- Email: [email protected]
Some properties of fuzzy metric spaces
Fuzzy Sets and Systems 115 (2000) 485–489
www.elsevier.com/locate/fss
Some properties of fuzzy metric spaces ValentÃn Gregori a , Salvador Romaguerab;∗ b
a Escuela Universitaria de GandÃa, Universidad Polità ecnica de Valencia, 46730 Grao de GandÃa, Valencia, Spain Escuela de Caminos, Departamento de MatemÃatica Aplicada, Universidad PolitÃecnica de Valencia, 46071 Valencia, Spain
Received March 1998; received in revised form May 1998
Abstract We prove that the topology induced by any (complete) fuzzy metric space (in the sense of George and Veeramani) is (completely) metrizable. We also show that every separable fuzzy metric space admits a precompact fuzzy metric and that a fuzzy metric space is compact if and only if it is precompact and complete. Finally, we generalize, among other classical c 2000 Elsevier Science B.V. All rights reserved. results, the Niemytzki–Tychono theorem to fuzzy metric spaces. Keywords: Topology; Fuzzy metric space; Complete; Compact; Precompact
1. Introduction One of the main problems in the theory of fuzzy topological spaces is to obtain an appropriate and consistent notion of a fuzzy metric space. Many authors have investigated this question and several di erent notions of a fuzzy metric space have been deÿned and studied. In particular, and modifying the concept of metric fuzziness introduced by Kramosil and Michalek [6], George and Veeramani [2,3], have recently studied an interesting notion of fuzzy metric space. In particular, they proved that every fuzzy metric space generates a Hausdor ÿrst countable topology. Here, we obtain a somewhat surprising stronger result. In fact, we prove (Theorem 1) that the topology generated by any fuzzy metric space is metrizable. We also show that if the fuzzy metric space is complete, then the generated topology is completely metrizable. ∗ Corresponding author. E-mail address: [email protected] (S. Romaguera).
In the light of these results we think that George and Veeramani’s deÿnition is an appropriate notion of metric fuzziness in the sense that it provides rich topological structures which can be obtained, in many cases, from classical theorems. In particular, the celebrated Niemytzki–Tychono theorem is generalized to the fuzzy setting (Theorem 6). We also introduce a notion of precompact fuzzy metric space which provides several satisfactory results in this context. For instance, we show that every separable fuzzy metric space admits a compatible precompact fuzzy metric and that a fuzzy metric space is compact if and only if it is precompact and complete. Throughout this paper the letter N will denote the set of all positive integers. Our basic reference for general topology is [1]. According to [7] a binary operation ∗ : [0; 1] × [0; 1] → [0; 1] is a continuous t-norm if ∗ satisÿes the following conditions: (i) ∗ is associative and commutative; (ii) ∗ is continuous;
c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 2 8 1 - 4
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V. Gregori, S. Romaguera / Fuzzy Sets and Systems 115 (2000) 485–489
(iii) a ∗ 1 = a for every a ∈ [0; 1]; (iv) a ∗ b6c ∗ d whenever a6c and b6d, and a; b; c; d ∈ [0; 1]. According to [2,3], a fuzzy metric space is an ordered triple (X; M; ∗) such that X is a (nonempty) set, ∗ is a continuous t-norm and M is a fuzzy set of X × X × (0; +∞) satisfying the following conditions, for all x; y; z ∈ 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)6M (x; z; t + s); (v) M (x; y; ·) : (0; +∞) → [0; 1] is continuous. If (X; M; ∗) is a fuzzy metric space, we will say that M is a fuzzy metric on X: Let (X; d) be a metric space. Deÿne a ∗ b = ab for every a; b ∈ [0; 1], and let Md be the function deÿned on X × X × (0; +∞) by Md (x; y; t) =
t : t + d(x; y)
Then (X; Md ; ∗) is a fuzzy metric space, and Md is called the fuzzy metric induced by d (see [2]). George and Veeramani proved in [2] that every fuzzy metric M on X generates a topology M on X which has as a base the family of open sets of the form {B(x; r; t): x ∈ X; 0¡r¡1; t¿0}, where B(x; r; t) = {y ∈ X : M (x; y; t)¿1−r} for every r, with 0¡r¡1, and t¿0. They proved that (X; M ) is a Hausdor ÿrst countable topological space. Moreover, if (X; d) is a metric space, then the topology generated by d coincides with the topology Md generated by the induced fuzzy metric Md . We say that a topological space (X; ) admits a compatible fuzzy metric if there is a fuzzy metric M on X such that = M . Thus, by results of George and Veeramani mentioned above, every metrizable topological space admits a compatible fuzzy metric. Conversely, we shall prove that every fuzzy metric space is metrizable. 2. The results A classical result in the theory of metrizable topological spaces is the celebrated Kelley metrization lemma [5] which is stated as follows.
Lemma 1. A T1 topological space (X; ) is metrizable if and only if it admits a compatible uniformity with a countable base. Lemma 2 (George and Veeramani [2]). Let (X; M; ∗) be a fuzzy metric space. Then M is a Hausdor topology and for each x ∈ X; {B(x; 1=n; 1=n): n ∈ N} is a neighborhood base at x for the topology M . Theorem 1. Let (X; M; ∗) be a fuzzy metric space. Then; (X; M ) is a metrizable topological space. Proof. For each n ∈ N deÿne Un = {(x; y) ∈ X × X : M (x; y; 1=n)¿1 − (1=n)}: We shall prove that {Un : n ∈ N} is a base for a uniformity U on X whose induced topology coincides with M : We ÿrst note that for each n ∈ N, {(x; x): x ∈ X } ⊆ Un ; Un+1 ⊆ Un and Un = Un−1 : On the other hand, for each n ∈ N, there is, by the continuity of ∗, an m ∈ N such that m¿2n and (1 − (1=m)) ∗ (1 − (1=m))¿1 − (1=n): Then, Um ◦ Um ⊆ Un : Indeed, let (x; y) ∈ Um and (y; z) ∈ Um . Since M (x; y; ·) is nondecreasing [4], M (x; z; 1=n)¿M (x; z; 2=m): So M (x; z; 1=n) ¿M (x; y; 1=m) ∗ M (y; z; 1=m) ¿(1 − (1=m)) ∗ (1 − (1=m)) ¿1 − (1=n): Therefore (x; z) ∈ Un : Thus {Un : n ∈ N} is a base for a uniformity U on X: Since for each x ∈ X and each n ∈ N; Un (x) = {y ∈ X : M (x; y; 1=n)¿1 − (1=n)} = B(x; 1=n; 1=n), we deduce, from Lemma 2, that the topology induced by U coincides with M . By Lemma 1, (X; M ) is a metrizable topological space. Corollary 1. A topological space is metrizable if and only if it admits a compatible fuzzy metric. Proof. Suppose that (X; ) is a metrizable space. Let d be a metric on X compatible with . Then, the fuzzy
V. Gregori, S. Romaguera / Fuzzy Sets and Systems 115 (2000) 485–489
metric Md , induced by d, is compatible with [2]. The converse follows immediately from Theorem 1. Corollary 2 (George and Veeramani [3]). Every seprable fuzzy metric space is second countable. Proof. Let (X; M; ∗) be a separable fuzzy metric space. By Theorem 1, (X; M ) is a separable metrizable space. So, it is second countable [1]. Let us recall that a metrizable topological space (X; ) is said to be completely metrizable if it admits a complete metric [1]. On the other hand, a fuzzy metric space (X; M; ∗) is called complete [3] if every Cauchy sequence is convergent, where a sequence (xn )n ∈ N is Cauchy provided that for each r, with 0¡r¡1, and each t¿0, there exists an n0 ∈ N such that M (xn ; xm ; t)¿1 − r for every n; m¿n0 . If (X; M; ∗) is a complete fuzzy metric space, we say that M is a complete fuzzy metric on X. Theorem 2. Let (X; M; ∗) be a complete fuzzy metric space. Then; (X; M ) is completely metrizable. Proof. It follows from the proof of Theorem 1 that {Un : n ∈ N} is a base for a uniformity U on X compatible with M , where Un = {(x; y) ∈ X × X : M (x; y; 1=n)¿1 − (1=n)} for every n ∈ N. Then there exists a metric d on X whose induced uniformity coincides with U. We want to show that the metric d is complete. Indeed, given a Cauchy sequence (xn )n ∈ N in (X; d); we shall prove that (xn )n∈N is Cauchy in (X; M; ∗): To this end, ÿx r; t, with 0¡r¡1 and t¿0. Choose a k ∈ N such that 1=k6 min{t; r}: Then, there is n0 ∈ N such that (xn ; xm ) ∈ Uk for every n; m¿n0 . Consequently, for each n; m¿n0 , M (xn ; xm ; t)¿M (xn ; xm ; 1=k) ¿ 1 − (1=k)¿1 − r: We have shown that (xn )n∈N is a Cauchy sequence in the complete fuzzy metric space (X; M; ∗), so it is convergent with respect to M . Hence, d is a complete metric on X: We conclude that (X; M ) is completely metrizable. Corollary 3. A topological space is completely metrizable if and only if it admits a compatible complete fuzzy metric.
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Proof. Suppose that (X; ) is a completely metrizable space. Let d be a complete metric on X compatible with . By [3, Result 2.9], the fuzzy metric Md induced by d is complete, and it is compatible with . The converse follows immediately from Theorem 2. Since every completely metrizable space is a Baire space [1], we deduce from Theorem 2 the following. Corollary 4 (George and Veeramani [2]). Every complete fuzzy metric space is a Baire space. Deÿnition 1. A fuzzy metric space (X; M; ∗) is called precompact if for each r, with 0¡r¡1, and each S t¿0; there is a ÿnite subset A of X, such that X = a ∈ A B(a; r; t). In this case, we say that M is a precompact fuzzy metric on X. A fuzzy metric space (X; M; ∗) is called compact if (X; M ) is a compact topological space. Lemma 3. A fuzzy metric space is precompact if and only if every sequence has a Cauchy subsequence. Proof. Suppose that (X; M; ∗) is a precompact fuzzy metric space. Let (xn )n∈N be a sequence in X: For each m ∈ NS there is a ÿnite subset Am of X such that X = a∈Am B(a; 1=m; 1=m). Hence, for m = 1, there exists an a1 ∈ A1 and a subsequence (x1(n) )n∈N of (xn )n∈N such that x1(n) ∈ B(a1 ; 1; 1) for every n ∈ N. Similarly, there exist an a2 ∈ A2 and a subsequence (x2(n) )n∈N of (x1(n) )n∈N such that x2(n) ∈ B(a2 ; 12 ; 12 ) for every n ∈ N. Following this process, for m ∈ N; m¿1; there is an am ∈ Am and a subsequence (xm(n) )n∈N of (x(m−1)(n) )n∈N such that xm(n) ∈ B(am ; 1=m; 1=m) for every n ∈ N. Now, consider the subsequence (xn(n) )n∈N of (xn )n∈N . Given r; with 0¡r¡1, and t¿0, there is an n0 ∈ N such that (1 − (1=n0 )) ∗ (1 − (1=n0 ))¿1 − r and 2=n0 ¡t: Then, for every k; m¿n0 ; we have M (xk(k) ; xm(m) ; t) ¿M (xk(k) ; xm(m) ; 2=n0 ) ¿M (xk(k) ; an0 ; 1=n0 ) ∗ M (an0 ; xm(m) ; 1=n0 ) ¿(1 − (1=n0 )) ∗ (1 − (1=n0 ))¿1 − r: Hence (xn(n) )n∈N is a Cauchy sequence in (X; M; ∗).
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Conversely, suppose that (X; M; ∗) is a nonprecompact fuzzy metric space. Then there exist r, with 0¡r¡1, and t¿0 S such that for each ÿnite subset A of X , X 6= a∈A B(a; r; t). Fix x1 ∈ X . There is x2 ∈ X \B(x1 ; r; t). Moreover, there is S2 x3 ∈ X \( k=1 B(xk ; r; t)): Following this process, we construct a sequenceS(xn )n∈N of distinct points in n X , such that xn+1 ∈= k=1 B(xk ; r; t) for every n ∈ N. Therefore (xn )n∈N has no Cauchy subsequence. This completes the proof.
sequence (xk(n) )n∈N of (xn )n∈N that converges to x with respect to M . Thus, given r, with 0¡r¡1, and t¿0 there is an n0 ∈ N such that for each n¿n0 , M (x; xk(n) ; t=2)¿1 − s, where s¿0 satisÿes (1 − s) ∗ (1 − s)¿1 − r. On the other hand, there is n1 ¿k(n0 ) such that for each n; m¿n1 , M (xn ; xm ; t=2)¿1 − s. Therefore, for each n¿n1 , we have
Theorem 3. A fuzzy metric space (X; M; ∗) is separable if and only if (X; M ) admits a compatible precompact fuzzy metric.
We conclude that the Cauchy sequence (xn )n∈N converges to x.
Proof. Suppose that (X; M; ∗) is a separable fuzzy metric space. By Theorem 1 and Corollary 2 (X; M ) is a separable metrizable space, so, it admits a compatible precompact metric d [1]. We shall show that, then, the fuzzy metric Md , induced by d, is precompact: Indeed, let (xn )n∈N be a sequence in X. By precompactness of d; (xn )n∈N has a Cauchy subsequence (xk(n) )n∈N in (X; d). Given r, with 0¡r¡1, and t¿0, put = t=(1 − r) − t. Then, there is n0 ∈ N such that d(xk(n) ; xk(m) )¡ for every n; m¿n0 . Therefore Md (xk(n) ; xk(m) ; t)¿
t =1 − r t+
for every n; m¿n0 : We have shown that (xk(n) )n∈N is a Cauchy sequence in the fuzzy metric space (X; Md ; ∗). By Lemma 3 (X; Md ; ∗) is precompact. Conversely, suppose that (X; M ) admits a compatible precompact fuzzy metric L. Then, for each n ∈ NS there is a ÿnite subset AS n of X such that ∞ X = a∈An B(a; 1=n; 1=n). Put A = n = 1 An . We shall show that the countable subset A of X is dense in X : Indeed, let x ∈ X and let B(x; 1=m; 1=m) be a basic neighborhood of x. Then, there exists a ∈ Am such that x ∈ B(a; 1=m; 1=m): Thus, A is dense in X. We conclude that (X; L; ∗) is separable, i.e. (X; M ) is a separable topological space. Lemma 4. Let (X; M; ∗) be a fuzzy metric space. If a Cauchy sequence clusters to a point x ∈ X; then the sequence converges to x. Proof. Let (xn )n∈N be a Cauchy sequence in (X; M; ∗) having a cluster point x ∈ X. Then, there is a sub-
M (x; xn ; t) ¿ M (x; xk(n) ; t=2) ∗ M (xk(n) ; xn ; t=2) ¿ (1 − s) ∗ (1 − s)¿1 − r:
Theorem 4. A fuzzy metric space is compact if and only if it is precompact and complete. Proof. Suppose that (X; M; ∗) is a compact fuzzy metric space. For each r; with 0¡r¡1, and each t¿0; the open cover {B(x; r; t): x ∈ X } of X, has a ÿnite subcover. Hence (X; M; ∗) is precompact. On the other hand, every Cauchy sequence (xn )n∈N in (X; M; ∗) has a cluster point y ∈ X. By Lemma 4 (xn )n∈N converges to y. Thus (X; M; ∗) is complete. Conversely, let (xn )n∈N be a sequence in X. From Lemma 3 and the completeness of (X; M; ∗), it follows that (xn )n∈N has a cluster point. Since, by Theorem 1, (X; M ) is metrizable and every sequentially compact metrizable space is compact, we conclude that (X; M; ∗) is compact. Theorem 5. A metrizable topological space is compact if and only if every compatible fuzzy metric is precompact. Proof. Suppose that (X; ) is a compact metrizable space. By Theorem 4, every compatible fuzzy metric is precompact. Conversely, let d be any metric on X compatible with . Consider the fuzzy metric Md , induced by d. By hypothesis, Md is precompact. Let (xn )n∈N be any sequence in X. By Lemma 3 (xn )n∈N has a Cauchy subsequence (xk(n) )n∈N in (X; Md ; ∗). Given , with 0¡¡ 12 , there is n0 ∈ N such that Md (xk(n) ; xk(m) ; 12 )¿1 − for every n; m¿n0 : Then, an easy computation shows that d(xk(n) ; xk(m) )¡= 2(1 − )¡ for every n; m¿n0 . Thus, we have proved that (xn )n∈N has a Cauchy subsequence in (X; d), so,
V. Gregori, S. Romaguera / Fuzzy Sets and Systems 115 (2000) 485–489
every metric d on X compatible with is precompact. By [1, 4.3. E(c)], (X; ) is compact. Theorem 6. A metrizable topological space is compact if and only if every compatible fuzzy metric is complete. Proof. Suppose that (X; ) is a compact metrizable space. By Theorem 4, every compatible fuzzy metric is complete. Conversely, let d be any metric on X compatible with . Consider the fuzzy metric Md , induced by d. By hypothesis, Md is complete. So, by [3, Result 2.9], d is complete. Applying the Niemytzki–Tychono theorem [1, 4.3.E(d)], we deduce that (X; ) is compact.
Acknowledgements The authors acknowledge the support of the DGES, under grant PB95-0737.
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References [1] R. Engelking, General Topology, PWN-Polish Sci. Publ., Warsaw, 1977. [2] A. George, P.V. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994) 395–399. [3] A. George, P.V. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems 90 (1997) 365–368. [4] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1989) 385–389. [5] J.L. Kelley, General Topology, Van Nostrand, 1955. [6] O. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975) 326 –334. [7] B. Schweizer, A. Sklar, Statistical metric spaces, Paciÿc J. Math. 10 (1960) 314 –334.
www.elsevier.com/locate/fss
Some properties of fuzzy metric spaces ValentÃn Gregori a , Salvador Romaguerab;∗ b
a Escuela Universitaria de GandÃa, Universidad Polità ecnica de Valencia, 46730 Grao de GandÃa, Valencia, Spain Escuela de Caminos, Departamento de MatemÃatica Aplicada, Universidad PolitÃecnica de Valencia, 46071 Valencia, Spain
Received March 1998; received in revised form May 1998
Abstract We prove that the topology induced by any (complete) fuzzy metric space (in the sense of George and Veeramani) is (completely) metrizable. We also show that every separable fuzzy metric space admits a precompact fuzzy metric and that a fuzzy metric space is compact if and only if it is precompact and complete. Finally, we generalize, among other classical c 2000 Elsevier Science B.V. All rights reserved. results, the Niemytzki–Tychono theorem to fuzzy metric spaces. Keywords: Topology; Fuzzy metric space; Complete; Compact; Precompact
1. Introduction One of the main problems in the theory of fuzzy topological spaces is to obtain an appropriate and consistent notion of a fuzzy metric space. Many authors have investigated this question and several di erent notions of a fuzzy metric space have been deÿned and studied. In particular, and modifying the concept of metric fuzziness introduced by Kramosil and Michalek [6], George and Veeramani [2,3], have recently studied an interesting notion of fuzzy metric space. In particular, they proved that every fuzzy metric space generates a Hausdor ÿrst countable topology. Here, we obtain a somewhat surprising stronger result. In fact, we prove (Theorem 1) that the topology generated by any fuzzy metric space is metrizable. We also show that if the fuzzy metric space is complete, then the generated topology is completely metrizable. ∗ Corresponding author. E-mail address: [email protected] (S. Romaguera).
In the light of these results we think that George and Veeramani’s deÿnition is an appropriate notion of metric fuzziness in the sense that it provides rich topological structures which can be obtained, in many cases, from classical theorems. In particular, the celebrated Niemytzki–Tychono theorem is generalized to the fuzzy setting (Theorem 6). We also introduce a notion of precompact fuzzy metric space which provides several satisfactory results in this context. For instance, we show that every separable fuzzy metric space admits a compatible precompact fuzzy metric and that a fuzzy metric space is compact if and only if it is precompact and complete. Throughout this paper the letter N will denote the set of all positive integers. Our basic reference for general topology is [1]. According to [7] a binary operation ∗ : [0; 1] × [0; 1] → [0; 1] is a continuous t-norm if ∗ satisÿes the following conditions: (i) ∗ is associative and commutative; (ii) ∗ is continuous;
c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 2 8 1 - 4
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V. Gregori, S. Romaguera / Fuzzy Sets and Systems 115 (2000) 485–489
(iii) a ∗ 1 = a for every a ∈ [0; 1]; (iv) a ∗ b6c ∗ d whenever a6c and b6d, and a; b; c; d ∈ [0; 1]. According to [2,3], a fuzzy metric space is an ordered triple (X; M; ∗) such that X is a (nonempty) set, ∗ is a continuous t-norm and M is a fuzzy set of X × X × (0; +∞) satisfying the following conditions, for all x; y; z ∈ 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)6M (x; z; t + s); (v) M (x; y; ·) : (0; +∞) → [0; 1] is continuous. If (X; M; ∗) is a fuzzy metric space, we will say that M is a fuzzy metric on X: Let (X; d) be a metric space. Deÿne a ∗ b = ab for every a; b ∈ [0; 1], and let Md be the function deÿned on X × X × (0; +∞) by Md (x; y; t) =
t : t + d(x; y)
Then (X; Md ; ∗) is a fuzzy metric space, and Md is called the fuzzy metric induced by d (see [2]). George and Veeramani proved in [2] that every fuzzy metric M on X generates a topology M on X which has as a base the family of open sets of the form {B(x; r; t): x ∈ X; 0¡r¡1; t¿0}, where B(x; r; t) = {y ∈ X : M (x; y; t)¿1−r} for every r, with 0¡r¡1, and t¿0. They proved that (X; M ) is a Hausdor ÿrst countable topological space. Moreover, if (X; d) is a metric space, then the topology generated by d coincides with the topology Md generated by the induced fuzzy metric Md . We say that a topological space (X; ) admits a compatible fuzzy metric if there is a fuzzy metric M on X such that = M . Thus, by results of George and Veeramani mentioned above, every metrizable topological space admits a compatible fuzzy metric. Conversely, we shall prove that every fuzzy metric space is metrizable. 2. The results A classical result in the theory of metrizable topological spaces is the celebrated Kelley metrization lemma [5] which is stated as follows.
Lemma 1. A T1 topological space (X; ) is metrizable if and only if it admits a compatible uniformity with a countable base. Lemma 2 (George and Veeramani [2]). Let (X; M; ∗) be a fuzzy metric space. Then M is a Hausdor topology and for each x ∈ X; {B(x; 1=n; 1=n): n ∈ N} is a neighborhood base at x for the topology M . Theorem 1. Let (X; M; ∗) be a fuzzy metric space. Then; (X; M ) is a metrizable topological space. Proof. For each n ∈ N deÿne Un = {(x; y) ∈ X × X : M (x; y; 1=n)¿1 − (1=n)}: We shall prove that {Un : n ∈ N} is a base for a uniformity U on X whose induced topology coincides with M : We ÿrst note that for each n ∈ N, {(x; x): x ∈ X } ⊆ Un ; Un+1 ⊆ Un and Un = Un−1 : On the other hand, for each n ∈ N, there is, by the continuity of ∗, an m ∈ N such that m¿2n and (1 − (1=m)) ∗ (1 − (1=m))¿1 − (1=n): Then, Um ◦ Um ⊆ Un : Indeed, let (x; y) ∈ Um and (y; z) ∈ Um . Since M (x; y; ·) is nondecreasing [4], M (x; z; 1=n)¿M (x; z; 2=m): So M (x; z; 1=n) ¿M (x; y; 1=m) ∗ M (y; z; 1=m) ¿(1 − (1=m)) ∗ (1 − (1=m)) ¿1 − (1=n): Therefore (x; z) ∈ Un : Thus {Un : n ∈ N} is a base for a uniformity U on X: Since for each x ∈ X and each n ∈ N; Un (x) = {y ∈ X : M (x; y; 1=n)¿1 − (1=n)} = B(x; 1=n; 1=n), we deduce, from Lemma 2, that the topology induced by U coincides with M . By Lemma 1, (X; M ) is a metrizable topological space. Corollary 1. A topological space is metrizable if and only if it admits a compatible fuzzy metric. Proof. Suppose that (X; ) is a metrizable space. Let d be a metric on X compatible with . Then, the fuzzy
V. Gregori, S. Romaguera / Fuzzy Sets and Systems 115 (2000) 485–489
metric Md , induced by d, is compatible with [2]. The converse follows immediately from Theorem 1. Corollary 2 (George and Veeramani [3]). Every seprable fuzzy metric space is second countable. Proof. Let (X; M; ∗) be a separable fuzzy metric space. By Theorem 1, (X; M ) is a separable metrizable space. So, it is second countable [1]. Let us recall that a metrizable topological space (X; ) is said to be completely metrizable if it admits a complete metric [1]. On the other hand, a fuzzy metric space (X; M; ∗) is called complete [3] if every Cauchy sequence is convergent, where a sequence (xn )n ∈ N is Cauchy provided that for each r, with 0¡r¡1, and each t¿0, there exists an n0 ∈ N such that M (xn ; xm ; t)¿1 − r for every n; m¿n0 . If (X; M; ∗) is a complete fuzzy metric space, we say that M is a complete fuzzy metric on X. Theorem 2. Let (X; M; ∗) be a complete fuzzy metric space. Then; (X; M ) is completely metrizable. Proof. It follows from the proof of Theorem 1 that {Un : n ∈ N} is a base for a uniformity U on X compatible with M , where Un = {(x; y) ∈ X × X : M (x; y; 1=n)¿1 − (1=n)} for every n ∈ N. Then there exists a metric d on X whose induced uniformity coincides with U. We want to show that the metric d is complete. Indeed, given a Cauchy sequence (xn )n ∈ N in (X; d); we shall prove that (xn )n∈N is Cauchy in (X; M; ∗): To this end, ÿx r; t, with 0¡r¡1 and t¿0. Choose a k ∈ N such that 1=k6 min{t; r}: Then, there is n0 ∈ N such that (xn ; xm ) ∈ Uk for every n; m¿n0 . Consequently, for each n; m¿n0 , M (xn ; xm ; t)¿M (xn ; xm ; 1=k) ¿ 1 − (1=k)¿1 − r: We have shown that (xn )n∈N is a Cauchy sequence in the complete fuzzy metric space (X; M; ∗), so it is convergent with respect to M . Hence, d is a complete metric on X: We conclude that (X; M ) is completely metrizable. Corollary 3. A topological space is completely metrizable if and only if it admits a compatible complete fuzzy metric.
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Proof. Suppose that (X; ) is a completely metrizable space. Let d be a complete metric on X compatible with . By [3, Result 2.9], the fuzzy metric Md induced by d is complete, and it is compatible with . The converse follows immediately from Theorem 2. Since every completely metrizable space is a Baire space [1], we deduce from Theorem 2 the following. Corollary 4 (George and Veeramani [2]). Every complete fuzzy metric space is a Baire space. Deÿnition 1. A fuzzy metric space (X; M; ∗) is called precompact if for each r, with 0¡r¡1, and each S t¿0; there is a ÿnite subset A of X, such that X = a ∈ A B(a; r; t). In this case, we say that M is a precompact fuzzy metric on X. A fuzzy metric space (X; M; ∗) is called compact if (X; M ) is a compact topological space. Lemma 3. A fuzzy metric space is precompact if and only if every sequence has a Cauchy subsequence. Proof. Suppose that (X; M; ∗) is a precompact fuzzy metric space. Let (xn )n∈N be a sequence in X: For each m ∈ NS there is a ÿnite subset Am of X such that X = a∈Am B(a; 1=m; 1=m). Hence, for m = 1, there exists an a1 ∈ A1 and a subsequence (x1(n) )n∈N of (xn )n∈N such that x1(n) ∈ B(a1 ; 1; 1) for every n ∈ N. Similarly, there exist an a2 ∈ A2 and a subsequence (x2(n) )n∈N of (x1(n) )n∈N such that x2(n) ∈ B(a2 ; 12 ; 12 ) for every n ∈ N. Following this process, for m ∈ N; m¿1; there is an am ∈ Am and a subsequence (xm(n) )n∈N of (x(m−1)(n) )n∈N such that xm(n) ∈ B(am ; 1=m; 1=m) for every n ∈ N. Now, consider the subsequence (xn(n) )n∈N of (xn )n∈N . Given r; with 0¡r¡1, and t¿0, there is an n0 ∈ N such that (1 − (1=n0 )) ∗ (1 − (1=n0 ))¿1 − r and 2=n0 ¡t: Then, for every k; m¿n0 ; we have M (xk(k) ; xm(m) ; t) ¿M (xk(k) ; xm(m) ; 2=n0 ) ¿M (xk(k) ; an0 ; 1=n0 ) ∗ M (an0 ; xm(m) ; 1=n0 ) ¿(1 − (1=n0 )) ∗ (1 − (1=n0 ))¿1 − r: Hence (xn(n) )n∈N is a Cauchy sequence in (X; M; ∗).
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Conversely, suppose that (X; M; ∗) is a nonprecompact fuzzy metric space. Then there exist r, with 0¡r¡1, and t¿0 S such that for each ÿnite subset A of X , X 6= a∈A B(a; r; t). Fix x1 ∈ X . There is x2 ∈ X \B(x1 ; r; t). Moreover, there is S2 x3 ∈ X \( k=1 B(xk ; r; t)): Following this process, we construct a sequenceS(xn )n∈N of distinct points in n X , such that xn+1 ∈= k=1 B(xk ; r; t) for every n ∈ N. Therefore (xn )n∈N has no Cauchy subsequence. This completes the proof.
sequence (xk(n) )n∈N of (xn )n∈N that converges to x with respect to M . Thus, given r, with 0¡r¡1, and t¿0 there is an n0 ∈ N such that for each n¿n0 , M (x; xk(n) ; t=2)¿1 − s, where s¿0 satisÿes (1 − s) ∗ (1 − s)¿1 − r. On the other hand, there is n1 ¿k(n0 ) such that for each n; m¿n1 , M (xn ; xm ; t=2)¿1 − s. Therefore, for each n¿n1 , we have
Theorem 3. A fuzzy metric space (X; M; ∗) is separable if and only if (X; M ) admits a compatible precompact fuzzy metric.
We conclude that the Cauchy sequence (xn )n∈N converges to x.
Proof. Suppose that (X; M; ∗) is a separable fuzzy metric space. By Theorem 1 and Corollary 2 (X; M ) is a separable metrizable space, so, it admits a compatible precompact metric d [1]. We shall show that, then, the fuzzy metric Md , induced by d, is precompact: Indeed, let (xn )n∈N be a sequence in X. By precompactness of d; (xn )n∈N has a Cauchy subsequence (xk(n) )n∈N in (X; d). Given r, with 0¡r¡1, and t¿0, put = t=(1 − r) − t. Then, there is n0 ∈ N such that d(xk(n) ; xk(m) )¡ for every n; m¿n0 . Therefore Md (xk(n) ; xk(m) ; t)¿
t =1 − r t+
for every n; m¿n0 : We have shown that (xk(n) )n∈N is a Cauchy sequence in the fuzzy metric space (X; Md ; ∗). By Lemma 3 (X; Md ; ∗) is precompact. Conversely, suppose that (X; M ) admits a compatible precompact fuzzy metric L. Then, for each n ∈ NS there is a ÿnite subset AS n of X such that ∞ X = a∈An B(a; 1=n; 1=n). Put A = n = 1 An . We shall show that the countable subset A of X is dense in X : Indeed, let x ∈ X and let B(x; 1=m; 1=m) be a basic neighborhood of x. Then, there exists a ∈ Am such that x ∈ B(a; 1=m; 1=m): Thus, A is dense in X. We conclude that (X; L; ∗) is separable, i.e. (X; M ) is a separable topological space. Lemma 4. Let (X; M; ∗) be a fuzzy metric space. If a Cauchy sequence clusters to a point x ∈ X; then the sequence converges to x. Proof. Let (xn )n∈N be a Cauchy sequence in (X; M; ∗) having a cluster point x ∈ X. Then, there is a sub-
M (x; xn ; t) ¿ M (x; xk(n) ; t=2) ∗ M (xk(n) ; xn ; t=2) ¿ (1 − s) ∗ (1 − s)¿1 − r:
Theorem 4. A fuzzy metric space is compact if and only if it is precompact and complete. Proof. Suppose that (X; M; ∗) is a compact fuzzy metric space. For each r; with 0¡r¡1, and each t¿0; the open cover {B(x; r; t): x ∈ X } of X, has a ÿnite subcover. Hence (X; M; ∗) is precompact. On the other hand, every Cauchy sequence (xn )n∈N in (X; M; ∗) has a cluster point y ∈ X. By Lemma 4 (xn )n∈N converges to y. Thus (X; M; ∗) is complete. Conversely, let (xn )n∈N be a sequence in X. From Lemma 3 and the completeness of (X; M; ∗), it follows that (xn )n∈N has a cluster point. Since, by Theorem 1, (X; M ) is metrizable and every sequentially compact metrizable space is compact, we conclude that (X; M; ∗) is compact. Theorem 5. A metrizable topological space is compact if and only if every compatible fuzzy metric is precompact. Proof. Suppose that (X; ) is a compact metrizable space. By Theorem 4, every compatible fuzzy metric is precompact. Conversely, let d be any metric on X compatible with . Consider the fuzzy metric Md , induced by d. By hypothesis, Md is precompact. Let (xn )n∈N be any sequence in X. By Lemma 3 (xn )n∈N has a Cauchy subsequence (xk(n) )n∈N in (X; Md ; ∗). Given , with 0¡¡ 12 , there is n0 ∈ N such that Md (xk(n) ; xk(m) ; 12 )¿1 − for every n; m¿n0 : Then, an easy computation shows that d(xk(n) ; xk(m) )¡= 2(1 − )¡ for every n; m¿n0 . Thus, we have proved that (xn )n∈N has a Cauchy subsequence in (X; d), so,
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every metric d on X compatible with is precompact. By [1, 4.3. E(c)], (X; ) is compact. Theorem 6. A metrizable topological space is compact if and only if every compatible fuzzy metric is complete. Proof. Suppose that (X; ) is a compact metrizable space. By Theorem 4, every compatible fuzzy metric is complete. Conversely, let d be any metric on X compatible with . Consider the fuzzy metric Md , induced by d. By hypothesis, Md is complete. So, by [3, Result 2.9], d is complete. Applying the Niemytzki–Tychono theorem [1, 4.3.E(d)], we deduce that (X; ) is compact.
Acknowledgements The authors acknowledge the support of the DGES, under grant PB95-0737.
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