NON ARCHIMEDEAN METRIC INDUCED FUZZY UNIFORM SPACES

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. Consequential characterizations of topological and uni- f'orm properties are derived.

From a global point of view embedding NA(1) in FUS also seems natural.E.g. the func- tot NA(1) TOP does not preserve products.NA(1)., although being coreflectively em- bedded 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 em- bedded in FNS).Thus in order to have a more faithful relation with TOP it seems suit- able to consider NA(1) as a subcategory of FUS.In particular we further study comple- teness 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.

PRELIMINARIES.
Most notions used are standard, we just recall some notations and some concepts specific to the context.

+
As always R 0 stands for the strictly positive real numbers, I := [0,I], I 0 :ffi ]0,i] and I := [0,I[.If X is a set and A c X, 1A stands for the characteristic function of A. If A I X and I XxX then <> I X is given by <A>(x) sup A(y) A (y,x).If l{x we we simply put <x> and obviously <x>(y) (x,y).Yiso o I XX is given by o (x,y) sup (x,z) ^(z,y).
Further stands for the prefilter stands for the prefilter [{ sup If d is a pseudometric on X then we put T d and U d resp.the associated topology and uniformity.
If d fulfils the strong (or ultrametrlc) triangle inequality we call it a non- Archimedean pseudometrlc.

FUS FTS where then t(]) T UNIF
TOP where then T(U) stands for the topology associated with U. filter ] [3], we recall also that its characteristic value is given by If [9]   In mal stands for the fuzzy topology associated with ] and For a pre- c() IFUSI and C is a prefilter on X then it is called a hyper Cauchy prefilter 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 X, e e I0, ] y e 2 (X) sup v<x> e, which is also equivalent to the fact xeY that (X,t (W)) is precompact.We first put together some elementary technical properties.
LEMMA 3.1.If X is a set and d a non-Archlmedean pseudometric on X, then d := i-d X X I has the following properties a.
x e X d(X,X) I; b. d is symmetric; c. d o d d' or equivalently (x,y,z) e X 3 d(X,Z) ^(z,y/ d(x,y).iXxX 2 If conversely has the properties a.
In the sequel, if no confusion can arise, we simply put resp.d instead of d resp.d.Conversely, if U is a fuzzy uniformity on X, having a singleton basis {}, then this function satisfies the conditions a, b, c in Lemma 3.1.2, and therefore U ](d) where d:= 1-@ is a non-Archimedean pseudometric.
PROOF.The first part follows from Lemma 3.1.1and the second part from the defi- nition of a basis of a fuzzy uniformity [4] and an application of Lemma 3.1.2 .
We now describe the general properties of U(d), where it is .alwayssupposed that d is a non-Archimedean pseudometric such that d & I.  5 for all e !X the closure and the interior of in t((d)) are given by (x) sup (y) h (y,x), (x) inf (y) V d(y,x).yeX yeX 6 A e t((d)) iff there exists a partition PA of X by means of balls, i.e. a subset y c X and a function p Y Ox; ) is Hausdorff iff d(x,y) > = for all x y; 8 (X,(d)) is WT 2 iff d is metric; 9 (X,(d)) is T 2 iff d(,y) for all x y, i.e. iff d is the discrete metric.

PROOF.
Immediate from the definition of ((d)) and from #-l(]l-r,l]) D 2 From Lma 3.i. 3.and the fact that (X,U(d)) is a fuzzy neighborhood space.
3 From and u u,0 4 This is a know property of general fuzzy uniform spaces. 5Immediate from Proposition 2.4 in [5]. 6If A e t((d)), x e X, A(x) = and d(x,y) < =, it follows from that A(y) =.

A(y) A(y)
inf A(t)Vd(t,y) =< A(x) V d(x,y) However, we also have This means that A-l(=) D{B(x,=)IA(x) =}, and as each two of these balls are either identical or disjoint, we can choose Y c X such that {(x,=)lx y is a disjoint fa- mily with A-l(=) as a union.Putting Y := U Y p := = iff x e y we are done.Conversely, if A e I X is such that the described partition exists and if = e If, we have which is clearly open in (t(U(d))).Since (X,t((d))) is a fuzzy neighborhood space, t(W(d)) is maximal for its level topologies [13], and therefore A e t(W(d)). 7This follows from the fact that, by , la(t(U(d)) is Hausdorff iff 0re]a,l Dr is the diagonal of X X. 8 This is nothing else than the definition of WT 2. 9 (X,U(d)) is T 2 iff (x,y) 0 for x y, so iff d(x,y) for x y.
REMARKS 3.4.I.If d and 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 i. Starting from an arbitrary d, we can define a family of fuzzy uniformities.Indeed, given the non-Archimedean pseudo- + metric d on X and E R0, we can define d ( d) A which is equivalent to d and consider U(d) := ," where .-'-dE (I-V 0.Even in this case the fuzzy + uniformities (dE), 0' are in general not equivalent to each other, (some interest- ing relations will be established in Propositions 3.5 and 4.), e.g. if X := E where then it is well known that d given by 0 Yn x =Yn d(x )n (Yn)n) := n n (min k x kyk otherwise is a non-Archimedean metric on X, and it is easily seen that (X,l(d)) and (X,(d ,)) are not isomorphic if '.
3. It is evident that the properties of U(d) can be obtained from the corresponding ones of B(d) by replacing everywhere d by d So for instance, it follows from Propo-E sition 3.4.1 that a basis for tu,a(O(d )) is given by {D'l=r < r} where D' {(x,y) ld(x,y) < r}.
r Since this translation of properties of (d) into properties of (de) is a simple exer- cise, while the formulation of the former is simpler, we shall continue to restrict ourselves mainly to the case d i. 4. From Proposition 3.3.6 it follows that all elements of t(U(d)) are l-Lipschitz, the converse however is not true.Consequently t(l) is strictly coarser than the structure A(1) of [7].sup + (d e) c u(Ud).The converse inclusion follows at once upon remarking that for ee 0 all e 0 we have 1D e (de) and that {De[e is a basis for U d- PROOF.Immediate from the fact that # -< (fxf)-l(') if and only if d' o (fxf) <. d.For concepts and results concerning convergence we refer to [2], [3].
PROPOSITION.4.3.If F is a filter on X then F x in (X,T d) if and only if lim (F) <x> in (X,t(W(d))).
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 (X,(de)), e e 0, are mutually isomorphic.
If Y c X is a finite subset such that X U B(x,E) then we have inf d(x,t) < for xeY xeY all t e X which is equivalent to sup <x> > i which by Theorem 2.2 in [6] proves xeY our claim.+ REMARK 5.2.Since for any e e 0 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.

COMPLETENESS.
For concepts and results concerning completeness and completions we refer to [9].The following result is an immediate consequence of Theorem 4. (,) the metric completion of (X,d) II.(,U()) the ultracompletion of (X,(d)) III. (X Ud)) the ultracompletlon of (X,u(Ud)) U Then we obtain the following collection of complete or ultracomplete spaces. IV.
The complete space (X (Ud))) U^U VI.
However, we prefer to explicitly describe the isomorphism which at the same time allows us to describe the points of too.
Given the non-Archimedean space (X,d), its metric completion (,) can be considered as the set of all equivalence classes of equivalent Cauchy sequences in X, equipped with the metric defined by (,) nlim d(x n,yn ), where (Xn) n and (Yn)n are arbitrary representatives of and respectively The ultracompletion [9] of (X,U(d)) is given by (,U[d)), where is the set of all ml- nimal hyper Cauchy prefilters on (X,U(d)), and where Ud) is the fuzzy uniformity gene- rated by {}, this function being defined by LEMMA 6.2.If (Xn) n is a Cauchy-sequence in (X,d), the sequence (<Xn>) n converges uni- formly to a mapping () X I t llm #(t,x n) which depends only on the equivalence nclass of (Xn)n, and which has the following properties PROOF.If t is the class of the constant sequence (tn=t)n, we know that (') nlim d(t,Xn is independent of the choice of (Xn) n e .
If e > 0 and n 0 is chosen such that p .> no, q n o ffi> (Xp,Xq) I E, then for x e X, p n o q >. n o either (X,Xp) < and then (X,Xq) (X,Xp), or @(X,Xp) >. 1-and then also #(x,xq) I-, so in any case l#(x,xp)-(X,Xq) e, which proves the uniform convergence.The property a follows by considering ()(Xn), and b and c follow by standard verification.
It follows from the foregoing lemma, that the prefilter r() {()}~'() is a minimal hyper Cauchy prefilter on (X,(d)), and so we obtain a mapping r .(Yn)n converges uniformly to y() e C, where is the equivalence class of (Xn)n; c. n ]q n n <-+ Pn where l im 0 0. n n PROOF.It follows from (HC3) that we can find a non-increasing sequence (Sn) n of elements of C such that for all n elN -n n 8n <_ + 2 By (HCl) we can find a sequence (Xn) n in X such that for all n e -n-I 2 & 8n(Xn)" Since (8n)n 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) which shows (Xn) n is a Cauchy sequence.
Further by (I) and ( 2) we have that for all n el and x e X (1) Thus it follows from (HC2) that for all n ]q n := sup #<Xk> C. kZn Since (<Xn>) n converges uniformly to () the same is true for (n)n and thus again by (HC2) we obtain that () 6.
Finally we still have that for all n and x,y X sup (Xk,X) V #(Xm,Y) fn(X) A -n(y) kn,m>-n We are now in a position to prove the isomorphism result.THEOREM 6.4.The map r (,u()) (,Ud)) is an isomorphism.
be repre-PROOF To see that r is into let x,y e X x and let (Xn) n and (Yn)n sentatives of and respectively.Then there exists e > 0 and n o e such that for all p,q n o d(x ,yq) which implies that for any p n o we have (9)(Xp) & -P whereas lim ()(x n) I. Thus () # () and therefore r() # r().
To see that r is onto, take C e , and consider () as constructed in Lemmas 6.2 and 6.3.It then follows that r() c C, and so r() by minimality.
s, tX the symmetry of , and the fact that () y() =< , we obtain d(r(),r() 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 (X,d) we have lim d(xn,Yn) sup llm(d(s,t)-d(Xn,S)Vd(Yn,t)).

SteX n
From the ultrametric property we obtain llm( n q, 3 x X Yn(Xn) n d.V n Yn Yn $ +pn where l im Pn =0.PROOF.Since => 2 was proved in Theorem 6.4 and Lemma 6.2, while 2" => and 3 => 2 are obvious, it is sufficient to prove 2 => 3 We can repeat the construction in Lemma 6.3 with n 8 for all n I.
(n)n has the properties a, c and d by the construction in Lemma 6.3.

follows from
The sequence As to b, this <n>(X) sup n(t) V (t,x) sup sup(k(t)^(t,x)) tX reX kn sup sup(k(t)A$(t,x)) sup <k>(X) kn rex kn sup Uk(X) n(X), kn where n <Xn >" Since the minimal hyper Cauchy prefilter generated by {nln } is coarser than C it coincides with C. REMARK 6.6.A characterization of minimal Cauchy filters, probably belonging to the folklore of the subject, and with a standard proof which we leave to the reader, is given by the following ((X,d) is a pseudometric space) F is a minimal Cauchy filter on (X,U d) if and only if F is a filter having a basis (Bn)nl which is a non-increas- ing chain of open balls Bn B(x n,rn) with the property~~n lim rn 0. An alternative me- thod for proving the isomorphism of (,U[d)) and (X,U(d)) can be based on this and on Theorem 6.5.Indeed, we consider as the set of minimal Cauchy filters 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).

CONNECTEDNESS
In [8] a number of connectedness concepts in G. Preuss' sense have been introduced and studied.
We recall that a space (X A) IFTSI is called 2 -connected if and only if there does not exist a non-empty proper subset A c X such that {alA,aIX\A} c a and is called D-connected if and only if it is 2a-connected for each a I 0. For the meaning of the notations 2 and D we refer to [8].PROPOSITION 7.1.For a e I 0 and A e 2X\{@,X} the following are equivalent {alA,alx\A} t((d)) 2 d(A,X\A) > . PROOF.[8].
This follows by straightforward verification using e.g.Proposition 3.4.6.
The following is an immediate consequence.d(x,y) := inf{alP=(x)=P=(y)} sup{aIP(x)'Pa(y)} (with inf I, sup 0).Clearly d(x,x) 0 and d(x,y) d(y,x).Further, if d(x,y) a' a" d(x,z), then for all > " we have P (z) Pa(x) Pa(y), and therefore d(y,z) & a".So d is a non-Archimedean pseudometric, and we only have to prove that U d u,=(U(d))" First, if a < 8, we can take r such that a < r < 8.If then d(x,y) < r, we have Ps(x) Ps(y), so (x,y) e U p p, and therefore D c U p p. From Proposi- pep8 r pep tion 9.2 it now follows that U P P e a,u(U(d)) whence U c =((d)) by at- pep = u, bitrariness of 8 > .Conversely, if = < r, we can take = < 8 < r and then d(x,y) >. r => P(x) Ps(y) => (x,y) tJ p p,

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: was shown that for any hyper Cauchy prefilter C, there exists a unique mini- hyper Cauchy prefilter C 0 c C. Moreover, if B is a basis for C and M a basis for then 0 {<>I eW' eB) ~.
e0PROOF.is evident, and for 2 it suffices to remark that if inf+ U(d) then V e d for all e eO.+ For 3 note that by Proposition 3.3.3for all e eO we have U(de c u (U d) i.e.
,W(d)) is ultracomplete.Given (X,d) we can now construct the following completions.I.

LEMMA 6 . 3 .
If C is a hyper Cauchy prefllter on (X,U(d)) then there exists a Cauchy se- quence (Xn) n in (X,d) such that a. n elq n := sup <Xk> e C; k>.n b.
< 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)) on the other hand since sup I is continuous if I XxX is equipped with the uniform topology and I with the usual one, we have sup lim(d(s,t)-d(Xn,S)Vd(Yn,t)) s,teX n lim sup (d(s,t)-d(XnJS)Vd(Yn,t)) n s,teX lim d(xn,Yn).nInorder to describe the points of in more detail we have the next result.THEOREM 6.5.The following are equivalent is a minimal hyper Cauchy prefilter on (X,U( whence D e U Again, by arbitrariness of r > =, we obtain pep r r a (U(d)) c U and so we are done.

First
Round of Reviews May 1, 2009 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 3.4.2and 3.4.3 in mind, the proof is obvious. m