Internat. J. Math. & Math. Sci.
VOL. 12 NO.
(1989) 47-60
NON ARCHIMEDEAN METRIC INDUCED
FUZZY UNIFORM SPACES
R. LOWEN
A.K. SRIVASTAVA"
P. WUYTS
Wiskundige Analyse
University of Antwerp, R.U.C.A.
Groenenborgerlaan 171
2020 Antwerpen, BELGIUM
(Received April 7, 1988)
ABSTRACT.
It is shown that the category of non-Archimedean metric spaces with l-Lip-
schitz maps can be embedded as a coreflectlve non-bireflective subcategory in the cate-
gory of fuzzy uniform spaces.
f’orm
Consequential characterizations of topological and uni-
properties are derived.
KEYWORDS AND PHRASES. Non-Archlmedean, coreflective, completion, fuzzy uniform space.
1980 MATHEMATICS SUBJECT CLASSIFICATION. 54E15, 54A40, 46P05.
I. INTRODUCTION.
We show that the category NA(1) of non-Archimedean metric spaces with metric boundand with morphisms the non-expanslve maps is coreflectively embedded in the category FUS of fuzzy uniform spaces [4], [9] in an extremely simple and natural way.
FNS [5] each space in NA(1) then moreover determines
Through the forgetful functor FUS
a non-topologically generated space in FNS, the topological modification (i.e. TOP-coreflection) of which is nothing else then the metric topology. This means that the dlaed by
gram
NA(1)
’emb’edd!ng
> FUS
forgetful
v
functor
TOP <’
is commutative.
coreflecion
v
forgetful
functor
FNS
From a local point of view, an interesting aspect of this situation is
that given (X,d) we can study this space first with the concepts available in FUS [4],
[9] and with those available in FNS [5], [6] before going to the topological space
permanent address
Banaras Hindu University
Department of Mathematics
Varanasi 221005
INDIA
R. LOWEN, A.K. SRIVASTAVA AND P. WUYTS
48
(X,T d) associated with (X,d).
E.g. we can "forget" at intermediate stages.
From a global point of view embedding NA(1) in FUS also seems natural. E.g. the functot NA(1)
TOP does not preserve products. NA(1)., although being coreflectively embedded in FUS, is not bireflectively embedded, in particular the embedding does not
preserve products, but it are precisely the products in FUS and in FNS which are mapped
onto the topological product in TOP (TOP is both coreflectively and bireflectively embedded in FNS). Thus in order to have a more faithful relation with TOP it seems suitable to consider NA(1) as a subcategory of FUS. In particular we further study completeness of NA(1)-objects in FUS, and we also give a fairly complete account of the most
important topological properties of NA(1)-obJects in FNS.
2. PRELIMINARIES.
Most notions used are standard, we just recall some notations and some concepts
specific to the context.
As always R+ stands for the strictly positive real numbers, I := [0,I],
0
I 0 :ffi ]0,i] and I := [0,I[.
If X is a set and A c X,
stands for the characteristic function of A.
and
IXxX then
I X is given by <A>(x)
sup A(y) A (y,x). If
XX is
we simply put
and obviously <x>(y)
o
1A
<>
(x,y).Yiso
<x>
o (x,y)
Further
sup (x,z)
^
I
IX
If A
l{x
we
given by
(z,y).
stands for the prefilter
stands for the prefilter [{ sup
If d is a pseudometric on X then we put T and U resp. the associated topology
d
d
and uniformity.
If d fulfils the strong (or ultrametrlc) triangle inequality we call it a nonArchimedean pseudometrlc.
The functors
U,
i,
t,
’ u are well-known
[3], [4], [5] but we recall the functors
UNIF determined by
FUS
U,
and t
T
FTS where then t(]) stands for the fuzzy topology associated with ] and
TOP where then T(U) stands for the topology associated with U. For a pre[3], we recall also that its characteristic value is given by
FUS
UNIF
filter ]
c()
:ffi
inf sup l(x)
eF xX
Ila
inf{a
}.
X,])
IFUSI and C is a prefilter on X then it is called a hyper Cauchy prefilter
[9] if it satisfies the conditions
If
(HCI)
(HC2)
(HC3)
In
c(C)
i
C
v
U
e e
I0
Pe
C
Pe
Pe
e
v.
9] it was shown that for any hyper Cauchy prefilter C, there exists a unique minic C. Moreover, if B is a basis for C and M a basis for
0
mal hyper Cauchy prefilter C
CATEGORY OF NON-ARCHIMEDEAN METRIC SPACES
49
then
{<>I eW’
0
~.
eB)
A fuzzy uniform space (X,U) is called ultracomplete [9] if for each minimal hyper
Cauchy prefilter C there exists x e X such that C U(x) where U(x) :=
which is equivalent to the fact that (X,I ()) is complete.
u
Finally, (X,) is called precompact [9] if it satisfies the condition
(X)
e e I0, ] y e 2
x e X,
sup v<x>
e, which is also equivalent to the fact
xeY
that (X,t (W)) is precompact.
U
3. DEFINITIONS AND FUNDAMENTAL PROPERTIES.
We first put together some elementary technical properties.
LEMMA 3.1.
If X is a set and d
d := i- d X X
d(X,X)
x e X
a.
b.
a non-Archlmedean pseudometric on X, then
I has the following properties
I;
is symmetric;
d
d o d d’
c.
or equivalently
(x,y,z)
2
(x,x)
X
a.
x
b.
is synetric
c.
o
then
3
iXxX
If conversely
d
:=
e
X
3
d(X,Z) ^ (z,y/
d(x,y).
has the properties
1
,
1-
is a non-Archimedean pseudometric on
X for which
d
I.
If d .< I is a non-Archimedean pseudometrlc on X, and if we put
D
{(x y) Id(x,y) < r}
B(x,r)
{yld(x,y) < r},
then
l(]r I])
PROOF.
D -r’
(#d<X>)’l(]r,l])
B(x l-r)
Straightforward.
In the sequel, if no confusion can arise, we simply put
resp.
d.
THEOREM 3.2.
If d
.<
resp. d instead of
.
is a non-Archimedean pseudometric on a set
d
X, then {}, with
:= 1- d, is a basis for a fuzzy uniformity ](d) on X, where
U(d) :=
{,}~
Conversely, if U is a fuzzy uniformity on X, having a singleton basis {}, then this
function
where d:=
satisfies the conditions a, b, c in Lemma 3.1.2
1-@
is a non-Archimedean pseudometric.
,
and therefore U
](d)
R. LOEN, A.K. SRIVASTAVA AND P.
50
The first part follows from Lemma 3.1.1
PROOF.
[4] and an application of Lemma 3.1.2
nition of a basis of a fuzzy uniformity
.
and the second part from the defi-
We now describe the general properties of U(d), where it is .always supposed that d
is a non-Archimedean pseudometric such that d & I.
PROPOSITION 3.3.
for all
U,
2
=
c
a basis for
u,
=(g(d))
is given by {D
X, the neighborhoodfilter
]=,I]} as a basis;
e
u(U(d))
U
=
r
Ire
]=,I]}
and therefore
Ud;
for all (=,x) e I
{B(x,r)Ir
3
I
e
=(g(d))
The following hold
and so
d
T(tu(U(d)))
N=(x)
of x in
t=(t(U(d)))
has
Td;
(t((d))) T( =((d)));
X
e !
5 for all
of
the closure
in t((d)) are given by
and the interior
V
(x)
h
d(y,x).
(y)
inf
(y)
(x)
(y,x),
sup
yeX
yeX
of X by means of balls, i.e. a subset
6 A e t((d)) iff there exists a partition
y c X and a function p
Y
Ox;
4
for all
e
I
PA
(t(U(d)))
7
is Hausdorff iff
8
(X,(d))
is
WT
9
(X,(d))
is
T 2 iff
2
3
4
5
6
=
d(x,y) >
for all x
y;
iff d is metric;
2
d(,y)
for all x
y, i.e. iff d is the discrete metric.
.
#-l(]l-r,l])
((d)) and from
PROOF.
Immediate from the definition of
and the fact that (X,U(d)) is a fuzzy neighborhood space.
From Lma 3.i.3
and
From
u
u,0
This is a know property of general fuzzy uniform spaces.
Immediate from Proposition 2.4 in [5].
If A e t((d)), x e X, A(x)
= and d(x,y) < =, it follows from
A(y)
that
A(y)
=.
inf A(t)Vd(t,y)
=< A(x)
V
d(x,y)
However, we also have
=
hence A(y)
A(y)
D
A(x)
(x)
inf A(t) V
d(t,x)
$
A(y) V d(x,y),
reX
=.
A-l(=)
D{B(x,=)IA(x)
=}, and as each two of these balls are either
y
is a disjoint faX such that {(x,=)lx
identical or disjoint, we can choose Y
we are done.
p := = iff x e y
mily with A-l(=) as a union. Putting Y := U Y
X
e
we have
exists and if
Conversely, if A e I is such that the described
This means that
c
partition
A
which is clearly open in
t(W(d))
I(]=,1]) U{B(X,Px)[X e O(t(U(d))).
Since
is maximal for its level topologies
= If,
]=,I]},
(X,t((d))) is a fuzzy neighborhood space,
[13], and therefore A e t(W(d)).
51
CATEGORY OF NON-ARCHIMEDEAN METRIC SPACES
7
9
is Hausdorff iff
0re]a,l
Dr
is
X.
the diagonal of X
8
, la(t(U(d))
This follows from the fact that, by
This is nothing else than the definition of WT
2.
(X,U(d)) is T iff (x,y) 0 for x y, so iff d(x,y)
y.
for x
2
and d’
REMARKS 3.4. I. If d
are equivalent, the fuzzy uniformities (d) and
U(d’) are nevertheless in general different.
2. In the foregoing it was always supposed that d
metric d on X and
E
+
R0,
,"
( d)
we can define d
Starting from an arbitrary d,
i.
Indeed, given the non-Archimedean pseudo-
we can define a family of fuzzy uniformities.
(I-
which is equivalent to d
A
and
Even in this case the fuzzy
U(d)
E
+ are in general
not equivalent to each other, (some interestuniformities (dE),
where
ing relations will be established in Propositions 3.5 and 4.), e.g. if X :=
then it is well known that d given by
:=
consider
where
d
.-’-
V 0.
0’
E
0
d(x
n )n (Yn)n)
is a non-Archimedean metric on
Yn
:=
(min k x kyk
x
n
=Yn
otherwise
X, and it is easily seen that (X,l(d)) and (X,(d E ,))
E
’.
are not isomorphic if
3. It is evident that the properties of
U(d)
can be obtained from the corresponding
B(d) by replacing everywhere d by d E So for instance, it follows from Proposition 3.4.1 that a basis for
)) is given by {D’l=r < r} where
D’r {(x,y) ld(x,y) < r}.
Since this translation of properties of (d) into properties of (de) is a simple exerones of
tu,a(O(d
cise, while the formulation of the former is simpler, we shall continue to restrict
ourselves mainly to the case d
4. From Proposition 3.3.6
i.
it follows that all elements of
converse however is not true.
t(U(d)) are l-Lipschitz, the
Consequently t(l) is strictly coarser than the structure
A(1) of [7].
PROPOSITION 3.5.
2
’inf+
(dE) c (dE,)
U(d e)
e0
3
=>
E
{i};
sup+ (d)
u(Ud).
e0
PROOF.
V
For 3
e
The following hold
is evident, and for 2
d for all e
eO.
note that by Proposition 3.3.3
sup
(d e)
all e
0 we have 1D E e (de)
ee+0
c
u(Ud).
inf+ U(d)
it suffices to remark that if
for all e
+
eO
we have
U(de
c
then
u (Ud) i.e.
The converse inclusion follows at once upon remarking that for
and that
{De[e
is a basis for U
d-
4. CONTINUITY AND CONVERGENCE
PROPOSITION
f
(X,d)
4.1.
A map f
(X’,d’)
(X,[d))
(X’,U(d’))
is uniformly continuous if and only
is l-Lipschitz, i.e. non-expanslve.
52
R. LOWEN, A.K. SRIVASTAVA AND P. WUYTS
(fxf)-l(’)
Immediate from the fact that # -<
PROOF.
if and only if d’ o (fxf) <. d.
Since in the case of the above result the local character of the Lipschitz condi-
tion has disappeared we reformulate the foregoing result in the general case.
With
Remarks 3.4.2 and 3.4.3 in mind, the proof is obvious.
(X’,U(d’)
(X,U(d))
A map f
ee’
(X’ d’) is e-locally --Lipschitz.
COROLLARY 4.2.
(X,d)
f
is uniformly continuous if and only if
m
For concepts and results concerning convergence we refer to [2], [3].
PROPOSITION.
If F is a filter on X then F
4.3.
<x>
lim (F)
in
x in (X,T
d)
if and only if
(X,t(W(d))).
As Theorem 5.3 in [7].
PROOF.
In spite of Remark 3.4.2 in special cases the spaces (X,(d)) and (X,(d ’)) can be
isomorphic.
If X is a non-Archimedean normed space then all
PROPOSITION 4.4.
(X,(de)),
e
e+0,
are mutually isomorphic.
As Theorem 5.2 in [7].
PROOF.
5. COMPACTNESS.
For concepts and results on compactness and precompactness we refer to [6], [9].
THEOREM 5. I.
2
3
The following are equivalent
(X,U(d)) is compact
(X,(d)) is precompact
(X,d) is totally bounded.
PROOF.
If Y
c
The implications 1
=>
=>
2
3
<x>
xeY
our claim.
REMARK 5.2.
Since for any e e
+
0
To show 3
=>
1
let
e
I 0.
U B(x,E) then we have inf d(x,t) <
for
xeY
xeY
> i
which by Theorem 2.2 in [6] proves
X is a finite subset such that X
all t e X which is equivalent to sup
are trivial.
we have that
de
is totally bounded if and only if
d is totally bounded, it follows from the foregoing result that either all spaces
(X,(d)) are compact
or none of them is.
6. COMPLETENESS.
For concepts and results concerning completeness and completions we refer to [9].
The following result is an immediate consequence of Theorem 4.5 in [9] and Proposition
3.3.3
THEOREM 6.1.
2
The following are equivalent
(X,d) is complete
(X,W(d)) is ultracomplete.
Given
(X,d)
I.
(,)
II.
(,U())
III. (X
we can now construct the following completions.
the metric completion of
U
the ultracompletion of
Ud))
(X,d)
(X,(d))
the ultracompletlon of
(X,u(Ud))
CATEGORY OF NON-ARCHIMKDEAN METRIC SPACES
53
Then we obtain the following collection of complete or ultracomplete spaces.
(tu((d)))
IV.
The complete space
V.
The complete space (X
VI.
VII.
The ultracomplete space
(X,U()).
The ultracomplete space
(X,u(U)).
U^ U
(Ud)))
Now it follows from [9] that
(,u(U)) and (X,uiUd)) are isomorphic
(,U), (,tue(d)) and (X*, u(uUd))) are
2
isomorphic
(,U())
Using the methods of [9] it can be shown conceptually that the remaining spaces
(,Ud))
and
are isomorphic too.
However, we prefer to explicitly describe the isomorphism which at the same time allows
too.
us to describe the points of
space (X,d), its metric completion
Given the non-Archimedean
with the metric
(,)
defined by
and
representatives of
can be considered
nlim d(xn,yn ), where (Xn) n and (Yn)n are arbitrary
respectively
(,U[d)),
where Ud)
The ultracompletion [9] of (X,U(d)) is given by
nimal hyper Cauchy prefilters on
rated by
(,)
of all equivalence classes of equivalent Cauchy sequences in X, equipped
as the set
{},
(X,U(d)), and
is the set of all ml-
where
is the fuzzy uniformity gene-
this function being defined by
(Cl,C 2)
inf{El]
EeC 1
O C
2
+}.
U
LEMMA 6.2. If (Xn) n is a Cauchy-sequence in (X,d), the sequence (<Xn>) n converges unit
I
X
llm #(t,xn) which depends only on the equivalence
formly to a mapping ()
of
class
a. sup
(x)(t)
,
() <=
()
c.
<y()>
PROOF.
(’)
If e
>
and which has the following properties
I,
rex
b.
n-
(Xn) n,
y().
nlim d(t,Xn
0 and n
0
is independent of the choice of
then for x e X, p
>. 1-
we know that
is chosen such that
p .>
@(X,Xp)
.
(tn=t) n,
(Xn) n e
If t is the class of the constant sequence
n
o
q
no,
q
>. n o
and then also
either
#(x,xq)
proves the uniform convergence.
(Xp,Xq)
(X,Xp) <
n o ffi>
I-
,
I
E,
(X,Xq) (X,Xp), or
l#(x,xp)-(X,Xq) e, which
and then
so in any case
The property a follows by considering
()(Xn),
and c follow by standard verification.
It follows from the foregoing lemma, that the prefilter r()
{()}~
minimal hyper Cauchy prefilter on (X,(d)), and so we obtain a mapping r
LEMMA 6.3.
quence
(Xn) n
.
’()
and b
is a
If C is a hyper Cauchy prefllter on (X,U(d)) then there exists a Cauchy sein
(X,d) such that
R. LOWEN, A.K. SRIVASTAVA AND P. WUYTS
54
b.
n
e C;
:= sup
k>.n
converges uniformly to y()
n elq
a.
(Yn)n
n n <-
n ]q
c.
<Xk>
+
C, where
e
where l im 0
n n
Pn
is the equivalence class of
0.
It follows from (HC3) that we can find a non-increasing sequence
elements of C such that for all n elN
PROOF.
n
By (HCl) we can find a sequence
8n
(Xn) n
2
Since
(8n)n
+
<_
2
(Sn) n
-n
of
(1)
e
in X such that for all n
-n-I &
(Xn)n;
(2)
8n(Xn)"
is non-increasing it follows that
2
-n-2 <-
8n(Xn+l)
and consequently
2
-n-1 <-
n(Xn)
^ n(Xn+l) #(Xn,Xn+ I)
(Xn)
is a Cauchy sequence.
n
Further by (I) and (2) we have that for all n el and x e X
which shows
Bk(X)
2
-k
.<
<-
.<
Bk(X) A (1-2 "k-l) 2
-k-1
Bk(X) A Bk(Xk) 2
(Xk,k).
-k-1
Thus it follows from (HC2) that for all n
n
Since
(<Xn>) n
:= sup
kZn
converges uniformly to
(HC2) we obtain that
()
C.
#<Xk>
()
the same is true for
(n)n
and thus again by
6.
Finally we still have that for all n
and x,y
X
(Xk,X) V #(Xm,Y)
-n-1
(Xk,X) ^ (Xm,Y) ^ ((Xk,Xm)+2
sup
kn,m>-n
fn(X) A -n(y)
<*
]q
sup
kn,m>-n
& (x,y)
+ 2 -n-l.
We are now in a position to prove the isomorphism result.
THEOREM 6.4.
The map
r
(,u())
(,Ud))
is an isomorphism.
To see that
PROOF
sentatives of
all p,q
n
o
and
d(x
P
r
is into
respectively.
,yq)
let x,y e X
x
and let
Then there exists e
which implies that for any p
>
(Xn) n
0 and n
n
o
o
and
(Yn)n
e
such that for
we have
(9)(Xp)
be repre-
&
-
CATEGORY OF NON-ARCHIMEDEAN METRIC SPACES
whereas lim
()(x n)
To see that
r
I.
Thus
is onto, take C e
r()
It then follows that
6.3.
r
To show that
)
r()
,
()
c
#
()
and therefore
and consider
C, and so
()
r()
r()
55
#
r().
by minimality.
is an isomorphism it now suffices to show that
d
o
(rr)
we first have
d(r(),r())
((()v())
sup
.
as constructed in Lemmas 6.2 and
Since
(()v())(s,t)-,(s,t)).
s, tX
Using distributivity in
(()vC))(s)
the symmetry of
,
(()v())(t)
^
y() =<
()
and the fact that
d(r(),r())
,
we obtain
(()(s)v()(t)-(s,t))
sup
s, tX
and from this it follows that in the end we have to show that for any pair of Cauchy
sequences (x
n)n
and
(yn)n
in
d(xn,Yn)
lim
(X,d)
we have
llm(d(s,t)-d(Xn,S)Vd(Yn,t)).
sup
SteX n
From the ultrametric property we obtain
llm(d(s, t)-d(xn,s)vd(yn, t)
n-o
.<
lim(d(Xn, Yn)Vd(Xn, s)d(Yn, t)-d(Xn, s)d(Yn, t)
lim(d(xn,Yn)-d(Xn,S)Vd(Yn,t))
V 0
n-
lim d(x
n ,yn
I is continuous if I XxX is equipped with the
uniform topology and I with the usual one, we have
sup
I xxX
since sup
while on the other hand
lim(d(s,t)-d(Xn,S)Vd(Yn,t))
s,teX n
lim sup
n s,teX
lim
n
d(xn,Yn).
In order to describe the points of
THEOREM 6.5.
(d(s,t)-d(XnJS)Vd(Yn,t))
in more detail we have the next result.
The following are equivalent
is a minimal hyper Cauchy prefilter on
2
3
(X,U(d))
where 8 fulfils
%
,
a.
8
b.
<O>
8,
c. sup O
1;
is a prefilter with a basis
{nln
e} fulfilling
R. LOWEN, A.K. SRIVASTAVA AND P. WUYTS
56
a.
V n
b.
V n
c.
n
d.
V n
q, 3 xn
Yn(Xn)
Yn Yn +pn where
=> 2
X
$
PROOF.
3
<
Yn+l Yn
<yn > Yn
I
=>
Since
l im
Pn
=0.
was proved in Theorem 6.4 and
2
=>
are obvious, it is sufficient to prove 2
Lemma 6.2, while 2" =>
and
3
n
8 for all n
I. The sequence
We can repeat the construction in Lemma 6.3 with
has the properties a, c and d by the construction in Lemma 6.3. As to b, this
(n)n
follows from
<n>(X)
where
sup
tX
n(t)
sup sup(k(t)^(t,x))
reX kn
(t,x)
V
sup
kn
sup(k(t)A$(t,x))
rex
sup
kn
Uk(X)
sup
kn
<k>(X)
n(X),
n <Xn>"
Since the minimal hyper Cauchy prefilter generated by
{nln
} is coarser than C it
coincides with C.
A characterization of minimal Cauchy filters, probably belonging to the
REMARK 6.6.
folklore of the subject, and with a standard proof which we leave to the reader, is
((X,d) is a pseudometric space)
(X,U d) if and only if F is a filter having a basis
given by the following
on
ing chain of open balls
Bn
B(x
n,rn)
with the
F is a minimal Cauchy filter
(Bn)nl
property~~nlim rn
which is a non-increas-
0.
An alternative me-
thod for proving the isomorphism of
(,U[d))
Theorem 6.5.
as the set of minimal Cauchy filters on
Indeed, we consider
and (X,U(d)) can be based on this and on
(X,d), and
the foregoing then allows a bijection between minimal hyper Cauchy prefilters on
(X,U(d)) and minimal Cauchy filters on (X,d).
7. CONNECTEDNESS
In [8] a number of connectedness concepts in G. Preuss’ sense have been introduced
and studied.
IFTSI
We recall that a space (X A)
does not exist a non-empty proper subset
D-connected if and only if it is
notations 2
{alA,alx\A}
d(A,X\A)
PROOF.
c
2a-connected
X such that
for each a
{alA,aIX\A}
I 0.
c
a and
is called
For the meaning of the
and D we refer to [8].
PROPOSITION 7.1.
2
is called 2 -connected if and only if there
A
>
.
For a
e
I 0 and A
e
2X\{@,X}
the following are equivalent
t((d))
This follows by straightforward verification using e.g. Proposition 3.4.6.
[8].
The following is an immediate consequence.
57
CATEGORY OF NON-ARCHIMFFJq METRIC SPACES
THEOREM 7.2.
10
2
The following hold
(X,t(U(d))) is 2 -connected if and only if there exists no non-empty proper subset
A X such that d(A,X\A)
;
(X,t(U(d))) is D-connected if and only if there exists no non-empty proper subset of
A c X such that d(A,X\A) > 0.
8. CATEGORICAL CONSIDERATIONS
Let NA(1) stand for the category of non-Archlmedean pseudometrlc spaces (X,d) where
d <.
and with morphisms l-Lipschltz or non-expanslve maps. We already know that the
functor
> FUS
> (X,II(d))
NA(
(X,d)
which leaves morphisms unaltered is a full embedding.
NA(1) to be
a full subcategory of
NA(1)
PROOF.
Obviously
Given
%o%
IFUSI
,
put
:= inf v(x,y).
hii(x,y)
%
d
:= 1
Since
D
%
it is also immediately clear that
(x,o)
(x,o)
i<x
is uniformly continuous.
%(x,x)
I. Thus {%} generates
x e X
clearly is a non-Archimedean pseudometrlc, we
is symmetric and
Since
U().
moreover have
Now, given (Y,W)
e
INA(1)I
we can choose a non-Archimedean pseudometric d $
9
e
d
I) In [15]
&
o
and it then fol-
(fxf) which proves that
(Y,W)--> (x,u)
it was shown that for
0’ i’ Zl
0 (x’y)
#d
{d}
m
if also uniformly continuous,
REMARKS 8.2.
such that W
(fxf) and thus also
$ 9 o
f
separation functions
and a uniformly continuous map
(Y,W)--> (x,)
f
lows that for all
FUS.
is a bicoreflective subcategory of
(X,I])e
a fuzzy uniformity ]
FUS.
[I]).
nice subcategory (see also
THEOREM 8.1.
Consequently we may consider
We shall now prove that NA(1) actually is a very
and
2
on
(X,t())
Zl’(x,y)
Zl (x’y)
(X,)
e
IFUS
the
TO-, TI-,
and T
2-
are given by
2(x,y)=
1
inf
v(x,y)
Thus we simply have
ZO
2) It
is easily seen that
i
Zl =z2
d.
NA(1) is not a reflective subcategory of FUS.
is a non-finite collection in
INA(1)I
is the fuzzy uniformity generated by
then their product is given by
(Xj,l(dj))jej
Xj,l]) where
If
N
jeJ
58
R. LOWEN, A.K. SRIVASTAVA AND P. WUYTS
{$KIK
SK
and where
J, K finite)
is defined by
n x j)--> I
SK:(n
Xj)
jeJ
jJ
((xj)j,(yj)j)-->
inf
Sdk(Xk,Yk).
Clearly, then
NA(1)
is however closed for finite
products in FUS.
9. DETERMINATION OF U(d) BY ITS LEVEL UNIFORMITIES
We recall [i0], [Ii] that a uniformity U on X is called non-Archimedean if there
p pip e } is a basis for U.
U
exists a collection @ of partitions of X such that
PeP
In the sequel, if P is a partition of X, we shall write P(x) for the member of P
that contains x e X.
is a non-Archlmedean uniformity on X
PROPOSITION 9.1.
U =(U d)
:-- {P
where P := {B(x,r)Ix e X}.
=},
>
r
r
generated by
Ir
PROOF.
Since
{(y,x)Id(y,z)<r
d(x,y)Vd(x,z)<r}
U,
e(Ud)__
B(x,r), we have P r (x)
P r (x)
Dr
P r (x)
{(y,z)13
x e X
and it follows from Proposition 3.3.1
that
is non-Archimedean.
The rest of the theorem is a reformulation in this particular case of well-known relations
r
<
[12] between diagonal and covering uniformities and the fact that P r < P s if
s and P
,
P.
An immediate consequence of this is the next result.
PROPOSITION 9.2
There exists a family (P)
=
condition
eI
of partitions of X, satisfying the
I
<8">P <P
and such that
u,((d))
is generated by the family
of coverings, i.e. such
that
u
PIS
P
Pep
is a basis for
> }
=(](d)).
But we also have the converse
If (P)
THEOREM 9 3
=
=eI
is a family of partitions of
a<8:> P
then there is a non-Archimedean pseudometric d
uniformity U CZ
PROOF.
generated by
(Ps)B>
X, satisfying the condition
< P
<.
(I)
on X, such that for each
is the c,-level-uniformity
U
We first remark that by (i)
Pa(x)
We can therefore define d
Pa(y)
X
X
and a
I by
< 8 =>
Ps(x) Ps(y).
((d))
=
e
I
of (d).
the
CATEGORY OF NON-ARCHIMEDEAN METRIC SPACES
d(x,y) :=
Clearly d(x,x)
0).
I, sup
(with inf
inf{alP=(x)=P=(y)}
>
then for all
"
we have P (z)
so
(x,y)
sup{aIP(x)’Pa(y)}
0 and
a"
a’
d(x,y)
d(y,x).
d(x,y)
Further, if
d(x,z),
Pa(y),
Pa(x)
59
and therefore d(y,z) & a".
So d
u,=(U(d))"
is a non-Archimedean pseudometric, and we only have to prove that U d
< 8, we can take r such that a < r < 8. If then d(x,y) < r, we have
First, if a
Ps(x) Ps(y),
e
tion 9.2 it now follows that
8 >
bitrariness of
so
t3
p
p
c
pep
(U(d))
c
U
.
p, and
p
U
pep8
U
pep
P
P
Conversely, if
=
<
e
therefore D
r, we can take
P(x) Ps(y)
D r whence D r
e
p
U
c
pep U
whence
a,u(U(d))
d(x,y) >. r =>
Ua
r
=> (x,y)
=
< 8 <
tJ
=
p.
c
From Proposiu,
=((d))
by
at-
r and then
p
Again, by arbitrariness of r
p,
> =,
we obtain
and so we are done.
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bra V (1982), 279-383.
Gen.
TQpol. Rel. Modern Anal. and Ale-
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Internat. J. Math. & Math. Sci. 8 (1985), 497-511.
3. LOWEN, R. Convergence in fuzzy topological spaces,
149-160.
4. LOWEN, R.
5. LOWEN, R.
6. LOWEN, R.
General Topol. ADpl. I0 (1979),
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On
Preuss’
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Completeness, Compactness and Precompactness in Fuzzy Uniform
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Appl. 92
I0. SCHIKHOF, W.
ii. VAN ROOIJ, A.
12. WILLARD, S.
(1983), 342-371.
Ultrametric calculus, Cambridge U.P., Cambridge, 1984.
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