzzy (probabilistic) pseudo-metric (or fuzzy p-metric, in short) on a set X is a function d :X2---~ R*(1) which fulfills the following conditions: Vx, y, z e X; (FM1) d(x, x) = 0; (FM2) d(x, y) = d(y, x) (symmetry); (FM3) d (x, y) (~ d ( y , z) >>-d (x , z) (triangle inequality). If, in addition, d satisfies (FM4) x 4 : y ~ d(x, y)>O, then d is called a fuzzy (probabilistic) metric. The pair (X, d) is called a fuzzy pseudo-metric space. Definition 1.12 [30]. If space, then the fuzzy centre x e X and radius of X given for all y • X
(X, d) is a fuzzy p-metric open ball B(x;r), with r > O, is the fuzzy subset by
B(x; r)(y) = LrId(x, Y)I ( = 1 -
d(x, y)(r)).
Theorem 1.13 [11] (see also [26]). Let (X, d) be
a fuzzy p-metric space. Then the family °F(d) = {~p, e IX~: r > O}
is a fuzzy uniform basis in X, in the sense of Lowen [19], where W,(x,y)=Lr[d(x,y)],
r > 0 , x, y e X .
The fuzzy uniformity a//(d)= ~ ( d ) is called the fuzzy metric uniformity induced by d. Definition 1.14 [11]. A fuzzy uniformity ~¢"on a set X is said to be fuzzy p-metrizable if there exists a fuzzy p-metric d on X such that ~¢"= ~ ( d ) .
if s > 0 . Theorem 1.15 [11]. A fuzzy uniformity is fuzzy
Definition 1.9 [11, 22, 35, 36]. Addition ~ of fuzzy real numbers is defined as follows: For all r/, ~ e ~*(I), 7/~) ~ e •*(I) is given by ( r / ~ ~)(s) = sup{r/(b) ^ ~(s - b):
b e[O, s]}, s~>O. Proposition 1.10 [26]. / f v, /~j e R*(1) for ] e J,
p-metrizable if and only if it has a countable basis. In a fuzzy p-metric space (X, d), we follow [26] in denoting the fuzzy neighbourhood topology t(°ll(d)) (induced on X in accordance to [19, 20]) simply by t(d). The fns (X, t(d)) is said to be a fuzzy p-metric neighbourhood space [26].
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Theorem 1.16 [26]. A fuzzy p-metric neighbourhood space (X, t(d)) has a fuzzy neighbourhood basis ~a = (~a(x) = {B(x; r): r > O})x~X. Definition 1.17 [39]. A fns (X, t(X)) is said to be WNT2 if for all x C y in X, there exists (#, v) • Z(x) x Z(y) such that hgt(# ^ v) < 1.
Theorem 1.18 [11] (see also [26]). A fuzzy p-metric neighbourhood space (X, t(d)) is WNT2 if and only if d is a fuzzy metric. The following e06-characterization of continuity of functions between fuzzy p-metric neighbourhood spaces is an immediate consequence of Theorem 1.16 and the relevant definitions:
Theorem 2.2. A function f : (X, t(X))--> (r, t(,Y,')), between fuzzy neighbourhood spaces, is: (i) open if and only if for all oc • I0,1, / i s open: (X, t,(t(X)))--~ (Y, t,~(t(Z"))); (ii) continuous if and only if for all o: • Io.1, f is continuous: (X, t~(t(X)))--, (Y, t~,(t(,~'))). Proof. (i) Suppose that f satisfies the stated condition, and let # • t(X). Then # ~ • t~(t(,~)) for all o~• Io,1, and hence by Proposition 2.1, [f(#)]~ = f (#~') • t~(t(X')). Also,
Theorem 1.19 [6]. A function
[f(#)]o= I,_J [f(#)]'~
f :(x, t(d))~ (Y, t(d')),
O~E/N 1
between fuzzy p-metric neighbourhood spaces, is continuous at a point x • X if and only if for all e > 0 and 0 • 1o there exists 6 = 6~o > 0 such that d(x, z)(6) + 0 >I d'(f(x), f(z))(e)
Recall that a function, between fuzzy topological spaces, is said to be open if it sends open fuzzy subsets onto open fuzzy subsets. For fuzzy neighbourhood spaces, we get:
(1.1)
for all z • X.
Proposition 1.20 [31]. A descending chain {~,~: a:ell} of topologies on a set X is the indexed family of level topologies of some fuzzy topology r on X iff for all ol • 11, Z~ = s u p { Z / y • ]a~, 1[}. In this case, the finest such v exists, and it is the only fuzzy topology on X making (X, v) a fns with the given level topologies.
• sup{t~(t(,~')): te • loA} = t0(t(~7')), where the last equality holds by Proposition 1.20. Now, from the maximality property of the fuzzy topology of a fns with respect to its level topologies [38], it follows that f ( g ) ¢ t(X'). This proves that f is open. The inverse implication follows directly from Proposition 2.1. (ii) The proof is very similar to the above one. From the above characterization of continuity in FNS, it is easy to establish that a fuzzy neighbourhood system on a group is compatible with the group structure if and only if the corresponding e-level spaces are topological groups for all 0 < a~< 1.
2. Open functions in FNS Definition 2.3. A function We shall repeatedly use the following easily established result.
f : (X, t(X))--~ (Y, t(X')), between fns's, is said to be N-open at x • X if
Proposition 2.1. Let f :X--->Y be a function, # • I x and tr • 11. Then: (i) f ( # ~ ) = [f(#)1% (ii) f ( # ^ a) = f ( # ) ^ a.
f ( # ) • X'(f(x))
for all # ¢ X(x).
It is said to be N-open if it is N-open at all points of X.
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237
In the above definition, one can easily see that X can be replaced by any of its fuzzy neighbourhood bases.
Applying Proposition 2.4 again, we find that for all a e ll,
Proposition 2.4. Let (X, t(X)) be a f u z z y neigh-
But this implies, by Proposition 2.5, that f ( # ) e Y.'(f(x)), which proves that f is N-open. (ii) ~ (iii): Suppose f is N-open. For M e 2 x and x • X, put y = M°(x). Then from Proposition 2.4(ii),
bourhood space, and (x, M) • X x 2 x. Then for all tr~ 11: (i) (M °) ~ = intL,o(z)) (M), where o is the f u z z y interior operator of (X, t(X)), (ii) M v a • 2~(x) if and only if M°(x) >>-1 - ol.
[f(#)] ~ v a e Z ' ( f ( x ) ) .
M v (1 - y) e Z(x). By the N-openness of f,
Proof. (i) This follows from Theorem 1.6 by taking complements. See also [31, Proposition 3.2]. (ii) We have
f ( M ) v (1 -- y) • ,~'(f(x)).
M v a e X ( x ) iff ( X - M ) - ( x ) <~ o¢
[f(M)]°(f(x)) >t r = M°(x) •
(by [28, Proposition 4.1])
iff M°(x) >i 1
-
ol,
because M°(x) = 1 - ( X - M ) - ( x ) .
Proposition 2.5. Let ( X , t ( X ) ) be a f u z z y neighbourhood space, x • X and # • I x. Then e X ( x ) if and only if for all ol•11, g ° ' v
So, from Proposition 2.4(ii) again,
This proves that [ f ( M ) ] ° ~ f ( M ° ) . Hence, (iii) holds. (iii) ~ (i): Suppose f satisfies (iii), and let # • I x. Then f(#°)=f(sup
\oral1
[a A (#'~)°])
(from Theorem 1.7)
a • Z(x).
= sup [a ^ f ( ( # 0,)o)]
Proof. This follows directly from [32, Proposi-
oc~/l
tion 2.9], because X(x) is a prefilter.
(applying Proposition 2. l(ii))
We are now in a position to prove:
Theorem 2.6. Let f :(X, t(Z,))--~ (Y, t(Z')) be a function between f u z z y neighbourhood spaces. Then the following are equivalent statements: (i) f is open. (ii) f is N-open. (iii) For all M • 2 x, f ( M °) <~[f(M)] °.
~
(by (iii)) = sup [a A ([f(#)]~)°] ~elt
(from Proposition 2.1(i)) = if0,)] o
(from Theorem 1.7 again). This proves that f is open.
Proof. (i) ~ (ii): Suppose f is open. Let x • X, #•X(x) and tr•11. Then by the above proposition, #~ v a • Z(x). So, by Proposition 2.4, (/~'~)°(x)/> 1 - ol. Consequently, we get by Proposition 2.1(i) and the openness of f,
3. N-locally compact fuzzy neighbourhood spaces
([f(#)l °3°(f(x)) = ( f ( # ~))°(f(x))
(sup v)(x) > a~ for all x • X.
>~f((~ ~)°)(f(x)) i> (# ~)°(x)/> 1 - ol.
An o~-shading [9], where ol•11, of a nonempty set X is a family v ~ I x which satisfies
A fuzzy topological space (X, r) is called re-compact [9] if every open tr-shading of X
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includes a finite a~-subshading of X. A crisp subset of (X, ~) is called o~-compact if it is e-compact in its relative fuzzy topology.
Proposition 3.1. (i) [18]. A f t s is ol-compact if and only if its ol-level space is compact. (ii) [21] A fns is compat (in the sense of Lowen [16]) iff its ol-level spaces are compact for all 0 < a~< 1. Theorem 3.2. Let (X, t(X)) be a fuzzy neighbourhood space, and let (x, M, o0 e X x 2x x IoA. Then M is an t~(t(X))-neighbourhood of x if and only if there exists y e ]0, 1 - a~[ such that M v ~, eX(x). Proof. M is an t~(t(X))-neighbourhood of x iff x e int,~(t(r)) (M) iff x • (M °) ~ (by Proposition 2.4(i)) iff there exists ~ e ]0, 1 - re[ such that M°(x)/> 1 - 7 iff there exists y e ]0, 1 - re[ such that m v It • X(x) (by Proposition 2.4(ii)). Definition 3.3. A fuzzy neighbourhood space (X, t(X)) is said to be N-locally compact at x ~ X if (*) for all o~e I0,1 there exists (M, y) e 2x x ]0, 1-o~[ such that M is o~-compact and M y It e X(x). It is called N-locally compact if it is N-locally compact at all points of X. The above definition is designed to fulfill the following criterion:
Theorem 3.4. A fuzzy neighbourhood space is N-locally compact at a point if and only if its ol-level spaces, for all 0 < o~< 1, are locally compact at that point. Proof. By Propositions 3.1, 3.2, the assertion (*) is equivalent to saying that M is a compact neighbourhood of x in the o~-level space (X, t~(t(X))). Hence, the theorem follows. Level-topologies criteria, identical to the above one, have been established in FNS for compactness (by Lowen, cf. Proposition 3.1(ii)), N-regularity, N-normality and the NT2 property [27], fuzzy metrizability [6], and for openness and continuity of functions (Theorem 2.2). This level-topologies criterion is more demanding
than Lowen's criterion for a fuzzy topological property to be a good extension [18]. This is because, given a topological space (X, ~,), all the level topologies of its generated fts (X, o9(~)) are equal to 3,. Therefore, we have: Corollary 3.5. N-local compactness is a good extension.
Proposition 3.6. A compact fuzzy neighbourhood space is N-locally compact. Proof. This follows directly from the definition of N-local compactness; by taking M = 1 in (*), and by noticing that 1 is compact iff it is e-compact for all tre I0,1 (Proposition 3.1(ii)).
Proposition 3.7. The image of an N-locally compact fuzzy neighbourhood space, under an open continuous function between fuzzy neighbourhood spaces, is N-locally compact. Proof. In [9], it is shown that the continuous image of an a~-compact fts is e-compact. The assertion follows from this and the equivalence between openness and N-openness of functions in FNS (Theorem 2.6). The product fts of a family of fns's is also a fns [12, 24]. We then have:
Theorem 3.8. The product fuzzy neighbourhood space of an indexed family ~f of fuzzy neighbourhood spaces is N-locally compact if and only if ~ satisfies: (A1) all the fns's in ~ are N-locally compact, and (A2) for every oc e Io,1, the number of indices corresponding to non-m-compact fns's in ~ is finite. Proof. Let ~,={(Xj, r~):jeJ} and ( X , r ) = II)~j(Xj, r)). Since the re-level topology of a product fuzzy topology is the product of the corresponding o~-level topologies (by [18, Theorem 2.1]), we get the following equivalences: (X, r) is N-locally compact iff for all t r e lo,1,
[I (xj, jeJ
=
(x,
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239
is locally compact (by Theorem 3.4) iff (A1) and (A2) hold for ~ (by a known theorem in General Topology, Proposition 3.1(i) and Theorem 3.4).
Proof. Let (X, -) be a fns, and denote its 0-level
Characterizations for the Wuyts-Lowen weak Hausdorff axiom WNT2 in FNS (see Definition 1.17) are established in the next two theorems.
C-(x) v (X - C)-(y) < 1
Theorem 3.9. For a f u z z y neighbourhood space ( X , t ( X ) ) , the following two statements are equivalent: (i) (X, t(X)) is WNT2. (ii) For all x :/: y in X, there exists C ~ 2 x such that C - ( x ) v ( X - C ) - ( y ) < 1.
topology by Z. We have the following equivalences: The fns (X, -) is WNT2 iff for all x :/:y in X, there exists C 6 2x such that (by the preceding theorem) iff for all x :/: y in X, there exists C e 2x such that x @(C-)1. = the A-closure of C, and y $ ((X - C)-)v = the Z-closure of X - C (by Theorem 1.6) iff the topological space (X, A) is T2.
4. N-locally compact fuzzy neighbourhood groups
Proof. (i) ~ (ii): Suppose (X, t(X)) is WNT2, and choose any x :~y in X. Then there exists (#, v, re) e X(x) x Z ( y ) x I1 such that /MA V ~ a ,
equivalently,
p,~A v'~ = 0. Take C = X - I z ~. Then X - C = l z ~ _ X - v From [32, Theorem 2.10], we get C-(x)
= (x
-
~.
<
and ( X - C ) - ( y ) <~ ( X - v ~ ' ) - ( y ) <~ tr < 1.
This proves (ii). (ii)ff (i): Suppose (ii) holds, and choose any x:/:y in X. Then there exists (C, a 0 ~ 2 x X ll such that C - ( x ) v ( X - C ) - ( y ) = o:.
From [28, Proposition 4.1], we get = (x - c) v a e X(x),
Vl = C v a ~ ~'(y) and ~l A
V1 =
I[Z.
This proves that (X, t(X)) is WNTE.
Theorem 3.10. A f u z z y neighbourhood space is WNTE if and only if its zero-level space is a Hausdorff topological space.
A fuzzy topological space (X, r) is said to be homogenous if for every (x, y) ~ X 2 there exists a homeomorphism of (X, r) onto itself which sends x onto y. Since all left (and right) translations in a fuzzy neighbourhood group are homeomorphisms of that fuzzy neighbourhood group onto itself [1, Proposition 2.6], we deduce that fuzzy neighbourhood groups are homogeneous fuzzy neighbourhood spaces.
Theorem 4.1. A homogeneous f u z z y neighbourhood space is N-locally compact if and only if it is N-locally compact at one point.
Proof. Every homeomorphism between fuzzy neighbourhood spaces is continuous. Hence, it takes c~-compact subsets onto o~-compact subsets. It is also open, and so it is N-open (Theorem 2.6). The above clearly entails that the validity of (*) at one point of a homogeneous fuzzy neighbourhood space is equivalent to its validity at all points of that space.
5. Fuzzy (probabilistic) metrizability of fuzzy neighbourhood groups Definition 5.1. A fuzzy p-metric d on a group (X, -) is said to be: left invariant if d(bx, b y ) = d ( x , y ) ,
b,x,y~X,
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right invariant if d(xb, y b ) = d ( x , y),
b,x, y • X ,
two-sided invariant (or, simply, bi-invariant) if it is both left and right invariant. Definition 5.2. Let (X, .) be a group. A function p : (X, .)--* R *(I) is called a fuzzy absolute value function (favf, in short) if the following holds for all x, y • X: (AV1) p(e)=l), where e is the neutral element of (X, .); (AVE) p(x -l) = p(x) (symmetry); (AV3) p(xy) <~p(x) ~ p(y) (subadditivity). The four notions defined above are strongly related to each other; as summarized in the next two theorems, the easy proofs of which are exact replicas of those of the corresponding classical results, see for instance [8, 14, 37]. So, we omit the proofs. Theorem 5.3. Let (X, .) be a group. (i) There is a well defined canonical 1-1 correspondence T between the class of all left invariant fuzzy p-metrics d on (X, .) to the class of all favf's on (X, .), given by T(d)(x) = d(x, e),
Theorem 5.4. Let d be a left invariant, or right invariant, fuzzy p-metric on a group (X, .), and let p be its fuzzy absolute value function. Then the following statements are equivalent: (i) d is bi-invariant. (ii) d(x, y) -- d(x -1, y-l), x, y • X. (iii) p ( x y ) = p ( y x ) , x, y • X . Consequently, a fuzzy p-metric on an Abelian group is left invariant iff it is right invariant iff it is bi-invariant.
Proposition 5.5. Let d be a left invariant fuzzy p-metric on a group (X, .), and let d' be its right invariant fuzzy p-metric. Then the inversion mapping g of (X, .) is a fuzzy homeomorphism g: (X, t(d))--~ (X, t(d')). Also, the two fns's (X, t(d)) and (X, t(d')) will coincide if and only if g is continuous as a function from one of them onto itself.
x • X.
The inverse transformation T -1 is given on a f a v f p on (X, .) by T-l(p)(x, y ) = p ( y - l x ) ,
x • X.
The inverse transformation S -1 is given on a f a v f p on (X,.) by S-l(p)(x, y)=p(xy-1),
Proof. Since, for all x, y • X,
d'(g(x), g(y)) -- d'(x -1, y-l) = d(x, y),
x, y e X .
(ii) There is a well defined canonical 1-1 correspondence S between the class of all right invariant fuzzy p-metrics d' on (X, .) to the class of all favf's on (X, .), given by
S(d')(x) = d'(x, e),
We shall always be referring to one or more of the above canonical bijections when we speak of, say, a left invariant fuzzy p-metric d and its right invariant fuzzy p-metric d', or its fuzzy absolute value function p.
x, y e X .
(iii) The canonical 1-1 correspondence S -iv T from the class of all left invariant fuzzy p-metrics, on (X, .), is given by
S -iv T(d)(x, y) = d(x -1, y-l),
x, y • X.
The inverse transformation T - l o S is given on a right invariant fuzzy p-metric d' by T-l°S(d')(x,y)=d'(x-l,y-1),
x,y•X.
the requirement (1.1) of Theorem 1.19 is fulfilled for the function
g :(X, t(d))--* (X, t(d')), by always taking 6 = e . Therefore, by that theorem, g is continuous there. Similarly, g is continuous in the opposite direction. Therefore, it is a homeomorphism. Since g og = the identity function on X, the second assertion follows.
Proposition 5.6. Let d be a left invariant fuzzy p-metric on a group (X, .), and let d' be its right invariant fuzzy p-metric. Then ( X , . , t(d)) is a fuzzy neighbourhood group if and only if (X, ., t[d')) is a fuzzy neighbourhood group. In this case, those two fuzzy neighbourhood groups coincide.
T.M.G. Ahsanullah, N.N. Morsi / Metrizability of fuzzy neighbourhood groups Proof. If ( X , . , t(d)) is a fuzzy neighbourhood group then g is continuous: (X, t(d))--> (X, t(d))
which implies ( X , t ( d ) ) = ( X , t ( d ' ) ) (by the above proposition). Similar implications evidently hold with d and d' interchanged. In the sequel, it will make sense to deal with a fuzzy neighbourhood group ( X , . , t(p)), where p is a fuzzy absolute value function on (X, .). The new notation t(p) will stand for either t(d) or t(d'), where d and d' are the left and right invariant fuzzy p-metrics associated with p, respectively. By the above proposition, no ambiguity will ensue, as long as one end result is a fuzzy neighbourhood group. Proposition 5.7. Let d be a left,
or right, invariant fuzzy p-metric on a group (X, .), and let Zd=(,Y,d(X))x~X be its associated fuzzy neighbourhood system on X. Then "~d satisfies properties (a), (b) and (c) of Theorem 1.3. ProoL Let p : X - - - ~ * ( 1 ) , x ~ d ( x , e ) be the associated fuzzy absolute value function. We now prove that the fuzzy neighbourhood basis ~d of Zd, given by ~a(X)={B(x;r):r>O},
x•X,
satisfies the required three properties. Upon doing this, we would have established the same properties for Za; by applying the saturation operator (a) For all b • X and r > O , we get for all X •X, B(b;r)(x) = Lr[d(x, b)] = Lr[p(b-lx)] = B(e; r)(b-lx) = [b" B(e; r)](x). This demonstrates B(b; r) = ~b(B(e; r)). Hence ~d(b) = {~b(V): v • ~d(e)}, which is (a). (b) For all r > 0, we find for all x • X, B(e ; r)(x ) = Lr[p(x )] = Lr[p(x-~)]
241
This demonstrates g(B(e;r)) = B(e; r), which renders (b). (c) For all r, s > 0, we find for all x • X, [B(e; r). B(e; s)](x) = sup{B(e; r)(y) ^ B(e; s)(z):yz = x} = sup{Lr[p(y)] ^ Ls[p(z)] :yz = x} <~sup{ Lr+s[p(y ) ~p(z)] :y z = x } (by [30, Theorem 1.1(x)]) <~sup{Lr+~[p(yz)] :yz = x} (by the subadditivity of p) = Lr+s[p(x)] = B ( e ; r + s ) ( x ) .
This demonstrates B(e; r) • B(e; s) <~B(e; r + s), which renders (c). In the next theorem the following question is settled: Which left (or right) invariant fuzzy p-metrics on a group induce fuzzy neighbourhood systems compatible with the group structure? Theorem 5.8. Let d be a left, or right, invariant fuzzy p-metric on a group (X, .). Then the following are equivalent statements: (i) ( X , . , t(d)) is a fuzzy neighbourhood group. (ii) Every inner automorphism Intb, b e X, /s continuous at e in the fuzzy neighbourhood space (X, t(d)). (iii) For all b • X and e, 0 • Io, there exists 6 = 6b~O in Io such that for all x • X, p(x)(6) + 0 >~p(bxb-1)(e). Proof. (i)¢:>(ii): Statement (ii) coincides with property (d) of Theorem 1.3. Consequently, by that theorem and the preceding proposition, ( X , . , t(d)) is a fuzzy neighbourhood group iff it satisfies (ii). (ii) <=>(iii): Just apply Theorem 1.19.
Corollary 5.9. f f d is a bi-invariant fuzzy p-metric on a group (X, .), then (X, ., t(d)) is a fuzzy neighbourhood group. Proof. In this case,
(by the symmetry of p)
= B(e; r)(x-') = g(B(e; r))(x).
p(bxb -1) -- p(b-~bx) = p(x) (by Theorem 5.4). Thus, (iii) of the above
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theorem holds for (X,., t(d))7 by always taking 6=e. Corollary 5.10. If d is a left, or right, invariant fuzzy p-metric on an Abelian group (X, .), then (X, ., t(d)) is a fuzzy neighbourhood group. Proof. Conjugate Theorem 5.4 with Corollary 5.9. For the notions of the left fuzzy uniformity q/L, and the right fuzzy uniformity q/R, on a fuzzy neighbourhood group, we refer the reader to [1, Section 3]. Availing ourselves of the notations of [1], we get for invariant fuzzy p-metrics: Theorem 5.11. Let d be a left (resp. right) invariant fuzzy p-metric on a group (X, .). If (X, ., t(d)) is a fuzzy neighbourhood group, then d induces the left fuzzy uniformity q/L (resp. the right fuzzy uniformity q/R) of (X, ", t(d)).
Proof. Suppose d is left invariant. The other case has a very similar proof. The problem is to show that q/(d)
=
q/L.
The fuzzy uniformity qZ(d) has the fuzzy uniform basis {~Pr: r > 0} of Theorem 1.13. So, the fuzzy neighbourhood system 2'd of (X, t(d)) has basis at e [20] ~d(e) = {~Pr(e): r > 0}.
Now, for all x , y e X, invariance of d,
we get by the left
~Pr(X, y) = Lr[d(x, y)] = Lr[d(y-lx, e)] = lPr(e, y - i x ) = % ( e ) ( y - l x ) = (~Pr(e))L(X, Y).
Consequently, 6~d ~'-~{lpr : r > 0 ) -
~-- {(/pr(e))L: ~pr(e) E ~a(e))----q/L (by its definition in [1]). A fuzzy neighbourhood group is said to have bi-invariant fuzzy uniform structure if its left and right fuzzy uniformities coincide [2].
Corollary 5.12. Let d be a bi-invariant fuzzy p-metric on a group (X, .). Then the fuzzy neighbourhood group (X, ., t( d ) ) has b iinvariant fuzzy uniform structure.
Proof. By Theorem 5.11, q/L = q/(d) = q/R. We end this section by necessary and sufficient conditions for the invariant fuzzy p-metrizability of fuzzy neighbourhood groups. Theorem 5.13. Let ( X , . , t(,~)) be a fuzzy neighbourhood group. Then the following are equivalent statements: (a) t(~') is induced by a left invariant fuzzy p -metric. (b) t(,~) is induced by a right invariant fuzzy p-metric. (c) The left fuzzy uniformity q/L is fuzzy p -metrizable. (d) The right fuzzy uniformity q/R is fuzzy p -metrizable. (e) t(Z) is induced by a fuzzy p-metric. (f) The fuzzy neighbourhood system at e, Z(e), has a countable basis. Proof. (a) <=>(b): This is Proposition 5.6. (a) ~ (c) and (b) ~ (d): This follows from Theorem 5.11. (c) or ( d ) ~ (e): This follows from
t(q/L)
=
t(2~)
=
t(q/R).
(e)=>(f); When (e) holds, the fuzzy open balls with positive rational radii, centered, at e, form a countable basis for Z(e); by Theorem 1.16. ( f ) ~ ( c ) and (d): Suppose Z(e) has a countable basis. Then, by their definitions in [1], q/L and q/R will also have countable bases. Consequently, they are fuzzy p-metrizable by Theorem 1.15. (The 1-Hausdorff-separatedness requirement in [11] is easily seen to be needed only if q/L and q/R are to be fuzzy metrizable.) ( c ) ~ ( a ) : Suppose q/L is fuzzy p-metrizable; that is q/L = q/(d) for some fuzzy p-metric d on X. Then also
%
=
q/(d,),
where d l = d A i is the fuzzy p-metric on X obtained by truncating d at i.
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which proves that t(X) is induced by the left invariant fuzzy p-metric d.
Define a function d:X2---> R*(I) by
d(x, y) = V dl(bx, by), beX
(x, y) • X 2.
By direct verification, d is a left invariant fuzzy p-metric on iX'. We show that °/re = °-/t(d): Let °//'(dl)={l~Vr:r>0 }
and
°t/'(d)={lpr:r>O}
be the defining fuzzy uniform bases of q/(dl) = °//L and °//(d), respectively, given as in Theorem 1.13. Then we get for all r > 0 and (x, y) • X 2
~O,(x, y) = 1 - d(x, y)(r) = A [ 1 - d~(bx, by)(r)]; beX
that is ~P,(x, Y) = A ~XOr(bx, by). b~X
(5.1)
From which we get lp, ~< l~pr for all r > 0. This gives a//(d) _~ ~ ( d l ) = 0~L.
(5.2)
On the other hand, a//L has a fuzzy uniform basis consisting of left invariant fuzzy vicinities [1]. Consequently, given any r, 0 > 0, there is a left invariant A e °~L such that
l~O,>~A - O , whence (5.1) gives for all (x, y) • X 2, b~X
>I A [A(bx, b y ) - O] beX
= A(x, y) - O,
q/(d) _ 0"//L. From (5.2) and (5.3),
Theorem 6.1. A quotient group of a fuzzy neighbourhood group is a fuzzy neighbourhood group under its quotient fuzzy topology. Proof. By [3, Theorem 4.8], a quotient group of a fuzzy topological group is a fuzzy topological group under its quotient fuzzy topology, while a quotient fts of a fns is also a fns ([29, Corollary 2.1]; a description of a quotient fuzzy neighbourhood system is given in [5]). This proves the theorem. By the above theorem, we are allowed to speak of a quotient fuzzy neighbourhood group of some fuzzy neighbourhood group.
Theorem 6.2. Let ( X , . , t(~,)) be a fuzzy neighbourhood group, and let (Y, ., t(X')) be its quotient fuzzy neighbourhood group under a group epimorphism f : (X, . ) ---~( Y, .). Then (i) f is continuous and open: (X, t(X))--* ( r , t(X')). (ii) 2:'(e') = {v e l V : f - l ( v ) e 2:(e)}, where e and e' are the neutral elements of (X, .) and ( Y, .), respectively. (iii) If (X, t(Z)) is N-locally compact, then so is (Y, t(X')). Proof. (i) This is proved in [3, Proposition 4.2] for fuzzy topological groups in general. (ii) Put v = {v e I V : f - l ( v ) e X(e)}. Then by the continuity o f f , X'(e') ~_ v. By the openness, and hence N-openness, of f, v G X'(e'). Hence, equality holds. (iii) The assertion follows from (i) and Proposition 3.7.
~Pr(X, y) = A ~r(bx, by)
which renders ~Vr~>A- 0. This proves opposite inclusion
6. Quotient fuzzy neighbourhood groups
the (5.3)
Theorem 6.3. Let p : X - * ~ * ( I ) be a fuzzy absolute value function on a group (X, .), and let f:(X,.)--~(Y,.) be a group epimorphism. Define a function pf : Y---~ •*(I) by
0U(d) = % .
pf(y) =inf{p(x):x e X , f(x) = y } ,
Finally,
Then pf is a fuzzy absolute value function on the group (Y, .). Moreover, if (X, ., t(p)) is a fuzzy
t( d) = t( ~ ( d) ) = t( °UO = t( Z),
y e Y.
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T.M.G. Ahsanullah, N.N. Morsi / Metrizability of fuzzy neighbourhood groups
neighbourhood group, then ( Y , . , t(p:)) is its quotient fuzzy neighbourhood group under f. Proof. We have: (AV1) pf(e') = 0, because p(e) = O. (AV2) for all y • Y,
f(Bd(x; r))(f(z)) = sup {Bd(x; r)(Zl):f(z,) = f ( z ) }
= sup{Lr[d(x, Zl)]:f(z,) = f(z)} f(e) = e'
and
PI(Y-I) = inf{p(x): f ( x ) = y - l } = inf{p(x-l): f ( x ) = y} = inf{p(x): f ( x ) = y} = PI(Y),
= sup{Lr[d(x,, z,)]: f(xl) = f ( x ) , f(zl) = f ( z ) } (by a simple manipulation of the right invariance of d and the fact that f is a group homomorphism)
= Lr[inf{d(Xl, Zl)."f ( x | ) = f ( x ) , f(zl) = f ( z ) } ] (by [30, Theorem 1.1(viii)]) =
Lr[d'(f(x), f(z))]
because p(x -1) = p ( x ) for all x • X. (AV3) Since f is an epimorphism, then for all x • X, y, w • Y, f ( x ) = yw iff x = XlX2 for some xl, x 2 • X with f(xl) = y and f(x2) = w. Consequently,
This proves that f(Bd(X;r)) = Bd,(f(x);r), which demonstrates that f is N-open. Hence, it is open. On the other hand,
p/( yw ) = inf{p(x): f (x ) = yw }
( f x f)-l(~p~')(x, z) = ~)(f(x), f ( z ) )
=Bd,(f(x);r)(f(z))
(using (6.2)).
= L,[d'(f(x), f(z))] >! Lr[d(x, z)] (by (6.2))
= inf{p(xlx2): f(xl) = y, f(x2) = w}. (6.1) Therefore,
: ~r(X, Z).
p1(yw) <- inf{p(xt) ~)p(x2): f (xl) = y, f(x2) = w} (by the subadditivity of p) = inf{p(x0: f ( x O = y} inf{p(x2): f(x2) = w} (by twice applying Proposition 1.10)
This proves that ( f xf)-l(~p))/> lp~, which demonstrates that f is uniformly continuous: (X, q/(d))--, (Y, °//(d')). The above, together with Theorem 6.1, complete the proof of the second assertion of the theorem.
=Pr(Y) ~Pr(W). This completes the proof that Pr is a fuzzy absolute value function on (¥, .). Let d:X2---~R*(I) and d':y2--->~*(1) be the right invariant fuzzy p-metrics on X and ¥, respectively, defined by
d(x,z)=p(xz-'),
( x , z ) • X 2,
and d ' ( y , w) = p f ( y w - 1 ) ,
(y, w) • y2.
Then we get by (6.1) for all (y, w) • 1:2,
d'(y, w) =p1(yw -1) = inf{p(xz-l): f ( x ) = y, f ( z ) = w}. So,
d'(y, w) = inf{d(x, z): f ( x ) =y, f ( z ) = w}. (6.2) Now, for all x, z • X and r > 0,
References [1] T.M.G. Ahsanullah, On fuzzy neighbourhood groups, J. Math. Anal. Appl. 130 (1988) 237-251. [2] T.M.G. Ahsanullah, On the small invariant fuzzy neighbourhood groups, Bull. Calcutta Math. Soc. (to appear). [3] T.M.G. Ahsanullah, Quotient fuzzy topological groups, Ganit (J. Bang. Math. Soc.) 5 (1985) 37-42. [4] T.M.G. Ahsanullah, Some properties of the level spaces of fuzzy neighbourhood spaces and fuzzy neighbourhood groups, J. Bang. Acad. Sci. 13 (1989) 91-95. [5] A.A. Allam, K.M. Abd EI-Hakeim and N.N. Morsi, On fuzzy neighbourhood spaces, Fuzzy Sets and Systems 41 (1991) 201-212. [6] M.A. Amer and N.N. Morsi, Characterizations of some fuzzy topological notions in probabilistic metric spaces, Fuzzy Sets and Systems (submitted). [7] G. Artico and R. Moresco, Fuzzy proximities compatible with Lowen fuzzy uniformities, Fuzzy Sets and Systems 21 (1987) 85-98.
T.M.G. Ahsanullah, N.N. Morsi / Metrizability of.fuzzy neighbourhood groups [8] N. Bourbaki, General Topology, Vol. II (AddisonWesley, Reading, MA, 1966). [9] T.E. Gantner, R.C. Steinlage and R.H. Warren, Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62 (1978) 547-562. [10] W. Geping, Fuzzy addition in the Lowen fuzzy real line, Fuzzy Sets and Systems 27 (1988) 303-315. [11] U. H6hle, Probabilistic metrization of fuzzy uniformities, Fuzzy Sets and Systems 8 (1982) 63-69. [12] A.K. Katsaras, Linear fuzzy neighbourhood spaces, Fuzzy Sets and Systems 16 (1985) 25-40. [13] A.K. Katsaras, Linear fuzzy neighbourhood spaces II, J. Math. Anal. Appl. 115 (1986) 560-573. [14] V,L. Klee, Invariant metrics in groups (solution of a problem of Banach), Proc. Amer. Math. Soc. 3 (1952) 484-487. [15] A.J. Klein, Fuzzy topologies with specified level topologies, Fuzzy Sets and Systems 21 (1987) 233-241. [16] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976) 621-633. [17] R. Lowen, Initial and final fuzzy topologies and the fuzzy Tychonoff theorem, J. Math. Anal. Appl. 58 (1977) 11-21. [18] R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446-454. [19] R. Lowen, Fuzzy uniform spaces, J. Math. Anal. Appl. 82 (1981) 370-385. [20] R. Lowen, Fuzzy neighborhood spaces, Fuzzy Sets and Systems 7 (1982) 165-189. [21] R. Lowen, Compactness notions in fuzzy neighborhood spaces, Manuscripta Math. 38 (1982) 265-287. [22] R. Lowen, On (R(L), ~9), Fuzzy Sets and Systems 10 (1983) 203-209. [23] R. Lowen, Compact Hausdorff fuzzy topological spaces are topological, Topology Appl. 12 (1981) 65-74. [24] R. Lowen, On the Existence of Natural Non-topological Fuzzy Topological Spaces (Heldermann Verlag, Berlin, 1985). [25] R. Lowen and P. Wuyts, Concerning the constants in fuzzy topology, J. Math. Anal. Appl. 129 (1988) 256-268.
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[26] A.S. Mashhour and N.N. Morsi, Fuzzy metric neighbourhood spaces, Fuzzy Sets and Systems 45 (1992) 367-388. [27] A.S. Mashhour and N.N. Morsi, On regularity axioms in fuzzy neighbourhood spaces, Fuzzy Sets and Systems 44 (1991) 265-271. [28] A.S. Mashhour, N.N. Morsi and O.A. EI-Tantawy, Fuzzy neighbourhood syntopogenous structures, Fuzzy Sets and Systems 43 (1991) 219-234. [29] N.N. Morsi, A short note on fuzzy neighbourhood spaces, Fuzzy Sets and Systems 23 (1987) 393-397. [30] N.N. Morsi, On fuzzy pseudo-normed vector spaces, Fuzzy Sets and Systems 27 (1988) 351-372. [31] N.N. Morsi, Dual fuzzy neighbourhood spaces I, Fuzzy Sets and Systems 44 (1991) 245-263. [32] N.N. Morsi, Nearness concepts in fuzzy neighbourhood spaces and in their fuzzy proximity spaces, Fuzzy Sets and Systems 31 (1989) 83-109. [33] N.N. Morsi, The Urysohn lemma for fuzzy neighbourhood spaces, Fuzzy Sets and Systems 39 (1991) 347-360. [34] N.N. Morsi and S.E. Yehia, Continuity of fuzzy multiplication in the N-Euclidean space, Fuzzy Sets and Systems 46 (1992) 97-106. [35] S.E. Rodabaugh, Fuzzy addition in the L-fuzzy real line, Fuzzy Sets and Systems 8 (1982) 39-52. [36] B. Schweizer and A. Sklar, Probabilistic Metric Spaces (North-Holland, Amsterdam, 1983). [37] A. Wilansky, Topology for Analysis (Ginn, Wattham, MA, 1970). [38] P. Wuyts, On the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level topologies, Fuzzy Sets and Systems 12 (1984) 71-85. [39] P. Wuyts and R. Lowen, On separation axioms in fuzzy topological spaces, fuzzy neighborhood spaces, and fuzzy uniform spaces, J. Math. Anal. Appl. 93 (1983) 27-41. [40] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [41] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning I, Inform. Sci. 8 (1975) 199-249.
233
Invariant probabilistic metrizability of fuzzy neighbourhood groups T.M.G. Ahsanullah* Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh-11451, Saudi Arabia
N e h a d N. M o r s i Department of Mathematics, Military Technical College, Kobry EI-Qubba, Cairo, Egypt Received March 1990 Revised November 1990
Abstract: We introduce the notions of N-local compactness and N-openness in fuzzy neighbourhood spaces and in fuzzy neighbourhood groups, together with some characterizations for them. In particular, we prove that a fuzzy neighbourhood space is N-locally compact if and only if its or-level spaces, for all 0 < ot < 1, are locally compact. A similar criterion is established for N-open functions. We also introduce the notion of fuzzy absolute value functions on groups. We show that there exists a 1-1 correspondence between invariant fuzzy (probabilistic) pseudo-metrics on groups and fuzzy absolute value functions. We establish theorems on fuzzy (probabilistic) pseudo-metrizability of fuzzy neighbourhood groups. Finally, fuzzy (probabilistic) metrizability of quotient fuzzy neighbourhood groups is also taken into account.
AMS Subject Classification: 54A40. Keywords: Fuzzy neighbourhood space; fuzzy uniform space; fuzzy neighbourhood group, N-local compactness; N-open function; fuzzy real numbers; fuzzy (probabilistic) metric; fuzzy absolute value function; fuzzy metric neighbourhood space; o~-Ievel space, WNT2-space.
Introduction
The compatibility of Lowen's well known fuzzy neighbourhood systems with the group structure has been studied in [1]. It was also shown there that a fuzzy neighbourhood group (fng) is fuzzy uniformizable in the sense of Lowen [19]. However, other open questions remain unsolved: First, how to provide some * Permanent Address: Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh.
kind of fuzzy metrizability of fng's which would be a good extension of its classical counterpart? Second, does there exist an accompanying notion of local compactness for fuzzy neighbourhood spaces (fns's) which would also be a good extension? In 1982, H6hle [11] established a famous metrization theorem, which states: "A fuzzy T-uniformity is probabilistic metrizable if and only if it is Hausdorff separated and has a countable basis of fuzzy vicinities". His Tuniformity coincides with the Lowen fuzzy uniformity when the triangular norm T is Min, which is the one adopted throughout this paper. A probabilistic metric induces a fuzzy uniformity. This in turn induces a fuzzy neighbourhood space, said in [26] to be a fuzzy metric neighbourhood space. As probabilistic metrics have managed to occupy a prominent place in the theory of fuzzy neighbourhood spaces [6, 11, 13, 26, 33, 34], it now seems reasonable to also term them fuzzy metrics, especially whenever their fuzzy topological aspects are being considered. Taking fuzzy metric neighbourhood spaces as our starting point, we provide answers to the two questions listed above: In Section 2, we study open functions between fns's. We show that openness of functions is equivalent to a new notion of N-openness (neighbourhood openness) of functions, and that a function between fns's is open iff it is open in the entering re-level topologies for all 0 < tr < 1. Section 3 is devoted to studying a notion of N-local compactness in fuzzy neighbourhood spaces. We prove that this notion enjoys a level-topologies criterion identical to the one above. It is therefore a good extension. In Section 4, we present a characterization theorem for N-local compactness of fuzzy neighbourhood groups. Invariant fuzzy metrizability of fng's is considered in Section 5, with the help of a notion of fuzzy absolute value functions. Employing a canonical transformation
0165-0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved
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T.M.G. Ahsanullah, N.N. Morsi / Metrizability of fuzzy neighbourhood groups
between those two notions, several characterizations for the fuzzy metrization of fng's are provided. Finally, in Section 6, we investigate the fuzzy metrizability of quotient fng's.
for all v e / x and x • X. The pair (X, t(2")) is called a fuzzy neighbourhood space (fns), and X is referred to as its fuzzy neighbourhood system. We denote by FNS the category of fns's and their continuous maps.
1. Preliminaries
Definition 1.2 [1]. Let (X, -) be a fuzzy neighbourhood system on triple (X, .,t(2")) is said to neighbourhood group (fng) if conditions are fulfilled: (FNG1) the mapping
Let X be a set, I0=10, 11, I1=[0, 1[, Io,1=]0,1[
and
I=[0,1].
A fuzzy subset of X is an element of I x (={#:X--->I}). We denote by a • I x the constant fuzzy subset of X having value tr • L The power set 2x. of X is considered as a subfamily of IX; by identifying the subsets of X with their characteristic functions. The height of a fuzzy set # e I x is the real number hgt(#) = sup #(x). xEX
We follow the definition of fuzzy topological spaces (fts's) due to Lowen [16]. (See also [25].) We say that (X, -) is a fts to indicate that - is the fuzzy closure operator of that fts. We refer to [19] and [20] for fuzzy uniformities and fuzzy neighbourhood systems, respectively. However, we recall here: Definition 1.1 [20]. A family 2' = (2"(X))xEx of prefilters in a set X is said to be a f u z z y neighbourhood system on X if the following
conditions are satisfied: (N1) for all x e X and for all # • 2 " ( x ) , #(x) = 1; (N2) for all x e X and for every family (#o)o~o of elements of 2"(x), sup (Uo - O) e X(x); 0~!o
(N3) for all x • X, for all # ~ X(x) and for all 0 ~ Io, there exists a family (#~zEx ~ [L~x2"(z) such that for all y, z • X, #O(z) ^ #O(y) _ 0 ~<#(y). The elements of 2"(x) are called the fuzzy neighbourhoods of x. A fuzzy topology t(X) on X is given by the fuzzy closure operator -: v - ( x ) = inf hgt(v ^ #), u~(x)
group and ~7 a X. Then the be a f u z z y the following
m : ( X x X, t(X) x t(2"))--~ (X, t(2")): (x, y) ~--~xy
is continuous, (FNG2) the mapping g : ( X , t(2"))--~ (X, t(2")):
x t'-"-)x - 1
is continuous. Theorem 1.3 [1]. Let (X, .) be a group and X a fuzzy neighbourhood system on X. Then ( X , . , t(X)) is a fuzzy neighbourhood group if and only if the following conditions are satisfied: (a) for all b ~ X,
X(b) = {~b(#): # ~ X(e)} (alternatively, X(b) = {9~b(#): # e Z(e)}), where ~b, ~tb :X--'~X are the left and right translations on X by the element b, respectively; (b) for all ~ • X(e) and for all 0 > O, there exists v e X ( e ) such that v - 0 ~ g ( # ) ; i.e. the inversion map g : x ~ x -t is continuous at the neutral element e; (c) for all # ~ X(e) and for all 0 > O, there exists v ~ X(e) such that v . v - 0 <- #; i.e. the multiplication m :(x, y) ~ x y is continuous at (e, e); (d) for all # ~ X(e), for all 0 > 0 and for all x ~ X, there exists v ~ X(e) such that x • v . x -1 0 <~#; i.e. the inner automorphism Intx : z xzx -~ is continuous at e. (Notice that, according to our notation, {x} and lx are the same thing.) Definition 1.4 [40]. Let X be a set, # e I x and oce L The oc-cut (respectively c~*-cut) of # is the
crisp subset of X u ~ = {x ~ x : u ( x ) > a }
T.M.G. AhsanuUah, N.N. Morsi / Metrizability of.fuzzy neighbourhood groups
(respectively
then
U~. = {x e X: U(x) >I a}).
v ~ inf #j = inf (v ~/uj).
Definition 1.5 [18]. Let (X, r) be a fts, and let oce I1. The m-level topology of (X, t) is the ordinary topology on X ~ ( r ) = {U~: U ~ r}.
The topological space (X, t~(t)) is called the re-level space of the fts (X, t). Theorem 1.6 [4, 15, 31]. Let - be the fuzzy closure operator of a fuzzy neighbourhood space (X, t(Z)), and let (M, o:) e 2x x I 1. Then (M-)(1-,~). = the to,(t(X))-closure of M. 1.7 [31]. Let o be the fuzzy interior operator of a fuzzy neighbourhood space (X, t(Z)), and let tz e I x. Then
Theorem
# ° = sup [a ^ (# ~)o]. a~¢h
We denote by R* the set of nonnegative real numbers. Definition 1.8 [9, 11, 22, 35, 36]. A nonnegative fuzzy real number ~7 is a descending, left continuous real function: R*-+I, with 7/(0)= 1 and with infimum 0. The set of all nonnegative fuzzy real numbers is denoted by ~*(I). Nonnegative fuzzy real numbers are ordered by their natural ordering as real functions on R * Under this order, arbitrary subsets of R*(I) have their infima in R*(1). Hence, subsets of R*(I) which are bounded above also have their suprema in R*(I). The bottom element 0 of R*(1) is given by 1 ifs =0, 0(s)=
0
j~J
235
jeJ
Definition 1.11 [11,36] (see also [26,30]). A fuzzy (probabilistic) pseudo-metric (or fuzzy p-metric, in short) on a set X is a function d :X2---~ R*(1) which fulfills the following conditions: Vx, y, z e X; (FM1) d(x, x) = 0; (FM2) d(x, y) = d(y, x) (symmetry); (FM3) d (x, y) (~ d ( y , z) >>-d (x , z) (triangle inequality). If, in addition, d satisfies (FM4) x 4 : y ~ d(x, y)>O, then d is called a fuzzy (probabilistic) metric. The pair (X, d) is called a fuzzy pseudo-metric space. Definition 1.12 [30]. If space, then the fuzzy centre x e X and radius of X given for all y • X
(X, d) is a fuzzy p-metric open ball B(x;r), with r > O, is the fuzzy subset by
B(x; r)(y) = LrId(x, Y)I ( = 1 -
d(x, y)(r)).
Theorem 1.13 [11] (see also [26]). Let (X, d) be
a fuzzy p-metric space. Then the family °F(d) = {~p, e IX~: r > O}
is a fuzzy uniform basis in X, in the sense of Lowen [19], where W,(x,y)=Lr[d(x,y)],
r > 0 , x, y e X .
The fuzzy uniformity a//(d)= ~ ( d ) is called the fuzzy metric uniformity induced by d. Definition 1.14 [11]. A fuzzy uniformity ~¢"on a set X is said to be fuzzy p-metrizable if there exists a fuzzy p-metric d on X such that ~¢"= ~ ( d ) .
if s > 0 . Theorem 1.15 [11]. A fuzzy uniformity is fuzzy
Definition 1.9 [11, 22, 35, 36]. Addition ~ of fuzzy real numbers is defined as follows: For all r/, ~ e ~*(I), 7/~) ~ e •*(I) is given by ( r / ~ ~)(s) = sup{r/(b) ^ ~(s - b):
b e[O, s]}, s~>O. Proposition 1.10 [26]. / f v, /~j e R*(1) for ] e J,
p-metrizable if and only if it has a countable basis. In a fuzzy p-metric space (X, d), we follow [26] in denoting the fuzzy neighbourhood topology t(°ll(d)) (induced on X in accordance to [19, 20]) simply by t(d). The fns (X, t(d)) is said to be a fuzzy p-metric neighbourhood space [26].
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Theorem 1.16 [26]. A fuzzy p-metric neighbourhood space (X, t(d)) has a fuzzy neighbourhood basis ~a = (~a(x) = {B(x; r): r > O})x~X. Definition 1.17 [39]. A fns (X, t(X)) is said to be WNT2 if for all x C y in X, there exists (#, v) • Z(x) x Z(y) such that hgt(# ^ v) < 1.
Theorem 1.18 [11] (see also [26]). A fuzzy p-metric neighbourhood space (X, t(d)) is WNT2 if and only if d is a fuzzy metric. The following e06-characterization of continuity of functions between fuzzy p-metric neighbourhood spaces is an immediate consequence of Theorem 1.16 and the relevant definitions:
Theorem 2.2. A function f : (X, t(X))--> (r, t(,Y,')), between fuzzy neighbourhood spaces, is: (i) open if and only if for all oc • I0,1, / i s open: (X, t,(t(X)))--~ (Y, t,~(t(Z"))); (ii) continuous if and only if for all o: • Io.1, f is continuous: (X, t~(t(X)))--, (Y, t~,(t(,~'))). Proof. (i) Suppose that f satisfies the stated condition, and let # • t(X). Then # ~ • t~(t(,~)) for all o~• Io,1, and hence by Proposition 2.1, [f(#)]~ = f (#~') • t~(t(X')). Also,
Theorem 1.19 [6]. A function
[f(#)]o= I,_J [f(#)]'~
f :(x, t(d))~ (Y, t(d')),
O~E/N 1
between fuzzy p-metric neighbourhood spaces, is continuous at a point x • X if and only if for all e > 0 and 0 • 1o there exists 6 = 6~o > 0 such that d(x, z)(6) + 0 >I d'(f(x), f(z))(e)
Recall that a function, between fuzzy topological spaces, is said to be open if it sends open fuzzy subsets onto open fuzzy subsets. For fuzzy neighbourhood spaces, we get:
(1.1)
for all z • X.
Proposition 1.20 [31]. A descending chain {~,~: a:ell} of topologies on a set X is the indexed family of level topologies of some fuzzy topology r on X iff for all ol • 11, Z~ = s u p { Z / y • ]a~, 1[}. In this case, the finest such v exists, and it is the only fuzzy topology on X making (X, v) a fns with the given level topologies.
• sup{t~(t(,~')): te • loA} = t0(t(~7')), where the last equality holds by Proposition 1.20. Now, from the maximality property of the fuzzy topology of a fns with respect to its level topologies [38], it follows that f ( g ) ¢ t(X'). This proves that f is open. The inverse implication follows directly from Proposition 2.1. (ii) The proof is very similar to the above one. From the above characterization of continuity in FNS, it is easy to establish that a fuzzy neighbourhood system on a group is compatible with the group structure if and only if the corresponding e-level spaces are topological groups for all 0 < a~< 1.
2. Open functions in FNS Definition 2.3. A function We shall repeatedly use the following easily established result.
f : (X, t(X))--~ (Y, t(X')), between fns's, is said to be N-open at x • X if
Proposition 2.1. Let f :X--->Y be a function, # • I x and tr • 11. Then: (i) f ( # ~ ) = [f(#)1% (ii) f ( # ^ a) = f ( # ) ^ a.
f ( # ) • X'(f(x))
for all # ¢ X(x).
It is said to be N-open if it is N-open at all points of X.
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237
In the above definition, one can easily see that X can be replaced by any of its fuzzy neighbourhood bases.
Applying Proposition 2.4 again, we find that for all a e ll,
Proposition 2.4. Let (X, t(X)) be a f u z z y neigh-
But this implies, by Proposition 2.5, that f ( # ) e Y.'(f(x)), which proves that f is N-open. (ii) ~ (iii): Suppose f is N-open. For M e 2 x and x • X, put y = M°(x). Then from Proposition 2.4(ii),
bourhood space, and (x, M) • X x 2 x. Then for all tr~ 11: (i) (M °) ~ = intL,o(z)) (M), where o is the f u z z y interior operator of (X, t(X)), (ii) M v a • 2~(x) if and only if M°(x) >>-1 - ol.
[f(#)] ~ v a e Z ' ( f ( x ) ) .
M v (1 - y) e Z(x). By the N-openness of f,
Proof. (i) This follows from Theorem 1.6 by taking complements. See also [31, Proposition 3.2]. (ii) We have
f ( M ) v (1 -- y) • ,~'(f(x)).
M v a e X ( x ) iff ( X - M ) - ( x ) <~ o¢
[f(M)]°(f(x)) >t r = M°(x) •
(by [28, Proposition 4.1])
iff M°(x) >i 1
-
ol,
because M°(x) = 1 - ( X - M ) - ( x ) .
Proposition 2.5. Let ( X , t ( X ) ) be a f u z z y neighbourhood space, x • X and # • I x. Then e X ( x ) if and only if for all ol•11, g ° ' v
So, from Proposition 2.4(ii) again,
This proves that [ f ( M ) ] ° ~ f ( M ° ) . Hence, (iii) holds. (iii) ~ (i): Suppose f satisfies (iii), and let # • I x. Then f(#°)=f(sup
\oral1
[a A (#'~)°])
(from Theorem 1.7)
a • Z(x).
= sup [a ^ f ( ( # 0,)o)]
Proof. This follows directly from [32, Proposi-
oc~/l
tion 2.9], because X(x) is a prefilter.
(applying Proposition 2. l(ii))
We are now in a position to prove:
Theorem 2.6. Let f :(X, t(Z,))--~ (Y, t(Z')) be a function between f u z z y neighbourhood spaces. Then the following are equivalent statements: (i) f is open. (ii) f is N-open. (iii) For all M • 2 x, f ( M °) <~[f(M)] °.
~
(by (iii)) = sup [a A ([f(#)]~)°] ~elt
(from Proposition 2.1(i)) = if0,)] o
(from Theorem 1.7 again). This proves that f is open.
Proof. (i) ~ (ii): Suppose f is open. Let x • X, #•X(x) and tr•11. Then by the above proposition, #~ v a • Z(x). So, by Proposition 2.4, (/~'~)°(x)/> 1 - ol. Consequently, we get by Proposition 2.1(i) and the openness of f,
3. N-locally compact fuzzy neighbourhood spaces
([f(#)l °3°(f(x)) = ( f ( # ~))°(f(x))
(sup v)(x) > a~ for all x • X.
>~f((~ ~)°)(f(x)) i> (# ~)°(x)/> 1 - ol.
An o~-shading [9], where ol•11, of a nonempty set X is a family v ~ I x which satisfies
A fuzzy topological space (X, r) is called re-compact [9] if every open tr-shading of X
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includes a finite a~-subshading of X. A crisp subset of (X, ~) is called o~-compact if it is e-compact in its relative fuzzy topology.
Proposition 3.1. (i) [18]. A f t s is ol-compact if and only if its ol-level space is compact. (ii) [21] A fns is compat (in the sense of Lowen [16]) iff its ol-level spaces are compact for all 0 < a~< 1. Theorem 3.2. Let (X, t(X)) be a fuzzy neighbourhood space, and let (x, M, o0 e X x 2x x IoA. Then M is an t~(t(X))-neighbourhood of x if and only if there exists y e ]0, 1 - a~[ such that M v ~, eX(x). Proof. M is an t~(t(X))-neighbourhood of x iff x e int,~(t(r)) (M) iff x • (M °) ~ (by Proposition 2.4(i)) iff there exists ~ e ]0, 1 - re[ such that M°(x)/> 1 - 7 iff there exists y e ]0, 1 - re[ such that m v It • X(x) (by Proposition 2.4(ii)). Definition 3.3. A fuzzy neighbourhood space (X, t(X)) is said to be N-locally compact at x ~ X if (*) for all o~e I0,1 there exists (M, y) e 2x x ]0, 1-o~[ such that M is o~-compact and M y It e X(x). It is called N-locally compact if it is N-locally compact at all points of X. The above definition is designed to fulfill the following criterion:
Theorem 3.4. A fuzzy neighbourhood space is N-locally compact at a point if and only if its ol-level spaces, for all 0 < o~< 1, are locally compact at that point. Proof. By Propositions 3.1, 3.2, the assertion (*) is equivalent to saying that M is a compact neighbourhood of x in the o~-level space (X, t~(t(X))). Hence, the theorem follows. Level-topologies criteria, identical to the above one, have been established in FNS for compactness (by Lowen, cf. Proposition 3.1(ii)), N-regularity, N-normality and the NT2 property [27], fuzzy metrizability [6], and for openness and continuity of functions (Theorem 2.2). This level-topologies criterion is more demanding
than Lowen's criterion for a fuzzy topological property to be a good extension [18]. This is because, given a topological space (X, ~,), all the level topologies of its generated fts (X, o9(~)) are equal to 3,. Therefore, we have: Corollary 3.5. N-local compactness is a good extension.
Proposition 3.6. A compact fuzzy neighbourhood space is N-locally compact. Proof. This follows directly from the definition of N-local compactness; by taking M = 1 in (*), and by noticing that 1 is compact iff it is e-compact for all tre I0,1 (Proposition 3.1(ii)).
Proposition 3.7. The image of an N-locally compact fuzzy neighbourhood space, under an open continuous function between fuzzy neighbourhood spaces, is N-locally compact. Proof. In [9], it is shown that the continuous image of an a~-compact fts is e-compact. The assertion follows from this and the equivalence between openness and N-openness of functions in FNS (Theorem 2.6). The product fts of a family of fns's is also a fns [12, 24]. We then have:
Theorem 3.8. The product fuzzy neighbourhood space of an indexed family ~f of fuzzy neighbourhood spaces is N-locally compact if and only if ~ satisfies: (A1) all the fns's in ~ are N-locally compact, and (A2) for every oc e Io,1, the number of indices corresponding to non-m-compact fns's in ~ is finite. Proof. Let ~,={(Xj, r~):jeJ} and ( X , r ) = II)~j(Xj, r)). Since the re-level topology of a product fuzzy topology is the product of the corresponding o~-level topologies (by [18, Theorem 2.1]), we get the following equivalences: (X, r) is N-locally compact iff for all t r e lo,1,
[I (xj, jeJ
=
(x,
T.M.G. Ahsanullah, N.N. Morsi / Metrizability of fuzzy neighbourhood groups
239
is locally compact (by Theorem 3.4) iff (A1) and (A2) hold for ~ (by a known theorem in General Topology, Proposition 3.1(i) and Theorem 3.4).
Proof. Let (X, -) be a fns, and denote its 0-level
Characterizations for the Wuyts-Lowen weak Hausdorff axiom WNT2 in FNS (see Definition 1.17) are established in the next two theorems.
C-(x) v (X - C)-(y) < 1
Theorem 3.9. For a f u z z y neighbourhood space ( X , t ( X ) ) , the following two statements are equivalent: (i) (X, t(X)) is WNT2. (ii) For all x :/: y in X, there exists C ~ 2 x such that C - ( x ) v ( X - C ) - ( y ) < 1.
topology by Z. We have the following equivalences: The fns (X, -) is WNT2 iff for all x :/:y in X, there exists C 6 2x such that (by the preceding theorem) iff for all x :/: y in X, there exists C e 2x such that x @(C-)1. = the A-closure of C, and y $ ((X - C)-)v = the Z-closure of X - C (by Theorem 1.6) iff the topological space (X, A) is T2.
4. N-locally compact fuzzy neighbourhood groups
Proof. (i) ~ (ii): Suppose (X, t(X)) is WNT2, and choose any x :~y in X. Then there exists (#, v, re) e X(x) x Z ( y ) x I1 such that /MA V ~ a ,
equivalently,
p,~A v'~ = 0. Take C = X - I z ~. Then X - C = l z ~ _ X - v From [32, Theorem 2.10], we get C-(x)
= (x
-
~.
<
and ( X - C ) - ( y ) <~ ( X - v ~ ' ) - ( y ) <~ tr < 1.
This proves (ii). (ii)ff (i): Suppose (ii) holds, and choose any x:/:y in X. Then there exists (C, a 0 ~ 2 x X ll such that C - ( x ) v ( X - C ) - ( y ) = o:.
From [28, Proposition 4.1], we get = (x - c) v a e X(x),
Vl = C v a ~ ~'(y) and ~l A
V1 =
I[Z.
This proves that (X, t(X)) is WNTE.
Theorem 3.10. A f u z z y neighbourhood space is WNTE if and only if its zero-level space is a Hausdorff topological space.
A fuzzy topological space (X, r) is said to be homogenous if for every (x, y) ~ X 2 there exists a homeomorphism of (X, r) onto itself which sends x onto y. Since all left (and right) translations in a fuzzy neighbourhood group are homeomorphisms of that fuzzy neighbourhood group onto itself [1, Proposition 2.6], we deduce that fuzzy neighbourhood groups are homogeneous fuzzy neighbourhood spaces.
Theorem 4.1. A homogeneous f u z z y neighbourhood space is N-locally compact if and only if it is N-locally compact at one point.
Proof. Every homeomorphism between fuzzy neighbourhood spaces is continuous. Hence, it takes c~-compact subsets onto o~-compact subsets. It is also open, and so it is N-open (Theorem 2.6). The above clearly entails that the validity of (*) at one point of a homogeneous fuzzy neighbourhood space is equivalent to its validity at all points of that space.
5. Fuzzy (probabilistic) metrizability of fuzzy neighbourhood groups Definition 5.1. A fuzzy p-metric d on a group (X, -) is said to be: left invariant if d(bx, b y ) = d ( x , y ) ,
b,x,y~X,
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right invariant if d(xb, y b ) = d ( x , y),
b,x, y • X ,
two-sided invariant (or, simply, bi-invariant) if it is both left and right invariant. Definition 5.2. Let (X, .) be a group. A function p : (X, .)--* R *(I) is called a fuzzy absolute value function (favf, in short) if the following holds for all x, y • X: (AV1) p(e)=l), where e is the neutral element of (X, .); (AVE) p(x -l) = p(x) (symmetry); (AV3) p(xy) <~p(x) ~ p(y) (subadditivity). The four notions defined above are strongly related to each other; as summarized in the next two theorems, the easy proofs of which are exact replicas of those of the corresponding classical results, see for instance [8, 14, 37]. So, we omit the proofs. Theorem 5.3. Let (X, .) be a group. (i) There is a well defined canonical 1-1 correspondence T between the class of all left invariant fuzzy p-metrics d on (X, .) to the class of all favf's on (X, .), given by T(d)(x) = d(x, e),
Theorem 5.4. Let d be a left invariant, or right invariant, fuzzy p-metric on a group (X, .), and let p be its fuzzy absolute value function. Then the following statements are equivalent: (i) d is bi-invariant. (ii) d(x, y) -- d(x -1, y-l), x, y • X. (iii) p ( x y ) = p ( y x ) , x, y • X . Consequently, a fuzzy p-metric on an Abelian group is left invariant iff it is right invariant iff it is bi-invariant.
Proposition 5.5. Let d be a left invariant fuzzy p-metric on a group (X, .), and let d' be its right invariant fuzzy p-metric. Then the inversion mapping g of (X, .) is a fuzzy homeomorphism g: (X, t(d))--~ (X, t(d')). Also, the two fns's (X, t(d)) and (X, t(d')) will coincide if and only if g is continuous as a function from one of them onto itself.
x • X.
The inverse transformation T -1 is given on a f a v f p on (X, .) by T-l(p)(x, y ) = p ( y - l x ) ,
x • X.
The inverse transformation S -1 is given on a f a v f p on (X,.) by S-l(p)(x, y)=p(xy-1),
Proof. Since, for all x, y • X,
d'(g(x), g(y)) -- d'(x -1, y-l) = d(x, y),
x, y e X .
(ii) There is a well defined canonical 1-1 correspondence S between the class of all right invariant fuzzy p-metrics d' on (X, .) to the class of all favf's on (X, .), given by
S(d')(x) = d'(x, e),
We shall always be referring to one or more of the above canonical bijections when we speak of, say, a left invariant fuzzy p-metric d and its right invariant fuzzy p-metric d', or its fuzzy absolute value function p.
x, y e X .
(iii) The canonical 1-1 correspondence S -iv T from the class of all left invariant fuzzy p-metrics, on (X, .), is given by
S -iv T(d)(x, y) = d(x -1, y-l),
x, y • X.
The inverse transformation T - l o S is given on a right invariant fuzzy p-metric d' by T-l°S(d')(x,y)=d'(x-l,y-1),
x,y•X.
the requirement (1.1) of Theorem 1.19 is fulfilled for the function
g :(X, t(d))--* (X, t(d')), by always taking 6 = e . Therefore, by that theorem, g is continuous there. Similarly, g is continuous in the opposite direction. Therefore, it is a homeomorphism. Since g og = the identity function on X, the second assertion follows.
Proposition 5.6. Let d be a left invariant fuzzy p-metric on a group (X, .), and let d' be its right invariant fuzzy p-metric. Then ( X , . , t(d)) is a fuzzy neighbourhood group if and only if (X, ., t[d')) is a fuzzy neighbourhood group. In this case, those two fuzzy neighbourhood groups coincide.
T.M.G. Ahsanullah, N.N. Morsi / Metrizability of fuzzy neighbourhood groups Proof. If ( X , . , t(d)) is a fuzzy neighbourhood group then g is continuous: (X, t(d))--> (X, t(d))
which implies ( X , t ( d ) ) = ( X , t ( d ' ) ) (by the above proposition). Similar implications evidently hold with d and d' interchanged. In the sequel, it will make sense to deal with a fuzzy neighbourhood group ( X , . , t(p)), where p is a fuzzy absolute value function on (X, .). The new notation t(p) will stand for either t(d) or t(d'), where d and d' are the left and right invariant fuzzy p-metrics associated with p, respectively. By the above proposition, no ambiguity will ensue, as long as one end result is a fuzzy neighbourhood group. Proposition 5.7. Let d be a left,
or right, invariant fuzzy p-metric on a group (X, .), and let Zd=(,Y,d(X))x~X be its associated fuzzy neighbourhood system on X. Then "~d satisfies properties (a), (b) and (c) of Theorem 1.3. ProoL Let p : X - - - ~ * ( 1 ) , x ~ d ( x , e ) be the associated fuzzy absolute value function. We now prove that the fuzzy neighbourhood basis ~d of Zd, given by ~a(X)={B(x;r):r>O},
x•X,
satisfies the required three properties. Upon doing this, we would have established the same properties for Za; by applying the saturation operator (a) For all b • X and r > O , we get for all X •X, B(b;r)(x) = Lr[d(x, b)] = Lr[p(b-lx)] = B(e; r)(b-lx) = [b" B(e; r)](x). This demonstrates B(b; r) = ~b(B(e; r)). Hence ~d(b) = {~b(V): v • ~d(e)}, which is (a). (b) For all r > 0, we find for all x • X, B(e ; r)(x ) = Lr[p(x )] = Lr[p(x-~)]
241
This demonstrates g(B(e;r)) = B(e; r), which renders (b). (c) For all r, s > 0, we find for all x • X, [B(e; r). B(e; s)](x) = sup{B(e; r)(y) ^ B(e; s)(z):yz = x} = sup{Lr[p(y)] ^ Ls[p(z)] :yz = x} <~sup{ Lr+s[p(y ) ~p(z)] :y z = x } (by [30, Theorem 1.1(x)]) <~sup{Lr+~[p(yz)] :yz = x} (by the subadditivity of p) = Lr+s[p(x)] = B ( e ; r + s ) ( x ) .
This demonstrates B(e; r) • B(e; s) <~B(e; r + s), which renders (c). In the next theorem the following question is settled: Which left (or right) invariant fuzzy p-metrics on a group induce fuzzy neighbourhood systems compatible with the group structure? Theorem 5.8. Let d be a left, or right, invariant fuzzy p-metric on a group (X, .). Then the following are equivalent statements: (i) ( X , . , t(d)) is a fuzzy neighbourhood group. (ii) Every inner automorphism Intb, b e X, /s continuous at e in the fuzzy neighbourhood space (X, t(d)). (iii) For all b • X and e, 0 • Io, there exists 6 = 6b~O in Io such that for all x • X, p(x)(6) + 0 >~p(bxb-1)(e). Proof. (i)¢:>(ii): Statement (ii) coincides with property (d) of Theorem 1.3. Consequently, by that theorem and the preceding proposition, ( X , . , t(d)) is a fuzzy neighbourhood group iff it satisfies (ii). (ii) <=>(iii): Just apply Theorem 1.19.
Corollary 5.9. f f d is a bi-invariant fuzzy p-metric on a group (X, .), then (X, ., t(d)) is a fuzzy neighbourhood group. Proof. In this case,
(by the symmetry of p)
= B(e; r)(x-') = g(B(e; r))(x).
p(bxb -1) -- p(b-~bx) = p(x) (by Theorem 5.4). Thus, (iii) of the above
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theorem holds for (X,., t(d))7 by always taking 6=e. Corollary 5.10. If d is a left, or right, invariant fuzzy p-metric on an Abelian group (X, .), then (X, ., t(d)) is a fuzzy neighbourhood group. Proof. Conjugate Theorem 5.4 with Corollary 5.9. For the notions of the left fuzzy uniformity q/L, and the right fuzzy uniformity q/R, on a fuzzy neighbourhood group, we refer the reader to [1, Section 3]. Availing ourselves of the notations of [1], we get for invariant fuzzy p-metrics: Theorem 5.11. Let d be a left (resp. right) invariant fuzzy p-metric on a group (X, .). If (X, ., t(d)) is a fuzzy neighbourhood group, then d induces the left fuzzy uniformity q/L (resp. the right fuzzy uniformity q/R) of (X, ", t(d)).
Proof. Suppose d is left invariant. The other case has a very similar proof. The problem is to show that q/(d)
=
q/L.
The fuzzy uniformity qZ(d) has the fuzzy uniform basis {~Pr: r > 0} of Theorem 1.13. So, the fuzzy neighbourhood system 2'd of (X, t(d)) has basis at e [20] ~d(e) = {~Pr(e): r > 0}.
Now, for all x , y e X, invariance of d,
we get by the left
~Pr(X, y) = Lr[d(x, y)] = Lr[d(y-lx, e)] = lPr(e, y - i x ) = % ( e ) ( y - l x ) = (~Pr(e))L(X, Y).
Consequently, 6~d ~'-~{lpr : r > 0 ) -
~-- {(/pr(e))L: ~pr(e) E ~a(e))----q/L (by its definition in [1]). A fuzzy neighbourhood group is said to have bi-invariant fuzzy uniform structure if its left and right fuzzy uniformities coincide [2].
Corollary 5.12. Let d be a bi-invariant fuzzy p-metric on a group (X, .). Then the fuzzy neighbourhood group (X, ., t( d ) ) has b iinvariant fuzzy uniform structure.
Proof. By Theorem 5.11, q/L = q/(d) = q/R. We end this section by necessary and sufficient conditions for the invariant fuzzy p-metrizability of fuzzy neighbourhood groups. Theorem 5.13. Let ( X , . , t(,~)) be a fuzzy neighbourhood group. Then the following are equivalent statements: (a) t(~') is induced by a left invariant fuzzy p -metric. (b) t(,~) is induced by a right invariant fuzzy p-metric. (c) The left fuzzy uniformity q/L is fuzzy p -metrizable. (d) The right fuzzy uniformity q/R is fuzzy p -metrizable. (e) t(Z) is induced by a fuzzy p-metric. (f) The fuzzy neighbourhood system at e, Z(e), has a countable basis. Proof. (a) <=>(b): This is Proposition 5.6. (a) ~ (c) and (b) ~ (d): This follows from Theorem 5.11. (c) or ( d ) ~ (e): This follows from
t(q/L)
=
t(2~)
=
t(q/R).
(e)=>(f); When (e) holds, the fuzzy open balls with positive rational radii, centered, at e, form a countable basis for Z(e); by Theorem 1.16. ( f ) ~ ( c ) and (d): Suppose Z(e) has a countable basis. Then, by their definitions in [1], q/L and q/R will also have countable bases. Consequently, they are fuzzy p-metrizable by Theorem 1.15. (The 1-Hausdorff-separatedness requirement in [11] is easily seen to be needed only if q/L and q/R are to be fuzzy metrizable.) ( c ) ~ ( a ) : Suppose q/L is fuzzy p-metrizable; that is q/L = q/(d) for some fuzzy p-metric d on X. Then also
%
=
q/(d,),
where d l = d A i is the fuzzy p-metric on X obtained by truncating d at i.
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which proves that t(X) is induced by the left invariant fuzzy p-metric d.
Define a function d:X2---> R*(I) by
d(x, y) = V dl(bx, by), beX
(x, y) • X 2.
By direct verification, d is a left invariant fuzzy p-metric on iX'. We show that °/re = °-/t(d): Let °//'(dl)={l~Vr:r>0 }
and
°t/'(d)={lpr:r>O}
be the defining fuzzy uniform bases of q/(dl) = °//L and °//(d), respectively, given as in Theorem 1.13. Then we get for all r > 0 and (x, y) • X 2
~O,(x, y) = 1 - d(x, y)(r) = A [ 1 - d~(bx, by)(r)]; beX
that is ~P,(x, Y) = A ~XOr(bx, by). b~X
(5.1)
From which we get lp, ~< l~pr for all r > 0. This gives a//(d) _~ ~ ( d l ) = 0~L.
(5.2)
On the other hand, a//L has a fuzzy uniform basis consisting of left invariant fuzzy vicinities [1]. Consequently, given any r, 0 > 0, there is a left invariant A e °~L such that
l~O,>~A - O , whence (5.1) gives for all (x, y) • X 2, b~X
>I A [A(bx, b y ) - O] beX
= A(x, y) - O,
q/(d) _ 0"//L. From (5.2) and (5.3),
Theorem 6.1. A quotient group of a fuzzy neighbourhood group is a fuzzy neighbourhood group under its quotient fuzzy topology. Proof. By [3, Theorem 4.8], a quotient group of a fuzzy topological group is a fuzzy topological group under its quotient fuzzy topology, while a quotient fts of a fns is also a fns ([29, Corollary 2.1]; a description of a quotient fuzzy neighbourhood system is given in [5]). This proves the theorem. By the above theorem, we are allowed to speak of a quotient fuzzy neighbourhood group of some fuzzy neighbourhood group.
Theorem 6.2. Let ( X , . , t(~,)) be a fuzzy neighbourhood group, and let (Y, ., t(X')) be its quotient fuzzy neighbourhood group under a group epimorphism f : (X, . ) ---~( Y, .). Then (i) f is continuous and open: (X, t(X))--* ( r , t(X')). (ii) 2:'(e') = {v e l V : f - l ( v ) e 2:(e)}, where e and e' are the neutral elements of (X, .) and ( Y, .), respectively. (iii) If (X, t(Z)) is N-locally compact, then so is (Y, t(X')). Proof. (i) This is proved in [3, Proposition 4.2] for fuzzy topological groups in general. (ii) Put v = {v e I V : f - l ( v ) e X(e)}. Then by the continuity o f f , X'(e') ~_ v. By the openness, and hence N-openness, of f, v G X'(e'). Hence, equality holds. (iii) The assertion follows from (i) and Proposition 3.7.
~Pr(X, y) = A ~r(bx, by)
which renders ~Vr~>A- 0. This proves opposite inclusion
6. Quotient fuzzy neighbourhood groups
the (5.3)
Theorem 6.3. Let p : X - * ~ * ( I ) be a fuzzy absolute value function on a group (X, .), and let f:(X,.)--~(Y,.) be a group epimorphism. Define a function pf : Y---~ •*(I) by
0U(d) = % .
pf(y) =inf{p(x):x e X , f(x) = y } ,
Finally,
Then pf is a fuzzy absolute value function on the group (Y, .). Moreover, if (X, ., t(p)) is a fuzzy
t( d) = t( ~ ( d) ) = t( °UO = t( Z),
y e Y.
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neighbourhood group, then ( Y , . , t(p:)) is its quotient fuzzy neighbourhood group under f. Proof. We have: (AV1) pf(e') = 0, because p(e) = O. (AV2) for all y • Y,
f(Bd(x; r))(f(z)) = sup {Bd(x; r)(Zl):f(z,) = f ( z ) }
= sup{Lr[d(x, Zl)]:f(z,) = f(z)} f(e) = e'
and
PI(Y-I) = inf{p(x): f ( x ) = y - l } = inf{p(x-l): f ( x ) = y} = inf{p(x): f ( x ) = y} = PI(Y),
= sup{Lr[d(x,, z,)]: f(xl) = f ( x ) , f(zl) = f ( z ) } (by a simple manipulation of the right invariance of d and the fact that f is a group homomorphism)
= Lr[inf{d(Xl, Zl)."f ( x | ) = f ( x ) , f(zl) = f ( z ) } ] (by [30, Theorem 1.1(viii)]) =
Lr[d'(f(x), f(z))]
because p(x -1) = p ( x ) for all x • X. (AV3) Since f is an epimorphism, then for all x • X, y, w • Y, f ( x ) = yw iff x = XlX2 for some xl, x 2 • X with f(xl) = y and f(x2) = w. Consequently,
This proves that f(Bd(X;r)) = Bd,(f(x);r), which demonstrates that f is N-open. Hence, it is open. On the other hand,
p/( yw ) = inf{p(x): f (x ) = yw }
( f x f)-l(~p~')(x, z) = ~)(f(x), f ( z ) )
=Bd,(f(x);r)(f(z))
(using (6.2)).
= L,[d'(f(x), f(z))] >! Lr[d(x, z)] (by (6.2))
= inf{p(xlx2): f(xl) = y, f(x2) = w}. (6.1) Therefore,
: ~r(X, Z).
p1(yw) <- inf{p(xt) ~)p(x2): f (xl) = y, f(x2) = w} (by the subadditivity of p) = inf{p(x0: f ( x O = y} inf{p(x2): f(x2) = w} (by twice applying Proposition 1.10)
This proves that ( f xf)-l(~p))/> lp~, which demonstrates that f is uniformly continuous: (X, q/(d))--, (Y, °//(d')). The above, together with Theorem 6.1, complete the proof of the second assertion of the theorem.
=Pr(Y) ~Pr(W). This completes the proof that Pr is a fuzzy absolute value function on (¥, .). Let d:X2---~R*(I) and d':y2--->~*(1) be the right invariant fuzzy p-metrics on X and ¥, respectively, defined by
d(x,z)=p(xz-'),
( x , z ) • X 2,
and d ' ( y , w) = p f ( y w - 1 ) ,
(y, w) • y2.
Then we get by (6.1) for all (y, w) • 1:2,
d'(y, w) =p1(yw -1) = inf{p(xz-l): f ( x ) = y, f ( z ) = w}. So,
d'(y, w) = inf{d(x, z): f ( x ) =y, f ( z ) = w}. (6.2) Now, for all x, z • X and r > 0,
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