19.dvi @ Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 2, No. 1, 2001pp. 63 - 75 A contribution to the study of fuzzy metricspacesAlmanzor SapenaAbstract. We give some examples and properties of fuzzymetric spaces, in the sense of George and Veeramani, and charac-terize the T0 topological spaces which admit a compatible unifor-mity that has a countable transitive base, in terms of the fuzzytheory.2000 AMS Classi�cation: 54A40Keywords: Fuzzy metric, precompact, strongly zero-dimensional.1. IntroductionOne of the main problems in the theory of fuzzy topological spaces is to obtainan appropriate and consistent notion of a fuzzy metric space. Many authorshave investigated this question and several notions of a fuzzy metric space havebeen de�ned and studied. In particular, and modifying the concept of metricfuzziness introduced by Kramosil and Michalek [9] (which is a generalizationof the concept of probabilistic metric space introduced by K. Menger [10] tothe fuzzy setting), George and Veeramani [4, 5], have studied a notion of fuzzymetric space. In a previous paper [7], Gregori and Romaguera proved that theclass of fuzzy metric spaces, in George and Veeramani's sense, coincides withthe class of metric spaces. In the light of the results obtained in [7], we thinkthat the George and Veeramani's de�nition is an appropriate notion of metricfuzziness in the sense that it provides rich fuzzy topological structures which canbe obtained, in many cases, from classical theorems. On the other hand, metricspaces can be studied from the point of view of fuzzy theory. Unfortunately,not much examples of such spaces have been given. In this paper we give newexamples of fuzzy metric spaces and study some properties of these spaces.The structure of the paper is as follows. After preliminaries, in section 3,we construct new fuzzy metrics from a given one, and study some questionsrelative to boundedness. In Section 4 we give new examples of fuzzy metrics.In Section 5 we study a property of Cauchy sequences in standard fuzzy metricspaces, and �nally, in Section 6, we de�ne the concept of non-Archimedean 64 A. Sapenafuzzy metric space and prove that the family of these spaces agrees with theclass of non-Archimedean metric spaces, so it provides a characterization of theT0 topological spaces which admit a compatible uniformity, that has a countabletransitive base, in the fuzzy setting.2. PreliminariesThroughout this paper the letters N and R will denote the set of all posi-tive integers and real numbers, respectively. Our basic reference for GeneralTopology is [2].According to [11] a binary operation � : [0;1]�[0;1] �! [0;1] is a continuoust-norm if � satis�es the following conditions:(i) � is associative and commutative(ii) � is continuous(iii) a � 1 = a for every a 2 [0;1](iv) a � b � c � d whenever a � c and b � c, for all a;b;c;d 2 [0;1]According to [4],[5], a fuzzy metric space is an ordered triple (X;M;�) suchthat X is a non-empty set, � is a continuous t-norm and M is a fuzzy set ofX � X�]0;+1[ satisfying the following conditions, for all x;y;z 2 X, s;t > 0:(i) M(x;y;t) > 0(ii) M(x;y;t) = 1 if and only if x = y(iii) M(x;y;t) = M(y;x;t)(iv) M(x;y;t) � M(y;z;s) � M(x;z;t + s) (triangular inequality)(v) M(x;y; �) : ]0;+1[�! [0;1] is continuous.If (X;M;�) is a fuzzy metric space, we will say that (M;�), or M (if it is notnecessary to mention �), is a fuzzy metric on X.Lemma 2.1. [6] M(x;y; �) is nondecreasing for all x;y 2 X.Lemma 2.2. [1] Let (X;M;�) be a fuzzy metric space.(i) If M(x;y;t) > 1 � r for x;y 2 X, t > 0, 0 < r < 1, we can �nd a t0,0 < t0 < t such that M(x;r;t0) > 1 � r.(ii) For any r1 > r2, we can �nd a r3 such that r1 � r3 � r2, and for anyr4 we can �nd a r5 such that r5 � r5 � r4, (r1; r2; r3; r4; r5 2]0;1[).Let (X;d) be a metric space. De�ne a � b = ab for every a;b 2 [0;1]; and letMd be the function on X � X�]0;+1[ de�ned byMd(x;y;t) = tt + d(x;y)Then (X;Md;�) is a fuzzy metric space, and Md is called the standard fuzzymetric induced by d (see [4]).George and Veeramani proved that every fuzzy metric M on X generates aHausdor� topology �M on X which has as a base the family of open sets of theform: fBM(x;r;t) : x 2 X;0 < r < 1; t > 0g A contribution to the study of fuzzy metric spaces 65where BM(x;r;t) = fy 2 X : M(x;y;t) > 1 � rgfor every r 2]0;1[, and t > 0. (We will write B(x;r;t) when confusion is notpossible).De�nition 2.3. A sequence fxng in a fuzzy metric space (X;M;�) is calleda Cauchy sequence [5], if for each " > 0, t > 0 there exists n0 2 N such thatM(xn;xm; t) > 1 � ", for all m;n � n0.A subset A of X is said to be F-bounded if there exist t > 0 and r 2]0;1[such that M(x;y;t) > 1 � r for all x;y 2 A.Proposition 2.4 ([4]). If (X;d) is a metric space, then:(i) The topology �d on X generated by d coincides with the topology �Mdgenerated by the standard fuzzy metric Md.(ii) fxng is a d�Cauchy sequence (i.e., a Cauchy sequence in (X;d)) ifand only if it is a Cauchy sequence in (X;Md;�).(iii) A � X is bounded in (X;d) if and only if it is F-bounded in (X;Md;�).We say that a topological space (X;�) is fuzzy metrizable if there exists afuzzy metric M on X such that � = �M. In [7] it is proved that a topologicalspace is fuzzy metrizable if and only if it is metrizable.Unless explicit mention we will suppose R endowed with the usual topology.3. Some properties of fuzzy metric spacesFrom now on we will denote by Ti (i = 1;2;3) the following continuoust-norms: T1(x;y) = minfx;ygT2(x;y) = xyT3(x;y) = maxf0;x + y � 1gThe following inequalities are satis�ed:T3(x;y) � T2(x;y) � T1(x;y)and T(x;y) � T1(x;y)for each continuous t-norm T .In consequence the following lemma holds.Lemma 3.1. Let X be a non-empty set. If (M;T) is a fuzzy metric on X andT 0 is a continuous t-norm such that T 0 � T, then (M;T 0) is a fuzzy metric onX.Next two properties give methods for constructing F-bounded fuzzy metricsfrom a given fuzzy metric. 66 A. SapenaProposition 3.2. Let (X;M;�) be a fuzzy metric space and k 2]0;1[. De�neN(x;y;t) = maxfM(x;y;t);kg; for each x;y 2 X, t > 0:Then (N;�) is an F-bounded fuzzy metric on X, which generates the sametopology that M.Proof. It is straightforward. �Proposition 3.3. Let i 2 f1;2;3g and k > 0. Suppose that (X;M;Ti) is afuzzy metric space, and de�ne:N(x;y;t) = k + M(x;y;t)1 + k for all x;y 2 X;t > 0:Then, (N;Ti) is an F-bounded fuzzy metric on X, which generates the sametopology that M.Proof. We prove this proposition for the case i = 2. For seeing that (N;Ti) isa fuzzy metric on X, we only show the triangular inequality.Now, it is an easy exercise to verify that the following relationk + a1 + k � k + b1 + k � k + ab1 + kholds, for all a;b 2 [0;1].Therefore,k + M(x;y;t)1 + k � k + M(y;z;s)1 + k � k + M(x;y;t) � M(y;z;s)1 + k� k + M(x;z;t + s)1 + kClearly k1+k is a lower bound of N(x;y;t), for all x;y 2 X, t > 0.Finally, for t > 0;r 2]0;1[ it is satis�ed thatBM(x;r;t) = BN(x; r1 + k;t)and BN(x;r;t) = BM(x;r(k + 1); t);and so �M = �N.The cases i = 1;3 are left as simple exercises. �Problem 3.4. If (M;�) is a fuzzy metric on X and k > 0; then, is�k + M(x;y;t)1 + k ;��a fuzzy metric on X?Proposition 3.5. Let (M1;�) and (M2;�) be two fuzzy metrics on X. De�ne:M(x;y;t) = M1(x;y;t) � M2(x;y;t)N(x;y;t) = minfM1(x;y;t);M2(x;y;t)gThen: A contribution to the study of fuzzy metric spaces 67(i) (M;�) is a fuzzy metric on X if a � b 6= 0 whenever a;b 6= 0.(ii) (N;�) is a fuzzy metric on X.(iii) The topologies generated by M and N are the same.Proof. The proofs of (i) and (ii) are straightforward.(iii) First we will prove that �N < �MLet A 2 �N; then 8x 2 A;9r 2]0;1[ such thatBN(x;r;t) = fy 2 X : N(x;y;t) > 1 � rg � AConsider BM(x;r;t) = fy 2 X : M(x;y;t) > 1 � rg:If z 2 BM(x;r;t); then M(x;z;t) > 1 � r, i.e.,M1(x;z;t) � M2(x;z;t) > 1 � r:Notice that M1(x;z;t) � M1(x;z;t) � M2(x;z;t) > 1 � r;and M2(x;z;t) � M1(x;z;t) � M2(x;z;t) > 1 � rso, N(x;z;t) = minfM1(x;z;t);M2(x;z;t)g > 1 � r:Then, BM(x;r;t) � BN(x;r;t) � A;thus A 2 �M and hence �N < �M.For seeing that �M < �N, let A 2 �M; then 8x 2 A;9r 2]0;1[ such thatBM(x;r;t) = fy 2 X : M(x;y;t) > 1 � rg � A:Let s 2]0;1[ such that (1 � s) � (1 � s) > 1 � r.ConsiderBN(x;s;t) = fy 2 X : N(x;y;t) > 1 � sg= fy 2 X : minfM1(x;y;t);M2(x;y;t)g > 1 � sgIf z 2 BN(x;s;t), then M1(x;z;t) > 1 � s and M2(x;z;t) > 1 � s.So, M1(x;z;t) � M2(x;z;t) > (1 � s) � (1 � s)> 1 � r:Then BN(x;s;t) � BM(x;r;t) � A, and hence �M < �N. �Remark 3.6. If we consider the fuzzy metric (Md;�) where d is the usualmetric on R and � is T3, it is easy to verify that M = Md � Md is not a fuzzymetric on R (compare with 2.10 of [5]).De�nition 3.7. [7] A fuzzy metric space (X;M;�) is called precompact if foreach r 2]0;1[; and t > 0, there exists a �nite subset A of X such that X =SfB(a;r;t) : a 2 Ag. In this case, we say that M is a precompact fuzzy metricon X. 68 A. SapenaIn [7] it is proved that a fuzzy metric space is precompact if and only if everysequence has a Cauchy subsequence. Using this fact, the proof of the followingproposition is straightforward.Proposition 3.8. Let (X;d) be a metric space and let Md be the standard fuzzymetric deduced from d. Then, d is a precompact metric if and only if Md is aprecompact fuzzy metric.Proposition 3.9. Let (X;M;�) be a precompact fuzzy metric space, and sup-pose a � b 6= 0 whenever a;b 6= 0. Then, (M;�) is F-bounded.Proof. (Compare with the end of the proof of [4, Theorem 3.9].)Let r 2]0;1[ and t > 0. By assumption there is a �nite subset A =fa1; : : : ;ang of X such that X = nSi=1 B(ai;r;t). Let� = minfM(ai;aj; t) : i;j = 1; : : : ;ng > 0:Let x;y 2 X. Then x 2 B(ai;r;t) and y 2 B(aj;r;t) for some i;j 2 f1; : : : ;ng.Therefore M(x;ai; t) > 1 � r and M(y;aj; t) > 1 � r. Now,M(x;y;3t) � M(x;ai; t) � M(ai;aj; t) � M(aj;y;t)� (1 � r) � � � (1 � r)> 1 � sfor some s 2]0;1[ by the assumption on �, and so M is F-bounded �Problem 3.10. Is each precompact fuzzy metric space F-bounded?Remark 3.11. The converse of the last proposition is false. In fact, the sub-space X of the Hilbert metric space (R1;d), formed by the points of unit weight(0; : : : ;0;1;0; : : : ;0), is not precompact and bounded (it has diameter p2), andthen by (iii) of Proposition 2.2, (X;Md;�) is F-bounded, and by Proposition3.8 Md is not precompact.4. Examples of fuzzy metric spacesIn this section we will see examples of fuzzy metrics where the t-norm isT1, and other fuzzy metrics (M;Ti);(i = 2;3) which are not fuzzy metrics inconsidering (M;Ti�1). Before, we need the following lemma.Lemma 4.1. Let (X;d) be a metric space and s;t > 0. The following inequalityholds, for all n � 1; d(x;z)(t + s)n � max �d(x;y)tn ; d(y;z)sn �Proof. We distinguish three cases:(1) d(x;z) � d(x;y)(2) d(x;z) � d(y;z)(3) d(x;z) > d(x;y) and d(x;z) > d(y;z)The inequality chosen is obvious in cases (1) and (2). Now, suppose (3) issatis�ed and distinguish two possibilities: A contribution to the study of fuzzy metric spaces 69(3.1) d(x;z) = d(x;y) + d(y;z)(3.2) d(x;z) < d(x;y) + d(y;z)Suppose (3.1) is satis�ed. Put d(x;y) = �d(x;z) with � 2]0;1[ and henced(y;z) = (1 � �)d(x;z):Now, to show the above inequality we have to prove that1(t + s)n � max � �tn ; 1 � �sn � :Therefore, consider the functions f(�) = tn� and g(�) = sn1�� which arestrictly decreasing and increasing, respectively. Now, the largest value of minntn� ; sn1��ois taken when f(�) = g(�), that is, for � = tntn+sn . Then,(t + s)n � tn + sn= f( tntn + sn)� min�tn� ; sn1 � ��and the chosen inequality is stated.The case (3.2) is a consequence of (3.1). �Example 4.2. Let (X;d) be a metric space, and denote B(x;r) the open ballcentered in x 2 X with radius r > 0.(i) For each n 2 N, (X;M;T1) is a fuzzy metric space where M is given byM(x;y;t) = 1ed(x;y)tn for all x;y 2 X, t > 0;and �M = �(d).(This example when n = 1 has been given in [4].)(ii) For each k;m 2 R+, n � 1, (X;M;T1) is a fuzzy metric space where Mis given by M(x;y;t) = ktnktn + md(x;y) for all x;y 2 X;t > 0;and �M = �(d).Proof. (i) It is easy to verify that (M;T1) satis�es all conditions of fuzzy metrics;in particular the triangular inequality is a consequence of the previous lemma.Now, for x 2 X;r 2]0;1[ and t > 0 we have thatBM(x;r;t) = B(x;�tn ln(1 � r));and B(x;r) = BM(x;1 � 1e rtn ; t);and hence �M = �(d). 70 A. Sapena(ii) We will only give a proof of the triangular inequality. Indeed, by theprevious lemma1 + md(x;z)k(t + s)n � max �1 + md(x;y)ktn ;1 + md(y;z)ksn �hence k(t + s)nk(t + s)n + md(x;z) � min� ktnktn + md(x;y); ksnksn + md(y;z)� ;and the triangular inequality is stated.Now, for x 2 X, t > 0 and r 2]0;1[ we have thatBM(x;r;t) = B(x; ktnrm(1 � r));and B(x;r) = BM(x; mrktn + mr;t);and hence �M = �(d). �Remark 4.3. The above expression of M cannot be generalized to n 2 R+(take the usual metric d on R, k = m = 1, n = 1=2). Nevertheless it is easy toverify that (M;T2) is a fuzzy metric on X, for n � 0. (Compare with 2.9-2.10of [5]).Next, we will give fuzzy metrics which cannot be deduced from a metric, inthe sense of last example, since they will not be fuzzy metrics for the t-normT1.Example 4.4. Let X be the real interval ]0;+1[ and a > 0. It is easy toverify that (X;M;T2) is a fuzzy metric space, where M is de�ned byM(x;y;t) = (�xy�a if x � y�yx�a if y � xfor all x;y 2 X, t > 0.(We notice that this example for X = N and a = 1 was given in [4]).Now, for x 2 X, t > 0 and r 2]0;1[, we haveB(x;r;t) = #(1 � r) 1a x; x(1 � r) 1a "and hence B(x;r;t) is an open interval of R, whose diameter converges to zeroas r ! 0. In consequence, �M is the usual topology of R relative to X.Finally, (X;M;T1) is not a fuzzy metric space. Indeed, for a = 1, if we takex = 1, y = 2 and z = 3, thenM(x;z;t + s) = 13< minf12; 23g= minfM(x;y;t);M(y;z;s)g: A contribution to the study of fuzzy metric spaces 71Next, we will give examples of fuzzy metric spaces for the t-norm T3 whichare not for the t-norm T2.Example 4.5. Let X be the real interval ]1;+1[ and consider the mappingM on X2�]0;+1[ given byM(a;b;t) = 1 � ( 1a ^ b � 1a _ b) for all a;b 2 X;t > 0:We will see that (X;M;T3) is a fuzzy metric space and (X;M;T2) is not.Further, the topology �M on X is the usual topology of R relative to X:For seeing that (M;T3) is a fuzzy metric we only prove the triangular in-equality, which becomes (when the left side of the inequality is distinct of zero)(4.1)�1 � � 1a ^ b � 1a _ b�� + �1 � � 1b ^ c � 1b _ c�� � 1 � 1 � � 1a ^ c � 1a _ c�For it, �rst, we distinguish 6 cases:(1) Suppose a < b < c. In this case, the inequality 4.1 becomes an equality.(2) Suppose a < c < b. In this case, the inequality 4.1 becomes:1b + 1b + 1a � 1a + 1c + 1cwhich is true, since 1b < 1c.(3) Suppose c < a < b. In this case, the inequality 4.1 becomes:1b + 1b + 1c � 1a + 1c + 1awhich is true, since 1b < 1a.(4) Suppose b < a < c. In this case, the inequality 4.1 becomes:1a + 1c + 1a � 1b + 1b + 1cwhich is true, since 1a < 1b.(5) Suppose b < c < a. In this case, the inequality 4.1 becomes:1a + 1c + 1c � 1b + 1b + 1awhich is true, since 1c < 1b(6) Suppose c < b < a. In this case, the inequality 4.1 becomes an equality.Now, if a = b, or a = c, or b = c, the inequality 4.1 is obvious, and thetriangular inequality is stated, so (M;T3) is a fuzzy metric.On the other hand, if we take a = 2, b = 3 and c = 10, thenM(a;b;t) � M(b;c;s) > M(a;c;t + s)and thus, (M;T2) is not a fuzzy metric.Finally, if we take x 2 X, r 2]0;1[ with r < 1x, and t > 0, it is easy to verifythat B(x;r;t) = i x1+rx; x1�rxh, then B(x;r;t) is an open interval or R whichdiameter converges to zero as r ! 1, and thus �M is the usual topology of Rrelative to X. 72 A. SapenaExample 4.6. Let X be the real interval ]2;+1[ and consider the mappingM on X2�]0;+1[ de�ned as followsM(a;b;t) = (1 if a = b1a + 1b if a 6= b, t > 0.It is easy to verify that (X;M;T3) is a fuzzy metric space. On the other handif we take a = 1000, b = 3 and c = 10000, thenM(a;b;t) � M(b;c;s) > M(a;c;t + s)and so (X;M;T2) is not a fuzzy metric space.Finally, the topology �M is the discrete topology on X. Indeed, for x 2 X,if we take r < 12 � 1x then B(x;r;t) = fxg.Next example is based in [8].Example 4.7. Let fA;Bg be a nontrivial partition or the real interval X =]2;+1[. De�ne the mapping M on X2�]0;+1[ as followsM(x;y;t) = (1 � � 1x^y � 1x_y� if x;y 2 A or x;y 2 B1x + 1y elsewhere.Then, imitating example 2 of [8], one can prove that (X;M;T3) is a fuzzymetric space, and by example 4.4, clearly (X;M;T2) is not a fuzzy metric space.From examples 4.4 and 4.5 it is deduced that an open base for the neighbor-hood system of a point x 2 X, is i x1+rx; x1�rxh\A if x 2 A, and i x1+rx; x1�rxh\B,with 0 < r < 12 � 1x, if x 2 B.5. Some properties of standard fuzzy metricsIn this section (X;d) will be a metric space, and Md the standard fuzzymetric deduced of d.Grosso modo, we can say that all properties of classical metrics can be trans-lated to standard fuzzy metrics. Now, an interesting question is to know whichof these properties can be generalized to any fuzzy metric. In this sense wewill see a new property which is satis�ed by standard fuzzy metrics. (Noticethat there is no signi�cative di�erence between the standard fuzzy metric Mdand the fuzzy metric ktnktn+md(x;y) of example 4.2, unless Md is the most simpleexpression depending of t).Proposition 5.1. Let fxng1n=1 and fyng1n=1 be two Cauchy sequences in (X,Md, T2) and let t > 0. Then, the sequence of real numbers fMd(xn;yn; t)g1n=1converges to some real number in ]0;1[.Proof. Suppose fxng1n=1 and fyng1n=1 are Cauchy sequences in (X;Md;T2). By(ii) of proposition 2.2, fxng1n=1 and fyng1n=1 are Cauchy sequences in (X;d)and then it is easy to verify that fd(xn;yng1n=1 is a Cauchy sequence in R. A contribution to the study of fuzzy metric spaces 73Now, let " > 0, t > 0. Then, there exists n0 2 N such thatjd(xn;yn) � d(xm;ym)j < "t for all m;n � n0:Hence,���� 1Md(xn;yn; t) � 1Md(xm;ym; t)���� = 1t jd(xn;yn) � d(xm;ym)j< ";for all m;n � n0, and therefore n 1Md(xn;yn;t)o is a Cauchy sequence in R, whichconverges to some k 2 R, and then the sequence fMd(xn;yn; t)g1n=1 convergesto 1k 2]0;1[, since k 6= 1 and Md(xn;yn; t) � 1, for all n 2 N. �Corollary 5.2. Let fxng1n=1 be a Cauchy sequence in the fuzzy metric space(X;Md;T2) and a 2 X. Then, the sequence of real numbers fMd(a;xn; t)g1n=1converges to some real number in ]0;1[.Problem 5.3. Let fxng1n=1 be a Cauchy sequence in the fuzzy metric space (X,M, �) and let a 2 X, t > 0. Does the sequence of real numbers fM(a;xn; t)g1n=1converge to some real number in ]0;1[?The last proposition is not true for any fuzzy metric space as shows thefollowing example.Example 5.4. Let fA;Bg be a partition of the real interval X =]2;+1[, suchthat f2n � 1g1n=2 � A and f2ng1n=1 � B, and consider the fuzzy metric space(X;M;T3) of example 4.6. It is easy to verify that both sequences are Cauchyin (X;M;T3). Now, if we put an = 2n � 1 and bn = 2n, for n = 2;3; : : : wehave M(an;bn; t) = � 12n � 1 + 12n� �! 0 as n �! 1:6. On non-Archimedean fuzzy metricsRecall that a metric d on X is called non-Archimedean ifd(x;z) � maxfd(x;y);d(y;z)g; for all x;y;z 2 X:Now, we give the following de�nition.De�nition 6.1. A fuzzy metric (M;�) on X is called non-Archimedean ifM(x;z;t) � minfM(x;y;t);M(y;z;t)g for all x;y;z 2 X, t > 0:Clearly, if M is a non-Archimedean fuzzy metric on X, then (M;T1) is afuzzy metric on X:Proposition 6.2. Let d be a metric on X and Md the corresponding stan-dard fuzzy metric. Then, d is non-Archimedean if and only if Md is non-Archimedean. 74 A. SapenaProof. Suppose d is non-Archimedean. Then,Md(x;z;t) = tt + d(x;z)� tt + maxfd(x;y);d(y;z)g= minfMd(x;y;t);Md(y;z;t)g:Conversely, if Md is non-Archimedean then,d(x;z) = t� 1Md(x;y;t) � 1�� t� 1minfMd(x;y;t);Md(y;z;t)g � 1�= maxfd(x;y);d(y;z)g : �Recall that a completely regular space is called strongly zero-dimensional ifeach zero-set is the countable intersection of sets that are open and closed, andthat a T0 topological space (X;�) is strongly zero-dimensional and metrizableif and only if there is a uniformity U compatible with � that has a countabletransitive base ([3, Theorem 6.8]).Theorem 6.3. A topological space (X;�) is strongly zero-dimensional andmetrizable if and only if (X;�) is non-Archimedeanly fuzzy metrizable.Proof. Suppose (X;�) is strongly zero-dimensional and metrizable. Then, from[3, Theorem 6.8], (X;�) is non-Archimedeanly metrizable and by Proposition6.2 it is non-Archimedeanly fuzzy metrizable.Conversely, suppose M is a compatible non-Archimedean fuzzy metric for(X;�). Now, for a fuzzy metric space (X;M;�) in [7] it is proved that thefamily fUn : n 2 Ng whereUn = �(x;y) 2 X � X : M(x;y; 1n) > 1 � 1n�is a base for a uniformity U on X which is compatible with �M. Now, we willsee that fUn : n 2 Ng is transitive.Indeed, if (x;y);(y;z) 2 Un thenM(x;z; 1n) � min�M(x;y; 1n);M(y;z; 1n)�> 1 � 1n;and thus (x;z) 2 Un, i.e., Un � Un � Un.Now, from the mentioned theorem of [3], the Hausdor� topological space(X;�) is a strongly zero-dimensional and metrizable space. � A contribution to the study of fuzzy metric spaces 75References[1] Deng Zi-De, Fuzzy pseudo metric spaces, J. Math. Anal. Appl. 86 (1982), 74{95.[2] R. Engelking,General Topology, PWN-Polish Sci. Publ, Warsaw, 1977.[3] P. Fletcher and W. Lindgren,Quasi Uniform Spaces, Marcel Dekker, New York, 1982.[4] A. George and P.V. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets andSystems 64 (1994), 395{399.[5] A. George and P.V. Veeramani, On some results of analysis for fuzzy metric spaces, FuzzySets and Systems 90 (1997), 365{368.[6] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1988),385{389.[7] V. Gregori and S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets andSystems, 115 (2000), 485{489.[8] V. Gregori and S. Romaguera, On completion of fuzzy metric spaces, preprint.[9] O. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11(1975), 326{334.[10] B. Schweizer and A. Sklar, Probabilistic metric spaces, Elsevier Science Publishing Co.,New York (1983).[11] B. Schweizer and A. Sklar, Statistical metric spaces, Paci�c J. Math. 10 (1960), 314{334.Received October 2000Revised version April 2001 Almanzor Sapena PieraDepartamento de Matem�atica AplicadaEscuela Polit�ecnica Superior de AlcoyPza. Ferr�andiz-Carbonell, 203801 Alcoy (Alicante), SpainE-mail address: alsapie@mat.upv.es