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On the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level-topologies
Fuzzy Sets and Systems 12 (1984) 71-85 North-Holland
71
O N T H E D E T E R M I N A T I O N OF F U Z Z Y T O P O L O G I C A L SPACES AND FUZZY NEIGHBOURHOOD SPACES BY THEIR LEVEL-TOPOLOGIES P. W U Y T S
Ri]ksuniversitair Centrum Antwerpen, Antwerp 2020, Belgium Received December 1982 Revised February 1983 In this paper we discuss the problem of the reconstruction of a fuzzy topological space or a fuzzy neighbourhood space from an a priori given family of level-topologies. Necessary and sufficient conditions for the existence of a solution are given, and it is proved that in the particular case of fuzzy neighbourhood spaces this solution is always unique.
Keywords: Fuzzy topological space, Fuzzy neighbourhood space.
1. In~oduction In [3] R. Lowen introduced for an arbitrary fuzzy topological space (X, A) the family {,. (A); e t e [ 0 , 1[} of its level-topologies, and showed how properties of these topologies can sometimes be used to characterize properties of (X, A). In [6] he showed that in the case of a fuzzy neighbourhood space these level-topologies always form a descending chain. The first of these results raises the question to know in how far a fuzzy topological space is determined by the family of its level-topologies, while the second one already shows that, at least in the special case of fuzzy neighbourhood spaces, the family of the level-topologies cannot be given arbitrarily a priori. In this paper we give a fairly complete answer to the questions raised above. Given a family ~: = { g r ; a ~ [0, 1[} of topologies on a set X we give a necessary and sufficient condition under which there exists at least one fuzzy topology A on X, having ,~ as its family of level-topologies (i.e. such that ~ ( a ) = g r for each tx = [0, 1D, and prove that the set of all these fuzzy topologies always has a maximum but in general no minimum. Further we give necessary and sufficient conditions under which there exists at least one fuzzy neighbourhood space having ~: as its family of level-topologies and we prove that in this case the solution is always unique and coincides with the above mentioned maximum solution. After having completed this study we became aware, by oral communication, of the fact that W. Fliish6h should have obtained results in this field, especially in the case of fuzzy neighbourhood spaces. At this moment we do not have any detail on his results, nor on the methods employed. 0165-0114/84/$3.00 ~ 1984, Elsevier Science Publishers B.V. (North-Holland)
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2. Definitions and preliminary results We recall a certain number of definitions and known results to be used in the sequel. For all other definitions and notations and for the proofs of these results we refer to [2], [4], [5]. Because of their fundamental importance for the rest of the paper, we treat in detail the relations between a function I ~ X and its level sets.
2.1. Fuzzy topological spaces. The closed unit interval is always denoted I, while Io = ]0, 1] and I~ = [0, 1[. If A is a subset of X, we always denote its characteristic function by 1A. Fuzzy topologies are always supposed to contain the constant functions. If X, A is a fuzzy toplogical space, its a-level-topology (a E I 0 is defined by
~,~(~): {x-'(]a, 1]); x eA}. For all a E I , /3 E I we write a+13 instead of ( a + / 3 ) A 1 and a - / 3
instead of
(a-/3)v0. 2.2. Fuzzy neighbourhood spaces. If ~ is a prefilterbasis on a set X, i.e. a nonempty subset of I x not containing 0 and such that for all h,/x c ~ there exists a v E ~ such that v<~h A it, we denote by [~1] the prefilter generated by it:
[~] ={~ E IX; 3X E ~ : x ~ . } , and ~ the family
A
It is known [5] that for each prefilterbasis ~ one has [~] = [~] and we denote this family by ~. A collection of prefilterbases ( ~ ( x ) ) ~ x is called a fuzzy neighbourhood basis iff the following conditions hold: (B1) VxEX, V l 3 s ~ ( x ) : / 3 ( x ) = 1;
(B2) Vx E X, V/3 E~(x), Ve E Io, :l(/3~)~x El'-[~x ~B(z): VyEX:
sup/3~(z) A/3~(y) ~/3(y) + e. z~X
A collection ( ~ ( x ) ) ~ x of prefilters is called a fuzzy neighbourhood system iff it has the properties (B1), (B2) and moreover (N) Vx E X : ~r(x) = ~ ( x ) , i.e. V(v~)~oE~r(x)~o: sup (v~ - e) E ~(x). If ( ~ ( x ) ) ~ x is a fuzzy neighbourhood system we shall say that (~(x))x~x is a basis for it iff it is a collection of prefilterbases and such that ~(x) = ~ ( x ) for ea~ach x E X. It is known [5] that if ( ~ ( x ) ) ~ x is a fuzzy neighbourhood basis, then (~l(x)),:~x is
Determination of [uzzy topologicalspaces by level-topologies
73
a fuzzy n e i g h b o u r h o o d system having (~(x))x~x as a basis, while also each basis of a fuzzy n e i g h b o u r h o o d system is a fuzzy n e i g h b o u r h o o d basis. F r o m a fuzzy n e i g h b o u r h o o d system (W(x))x~x we can d e d u c e [5] a fuzzy closure o p e r a t o r and t h e r e f o r e a fuzzy t o p o l o g y on X, which we shall always d e n o t e t(W). M a k i n g use of [5, Prop. 2.3 and 2.4], we can see easily that if (~(x))x~x is a basis for the fuzzy n e i g h b o u r h o o d system (W(x))x~x, the fuzzy interior )t ° of )t ~ I x in (X, t(W)) is given by
Vx~X:
,k°(x) = sup Ig ~ ( x )
inf ~ . v ( 1 - / 3 ) ( y ) . yeX
A fuzzy topological space (X, A) is said to be a fuzzy n e i g h b o u r h o o d space if there exists a fuzzy n e i g h b o u r h o o d system (W(x)).~x such that A = t(~), and it is k n o w n [5] that in that case the system ('//'(x)).~x is unique. M o r e o v e r , in this case [5, T h e o r . 6.4], the n e i g h b o u r h o o d filter of x in the level-toplogy t=(A) is given by *.o,("//'(x)) = {v-1(]/3, 1]); v e 'F'(x),/3 e [0, 1 - a [ } .
2.3. a-cuts of a function. G i v e n a function tz : X - - ~ / , /2 : 11 ~ 2 x by VaEIl:
ff.(a)=tx-l(]a,
we can define a m a p
1]),
and this m a p is always decreasing. It has also the p r o p e r t i e s ~
~
til~
and
VxeX:
tx(x)=l
or
~(x)
2.4. A left-inverse for ". G i v e n now a function to : I1 ~ 2 x, we can define a m a p to*:X---> I by to*(x) = sup{a; x e to(a)} = sup a 1,o~(x)
(sup ~ = 0).
OtEl I
As s u p { a ; x e to(a)} = m a x { a ; x e t0(a)} ¢~ sup{a; x e to(a)} e { a ; x ~ to(a)}, it follows i m m e d i a t e l y that, if t o * ( x ) ~ 1, we have to*(x) = m a x { a ; x e to(a)} ¢:> x ~ to(to*(x)). M o r e o v e r , the following relations derive i m m e d i a t e l y f r o m the definitions:
(a) Vto : I , ---, 2 ×, Va ~ I, : ~'~(a) = U~>,:, to(/3). (b) V/z : X ~ I : (t~)* = ~. (c) For any to : I1 ~ 2 x there exists a decreasing to': 11 ~ 2 x such that to'* = to*. Indeed, it is sufficient to take to'= ~ ' . 2.5. Lattice properties. The m a p (2x) t, ~ I X : t o - - ~ to* has the following p r o p e r ties: (a) If tol<~to2, then to*l <~to2*. (b) F o r any decreasing functions tol : 11 --~ 2 x, to2 : I1 --~ 2 x we have ( ~ ^ to2)* = to* ^'#2".
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Indeed, (~%^~0z)*~ 0 , h e n c e (qh ^ ~0z)*(x) I> )t - e and t h e r e f o r e
(~* ^ ~*)(x) ~ (,m ^ ~,~)*(x). (c) F o r any family ~ of functions I~ - + 2 x, ~#o = sup,~a,~, we h a v e q~* = (sup ~0)* = sup q~*. Indeed, if q~*(x)=X, w h e t h e r xeq%(A) or not, there exists for each A ' < A a )t'<~ X"~< A such that x ~ ~0o(~"), and hence a % , e q~ such that x e q~x,,(h"), w h e n c e ~*,,(x) ~ X " >~ X' and t h e r e f o r e sup,o~, ~ * ( x ) >/A'. By arbitrariness of h' < X it follows that s u p , ~ , q~*~ ~0o, while the o p p o s i t e , i n e q u a l i t y is trivial. (d) If ~1 : I , - - - . 2 ×, @z:I,---~2x, @', = ~ , ~% = ~ ' , then @ * A ¢#~ = (@lI A @;)*.
This follows at once f r o m (b) and the r e m a r k s in 2.4. 2.6. (LO)-property. W e shall say that a m a p ~o : I~ ~ 2 x has the ( L O ) - p r o p e r t y if it satisfies the following condition: (LO): V x e X : ~ 0 * ( x ) = l or q~*(x) X such that x e q~(iB) and thus X <13 <~q~*(x). (b) If q~ and q~2 are decreasing and have the ( L O ) - p r o p e r t y , then q~ = qhAq~2 also does. Indeed, if q3*(x) = q~*(X)A~O*(X)= 1, there is nothing to prove. If q~*(x) = A < 1, then for instance X=q~*(x)~
x¢ ~(x) = ,~(,~*(x)). (c) If q~ is a family of functions having the ( L O ) - p r o p e r t y , then ~0---sup,o~, ~o also does. Indeed, if q~*(x) = 1 then once m o r e there is nothing to prove. If ~o~(x) = a < 1, then q~*(x)<~a for all q~ e qb. If then we had x e ~0o(O~), t h e r e should exist a q~ c with x E ~o(a), so ~0*(x) ~ ~, hence ~0*(x) = a and x c ~o(q~*(x)).
3. Fuzzy topologies associated with a family of topologies In this section we will derive several fuzzy topologies f r o m a given family 1~ of topologies on a set X. Only one of t h e m will show to have level-topologies that are closely related to the given ~: and lead finally to the solution of the main p r o b l e m in this paper. B e t w e e n the others the first o n e h o w e v e r s e e m s to play a role in the case of fuzzy n e i g h b o u r h o o d spaces (cf. 5.2(b)), and the remaining ones are m e n t i o n e d b e c a u s e of the analogies with the first one.
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75
3.1. L e m m a . If ~; = { ~ ; a ~ I~} is a family of topologies on a set X, and if qr)0(~) = I(0 ~ l-I g r ; (0:1,---> 2 x decreasing}, then 4)~(o*) = ((0*; (0 e 4)o(~)}
is a fuzzy topology on X.
Proof. If (0 = X (constant function) we have (0* = 1, while for c e I~ we can define q~ by ( 0 ( a ) = X if a e [ O , c [ , (0(x)=0 if a ~ [ c , l [ . Then ( 0 " = c , and so (/)*(~-) contains all the constants. T h e rest of the proof follows immediately from 2.5(b) and 2.5(c). = {ff,,;a c I,} is a family of topologies on a set X, forming a 3.2. Lemma. If descending chain in the sense that
O~a~/3
=> g r a c g - ,
(3.1)
and if
4),(97) = 1-I 3"-,,c{(0; (0 : I, ----)2x }, (xEll
4)2(9;) = {(0 ~ (/),(9;); (0 satisfies (LO)}, then cI)*(~;) and cI)*(~) are fuzzy topologies on X. Proof. That both qb*(9;) and ~ * ( ~ ) contain the constant functions follows from the proof in 3.1. Further, (a) q~*(~:) is a fuzzy toplogy. If (0* = (0"A(0" with (01 c q~l(~:), (02~ qb:(~), then, with the notations of 2.5(d), we have (0* = ((0~ A (0~)*, while (0~ ~ qb:(~:), (0~ ~ qbl(,~ ) by the chain-condition on ~ , and therefore also (0~ A (0:z~ q51(~:). If qb c q)a(.~) then clearly (00 = s u p ~ o (0 (/h(~:) and (00*= sup~o~ (0*. (b) q)*(9:) is a fuzzy toplogy. With the same notations as in the preceding case, as (01 and (02 now have the (LO)-property it follows from 2.6(a) and 2.6(b) that (0~A(0~ also has that property, and on ground of 2.6(c) this is also the case for (00.
3.3. Proposition. If ~ = { ~ ; a c I,} is a family of topologies on a set X, and if q~(o~) = / c( 0~
I-I ~-,,; (0:1,--* 2 x is decreasing and satisfies (LO)}, ¢tEl I
then 4 ) * ( ~ ) = {(0*; (0 ~ 4)(s*)} is a fuzzy topology on X. If moreover a(~:) = {~ ~ IX; V a e 11 : ~ - ' ( ] a , 1]) e gr}, then A(~;) = cI)*(~;), and this fuzzy topology is the finest fuzzy topology A on X whose level-topologies are coarser than the given ~;, i.e. which is such that we have Va~Ii:
i,,,(A) c 3",~.
(3.2)
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P. Wuyts
Proo|. That zl(~:) is a fuzzy topology with the quoted extremum property is trivial. That also qb*(~) is a fuzzy topology follows from 2.5(b), 2.5(c), 2.6(b) and 2.6(c). If now ~ ~ Zl(~), then ¢ --/2 ~ ~(~'). As ~* = ~, it follows that zl (~') c ~*(~-). Conversely, take ~ o ~ ( ~ ) , a ~ I ~ . If x ~ o * - ~ ( ] a , 1]) then ~*(x)>ot and thus x ~ ~o(/3) for some /3 > a ; hence also, as ~o is decreasing, x ~ q~(a). If x ~ ~o(o~), we have ~o*(x)~ot. If ¢ * ( x ) = 1, then x ~ ~o*-~(]a, 1]); if ¢ * ( x ) < 1, then x~ ~o(~o*(x)), whence ~o*(x):/: a, i.e. ¢ * ( x ) > or. All by all we have ¢*-~(]a, 1])= ¢(ot)~ ~ r and therefore ~ * ( ~ ) c zl(~). 3.4. Remarks and counterexamples. (a) If St,, = {0, X} for a ~ Q (the other Sr~ being arbitrary), one verifies without difficulty that q~*(S~) contains only the constant functions, and the inclusion in (3.2) is therefore in general an inequality. (b) If we take X = I, if °//'(0) is the neighbourhood system of 0 e N for the usual topology and if =I{0}u{InV;V~'(0) and½~V} 3-= [ { O I U { I R V ; Vc~(O)}
if0~a~½, if ½ < a < l ,
then /z : I --->X, defined by fl-2x tx(x) =~½0
if0~x~¼, if x=½, if x ~ , 1 1 1], ~[U~,
belongs to q~*(3:). Indeed tx = ~o* if ~o is defined by = I[O, ¼]U{~} q~(ot) [ [ 0 , ½ ( I - a ) ]
if O ~ a ~½, if½
But tx~q~*(3:). Indeed, for every ~0Cq~o(3:) such that q~*=~, as ~o must be decreasing and not empty for ½ < o r < l , we must have ½e~o(½), so necessarily
½e q~(~*(½)). This means that in general q~*(3:)~ ~*(3:). (c) If 3: = {~r ; a ~ I1} is a family of topologies on X, we can now define some other families, derived from ~ , by ~l={ff=,< =
[-I ff,~;a~I1}, 0~13<:c~
~ 2 ~-~{~F~.~ =
n ~']x ; a E '1}, o~B~
All these families satisfy (3.1), and so we could, using Lemma 3.2, build up
Determination of fuzzy topologicalspaces by level-topologies
77
eight fuzzy topologies q~*(~k) (i e{1, 2}, k e{1, 2, 3, 4}). We shall see in 5.2 that if,,.< seems to play a role in the case of fuzzy neighbourhood spaces.
4. Fuzzy topologies with given level-topologies In this section we shall prove that under a suitable (necessary and sufficient) condition the inclusion in (3.2) becomes an equality. 4.1. (LT)-families and (LT)-property. If {~-,~; ote Ii} is a family of topologies on X, and if a c I~, G c g r , x e G, we say that {Ga; a
to { ~ r ; ~ I 3 . 4.2. Lelmma. If X, A is a f u z z y topological space, the family {~,~(A), t o i l }
of its
level-topologies has the (LT)-property. Proof. Immediate by considering the level-sets h-l(]a, 1]) for h ~ A, a ~ 11. 4.3. Theorem. If :~ = {ff,~; a ~ 11} is a family of topologies on a set X, there exist
f u z z y topologies A on X having ~; as their level-topologies, i.e. such that Vote/l:
t.~(A) = fie,,
(4.1)
if and only if ~r has the (LT)-property. Moreover, if this condition is satisfied, the f u z z y topology A(:~) = 49*(@-) is the finest of all f u z z y topologies on X for which (4.1) holds. Proof. In view of 3.3 and 4.2 we only have to prove that if,, c ~,,(A($~)) for all a ~ 11. Let therefore G c ~/,~ and x c G. With the notations of 4.1, we put pc(x) = sup{or; ::l(LT)-family (GB),~
U {G~; ,/
is an
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Each family of this kind can be considered as an element 4' of l-I {ff.~; a < 3' < pc(x)}. If then W is the set of all these elements, then sup 4'= (G~)~<~<~(x~
with G~ = U 4,(3")
is the 'largest' (LT)-family for (x, G, ct) on the interval ]a, pc(x)[. Given x, G, a, we shall denote this uniquely determined family by (G~)~<~<~(~). We then define an element q~ ~ ~ 0 ( ~ ) by
if 3 " = a , q~(3") =
-y
if ct < 3, < pc(x), if pc(x) ~<3' < 1,
where the first line can be omitted if a = 0 and the last one if pc(x) = 1. In order to obtain ~ * e q~*(,,~), we have to prove that VycX:
q~*(y)=l
or
q~*(y)
(4.2)
If y ~ G we have ,p*(y) = sup{,/;
y ~ ,p~(3,)} = ,~
and therefore (4.2) is satisfied. If y ~ G we have a ~<¢*(y)<~po(x), and three cases can be distinguished: (a) ¢*(y) = pc(x) = 1, and nothing has to be proved; (b) q ~ * ( y ) = p c ( x ) < l , in which case ¢ ~ ( ¢ * ( y ) ) = ¢ , and (4.2) is satisfied;
(c) ¢*~(y)= 8 < 0 ~ ( x ) . Suppose that, in this last case, y c q~(8). If then (L~)8<~<, is an (LT)-family for (y, q~(8), ~3), the family (L~)=<~a, ye~.(~,)}=
U
~.(3,)cG
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79
which proves that G is a neighbourhood of x in L~(qb*(~:)) and therefore, by arbitrariness of G ~ ~ r and x c G, that ~,, c L.(~*(~:)). 4.4. R e m a r k . With the notations of the preceding proof, it is not difficult to show that in fact p6(x) = max{or, q(LT)-family (Ga).
4.5. Counterexample. W e take n ~ N * and define a fuzzy topology A,, on X = [0, 1] as follows: h ~ A n ¢~ (1) 3 a ~ I : h ~ c t , and (2) if x ~ I n Q , x = p / q in smallest terms, then, for some s c N N[0, q~ - 1] we have s/q ~ <~a < ~ h ( p / q ) ~ ( s + 1)/q ~. A straightforward verification shows that An is indeed a fuzzy topology, and it is not difficult to see that A~ is nothing else than the one in [3, {}4]. Taking a fixed a e I~, we have (a) for x ~ I irrational, that h = 1. ~ A and thus h - l ( ] a , 1]) = {x}; (b) for x c l n O , if x = p / q in smallest terms and s ~ N n [ 0 , q " - l ] is such that s/ q ~ <~a < ( s + 1) / q ~, that h = ct + ( ( s + 1) / q n - a ) l . ~ An and therefore again h - ' ( ] a , 1]) = {x}. This proves that for all n e N* the level-topologies of A are all discrete. F u r t h e r m o r e we have k
:=> A , , c A k
while
a=na. is still a fuzzy topology. If then x = p / q ~ y = u/v are both rational and in lowest terms, and if ~0c A, q~(x) = a, ~0(y) =/3, there must be, for each n ~ N * , a 3'n ~ I , an s. ~[0, q " - l ] n N and a w~ e [ 0 , v " - l ] n N such that _
sn+l q.
_
qn and
wn wn + 1 --~<3"n ~ qo(y)=/3 ~ - ~)n i~n Then however - al
~ max(/3 - 3'., a - 3'.) ~
and hence a =/3 by arbitrariness of n, unless {x, y} n{0, 1} ¢ 0. This shows that ~ p e a is necessarily a constant on ] 0 , 1 [ N Q , and the level-topologies of A therefore cannot be discrete. Finally, if Ao is an arbitrary fuzzy topology on I coarser than all ( A ) . ~ . , we necessarily have a 0 c A and therefore the topologies ~(Ao) cannot be discrete.
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From all this it follows: (a) if there exists a fuzzy topology A on a set X, having a given family ~:={~;a~Ii} as its level-topologies (~,~(A)=~,~), this A in general is not unique; (b) the set of all fuzzy topologies, having the given family ~: as its leveltopologies, if non-void, does not contain in general a coarsest element; (c) if A ' c A" are fuzzy topologies on a set X, then Va c 11:
~,,(A ') = L~(A"),
but the converse in general does not hold. Indeed, if
(V(~EII: ~ ( A ' ) c ,
(A")) ~
A'cA"
were always true, this should imply the uniqueness of the fuzzy topology with given level-topologies.
5. Fuzzy neighbourhood spaces with given level-topologies It is known [5] that the level-topologies of a fuzzy neighbourhood space, apart from having the (LT)-property, also form a descending chain. We therefore must at least impose this supplementary condition on $: should ~ * ( ~ ) become a fuzzy neighbourhood space. W e shall prove that this also is sufficient and derive a uniqueness property. 5.1. Proposition. Let ~r = {~r~; a ~ I~} be a family of topologies on a set X. If ~ ( x ) is the set of open neighbourhoods of x in X, ~ , define Ad~(x) by
~(x)={tx:I1---~2x;
p~decreasing and Votell:P~(a)e
(-1
~(x)}.
/3eC0,l-a[
Then the family (~g(x))x~x, where ~ ( x ) = {~ *= sup a l~(~,; tz ~ ~ ( x ) } a~lt
is a fuzzy neighbourhood base on X. Moreover, if we denote by (Xs~(x))x~x the fuzzy neighbourhood system generated by (~;(x))x~x, the level-topologies of the corresponding fuzzy topological space are coarser than the given ~ , i.e. we have V~ et,:
~,,(t(.,v'.)) = ~r,~.
Proof. We divide the proof into three parts. (a) ~ ( x ) is a prefilter base, satisfying (B1). This is immediate. (b) ~s~(x) satisfies (B2). W e take ~ * ~ ~ ( x ) and e > 0. For each z c X we then choose a function ~,:I1--~ 2 x taking t ~ =ix if z = x, and otherwise as follows:
{
/x(~)
if a ~< ~ * ( z ) - e,
g(t~*(z)-e)
if a > / x * ( z ) - e
Determinationof fuzzy topologicalspacesby level-topologies
81
(recalling that we write a - / 3 for ( a - / 3 ) v0). This function is clearly decreasing. Moreover (1) if ot ~ t ~ * ( z ) - e , then still z s / z ( i z * ( z ) - e ) and therefore ~.~(a)= ~ ( / z * ( z ) - e) ~ N { ~ ( z ) ; / 3 e [0, 1 - tz*(z) + e[}, hence afortiori tx.~(a) ~ A { ~ ( z ) ; /3 e[0, 1 - ~ [ } . This proves that always p~, e.,R~z). Writing tx~ instead of ~ , for the sake of simplicity and taking a fixed y e X we then have either tx*(y)>I tx*(z)-e, hence /~ .(z) A/x ~(y) - t~*(z)/x/z*(y) ~
with y~t~(a0)
~*(y) = sup ot 1.¢~)(y)= sup ot 1.,(~)(y)= tz*(y) -~t~ o
ot " ( ~ o
and therefore /z*(z)/x/~*(y) ~
~(e, 0)e
M ~r?~(x) 13~[OA-o[
and /3~ = s u p o ~ ~pl.(~.,). Let us take e ' > 0 such that ~ + 2 e ' < l - c t . Then /z(e', 1 - a - e ' ) e Y ~ ( x ) for /3e[0,1-(1-a-e')[ and in particular p . ( e ' , l - c t - e ' ) ~ ° ~ ( x ) . If now z ~ p.(e', 1 - c t - e ' ) , then / 3 ~ ' ( z ) ~ l - a - e ' > 8 + e ' , so / 3 ~ ' ( z ) - e ' > 8 and therefore ~(z)>~5. This however means that i~(e',l-a-e')~v-~(]8,1]) and hence v-~(]ti, 1])e V,(x). 5.2. Remarks. (a) In the above proposition we supposed the sets /z(a) to be open in each of the (X, ffa) for / 3 ~ [ 0 , 1 - a [ . This condition however was introduced only for technical reasons. It is clear that if we ask only the condition txdecreasing
and
p~(ot)~
["1
°//'a(x),
13~[0.1--a[
(where ~ ( x ) is the neighbourhood filter of x in (X, ~-a)), we obtain a collection ~ ( x ) for which A/g(x)c A~s~(x)c [~g(x)], so that ~(x) = ~;(x) if ~ls~(x) = {tL*; t~ ~ Ats~(x)}. (b) If we define the topology ff~ on X by ~-~ = f-)0~<~ ff~, the condition
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82
p.(a) ~ ~a~[o.l-,~[ ~ a ( x ) means exactly t h a t / x ( a ) must be a neighbourhood of x in
(x, ~rL,). 5.3. Proposition. Let ~ = { ~ ; a ~ 11} be a family of topologies on a set X, having
the (LT)-property and forming a descending chain, i.e. such that
0~o~
~
~-e=~r.
Then the family of level-topologies of (X, t(W~)) is precisely ~, i.e. Vot~I~:
~,~(t ( J ~ ) ) = ~,~.
Proof. We have only to show that ~ r c ~(t(A%)). In order to do this, we take G ~ if',, x ~ G. By the (LT)-property of ,~ we can find a ~ > a, H ~ 3-~ such that x 6 H c G, and by the descending chain condition this implies that H~°l/'~(x) for /3~<& W e then define / x : I t - - ~ 2 x by ~ ( A ) = X for A < I - 6 and g , ( A ) = H for 1 - 6 ~ < A < 1 . Then immediately /z~A/~(x), hence ~ z * 6 ~ ( x ) . But p , * I H = 1, P~*IX \ H = 1 - 6 , whence
H = ~*-'(]1 - & 1]) = ~,-1(]/3, 12) if /3 = 1-¢3 < 1-c~. So H e ~,(N~(x)) and therefore also G e ~,(N~(x)). 5.4. Proposition. If (~(x)),~x is a fuzzy neighbourhood system on a set X and t(°l/) the associated fuzzy topology, if ~ = {or = ~,~(t(W)), a ~ It} is the family of its
level-topologies and (./¢s~(x)),~x the fuzzy neighbourhood system derived from it, then the two systems (°//(x)),~ x and (N~(x)),~x coincide; moreover, the fuzzy topology they generate is precisely ~ * ( f f ) , i.e. the finest fuzzy topology having ~; as its family of level-topologies. Proof. W e use the same notations as in 5.1 and 5.2, and divide the proof into four parts. (a) ~ ( x ) = N ~ ( x ) . Indeed, if v ~ ( x ) , we take /x = 13, so
~(,~) : ,,-'(]o~, 1])~
f3
~(x)
for a ~ I b while ~z is decreasing. Therefore v = tz* ~ [~(x)] = N~(x). (b) Representation of elements of A,~(x). We consider an element p. ~ A~s~(x) and take e > 0 . Then: (1) As ~ ( a ) c La(~(x)) for each/3 ~ [ 0 , 1 - a[, we can, by [5, Theor. 6.4] for each /3 ~[0, l - a [ find a X0 a [0, 1-/3[ and a v~ ~ °//'(x) such that ~z(a)= v-~l(]Xa, 1]). If now A~ < a , we can define v ; by
Ivy(z) u~(z)=[½(ol+l_/3)
if v~(z)~)to or v ~ ( z ) > a , if A a < v ~ ( z ) ~ < a .
Then v'~>~vo, so v'~°V(x), and m o r e o v e r ~z(a)= v ~' - l ' t~a, " 1]). This means that choosing A~, v~ as indicated above, we can always suppose that X~ I> a. (2) If we take then a / 3 e [ 0 , 1 - a [ satisfying ( 1 - / 3 ) - ~ < ½ e , we obtain
Determination of fuzzy topological spaces by level-topologies
83
a <~)te < 1 - / 3 < ot +½e for the c o r r e s p o n d i n g ha. This h o w e v e r m e a n s that for each o~ e I1 we can c h o o s e a u~ e ~ ( x ) and an a ' such that
a<~a'
and
I z ( a ) = v ~ , ~ ( ] a ',1]).
(5.1)
( c ) ) ¢ ' ~ ( x ) ~ ' V ( x ) . If p . * = s u p , ~ , c~l.t,~) with /z e./,~(x) and if e > 0 , we have for each at an a ' and a v,~ ~ ~V(x) satisfying (5.1). W e can choose a~, a2 . . . . . an+x such that 0 :
O~ 1 ~ 0 ~ 2 ~ "
Vke{1,2,..
" " <~Ot n ~O~n+ • ,
n+l}:
1 :
1,
ak--Ot~_~
<1
~e,
and put u~ =
inf u~ ¢ °V(x).
l~k~n
W e then have: (1) if z e ~ ( a ~ ) ,
then ~ * ( z ) > ~ o t , > l - ~ e , h e n c e v ~ ( z ) - e ~ l - e < ~ ¢ * ( z ) ; n - l } , then
(2) if Zelx(Ctk)\p.(ak+~) for k e { 1 , 2 , . . . , hence
v~ ( z ) - ~ ~ v ~ + , ( z ) - e ~ a ~ , + , - e < ak ~ t~*(Z); (3) if Z~ ~ ( a l ) , then v~,.(z) ~ a'l <½e, h e n c e v~ (z) - e ~ v,~,(z)- e ~ p~*(z). All by all this m e a n s that we have for each e > 0 a v~ e°V(x) such that v~-e~*, and therefore s u p ~ , , ( v ~ - e ) ~ * , h e n c e ~ * e ~ ( x ) , which proves that N~(x) c ~V(x). (d) t(W~) = t(°V) = qb*(~). As t(°V) is a fuzzy t o p o l o g y having ,~ as its family of level-topologies, we already have t(°V)c ~ * ( ~ ) . W e therefore have only to p r o v e that for each ~0 e qb(~), A = ~o* the function h is o p e n in (X, t(°V)). If h ° denotes.its fuzzy interior in (X, t(°V)) it is therefore sufficient to prove that for each x c X and each e > 0 we have h°(x) >1h ( x ) - e. T o do this we o b s e r v e first that by the first part of the t h e o r e m , ~lg(x) as defined in 5.1 is a basis for °V(x) and therefore )t°(x)=
sup
inf h v ( 1 - p . * ) ( y ) .
If now A(x) =/3 ~< e, we immediately obtain A(x) ~<)t°(x) + e. If on the contrary /3 > e, we define ~ : I~ --~ 2 x, recalling A = ~0*, by
/z(a)=
X
ifa~l-/3+e,
~(/3-e)
if a > l - / 3 + e .
T h e n /.L(a) e °V°,(x) for all a ' < 1 - a. In fact, this clearly h a p p e n s if a ~ 1 - / 3 + e, while for a > 1 - / 3 + e, we have a ' < 1 - ct < / 3 - e and so iz(a) = ~o(/3 - e) e F°,(x) since a ' < / 3 - e. For y e X, two cases now can occur: (1) A(y) I>/3 - e, h e n c e A v ( 1 - / z * ) ( y ) ~ > A ( x ) - e; (2) A(y) < / 3 - e, h e n c e y~ ¢(/3 - e). If then a ~< 1 - / 3 + e, we have a l~,(,~)(y) = a, hence 1 - a l ~ , t , ~ ) ( y ) = l - a > 1-(1-/3+e)=/3-e, while ff a > 1 - / 3 + e we have
84
P. Wuyts
al~(~)(y)=O, hence 1 - a l . c , , ) ( y ) = l . In any case ( 1 - t x * ) ( y ) = l - t z * ( y ) = inf,,~l,(1 - a 1.(,~(y)) = [3 - e and therefore )t v (1 - ~*)(y) >~/3 - e = )t(x) - e. All by all this means that for this particular /x we have inf A v ( 1 - ~L*)(y) ~>X(x)- e
y~X
and this ends the proof. 5.5. Theorem. Two fuzzy neighbourhood spaces X, A 1 and X, A2 on the same set X coincide, and hence derive from the same neighbourhood system, iff their families of level-topologies coincide, i.e. if for each et • Il we have ~,~(Al) = r~(A2). l ~ o o t . If ~: = ~,~(A1) = t,~(A2) and (T'k(x)).Ex(k e{1, 2}) are the neighbourhood systems defining Ak (k e{1, 2}), we have indeed T'~(x)= g'~(x)=T'2(x) for each xcX. 5.6. Theorem. If ~; = {ff ~ ; a c I~} is a family of topologies on a set X, there exists a fuzzy neighbourhood space (X, t(~)) determined by a fuzzy neighbourhood system (~(x))x~x and having ~ as the family of its level-topologies, if and only if the family if; has the (LT)-property and is a chain in the sense that ct < [3 implies ~ r ~ r a. Moreover, the fuzzy neighbourhood space, and so the system ('//'(x))x~x, is unique and it is the finest of all fuzzy topologies having the family ~; as their family of level-topologies. 5.7. Remark. The foregoing does not allow to conclude that a fuzzy topological space is a fuzzy neighbourhood space if and only if its level-topologies (which have automatically the (LT)-property) form a descending chain. The counterexample in 4.5 shows that this is not true: (X, I x) is a fuzzy neighbourhood space, but there are infinitely many fuzzy topologies having all their level-topologies discrete, and on account of 5.6 only one of them is a fuzzy neighbourhood space! 5.8. Conclusion. Considering the set Fuzx of all fuzzy topological spaces (X, A) with support 32, we can consider the relation AI~A 2 ~
V o ~ / I : i , ~ ( A 1 ) = I , ct(A2) ,
which is clearly an equivalence relation. The foregoing shows that in each equivalence-class their is always a finest element and at most one fuzzy neighbourhood space, which, if present, is precisely the finest element of its class. In view of the 'good' properties of fuzzy neighbourhood spaces one could wonder if more generally the maximum elements of these equivalence classes form an interesting subcategory of the category of fuzzy topological spaces. As another consequence of T h e o r e m 5.5 one could try to describe properties of a fuzzy neighbourhood space by equivalent properties of its level-topologies. This can e.g. be done for most of the separation properties in [7]. A further study of these properties in this sense will be the subject of a forthcoming paper.
Determination of fuzzy topological spaces by level-topologies
85
References [1] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980). [2] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976) 621--633. [3] R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446--454. [4] R. Lowen, Convergence in fuzzy topological spaces, Topology and Appl. 10 (1979) 147-160. [5] R. Lowen, Fuzzy neighborhood spaces, Fuzzy Sets and Systems, 7 (1982) 165-189. [6] C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis (Birkh~iuser Verlag, Basel, 1975). [7] 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.
71
O N T H E D E T E R M I N A T I O N OF F U Z Z Y T O P O L O G I C A L SPACES AND FUZZY NEIGHBOURHOOD SPACES BY THEIR LEVEL-TOPOLOGIES P. W U Y T S
Ri]ksuniversitair Centrum Antwerpen, Antwerp 2020, Belgium Received December 1982 Revised February 1983 In this paper we discuss the problem of the reconstruction of a fuzzy topological space or a fuzzy neighbourhood space from an a priori given family of level-topologies. Necessary and sufficient conditions for the existence of a solution are given, and it is proved that in the particular case of fuzzy neighbourhood spaces this solution is always unique.
Keywords: Fuzzy topological space, Fuzzy neighbourhood space.
1. In~oduction In [3] R. Lowen introduced for an arbitrary fuzzy topological space (X, A) the family {,. (A); e t e [ 0 , 1[} of its level-topologies, and showed how properties of these topologies can sometimes be used to characterize properties of (X, A). In [6] he showed that in the case of a fuzzy neighbourhood space these level-topologies always form a descending chain. The first of these results raises the question to know in how far a fuzzy topological space is determined by the family of its level-topologies, while the second one already shows that, at least in the special case of fuzzy neighbourhood spaces, the family of the level-topologies cannot be given arbitrarily a priori. In this paper we give a fairly complete answer to the questions raised above. Given a family ~: = { g r ; a ~ [0, 1[} of topologies on a set X we give a necessary and sufficient condition under which there exists at least one fuzzy topology A on X, having ,~ as its family of level-topologies (i.e. such that ~ ( a ) = g r for each tx = [0, 1D, and prove that the set of all these fuzzy topologies always has a maximum but in general no minimum. Further we give necessary and sufficient conditions under which there exists at least one fuzzy neighbourhood space having ~: as its family of level-topologies and we prove that in this case the solution is always unique and coincides with the above mentioned maximum solution. After having completed this study we became aware, by oral communication, of the fact that W. Fliish6h should have obtained results in this field, especially in the case of fuzzy neighbourhood spaces. At this moment we do not have any detail on his results, nor on the methods employed. 0165-0114/84/$3.00 ~ 1984, Elsevier Science Publishers B.V. (North-Holland)
P. Wuyts
72
2. Definitions and preliminary results We recall a certain number of definitions and known results to be used in the sequel. For all other definitions and notations and for the proofs of these results we refer to [2], [4], [5]. Because of their fundamental importance for the rest of the paper, we treat in detail the relations between a function I ~ X and its level sets.
2.1. Fuzzy topological spaces. The closed unit interval is always denoted I, while Io = ]0, 1] and I~ = [0, 1[. If A is a subset of X, we always denote its characteristic function by 1A. Fuzzy topologies are always supposed to contain the constant functions. If X, A is a fuzzy toplogical space, its a-level-topology (a E I 0 is defined by
~,~(~): {x-'(]a, 1]); x eA}. For all a E I , /3 E I we write a+13 instead of ( a + / 3 ) A 1 and a - / 3
instead of
(a-/3)v0. 2.2. Fuzzy neighbourhood spaces. If ~ is a prefilterbasis on a set X, i.e. a nonempty subset of I x not containing 0 and such that for all h,/x c ~ there exists a v E ~ such that v<~h A it, we denote by [~1] the prefilter generated by it:
[~] ={~ E IX; 3X E ~ : x ~ . } , and ~ the family
A
It is known [5] that for each prefilterbasis ~ one has [~] = [~] and we denote this family by ~. A collection of prefilterbases ( ~ ( x ) ) ~ x is called a fuzzy neighbourhood basis iff the following conditions hold: (B1) VxEX, V l 3 s ~ ( x ) : / 3 ( x ) = 1;
(B2) Vx E X, V/3 E~(x), Ve E Io, :l(/3~)~x El'-[~x ~B(z): VyEX:
sup/3~(z) A/3~(y) ~/3(y) + e. z~X
A collection ( ~ ( x ) ) ~ x of prefilters is called a fuzzy neighbourhood system iff it has the properties (B1), (B2) and moreover (N) Vx E X : ~r(x) = ~ ( x ) , i.e. V(v~)~oE~r(x)~o: sup (v~ - e) E ~(x). If ( ~ ( x ) ) ~ x is a fuzzy neighbourhood system we shall say that (~(x))x~x is a basis for it iff it is a collection of prefilterbases and such that ~(x) = ~ ( x ) for ea~ach x E X. It is known [5] that if ( ~ ( x ) ) ~ x is a fuzzy neighbourhood basis, then (~l(x)),:~x is
Determination of [uzzy topologicalspaces by level-topologies
73
a fuzzy n e i g h b o u r h o o d system having (~(x))x~x as a basis, while also each basis of a fuzzy n e i g h b o u r h o o d system is a fuzzy n e i g h b o u r h o o d basis. F r o m a fuzzy n e i g h b o u r h o o d system (W(x))x~x we can d e d u c e [5] a fuzzy closure o p e r a t o r and t h e r e f o r e a fuzzy t o p o l o g y on X, which we shall always d e n o t e t(W). M a k i n g use of [5, Prop. 2.3 and 2.4], we can see easily that if (~(x))x~x is a basis for the fuzzy n e i g h b o u r h o o d system (W(x))x~x, the fuzzy interior )t ° of )t ~ I x in (X, t(W)) is given by
Vx~X:
,k°(x) = sup Ig ~ ( x )
inf ~ . v ( 1 - / 3 ) ( y ) . yeX
A fuzzy topological space (X, A) is said to be a fuzzy n e i g h b o u r h o o d space if there exists a fuzzy n e i g h b o u r h o o d system (W(x)).~x such that A = t(~), and it is k n o w n [5] that in that case the system ('//'(x)).~x is unique. M o r e o v e r , in this case [5, T h e o r . 6.4], the n e i g h b o u r h o o d filter of x in the level-toplogy t=(A) is given by *.o,("//'(x)) = {v-1(]/3, 1]); v e 'F'(x),/3 e [0, 1 - a [ } .
2.3. a-cuts of a function. G i v e n a function tz : X - - ~ / , /2 : 11 ~ 2 x by VaEIl:
ff.(a)=tx-l(]a,
we can define a m a p
1]),
and this m a p is always decreasing. It has also the p r o p e r t i e s ~
~
til~
and
VxeX:
tx(x)=l
or
~(x)
2.4. A left-inverse for ". G i v e n now a function to : I1 ~ 2 x, we can define a m a p to*:X---> I by to*(x) = sup{a; x e to(a)} = sup a 1,o~(x)
(sup ~ = 0).
OtEl I
As s u p { a ; x e to(a)} = m a x { a ; x e t0(a)} ¢~ sup{a; x e to(a)} e { a ; x ~ to(a)}, it follows i m m e d i a t e l y that, if t o * ( x ) ~ 1, we have to*(x) = m a x { a ; x e to(a)} ¢:> x ~ to(to*(x)). M o r e o v e r , the following relations derive i m m e d i a t e l y f r o m the definitions:
(a) Vto : I , ---, 2 ×, Va ~ I, : ~'~(a) = U~>,:, to(/3). (b) V/z : X ~ I : (t~)* = ~. (c) For any to : I1 ~ 2 x there exists a decreasing to': 11 ~ 2 x such that to'* = to*. Indeed, it is sufficient to take to'= ~ ' . 2.5. Lattice properties. The m a p (2x) t, ~ I X : t o - - ~ to* has the following p r o p e r ties: (a) If tol<~to2, then to*l <~to2*. (b) F o r any decreasing functions tol : 11 --~ 2 x, to2 : I1 --~ 2 x we have ( ~ ^ to2)* = to* ^'#2".
P. Wuyts
74
Indeed, (~%^~0z)*~
(~* ^ ~*)(x) ~ (,m ^ ~,~)*(x). (c) F o r any family ~ of functions I~ - + 2 x, ~#o = sup,~a,~, we h a v e q~* = (sup ~0)* = sup q~*. Indeed, if q~*(x)=X, w h e t h e r xeq%(A) or not, there exists for each A ' < A a )t'<~ X"~< A such that x ~ ~0o(~"), and hence a % , e q~ such that x e q~x,,(h"), w h e n c e ~*,,(x) ~ X " >~ X' and t h e r e f o r e sup,o~, ~ * ( x ) >/A'. By arbitrariness of h' < X it follows that s u p , ~ , q~*~ ~0o, while the o p p o s i t e , i n e q u a l i t y is trivial. (d) If ~1 : I , - - - . 2 ×, @z:I,---~2x, @', = ~ , ~% = ~ ' , then @ * A ¢#~ = (@lI A @;)*.
This follows at once f r o m (b) and the r e m a r k s in 2.4. 2.6. (LO)-property. W e shall say that a m a p ~o : I~ ~ 2 x has the ( L O ) - p r o p e r t y if it satisfies the following condition: (LO): V x e X : ~ 0 * ( x ) = l or q~*(x)
x¢ ~(x) = ,~(,~*(x)). (c) If q~ is a family of functions having the ( L O ) - p r o p e r t y , then ~0---sup,o~, ~o also does. Indeed, if q~*(x) = 1 then once m o r e there is nothing to prove. If ~o~(x) = a < 1, then q~*(x)<~a for all q~ e qb. If then we had x e ~0o(O~), t h e r e should exist a q~ c with x E ~o(a), so ~0*(x) ~ ~, hence ~0*(x) = a and x c ~o(q~*(x)).
3. Fuzzy topologies associated with a family of topologies In this section we will derive several fuzzy topologies f r o m a given family 1~ of topologies on a set X. Only one of t h e m will show to have level-topologies that are closely related to the given ~: and lead finally to the solution of the main p r o b l e m in this paper. B e t w e e n the others the first o n e h o w e v e r s e e m s to play a role in the case of fuzzy n e i g h b o u r h o o d spaces (cf. 5.2(b)), and the remaining ones are m e n t i o n e d b e c a u s e of the analogies with the first one.
Determination of fuzzy topologicalspaces by level-topologies
75
3.1. L e m m a . If ~; = { ~ ; a ~ I~} is a family of topologies on a set X, and if qr)0(~) = I(0 ~ l-I g r ; (0:1,---> 2 x decreasing}, then 4)~(o*) = ((0*; (0 e 4)o(~)}
is a fuzzy topology on X.
Proof. If (0 = X (constant function) we have (0* = 1, while for c e I~ we can define q~ by ( 0 ( a ) = X if a e [ O , c [ , (0(x)=0 if a ~ [ c , l [ . Then ( 0 " = c , and so (/)*(~-) contains all the constants. T h e rest of the proof follows immediately from 2.5(b) and 2.5(c). = {ff,,;a c I,} is a family of topologies on a set X, forming a 3.2. Lemma. If descending chain in the sense that
O~a~/3
=> g r a c g - ,
(3.1)
and if
4),(97) = 1-I 3"-,,c{(0; (0 : I, ----)2x }, (xEll
4)2(9;) = {(0 ~ (/),(9;); (0 satisfies (LO)}, then cI)*(~;) and cI)*(~) are fuzzy topologies on X. Proof. That both qb*(9;) and ~ * ( ~ ) contain the constant functions follows from the proof in 3.1. Further, (a) q~*(~:) is a fuzzy toplogy. If (0* = (0"A(0" with (01 c q~l(~:), (02~ qb:(~), then, with the notations of 2.5(d), we have (0* = ((0~ A (0~)*, while (0~ ~ qb:(~:), (0~ ~ qbl(,~ ) by the chain-condition on ~ , and therefore also (0~ A (0:z~ q51(~:). If qb c q)a(.~) then clearly (00 = s u p ~ o (0 (/h(~:) and (00*= sup~o~ (0*. (b) q)*(9:) is a fuzzy toplogy. With the same notations as in the preceding case, as (01 and (02 now have the (LO)-property it follows from 2.6(a) and 2.6(b) that (0~A(0~ also has that property, and on ground of 2.6(c) this is also the case for (00.
3.3. Proposition. If ~ = { ~ ; a c I,} is a family of topologies on a set X, and if q~(o~) = / c( 0~
I-I ~-,,; (0:1,--* 2 x is decreasing and satisfies (LO)}, ¢tEl I
then 4 ) * ( ~ ) = {(0*; (0 ~ 4)(s*)} is a fuzzy topology on X. If moreover a(~:) = {~ ~ IX; V a e 11 : ~ - ' ( ] a , 1]) e gr}, then A(~;) = cI)*(~;), and this fuzzy topology is the finest fuzzy topology A on X whose level-topologies are coarser than the given ~;, i.e. which is such that we have Va~Ii:
i,,,(A) c 3",~.
(3.2)
76
P. Wuyts
Proo|. That zl(~:) is a fuzzy topology with the quoted extremum property is trivial. That also qb*(~) is a fuzzy topology follows from 2.5(b), 2.5(c), 2.6(b) and 2.6(c). If now ~ ~ Zl(~), then ¢ --/2 ~ ~(~'). As ~* = ~, it follows that zl (~') c ~*(~-). Conversely, take ~ o ~ ( ~ ) , a ~ I ~ . If x ~ o * - ~ ( ] a , 1]) then ~*(x)>ot and thus x ~ ~o(/3) for some /3 > a ; hence also, as ~o is decreasing, x ~ q~(a). If x ~ ~o(o~), we have ~o*(x)~ot. If ¢ * ( x ) = 1, then x ~ ~o*-~(]a, 1]); if ¢ * ( x ) < 1, then x~ ~o(~o*(x)), whence ~o*(x):/: a, i.e. ¢ * ( x ) > or. All by all we have ¢*-~(]a, 1])= ¢(ot)~ ~ r and therefore ~ * ( ~ ) c zl(~). 3.4. Remarks and counterexamples. (a) If St,, = {0, X} for a ~ Q (the other Sr~ being arbitrary), one verifies without difficulty that q~*(S~) contains only the constant functions, and the inclusion in (3.2) is therefore in general an inequality. (b) If we take X = I, if °//'(0) is the neighbourhood system of 0 e N for the usual topology and if =I{0}u{InV;V~'(0) and½~V} 3-= [ { O I U { I R V ; Vc~(O)}
if0~a~½, if ½ < a < l ,
then /z : I --->X, defined by fl-2x tx(x) =~½0
if0~x~¼, if x=½, if x ~ , 1 1 1], ~[U~,
belongs to q~*(3:). Indeed tx = ~o* if ~o is defined by = I[O, ¼]U{~} q~(ot) [ [ 0 , ½ ( I - a ) ]
if O ~ a ~½, if½
But tx~q~*(3:). Indeed, for every ~0Cq~o(3:) such that q~*=~, as ~o must be decreasing and not empty for ½ < o r < l , we must have ½e~o(½), so necessarily
½e q~(~*(½)). This means that in general q~*(3:)~ ~*(3:). (c) If 3: = {~r ; a ~ I1} is a family of topologies on X, we can now define some other families, derived from ~ , by ~l={ff=,< =
[-I ff,~;a~I1}, 0~13<:c~
~ 2 ~-~{~F~.~ =
n ~']x ; a E '1}, o~B~
All these families satisfy (3.1), and so we could, using Lemma 3.2, build up
Determination of fuzzy topologicalspaces by level-topologies
77
eight fuzzy topologies q~*(~k) (i e{1, 2}, k e{1, 2, 3, 4}). We shall see in 5.2 that if,,.< seems to play a role in the case of fuzzy neighbourhood spaces.
4. Fuzzy topologies with given level-topologies In this section we shall prove that under a suitable (necessary and sufficient) condition the inclusion in (3.2) becomes an equality. 4.1. (LT)-families and (LT)-property. If {~-,~; ote Ii} is a family of topologies on X, and if a c I~, G c g r , x e G, we say that {Ga; a
to { ~ r ; ~ I 3 . 4.2. Lelmma. If X, A is a f u z z y topological space, the family {~,~(A), t o i l }
of its
level-topologies has the (LT)-property. Proof. Immediate by considering the level-sets h-l(]a, 1]) for h ~ A, a ~ 11. 4.3. Theorem. If :~ = {ff,~; a ~ 11} is a family of topologies on a set X, there exist
f u z z y topologies A on X having ~; as their level-topologies, i.e. such that Vote/l:
t.~(A) = fie,,
(4.1)
if and only if ~r has the (LT)-property. Moreover, if this condition is satisfied, the f u z z y topology A(:~) = 49*(@-) is the finest of all f u z z y topologies on X for which (4.1) holds. Proof. In view of 3.3 and 4.2 we only have to prove that if,, c ~,,(A($~)) for all a ~ 11. Let therefore G c ~/,~ and x c G. With the notations of 4.1, we put pc(x) = sup{or; ::l(LT)-family (GB),~
U {G~; ,/
is an
P. Wuyts
78
Each family of this kind can be considered as an element 4' of l-I {ff.~; a < 3' < pc(x)}. If then W is the set of all these elements, then sup 4'= (G~)~<~<~(x~
with G~ = U 4,(3")
is the 'largest' (LT)-family for (x, G, ct) on the interval ]a, pc(x)[. Given x, G, a, we shall denote this uniquely determined family by (G~)~<~<~(~). We then define an element q~ ~ ~ 0 ( ~ ) by
if 3 " = a , q~(3") =
-y
if ct < 3, < pc(x), if pc(x) ~<3' < 1,
where the first line can be omitted if a = 0 and the last one if pc(x) = 1. In order to obtain ~ * e q~*(,,~), we have to prove that VycX:
q~*(y)=l
or
q~*(y)
(4.2)
If y ~ G we have ,p*(y) = sup{,/;
y ~ ,p~(3,)} = ,~
and therefore (4.2) is satisfied. If y ~ G we have a ~<¢*(y)<~po(x), and three cases can be distinguished: (a) ¢*(y) = pc(x) = 1, and nothing has to be proved; (b) q ~ * ( y ) = p c ( x ) < l , in which case ¢ ~ ( ¢ * ( y ) ) = ¢ , and (4.2) is satisfied;
(c) ¢*~(y)= 8 < 0 ~ ( x ) . Suppose that, in this last case, y c q~(8). If then (L~)8<~<, is an (LT)-family for (y, q~(8), ~3), the family (L~)=<~
U
~.(3,)cG
Delerminalion of fuzzy topological spaces by level-topologies
79
which proves that G is a neighbourhood of x in L~(qb*(~:)) and therefore, by arbitrariness of G ~ ~ r and x c G, that ~,, c L.(~*(~:)). 4.4. R e m a r k . With the notations of the preceding proof, it is not difficult to show that in fact p6(x) = max{or, q(LT)-family (Ga).
4.5. Counterexample. W e take n ~ N * and define a fuzzy topology A,, on X = [0, 1] as follows: h ~ A n ¢~ (1) 3 a ~ I : h ~ c t , and (2) if x ~ I n Q , x = p / q in smallest terms, then, for some s c N N[0, q~ - 1] we have s/q ~ <~a < ~ h ( p / q ) ~ ( s + 1)/q ~. A straightforward verification shows that An is indeed a fuzzy topology, and it is not difficult to see that A~ is nothing else than the one in [3, {}4]. Taking a fixed a e I~, we have (a) for x ~ I irrational, that h = 1. ~ A and thus h - l ( ] a , 1]) = {x}; (b) for x c l n O , if x = p / q in smallest terms and s ~ N n [ 0 , q " - l ] is such that s/ q ~ <~a < ( s + 1) / q ~, that h = ct + ( ( s + 1) / q n - a ) l . ~ An and therefore again h - ' ( ] a , 1]) = {x}. This proves that for all n e N* the level-topologies of A are all discrete. F u r t h e r m o r e we have k
:=> A , , c A k
while
a=na. is still a fuzzy topology. If then x = p / q ~ y = u/v are both rational and in lowest terms, and if ~0c A, q~(x) = a, ~0(y) =/3, there must be, for each n ~ N * , a 3'n ~ I , an s. ~[0, q " - l ] n N and a w~ e [ 0 , v " - l ] n N such that _
sn+l q.
_
qn and
wn wn + 1 --~<3"n ~ qo(y)=/3 ~ - ~)n i~n Then however - al
~ max(/3 - 3'., a - 3'.) ~
and hence a =/3 by arbitrariness of n, unless {x, y} n{0, 1} ¢ 0. This shows that ~ p e a is necessarily a constant on ] 0 , 1 [ N Q , and the level-topologies of A therefore cannot be discrete. Finally, if Ao is an arbitrary fuzzy topology on I coarser than all ( A ) . ~ . , we necessarily have a 0 c A and therefore the topologies ~(Ao) cannot be discrete.
P. Wuyts
80
From all this it follows: (a) if there exists a fuzzy topology A on a set X, having a given family ~:={~;a~Ii} as its level-topologies (~,~(A)=~,~), this A in general is not unique; (b) the set of all fuzzy topologies, having the given family ~: as its leveltopologies, if non-void, does not contain in general a coarsest element; (c) if A ' c A" are fuzzy topologies on a set X, then Va c 11:
~,,(A ') = L~(A"),
but the converse in general does not hold. Indeed, if
(V(~EII: ~ ( A ' ) c ,
(A")) ~
A'cA"
were always true, this should imply the uniqueness of the fuzzy topology with given level-topologies.
5. Fuzzy neighbourhood spaces with given level-topologies It is known [5] that the level-topologies of a fuzzy neighbourhood space, apart from having the (LT)-property, also form a descending chain. We therefore must at least impose this supplementary condition on $: should ~ * ( ~ ) become a fuzzy neighbourhood space. W e shall prove that this also is sufficient and derive a uniqueness property. 5.1. Proposition. Let ~r = {~r~; a ~ I~} be a family of topologies on a set X. If ~ ( x ) is the set of open neighbourhoods of x in X, ~ , define Ad~(x) by
~(x)={tx:I1---~2x;
p~decreasing and Votell:P~(a)e
(-1
~(x)}.
/3eC0,l-a[
Then the family (~g(x))x~x, where ~ ( x ) = {~ *= sup a l~(~,; tz ~ ~ ( x ) } a~lt
is a fuzzy neighbourhood base on X. Moreover, if we denote by (Xs~(x))x~x the fuzzy neighbourhood system generated by (~;(x))x~x, the level-topologies of the corresponding fuzzy topological space are coarser than the given ~ , i.e. we have V~ et,:
~,,(t(.,v'.)) = ~r,~.
Proof. We divide the proof into three parts. (a) ~ ( x ) is a prefilter base, satisfying (B1). This is immediate. (b) ~s~(x) satisfies (B2). W e take ~ * ~ ~ ( x ) and e > 0. For each z c X we then choose a function ~,:I1--~ 2 x taking t ~ =ix if z = x, and otherwise as follows:
{
/x(~)
if a ~< ~ * ( z ) - e,
g(t~*(z)-e)
if a > / x * ( z ) - e
Determinationof fuzzy topologicalspacesby level-topologies
81
(recalling that we write a - / 3 for ( a - / 3 ) v0). This function is clearly decreasing. Moreover (1) if ot ~
with y~t~(a0)
~*(y) = sup ot 1.¢~)(y)= sup ot 1.,(~)(y)= tz*(y) -~t~ o
ot " ( ~ o
and therefore /z*(z)/x/~*(y) ~
~(e, 0)e
M ~r?~(x) 13~[OA-o[
and /3~ = s u p o ~ ~pl.(~.,). Let us take e ' > 0 such that ~ + 2 e ' < l - c t . Then /z(e', 1 - a - e ' ) e Y ~ ( x ) for /3e[0,1-(1-a-e')[ and in particular p . ( e ' , l - c t - e ' ) ~ ° ~ ( x ) . If now z ~ p.(e', 1 - c t - e ' ) , then / 3 ~ ' ( z ) ~ l - a - e ' > 8 + e ' , so / 3 ~ ' ( z ) - e ' > 8 and therefore ~(z)>~5. This however means that i~(e',l-a-e')~v-~(]8,1]) and hence v-~(]ti, 1])e V,(x). 5.2. Remarks. (a) In the above proposition we supposed the sets /z(a) to be open in each of the (X, ffa) for / 3 ~ [ 0 , 1 - a [ . This condition however was introduced only for technical reasons. It is clear that if we ask only the condition txdecreasing
and
p~(ot)~
["1
°//'a(x),
13~[0.1--a[
(where ~ ( x ) is the neighbourhood filter of x in (X, ~-a)), we obtain a collection ~ ( x ) for which A/g(x)c A~s~(x)c [~g(x)], so that ~(x) = ~;(x) if ~ls~(x) = {tL*; t~ ~ Ats~(x)}. (b) If we define the topology ff~ on X by ~-~ = f-)0~<~ ff~, the condition
P. Wuyts
82
p.(a) ~ ~a~[o.l-,~[ ~ a ( x ) means exactly t h a t / x ( a ) must be a neighbourhood of x in
(x, ~rL,). 5.3. Proposition. Let ~ = { ~ ; a ~ 11} be a family of topologies on a set X, having
the (LT)-property and forming a descending chain, i.e. such that
0~o~
~
~-e=~r.
Then the family of level-topologies of (X, t(W~)) is precisely ~, i.e. Vot~I~:
~,~(t ( J ~ ) ) = ~,~.
Proof. We have only to show that ~ r c ~(t(A%)). In order to do this, we take G ~ if',, x ~ G. By the (LT)-property of ,~ we can find a ~ > a, H ~ 3-~ such that x 6 H c G, and by the descending chain condition this implies that H~°l/'~(x) for /3~<& W e then define / x : I t - - ~ 2 x by ~ ( A ) = X for A < I - 6 and g , ( A ) = H for 1 - 6 ~ < A < 1 . Then immediately /z~A/~(x), hence ~ z * 6 ~ ( x ) . But p , * I H = 1, P~*IX \ H = 1 - 6 , whence
H = ~*-'(]1 - & 1]) = ~,-1(]/3, 12) if /3 = 1-¢3 < 1-c~. So H e ~,(N~(x)) and therefore also G e ~,(N~(x)). 5.4. Proposition. If (~(x)),~x is a fuzzy neighbourhood system on a set X and t(°l/) the associated fuzzy topology, if ~ = {or = ~,~(t(W)), a ~ It} is the family of its
level-topologies and (./¢s~(x)),~x the fuzzy neighbourhood system derived from it, then the two systems (°//(x)),~ x and (N~(x)),~x coincide; moreover, the fuzzy topology they generate is precisely ~ * ( f f ) , i.e. the finest fuzzy topology having ~; as its family of level-topologies. Proof. W e use the same notations as in 5.1 and 5.2, and divide the proof into four parts. (a) ~ ( x ) = N ~ ( x ) . Indeed, if v ~ ( x ) , we take /x = 13, so
~(,~) : ,,-'(]o~, 1])~
f3
~(x)
for a ~ I b while ~z is decreasing. Therefore v = tz* ~ [~(x)] = N~(x). (b) Representation of elements of A,~(x). We consider an element p. ~ A~s~(x) and take e > 0 . Then: (1) As ~ ( a ) c La(~(x)) for each/3 ~ [ 0 , 1 - a[, we can, by [5, Theor. 6.4] for each /3 ~[0, l - a [ find a X0 a [0, 1-/3[ and a v~ ~ °//'(x) such that ~z(a)= v-~l(]Xa, 1]). If now A~ < a , we can define v ; by
Ivy(z) u~(z)=[½(ol+l_/3)
if v~(z)~)to or v ~ ( z ) > a , if A a < v ~ ( z ) ~ < a .
Then v'~>~vo, so v'~°V(x), and m o r e o v e r ~z(a)= v ~' - l ' t~a, " 1]). This means that choosing A~, v~ as indicated above, we can always suppose that X~ I> a. (2) If we take then a / 3 e [ 0 , 1 - a [ satisfying ( 1 - / 3 ) - ~ < ½ e , we obtain
Determination of fuzzy topological spaces by level-topologies
83
a <~)te < 1 - / 3 < ot +½e for the c o r r e s p o n d i n g ha. This h o w e v e r m e a n s that for each o~ e I1 we can c h o o s e a u~ e ~ ( x ) and an a ' such that
a<~a'
and
I z ( a ) = v ~ , ~ ( ] a ',1]).
(5.1)
( c ) ) ¢ ' ~ ( x ) ~ ' V ( x ) . If p . * = s u p , ~ , c~l.t,~) with /z e./,~(x) and if e > 0 , we have for each at an a ' and a v,~ ~ ~V(x) satisfying (5.1). W e can choose a~, a2 . . . . . an+x such that 0 :
O~ 1 ~ 0 ~ 2 ~ "
Vke{1,2,..
" " <~Ot n ~O~n+ • ,
n+l}:
1 :
1,
ak--Ot~_~
<1
~e,
and put u~ =
inf u~ ¢ °V(x).
l~k~n
W e then have: (1) if z e ~ ( a ~ ) ,
then ~ * ( z ) > ~ o t , > l - ~ e , h e n c e v ~ ( z ) - e ~ l - e < ~ ¢ * ( z ) ; n - l } , then
(2) if Zelx(Ctk)\p.(ak+~) for k e { 1 , 2 , . . . , hence
v~ ( z ) - ~ ~ v ~ + , ( z ) - e ~ a ~ , + , - e < ak ~ t~*(Z); (3) if Z~ ~ ( a l ) , then v~,.(z) ~ a'l <½e, h e n c e v~ (z) - e ~ v,~,(z)- e ~ p~*(z). All by all this m e a n s that we have for each e > 0 a v~ e°V(x) such that v~-e~*, and therefore s u p ~ , , ( v ~ - e ) ~ * , h e n c e ~ * e ~ ( x ) , which proves that N~(x) c ~V(x). (d) t(W~) = t(°V) = qb*(~). As t(°V) is a fuzzy t o p o l o g y having ,~ as its family of level-topologies, we already have t(°V)c ~ * ( ~ ) . W e therefore have only to p r o v e that for each ~0 e qb(~), A = ~o* the function h is o p e n in (X, t(°V)). If h ° denotes.its fuzzy interior in (X, t(°V)) it is therefore sufficient to prove that for each x c X and each e > 0 we have h°(x) >1h ( x ) - e. T o do this we o b s e r v e first that by the first part of the t h e o r e m , ~lg(x) as defined in 5.1 is a basis for °V(x) and therefore )t°(x)=
sup
inf h v ( 1 - p . * ) ( y ) .
If now A(x) =/3 ~< e, we immediately obtain A(x) ~<)t°(x) + e. If on the contrary /3 > e, we define ~ : I~ --~ 2 x, recalling A = ~0*, by
/z(a)=
X
ifa~l-/3+e,
~(/3-e)
if a > l - / 3 + e .
T h e n /.L(a) e °V°,(x) for all a ' < 1 - a. In fact, this clearly h a p p e n s if a ~ 1 - / 3 + e, while for a > 1 - / 3 + e, we have a ' < 1 - ct < / 3 - e and so iz(a) = ~o(/3 - e) e F°,(x) since a ' < / 3 - e. For y e X, two cases now can occur: (1) A(y) I>/3 - e, h e n c e A v ( 1 - / z * ) ( y ) ~ > A ( x ) - e; (2) A(y) < / 3 - e, h e n c e y~ ¢(/3 - e). If then a ~< 1 - / 3 + e, we have a l~,(,~)(y) = a, hence 1 - a l ~ , t , ~ ) ( y ) = l - a > 1-(1-/3+e)=/3-e, while ff a > 1 - / 3 + e we have
84
P. Wuyts
al~(~)(y)=O, hence 1 - a l . c , , ) ( y ) = l . In any case ( 1 - t x * ) ( y ) = l - t z * ( y ) = inf,,~l,(1 - a 1.(,~(y)) = [3 - e and therefore )t v (1 - ~*)(y) >~/3 - e = )t(x) - e. All by all this means that for this particular /x we have inf A v ( 1 - ~L*)(y) ~>X(x)- e
y~X
and this ends the proof. 5.5. Theorem. Two fuzzy neighbourhood spaces X, A 1 and X, A2 on the same set X coincide, and hence derive from the same neighbourhood system, iff their families of level-topologies coincide, i.e. if for each et • Il we have ~,~(Al) = r~(A2). l ~ o o t . If ~: = ~,~(A1) = t,~(A2) and (T'k(x)).Ex(k e{1, 2}) are the neighbourhood systems defining Ak (k e{1, 2}), we have indeed T'~(x)= g'~(x)=T'2(x) for each xcX. 5.6. Theorem. If ~; = {ff ~ ; a c I~} is a family of topologies on a set X, there exists a fuzzy neighbourhood space (X, t(~)) determined by a fuzzy neighbourhood system (~(x))x~x and having ~ as the family of its level-topologies, if and only if the family if; has the (LT)-property and is a chain in the sense that ct < [3 implies ~ r ~ r a. Moreover, the fuzzy neighbourhood space, and so the system ('//'(x))x~x, is unique and it is the finest of all fuzzy topologies having the family ~; as their family of level-topologies. 5.7. Remark. The foregoing does not allow to conclude that a fuzzy topological space is a fuzzy neighbourhood space if and only if its level-topologies (which have automatically the (LT)-property) form a descending chain. The counterexample in 4.5 shows that this is not true: (X, I x) is a fuzzy neighbourhood space, but there are infinitely many fuzzy topologies having all their level-topologies discrete, and on account of 5.6 only one of them is a fuzzy neighbourhood space! 5.8. Conclusion. Considering the set Fuzx of all fuzzy topological spaces (X, A) with support 32, we can consider the relation AI~A 2 ~
V o ~ / I : i , ~ ( A 1 ) = I , ct(A2) ,
which is clearly an equivalence relation. The foregoing shows that in each equivalence-class their is always a finest element and at most one fuzzy neighbourhood space, which, if present, is precisely the finest element of its class. In view of the 'good' properties of fuzzy neighbourhood spaces one could wonder if more generally the maximum elements of these equivalence classes form an interesting subcategory of the category of fuzzy topological spaces. As another consequence of T h e o r e m 5.5 one could try to describe properties of a fuzzy neighbourhood space by equivalent properties of its level-topologies. This can e.g. be done for most of the separation properties in [7]. A further study of these properties in this sense will be the subject of a forthcoming paper.
Determination of fuzzy topological spaces by level-topologies
85
References [1] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980). [2] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976) 621--633. [3] R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446--454. [4] R. Lowen, Convergence in fuzzy topological spaces, Topology and Appl. 10 (1979) 147-160. [5] R. Lowen, Fuzzy neighborhood spaces, Fuzzy Sets and Systems, 7 (1982) 165-189. [6] C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis (Birkh~iuser Verlag, Basel, 1975). [7] 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.