It is the purpose of this paper to put the "fuzzy real line" in a setting which proves to be advantageous to a more fun- damental study of that space. Actually there ...
manuscripta math. 39, 2 9 3 -
309 (1985)
manuscripta mathematica 9 Springer-Verlag 1985
THE ORDER A S P E C T
OF THE
Robert
FUZZY
REAL
LINE
Lowen
It is the p u r p o s e of this p a p e r to put the "fuzzy real line" in a setting which proves to be a d v a n t a g e o u s to a more fund a m e n t a l study of that space. A c t u a l l y there are three d i f f e r e n t fuzzy real lines to be found in the literature, m a i n l y d e f i n e d by U. HShle in [1], [2] and by B. H u t t o n in [4], [5], and a fourth one shall be a d d e d in this work. The m a i n result of this paper is the fact that three of the four spaces are h o m e o m o r p h i c to fuzzy t o p o l o g i c a l spaces the u n d e r l y i n g sets of w h i c h are, in each ease, the p r o b a b i l i t y m e a s u r e s on ~, and the fuzzy (resp. quasi fuzzy and translat i o n - c l o s e d fuzzy) topologies of w h i c h are d e t e r m i n e d by the left and right sections of a c a n o n i c a l fuzzy e x t e n s i o n of the strict order r e l a t i o n on ~. From this it will follow very f u n d a m e n t a l l y that it is the order of ~ and not the topology, w h i c h determines the fuzzy real line.
1. P r e l i m i n a r i e s The unit
interval
is d e n o t e d
If X is a set and A c X then A is d e n o t e d
1A.
then by f(x+)
If
f
and f(x-)
I, I 0 := ]0,1]
and
11
the c h a r a c t e r i s t i c
is any real we denote
valued
right
:= [0,1[.
function
function
and left
of
on
limit of
f
in x. Given stand fima stant
a set X, by a quasi a collection
of fuzzy
and a r b i t r a r y
suprema.
fuzzy
is stable (~+~)
fuzzy
sets we call
tion
closed
H~hle
[1],
fuzzy
(~-~)
A on X [9] we under-
sets on X, closed If m o r e o v e r
it a fuzzy
for t r a n s l a t i o n s ,
^ 1 e A and
topology
i.e.
topology
[1].
[9] and if it
it a transla-
In the t e r m i n o l o g y
[3], this is a p r o b a b i l i s t i c
293
in-
all con-
for all ~ e A and ~ e I :
v 0 e A, then we call
topology
for finite
A contains
topology
of U.
for the case
LOWEN L = [0,1] If TOP that
(see
[1]
for n o t a t i o n s ) .
denotes
the
category
of f u z z y
topological
l : FTS ~ TOP troduced called
in
the
and
I
of t o p o l o g i c a l
spaces
[9] and
[11].
The
metrizable
the B o r e l
on X and
o-algebra
M(X)
and the c o l l e c t i o n
res
is d e n o t e d
rate
if t h e r e
x e B.
D(X).
Throughout
If Y e B(X)
for w h i c h
map
it has
then
defined
this
P(Y)
tight
such
that
P(K)
ped w i t h
its n a t u r a l ,
i.e.
1A by
: M(X) Since
by its
collection
P
x"
of all
from
for all
[12].
discrete,
(1.1) [18].
~ e I 0 there
> 1 -s
Borel
exists
P e H [19].
If {0,1}
: X ~ {0,1}
identifying
on f.i.
isomorphism these
two 6A
extend
: P ~
is e q u i p -
a-algebra
is m e a s u r a b l e
then and
(1-P(A)) P0 + P ( A ) P 1
any p r o b a b i l i t y
value
is a n a t u r a l
on M(X),
1A
~ M({0,1})
@ : M({0,1})
We now
is d e n o t e d
if
an e x t e n s i o n
(1.1).
mined
the map
degene-
: X ~ Y is a m e a s u r a b l e
if for all
notions
has
is c a l l e d
measu-
= 1 if and o n l y
such a m e a s u r e
If f
The
is d e n o t e d
probability
P(B)
the
B(X)
extension
some b a s i c
for any A E B(X)
in-
(X,ICA)) is
on X.
on B(X)
P e M(X)
that
MY(x)
We r e c a l l
thus
the t o p o l o g y
that
functors
space
: M(X) ~ M(Y) -1 = P ( f ( B ) ) for all B e B(Y)
is c a l l e d
K c X, c o m p a c t ,
the
t h e n we d e n o t e
measures
= 1.
a natural
by f ( P ) ( B )
H c M(X)
work
t h e n we d e n o t e
P e M(X)
space,
x e X such
FTS
of A.
of all d e g e n e r a t e
Recall
exists
topological
T(X)
of all p r o b a b i l i t y
and
: TOP ~ FTS,
denote,
modification
If X is a s e p a r a b l e
collection
t h e n by ~
: FTS ~ TOP we
topological
spaces
on
between
sets,
: M(X)
{0,1}
is d e t e r -
the m a p p i n g
~ I : (1-~)P0 + ~ P 1
the t o p o l o g y
generated
{I},
measure
(1.2)
M({0,1})
~ ~
and
I.
(1.3) Then~
of X to the
294
upon
becomes,
~ I : P ~ P(A)
by
(1.2)
fuzzy
(1.4) topology
A(X)
LOWEN
E(x) The
space
tension again SMS
(M(X)~A(X))
:: {6GIG e T(X)}
is d e n o t e d
of a c o n t i n u o u s
continuous,
denote
the
subcategory
ble m e t r i z a b l e THEOREM
1.1.
map
f
we o b t a i n
simply
(1.5) M(X).
Since
: X ~ Y to f : M(X)
a functorial of T O P
the ex-
~ M(Y)
relationship.
consisting
of all
is Let
separa-
spaces.
T__he .COrrespondenc_~e E x t t o p
: SMS ~ FTS
defined
by Exttop(X)
:= M(X)
on objects
Exttop(f)
:= f
on morphisms
is a c o v a r i a n t
functor.
In
shown
[12]' it was
by ~
: X ~ M(X) ~ shall
2. T h r e e
fuzzy
to
as
The
"fuzzy
all n o n equal
real
real
to
= ~(x-)
"natural" by the
are
is t h a t
quasi
three
equal
l(x+)
fuzzy
as
on X a n d
which
are
referred
[4].
to
1.
Let
on ~
Let - be
I ~ B if and o n l y
topology
for all
R denote
with
infimum
the
eauivalen-
if
x e ~.
on RI ~ is the one
Then
the
generated
subbasis
for e a c h
x E ~
Lx([l]) In the
Let n o w
this D(~)
let
and
~(I)
quasi denote
B be the
and
usually
fuzzy the
U {RxlX e ~ }
[I] E RI~
:= i - l ( x - )
literature
ped with
and
spaces
functions
= ~(x+)
{LxlX e ~} where
of s t r u c t u r e s
map.
of B. H u t t o n
real-valued
R defined and
in M(X)
lines
supremum
on
this
embedded
line".
space
0 and
Regardless
denote
there
increasing
ce r e l a t i o n l(x-)
always
literature
coarsest
X is c a n o n i c a l l y
: x ~ Px"
on M(X),
In the
that
Rx([l])
denotes
:= l(x+).
the
space
RI~ e q u i p -
topology.
set of all
collection
distribution
of f u z z y
by
295
sets
functions
on D(~)
on
defined
LOWEN B :: {LxlX e ~ }
where for each x E m Lx(F) Let D0(R)
u {RxlX era}
and F E D(m)
= P(x)
and
Rx(F)
= 1 -F(x+).
denote D(m) equipped with F0(~)
:= the quasi fuz-
zy topology generated by B, then we have the following result, the proof of which is straightforward. PROPOSITION ~angnica!
2.1.
m(1)
and D0(m)
hgmepmorphism
: F ~ [l-F].
In view of the fact that D0(m) we shall refer to D0(m)
The in-between
is a more natural model than as the Hutton fuzzy real line.
space we obtain using the technique
rating a quasi fuzzy topology obtain a fuzzy topology suggested
in [21].
[9],
for the constants [14].
this work,
in order to
set again
:= the fuzzy topology
gene-
We shall denote this space simply D(~) and,
in
refer to it as the fuzzy real line.
The finest space is that of U. H6hle on D(m) the structure topology
of satu-
This space has also been
Thus we take as u n d e r l y i n g
D(~) but now equipped with F(m) rated by B.
and the
is given b~
D0(~) ~ ( I )
~(I),
ate h o m e g m g r p h i c
FI(~)
g e n e r a t e d by B.
[1],
[2] who considers
:: the t r a n s l a t i o n
closed fuzzy
This space we shall denote Dl(m)
and we shall refer to as the H6hle fuzzy real line. Remark that, D+(m)
U. H6hle usually only considers t h e
of D(m) consisting
for which
F(0)
of those d i s t r i b u t i o n
subspace
functions
F
= 0.
Eaeh of these three
spaces can be viewed as the "model" of a
certain concept of "fuzzy real line ~' in its c o r r e s p o n d i n g category. The results arguments,
of the fourth section will reveal apart from those of
maybe more natural
and canonical
D0(m).
296
fundamental
[16], why D(m) and Dl(m) spaces to consider than
are
LOWEN 3. The probabilistic
aspect
In classical probability
theory on ~ there is advantage
being able to work with both distribution bability measures on ~.
in
functions and pro-
We shall therefore translate the
concepts of the fuzzy real lines into the language of probability measures on ~. Let ~ denote the usual bijection from M(~) to D(~) i.e. : M(]R) -~ D(IR) : P -~ o.
p(P,R) then
For all
For all
v p(Q,P)
1 ~ This
since
:
p(P,R)
sup
3~ ( ^ - l i n e a r i t y ) p(P,Q)
P,Q e M(~)
1
~
2 ~ (weak A - t r a n s i t i v i t y ) p(P,Q)
For all
P is open,
4.2
[]
i.e.
it is
x M A ( ~ ). of the
definitions
topology.
consequence D
p not
is a set
theoretical
order
only
relation
304
on ~
but
also
a
LOWEN topological
one.
We shall now prove that AA(~) is indeed an "order fuzzy topology. Let us denote the left and right sections of p by pI(P)
: M(It) ~ I : Q ~ p(Q,p)
(4.10)
pr(P)
: M(IR) ~ I : Q ~ p(p,Q)
(4.11)
and the fuzzy topology generated by {p!(P)IP e M(Iq)} of all sections,
u
{pr(p)Ip e M(I9)}
(4.12)
by ^
