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Intuitionistic fuzzy metric spaces
Chaos, Solitons and Fractals 22 (2004) 1039–1046 www.elsevier.com/locate/chaos
Intuitionistic fuzzy metric spaces Jin Han Park
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Division of Mathematical Sciences, Pukyong National University, 599-1 Daeyeon 3-Dong, Nam-Gu, Pusan 608-737, South Korea Accepted 24 February 2004
Abstract Using the idea of intuitionistic fuzzy set due to Atanassov [Intuitionistic fuzzy sets. in: Sgurev V. (Ed.), VII ITKR’s Session, Sofia June, 1983; Fuzzy Sets Syst. 20 (1986) 87], we define the notion of intuitionistic fuzzy metric spaces as a natural generalization of fuzzy metric spaces due to George and Veeramani [Fuzzy Sets Syst. 64 (1994) 395] and prove some known results of metric spaces including Baire’s theorem and the Uniform limit theorem for intuitionistic fuzzy metric spaces. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction The concept of fuzzy topology may have very important applications in quantum particle physics particularly in connections with both string and ð1Þ theory which were given and studied by Elnaschie [8,9]. One of the most important problems in fuzzy topology is to obtain an appropriate concept of fuzzy metric space. This problem has been investigated by many authors [5,10,12,15,17] from different points of views. In particular, George and Veeramani [12] have introduced and studied a notion of fuzzy metric space with the help of continuous tnorms, which constitutes a slight but appealing modification of the one due to Kramosil and Michalek [17]. On the other hand, Atanassov [1] introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets [24], and later there has been much progress in the study of intuitionistic fuzzy sets by many authors [1–7,13,15]. In this paper, using the idea of intuitionistic fuzzy sets, we define the notion of intuitionistic fuzzy metric spaces with the help of continuous t-norms and continuous t-conorms as a generalization of fuzzy metric space due to George and Veeramani [12]. In Section 3, we define a Hausdorff topology on this intuitionistic fuzzy metric space and show that every metric induces an intuitionistic fuzzy metric. Further we introduce the notion of Cauchy sequences in an intuitionistic fuzzy metric space and prove the Baire’s theorem [21] for intuitionistic fuzzy metric spaces. In Section 4, we are finding a necessary and sufficient condition for an intuitionistic fuzzy metric space to be complete and show that every separable intuitionistic fuzzy metric space is second countable and that every subspace of an intuitionistic fuzzy metric space is separable. Finally, we prove the Uniform limit theorem [20] for intuitionistic fuzzy metric spaces.
2. Intuitionistic fuzzy metric spaces Definition 2.1 [22]. A binary operation : ½0:1 ½0; 1 ! ½0; 1 is continuous t-norm if is satisfying the following conditions:
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Tel.: +82-51-752-7457; fax: +82-51-628-7432. E-mail address: [email protected] (J.H. Park).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.02.051
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(a) (b) (c) (d)
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is commutative and associative; is continuous; a 1 ¼ a for all a 2 ½0; 1; a b 6 c d whenever a 6 c and c 6 d, and a; b; c; d 2 ½0; 1.
Definition 2.2 [22]. A binary operation } : ½0:1 ½0; 1 ! ½0; 1 is continuous t-conorm if } is satisfying the following conditions: (a) (b) (c) (d)
} is commutative and associative; } is continuous; a}0 ¼ a for all a 2 ½0; 1; a}b 6 c}d whenever a 6 c and c 6 d, and a; b; c; d 2 ½0; 1.
Note 2.3. The concepts of triangular norms (t-norms) and triangular conorms (t-conorms) are known as the axiomatic skeletons that we use for characterizing fuzzy intersections and unions, respectively. These concepts were originally introduced by Menger [19] in his study of statistical metric spaces. Several examples for these concepts were proposed by many authors (see [6,7,11,14,16,22,23]). Remark 2.4 (a) For any r1 ; r2 2 ð0; 1Þ with r1 > r2 , there exist r3 ; r4 2 ð0; 1Þ such that r1 r3 P r2 and r1 P r4 }r2 . (b) For any r5 2 ð0; 1Þ, there exist r6 ; r7 2 ð0; 1Þ such that r6 r6 P r5 and r7 }r7 6 r5 . Definition 2.5. A 5-tuple ðX ; M; N ; ; }Þ is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, is a continuous t-norm, } is a continuous t-conorm and M, N are fuzzy sets on X 2 ð0; 1Þ satisfying the following conditions: for all x; y; z 2 X , s; t > 0, (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)
Mðx; y; tÞ þ N ðx; y; tÞ 6 1; Mðx; y; tÞ > 0; Mðx; y; tÞ ¼ 1 if and only if x ¼ y; Mðx; y; tÞ ¼ Mðy; x; tÞ; Mðx; y; tÞ Mðy; z; sÞ 6 Mðx; z; t þ sÞ; Mðx; y; Þ : ð0; 1Þ ! ð0; 1 is continuous; N ðx; y; tÞ > 0; N ðx; y; tÞ ¼ 0 if and only if x ¼ y; N ðx; y; tÞ ¼ Nðy; x; tÞ; N ðx; y; tÞ}N ðy; z; sÞ P N ðx; z; t þ sÞ; N ðx; y; Þ : ð0; 1Þ ! ð0; 1 is continuous.
Then ðM; N Þ is called an intuitionistic fuzzy metric on X . The functions Mðx; y; tÞ and N ðx; y; tÞ denote the degree of nearness and the degree of non-nearness between x and y with respect to t, respectively. Remark 2.6. Every fuzzy metric space ðX ; M; Þ is an intuitionistic fuzzy metric space of the form ðX ; M; 1 M; ; }Þ such that t-norm and t-conorm } are associated [18], i.e. x}y ¼ 1 ðð1 xÞ ð1 yÞÞ for any x; y 2 X . Remark 2.7. In intuitionistic fuzzy metric space X , Mðx; y; Þ is non-decreasing and N ðx; y; Þ is non-increasing for all x; y 2 X . Example 2.8 (Induced intuitionistic fuzzy metric). Let ðX ; dÞ be a metric space. Denote a b ¼ ab and a}b ¼ minf1; a þ bg for all a; b 2 ½0; 1 and let Md and Nd be fuzzy sets on X 2 ð0; 1Þ defined as follows: Md ðx; y; tÞ ¼
htn
htn ; þ mdðx; yÞ
Nd ðx; y; tÞ ¼
dðx; yÞ ktn þ mdðx; yÞ
for all h; k; m; n 2 Rþ . Then ðX ; Md ; Nd ; ; }Þ is an intuitionistic fuzzy metric space.
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Remark 2.9. Note the above example holds even with the t-norm a b ¼ minfa; bg and the t-conorm a}b ¼ maxfa; bg and hence ðM; NÞ is an intuitionistic fuzzy metric with respect to any continuous t-norm and continuous t-conorm. In the above example by taking h ¼ k ¼ m ¼ n ¼ 1, we get t dðx; yÞ Md ðx; y; tÞ ¼ ; Nd ðx; y; tÞ ¼ : t þ dðx; yÞ t þ dðx; yÞ We call this intuitionistic fuzzy metric induced by a metric d the standard intuitionistic fuzzy metric. Example 2.10. Let X ¼ N. Define a b ¼ maxf0; a þ b 1g and a}b ¼ a þ b ab for all a; b 2 ½0; 1 and let M and N be fuzzy sets on X 2 ð0; 1Þ as follows: 8x < Mðx; y; tÞ ¼ y :y x
if x 6 y; if y 6 x;
8
if x 6 y; if y 6 x
for all x; y 2 X and t > 0. Then ðX ; M; N ; ; }Þ is an intuitionistic fuzzy metric space. Remark 2.11. Note that, in the above example, t-norm and t-conorm } are not associated. And there exists no metric d on X satisfying Mðx; y; tÞ ¼
t ; t þ dðx; yÞ
N ðx; y; tÞ ¼
dðx; yÞ ; t þ dðx; yÞ
where Mðx; y; tÞ and N ðx; y; tÞ are as defined in above example. Also note that the above functions ðM; N Þ is not an intuitionistic fuzzy metric with the t-norm and t-conorm defined as a b ¼ minfa; bg and a}b ¼ maxfa; bg. 3. Topology induced by an intuitionistic fuzzy metric Definition 3.1. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space, and let r 2 ð0; 1Þ, t > 0 and x 2 X . The set Bðx; r; tÞ ¼ fy 2 X : Mðx; y; tÞ > 1 r; N ðx; y; tÞ < rg is called the open ball with center x and radius r with respect to t. Theorem 3.2. Every open ball Bðx; r; tÞ is an open set. Proof. Let Bðx; r; tÞ be an open ball with center x and radius r with respect to t. Let y 2 Bðx; r; tÞ. Then Mðx; y; tÞ > 1 r and Nðx; y; tÞ < r. Since Mðx; y; tÞ > 1 r, there exists t0 2 ð0; tÞ such that Mðx; y; t0 Þ > 1 r and Nðx; y; t0 Þ < r. Put r0 ¼ Mðx; y; t0 Þ. Since r0 > 1 r, there exists s 2 ð0; 1Þ such that r0 > 1 s > 1 r. Now for given r0 and s such that r0 > 1 s, there exist r1 ; r2 2 ð0; 1Þ such that r0 r1 > 1 s and ð1 r0 Þ}ð1 r2 Þ 6 s. Put r3 ¼ maxfr1 ; r2 g and consider the open ball Bðy; 1 r3 ; t t0 Þ. We claim Bðy; 1 r3 ; t t0 Þ Bðx; r; tÞ. Now, let z 2 Bðy; 1 r3 ; t t0 Þ. Then Mðy; z; t t0 Þ > r3 and N ðy; z; t t0 Þ < r3 . Therefore Mðx; z; tÞ P Mðx; y; t0 Þ Mðy; z; t t0 Þ P r0 r3 P r0 r1 P 1 s > 1 r and Nðx; z; tÞ 6 N ðx; y; t0 Þ}N ðy; z; t t0 Þ 6 ð1 r0 Þ}ð1 r3 Þ 6 ð1 r0 Þ}ð1 r2 Þ 6 s < r: Thus z 2 Bðx; r; tÞ and hence Bðy; 1 r3 ; t t0 Þ Bðx; r; tÞ.
h
Remark 3.3. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space. Define sðM;N Þ ¼ fA X : for each x 2 A, there exist t > 0 and r 2 ð0; 1Þ such that Bðx; y; tÞ Ag. Then sðM;N Þ is a topology on X . Remark 3.4 (a) From Theorem 3.2 and Remark 3.3, every intuitionistic fuzzy metric ðN ; MÞ on X generates a topology sðM ;N Þ on X which has as a base the family of open sets of the form fBðx; r; tÞ : x 2 X ; r 2 ð0; 1Þ; t > 0g. (b) Since fBðx; 1n ; 1nÞ : n ¼ 1; 2; . . .g is a local base at x, the topology sðM;N Þ is first countable.
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Theorem 3.5. Every intuitionistic fuzzy metric space is Hausdorff. Proof. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space. Let x and y be two distinct points in X . Then 0 < Mðx; y; tÞ < 1 and 0 < N ðx; y; tÞ < 1. Put r1 ¼ Mðx; y; tÞ, r2 ¼ N ðx; y; tÞ and r ¼ maxfr1 ; 1 r2 g. For each r0 2 ðr; 1Þ, there exist r3 and r4 such that r3 r3 P r0 and ð1 r4 Þ}ð1 r4 Þ 6 1 r0 . Put r5 ¼ maxfr3 ; r4 g and consider the open balls Bðx; 1 r5 ; 2t Þ and Bðy; 1 r5 ; 2t Þ. Then clearly Bðx; 1 r5 ; 2t Þ \ Bðy; 1 r5 ; 2t Þ ¼ /. For if there exists z 2 Bðx; 1 r5 ; 2t Þ \ Bðy; 1 r5 ; 2t Þ, then t t M z; y; P r5 r5 P r3 r3 P r0 > r1 r1 ¼ Mðx; y; tÞ P M x; z; 2 2 and t t 6 ð1 r5 Þ}ð1 r5 Þ 6 ð1 r4 Þ}ð1 r4 Þ 6 1 r0 < r2 ; r2 ¼ N ðx; y; tÞ 6 N x; z; }N z; y; 2 2 which is a contradiction. Hence ðX ; M; N ; ; }Þ is Hausdorff.
h
Remark 3.6. Let ðX ; dÞ be a metric space. Let Mðx; y; tÞ ¼
t ; t þ dðx; yÞ
N ðx; y; tÞ ¼
dðx; yÞ ; kt þ dðx; yÞ
k 2 Rþ
be the intuitionistic fuzzy metric defined on X . Then the topology sd induced by the metric d and the topology sðM ;N Þ induced by the intuitionistic fuzzy metric ðM; N Þ are the same. Definition 3.7. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space. A subset A of X is said to be IF-bounded if there exist t > 0 and r 2 ð0; 1Þ such that Mðx; y; tÞ > 1 r and Nðx; y; tÞ < r for all x; y 2 A. Remark 3.8. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space induced by a metric d on X . Then A X is IF-bounded if and only if it is bounded. Theorem 3.9. Every compact subset A of an intuitionistic fuzzy metric space ðX ; M; N ; ; }Þ is IF-bounded. Proof. Let A be a compact subset of an intuitionistic fuzzy metric space X . Fix t > 0 and 0 < S r < 1. Consider an open cover fBðx; r; tÞ : x 2 Ag of A. Since A is compact, there exist x1 ; x2 ; . . . ; xn 2 A such that A ni¼1 Bðxi ; r; tÞ. Let x; y 2 A. Then x 2 Bðxi ; r; tÞ and y 2 Bðxj ; r; tÞ for some i; j. Thus we have Mðx; xi ; tÞ > 1 r, N ðx; xi ; tÞ < r, Mðy; xj ; tÞ > 1 r and Nðy; xj ; tÞ < r. Now, let a ¼ minfMðxi ; xj ; tÞ : 1 6 i; j 6 ng and b ¼ maxfN ðxi ; xj ; tÞ : i 6 i; j 6 ng. Then a > 0 and b > 0. Now, we have Mðx; y; 3tÞ P Mðx; xi ; tÞ Mðxi ; xj ; tÞ Mðxj ; y; tÞ P ð1 rÞ ð1 rÞ a > 1 s1
for some 0 < s1 < 1
and N ðx; y; 3tÞ 6 N ðx; xi ; tÞ}N ðxi ; xj ; tÞ}N ðxj ; y; tÞ 6 r}r}b < s2
for some 0 < s2 < 1:
Taking s ¼ maxfs1 ; s2 g and t0 ¼ 3t, we have Mðx; y; t0 Þ > 1 s and N ðx; y; t0 Þ < s for all x; y 2 A. Hence A is IF-bounded. h From Remark 3.8 and Theorems 3.5 and 3.9 we have the following: Remark 3.10. In an intuitionistic fuzzy metric space every compact set is closed and bounded. Theorem 3.11. Let ðX ; M; N ; ; }Þ be a fuzzy metric space and sðM;N Þ be the topology on X induced by the fuzzy metric. Then for a sequence fxn g in X , xn ! x if and only if Mðxn ; x; tÞ ! 1 and Nðxn ; x; tÞ ! 0 as n ! 1. Proof. Fix t > 0. Suppose xn ! x. Then for r 2 ð0; 1Þ, there exists n0 2 N such that xn 2 Bðx; r; tÞ for all n P n0 . Then 1 Mðxn ; x; tÞ < r and N ðxn ; x; tÞ < r and hence Mðxn ; x; tÞ ! 1 and Nðxn ; x; tÞ ! 0 as n ! 1.
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Conversely, if for each t > 0, Mðxn ; x; tÞ ! 1 and N ðxn ; x; tÞ ! 0 as n ! 1, then for r 2 ð0; 1Þ, there exists n0 2 N such that 1 Mðxn ; x; tÞ < r and N ðxn ; x; tÞ < r for all n P n0 . It follows that Mðxn ; x; tÞ > 1 r and N ðxn ; x; tÞ < r for all n P n0 . Thus xn 2 Bðx; r; tÞ for all n P n0 and hence xn ! x. h Definition 3.12. Let ðX ; M; N; ; }Þ be an intuitionistic fuzzy metric space. Then (a) a sequence fxn g in X is said to be Cauchy if for each > 0 and each t > 0, there exists n0 2 N such that Mðxn ; xm ; tÞ > 1 and N ðxn ; xm ; tÞ < for all n; m P n0 . (b) ðX ; M; N ; ; }Þ is called complete if every Cauchy sequence is convergent with respect to sðM;N Þ . Theorem 3.13. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space such that every Cauchy sequence in X has a convergent subsequence. Then ðX ; M; N ; ; }Þ is complete. Proof. Let fxn g be a Cauchy sequence and let fxin g be a subsequence of fxn g that converges to x. We prove that xn ! x. Let t > 0 and 2 ð0; 1Þ. Choose r 2 ð0; 1Þ such that ð1 rÞ ð1 rÞP 1 and r}r 6 . Since fxn g is Cauchy sequence, there is n0 2 N such that M xm ; xn ; 2t > 1 r and N xm ; xn ; 2t < r for all m; n P n0 . Since xni ! x, there is positive integer ip such that ip > n0 , M xip ; x; 2t > 1 r and N ðxip ; x; 2t Þ < r. Then, if n P n0 , t t M xip ; x; > ð1 rÞ ð1 rÞ P 1 Mðxn ; x; tÞ P M xn ; xip ; 2 2 and t t Nðxn ; x; tÞ 6 N xn ; xip ; }N xip ; x; < r}r 6 : 2 2 Therefore xn ! x and hence ðX ; M; N; ; }Þ is complete.
h
Theorem 3.14. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space and let A be a subset of X with the subspace intuitionistic fuzzy metric ðMA ; NA Þ ¼ ðMjA2 ð0;1Þ ; N jA2 ð0;1Þ Þ. Then ðA; MA ; NA ; ; }Þ is complete if and only if A is a closed subset of X . Proof. Suppose that A is a closed subset of X and let fxn g be a Cauchy sequence in ðA; MA ; NA ; ; }Þ. Then fxn g is a Cauchy sequence in X and hence there is a point X in X such that xn ! x. Then x 2 A ¼ A and thus fxn g converges in A. Hence ðA; MA ; NA ; ; }Þ is complete. Conversely, suppose that ðA; MA ; NA ; ; }Þ is complete and A is not closed. Let x 2 A n A. Then there is a sequence fxn g of points in A that converges to x and thus fxn g is a Cauchy sequence. Thus for each 0 < < 1 and each t > 0, there is n0 2 N such that Mðxn ; xm ; tÞ > 1 and N ðxn ; xm ; tÞ < for all n; m P n0 . Since fxn g is a sequence in A, Mðxn ; xm ; tÞ ¼ MA ðxn ; xm ; tÞ and Nðxn ; xm ; tÞ ¼ NA ðxn ; xm ; tÞ. Therefore fxn g is a Cauchy sequence in A. Since ðA; MA ; NA ; ; }Þ is complete, there is a y 2 A such that xn ! y. That is, for each 0 < < 1 and each t > 0, there is n0 2 N such that MA ðy; xn ; tÞ > 1 and NA ðy; xn ; tÞ < for all n P n0 . But since fxn g is a sequence in A and y 2 A, Mðy; xn ; tÞ ¼ MA ðy; xn ; tÞ and Nðy; xn ; tÞ ¼ NA ðy; xn ; tÞ. Hence fxn g converges in ðX ; M; N ; ; }Þ to both x and y. Since x 62 A and y 2 A, x 6¼ y. This is a contradiction. h Lemma 3.15. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space. If t > 0 and r; s 2 ð0; 1Þ such that ð1 sÞ ð1 sÞ P ð1 rÞ and s}s 6 r, then Bðx; s; 2t Þ Bðx; r; tÞ. Proof. Let y 2 Bðx; s; 2t Þ and let Bðy; s; 2t Þ be an open ball with center x and radius s. Since Bðy; s; 2t Þ \ Bðx; s; 2t Þ 6¼ /, there is a z 2 Bðy; s; 2t Þ \ Bðx; s; 2t Þ. Then we have t t M y; z; > ð1 sÞ ð1 sÞ P 1 r Mðx; y; tÞ P M x; z; 2 2 and t t < s}s 6 r: Nðx; y; tÞ 6 N x; z; }N y; z; 2 2 Hence z 2 Bðx; r; tÞ and thus Bðx; s; 2t Þ Bðx; r; tÞ.
h
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Theorem 3.16. A subset A of an intuitionistic fuzzy metric space ðX ; M; N; ; }Þ is nowhere dense if and only if every nonempty open set in X contains an open ball whose closure is disjoint from A. Proof. Let U be a nonempty open subset of X . Then there exists a nonempty open set V such that V U and V \ A 6¼ /. Let x 2 V . Then there exist r 2 ð0; 1Þ and t > 0 such that Bðx; r; tÞ V . Choose s 2 ð0; 1Þ such that ð1 sÞ ð1 sÞ P 1 r and s}s 6 r. By Lemma 3.15, Bðx; s; 2t Þ Bðx; r; tÞ. Thus Bðx; s; 2t Þ U and Bðx; s; 2t Þ \ A ¼ /. Conversely, suppose A is not nowhere dense. Then intðAÞ 6¼ /, so there exists a nonempty open set U such that U A. Let Bðx; r; tÞ be an open ball such that Bðx; r; tÞ U . Then Bðx; r; tÞ \ A 6¼ /. This is a contradiction. h Now, we shall prove Baire’s theorem for intuitionistic fuzzy metric space. Theorem 3.17 (Baire’s theorem). Let T fUn : n 2 Ng be a sequence of dense open subsets of a complete intuitionistic fuzzy metric space ðX ; M; N ; ; }Þ. Then n2N Un is also dense in X . Proof. Let V be a nonempty open set of X . Since U1 is dense in X , V \ U1 6¼ /. Let x1 2 V \ U1 . Since V \ U1 is open, there exist r1 2 ð0; 1Þ and t1 > 0 such that Bðx1 ; r1 ; t1 Þ V \ U1 . Choose r10 < r1 and t10 ¼ minðt1 ; 1Þ such that Bðx1 ; r10 ; t10 Þ V \ U1 . Since U2 is dense in X , Bðx1 ; r10 ; t10 Þ \ U2 6¼ /. Let x2 2 Bðx1 ; r10 ; t10 Þ \ U2 . Since Bðx1 ; r10 ; t10 Þ \ U2 is open, there exist r2 2 ð0; 12Þ and t2 > 0 such that Bðx2 ; r2 ; t2 Þ Bðx1 ; r10 ; t10 Þ \ U2 . Choose r20 < r2 and t20 ¼ minft2 ; 12g such that Bðx2 ; r20 ; t20 Þ Bðx1 ; r10 ; t10 Þ \ U2 . Continuing in this manner, we obtain a sequence fxn g in X and a sequence ftn0 g such that 0 < tn0 < 1n and 0 0 Bðxn ; rn0 ; tn0 Þ Bðxn1 ; rn1 ; tn1 Þ \ Un :
Now we claim that fxn g is a Cauchy sequence. For a given t > 0 and > 0, choose n0 2 N such that Then for n P n0 and m P n,
1 1 1 1 Mðxn ; xm ; tÞ P M xn ; xm ; P 1 > 1 ; N ðxn ; xm ; tÞ 6 N xn ; xm ; 6 < : n n n n
1 n0
< t and
1 n0
< .
0 0 Therefore fxn g is a Cauchy sequence. Since X is complete, there exists x 2 X such that xn ! x. Since xk 2 Bðx T n ; rn ; tn Þ for 0 0 0 ; t 0 Þ. Hence x 2 Bðx ; r 0 ; t 0 Þ Bðx k P n, we obtain x 2 Bðx ; r ; r ; t Þ \ U for all n. Therefore V \ ð U n n n1 n n1 n1 n2N n Þ 6¼ /. n n n n T Hence n2N Un is dense in X . h
Note 3.18. Since any complete intuitionistic fuzzy metric space cannot be represented as the union of a sequence of nowhere dense sets, it is not of the first category. Hence every complete intuitionistic fuzzy metric space is of the second category. Remark 3.19. Since every metric induces an intuitionistic fuzzy metric and intuitionistic fuzzy metric is a generalization of fuzzy metric, Baire’s theorem for complete metric space [21] and Baire’s theorem for complete fuzzy metric space [12] are particular cases of the above theorem. 4. Some properties of complete intuitionistic fuzzy metric spaces Definition 4.1. Let ðX ; N ; M; ; }Þ be an intuitionistic fuzzy metric space. A collection fFn gn2N is said to have intuitionistic fuzzy diameter zero if for each r 2 ð0; 1Þ and each t > 0, there exists n0 2 N such that Mðx; y; tÞ > 1 r and Nðx; y; tÞ < r for all x; y 2 Fn0 . Remark 4.2. A nonempty subset F of an intuitionistic fuzzy metric space X has intuitionistic fuzzy diameter zero if and only if F is a singleton set. Theorem 4.3. An intuitionistic fuzzy metric space ðX ; M; N; ; }Þ is complete if and only if every nested sequence fFn gn2N of nonempty closed sets with intuitionistic fuzzy diameter zero have nonempty intersection. Proof. First suppose that the given condition is satisfied. We claim that ðX ; M; N ; ; }Þ is complete. Let fxn g be a Cauchy sequence in X . Set Bn ¼ fxk : k P ng and Fn ¼ Bn , then we claim that fFn g has intuitionistic fuzzy diameter zero. For given s 2 ð0; 1Þ and t > 0, we choose r 2 ð0; 1Þ such thatð1 rÞ ð1 rÞ ð1 rÞ > 1 s and r}r}r < s. Since 1 1 fxn g is Cauchy there exists sequence, n0 2 N such that M xn ; xm ; 3 t > 1 r and N xn ; xm ; 3 t < r for all m; n P n0 . Therefore, M x; y; 13 t > 1 r and N x; y; 13 t < r for all x; y 2 Bn0 .
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yn0
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Let x; y 2 Fn0 . Then there exist sequences fx0n g and fyn0 g in Bn0 such that x0n ! x and yn0 ! y. Hence x0n 2 Bðx; r; 3t Þ and 2 Bðy; r; 3t Þ for sufficiently large n. Now we have t t t Mðx; y; tÞ P M x; x0n ; M x0n ; yn0 ; M yn0 ; y; > ð1 rÞ ð1 rÞ ð1 rÞ > 1 s 3 3 3
and t t t < r}r}r < s: Nðx; y; tÞ 6 N x; x0n ; }N x0n ; yn0 ; }N yn0 ; y; 3 3 3 Therefore, Mðx; y; tÞ > T 1 s and N ðx; y; tÞ < s for all x; y 2 Fn0 . Thus fFn g has intuitionistic fuzzy diameter zero and hence by hypothesis n2N Fn is nonempty. T Take x 2 n2N Fn . We show that xn ! x. Then, for r 2 ð0; 1Þ and t > 0, there exists n1 2 N such that Mðxn ; x; tÞ > 1 r and Nðxn ; x; tÞ < r for all n P n1 . Therefore, for each t > 0, Mðxn ; x; tÞ ! 1 and N ðxn ; x; tÞ ! 0 as n ! 1 and hence xn ! x. Therefore, ðX ; M; N ; ; }Þ is complete. Conversely, suppose that ðX ; M; N ; ; }Þ is complete and fFn gn2N is nested sequence of nonempty closed sets with intuitionistic fuzzy diameter zero. For each n 2 N, choose a point xn 2 Fn . We claim that fxn g is a Cauchy sequence. Since fFn g has intuitionistic fuzzy diameter zero, for t > 0 and r 2 ð0; 1Þ, there exists n0 2 N such that Mðx; y; tÞ > 1 r and Nðx; y; tÞ < r for all x; y 2 Fn0 . Since fFn g is nested sequence, Mðxn ; xm ; tÞ > 1 r and Nðxn ; xm ; tÞ < r for all n; m P n0 . Hence fxn g is a Cauchy sequence. Since ðX ; M; N; ; }Þ is complete, xn ! x for some x 2 X . Therefore, T x 2 Fn ¼ Fn for every n, and hence x 2 n2N Fn . This completes our proof. h T T Remark 4.4. The element x 2 n2N Fn is unique. For if there are two elements x; y 2 n2N Fn , since fFn g has intuitionistic fuzzy diameter zero, for each fixed t > 0, Mðx; y; tÞ > 1 1n and N ðx; y; tÞ < 1n for each n 2 N. This implies Mðx; y; tÞ ¼ 1 and N ðx; y; tÞ ¼ 0 and hence x ¼ y. Note that the topologies induced by the standard intuitionistic fuzzy metric and the corresponding metric are the same. So we have the following: Corollary 4.5. A metric space ðX ; dÞ is complete if and only if every nested sequence fFn gn2N of nonempty closed sets with diameter tending to zero have nonempty intersection. Theorem 4.6. Every separable intuitionistic fuzzy metric space is second countable. Proof. Let ðX ; M; N ; ; }Þ be the given separable intuitionistic fuzzy metric space. Let A ¼ fxn : n 2 Ng be a countable dense subset of X . Consider the family B ¼ fBðxj ; 1k ; 1k Þ : j; k 2 Ng. Then B is countable. We claim that B is a base for the family of all open sets in X . Let U be any open set in X and let x 2 U . Then there exist t > 0 and r 2 ð0; 1Þ such that Bðx; r; tÞ U . Since r 2 ð0; 1Þ, we can choose a s 2 ð0; 1Þ such that ð1 sÞ ð1 sÞ > 1 r and s}s < r. Take m 2 N such that m1 < minfs; 2t g. Since A is dense in X , there exists xj 2 A such that xj 2 Bðx; m1 ; m1 Þ. Now, if y 2 Bðxj ; m1 ; m1 Þ, then
t t 1 1 1 1 M y; xj ; P M x; xj ; M y; xj ; P 1 1 Mðx; y; tÞ P M x; xj ; 2 2 m m m m P ð1 sÞ ð1 sÞ > 1 r and
t t 1 1 1 1 Nðx; y; tÞ 6 N x; xj ; }N y; xj ; 6 N x; xj ; }N y; xj ; 6 } 6 s}s < r: 2 2 m m m m
Thus, y 2 Bðx; r; tÞ U and hence B is a base. h Remark 4.7. Since second countability is hereditary property and second countability implies separability, we obtain the following: Every subspace of a separable intuitionistic fuzzy metric space is separable. Definition 4.8. Let X be any nonempty set and ðY ; M; N ; ; }Þ be an intuitionistic fuzzy metric space. Then a sequence ffn g of functions from X to Y is said to converge uniformly to a function f from X to Y if given t > 0 and r 2 ð0; 1Þ, there exists n0 2 N such that Mðfn ðxÞ; f ðxÞ; tÞ > 1 r and Nðfn ðxÞ; f ðxÞ; tÞ < r for all n P n0 and for all x 2 X .
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Theorem 4.9 (Uniform limit theorem). Let fn : X ! Y be a sequence of continuous functions from a topological space X to an intuitionistic fuzzy metric space ðY ; M; N ; ; }Þ. If ffn g converges uniformly to f : X ! Y , then f is continuous. Proof. Let V be open set of Y and let x0 2 f 1 ðV Þ. We wish to find a neighborhood U of x0 such that f ðU Þ V . Since V is open, there exist t > 0 and r 2 ð0; 1Þ such that Bðf ðx0 Þ; r; tÞ V . Since r 2 ð0; 1Þ, we choose a s 2 ð0; 1Þ such that ð1 sÞ ð1 sÞ ð1 sÞ > 1 r and s}s}s < r. Since ff n g converges uniformly to f , given t > 0 and s 2 ð0; 1Þ, there exists n0 2 N such that M fn ðxÞ; f ðxÞ; 3t > 1 s and N fn ðxÞ; f ðxÞ; 3t < s for all n P n0 and for all x 2 X . Since fn continuous for all n 2 N, there exists a neighborhood U of x0 such that fn ðU Þ Bðfn ðx0 Þ; s; 3t Þ. Hence t M fn ðxÞ; fn ðx0 Þ; 3 > 1 s and N fn ðxÞ; fn ðx0 Þ; 3t < s for all x 2 U . Now
t t t M fn ðxÞ; fn ðx0 Þ; M fn ðx0 Þ; f ðx0 Þ; Mðf ðxÞ; f ðx0 Þ; tÞ P M f ðxÞ; fn ðxÞ; 3 3 3 P ð1 sÞ ð1 sÞ ð1 sÞ > 1 r and t t t N ðf ðxÞ; f ðx0 Þ; tÞ 6 N f ðxÞ; fn ðxÞ; }N fn ðxÞ; fn ðx0 Þ; }N fn ðx0 Þ; f ðx0 Þ; 6 s}s}s < r: 3 3 3 Thus, f ðxÞ 2 Bðf ðx0 Þ; r; tÞ V for all x 2 U . Hence f ðU Þ V and so f is continuous.
h
References [1] Atanassov K. Intuitionistic fuzzy sets. In: Sgurev V, editor. VII ITKR’s Session, Sofia June, 1983 (Central Sci. and Techn. Library, Bulg. Academy of Sciences, 1984). [2] Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;20:87–96. [3] Atanassov K. New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets Syst 1994;61:137–42. [4] C ß oker D. An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets Syst 1997;88:81–9. [5] Deng Zi-Ke. Fuzzy pseudo-metric spaces. J Math Anal Appl 1982;86:74–95. [6] Dombi J. A general class of fuzzy operators, the De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets Syst 1982;8:149–62. [7] Dubois D, Prade H. New results about properties and semantics of fuzzy set-theoretic operators. In: Wang PP, Chang SK, editors. Fuzzy sets: theory and applications to policy analysis and information systems. New York: Plenum Press; 1980. [8] Elnaschie MS. On the uncertainty of Cantorian geometry and two-slit experiment. Chaos, Soliton & Fractals 1998;9(3):517–29. [9] Elnaschie MS. On the verifications of heterotic strings theory and ð1Þ theory. Chaos, Soliton & Fractals 2000;11(2):2397–407. [10] Erceg MA. Metric spaces in fuzzy set theory. J Math Anal Appl 1979;69:205–30. [11] Frank MJ. On the simultaneous associativity of F ðx; yÞ and x þ y F ðx; yÞ. Aequationes Math 1979;19:194–226. [12] George A, Veeramani P. On some results in fuzzy metric spaces. Fuzzy Sets Syst 1994;64:395–9. [13] Grabiec M. Fixed points in fuzzy metric spaces. Fuzzy Sets Syst 1988;27:385–9. [14] Hamacher H. Uber logische Verknupfungen unscharfer Aussagen und deren Zugehorige Bewertungsfunktionen. In: Trappl R, Klir GJ, Ricciardi L, editors. Progress in cybernetics and systems research, vol. 3. Washington, DC: Hemisphere; 1978. [15] Kaleva O, Seikkala S. On fuzzy metric spaces. Fuzzy Sets Syst 1984;12:215–29. [16] Klement EP. Operations on fuzzy sets: an axiomatic approach. Inform Sci 1984;27:221–32. [17] Kramosil O, Michalek J. Fuzzy metric and statistical metric spaces. Kybernetica 1975;11:326–34. [18] Lowen R. Fuzzy set theory. Dordrecht: Kluwer Academic Publishers; 1996. [19] Menger K. Statistical metrics. Proc Nat Acad Sci 1942;28:535–7. [20] Munkres JR. Topology––a first course. New Jersey: Prentice-Hall; 1975. [21] Nagata J. Modern general topology. Amsterdam: North-Holland; 1974. [22] Schweizer B, Sklar A. Statistical metric spaces. Pacific J Math 1960;10:314–34. [23] Yager RR. On a general class of fuzzy connectives. Fuzzy Sets Syst 1980;4:235–42. [24] Zadeh LA. Fuzzy sets. Inform Control 1965;8:338–53.
Intuitionistic fuzzy metric spaces Jin Han Park
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Division of Mathematical Sciences, Pukyong National University, 599-1 Daeyeon 3-Dong, Nam-Gu, Pusan 608-737, South Korea Accepted 24 February 2004
Abstract Using the idea of intuitionistic fuzzy set due to Atanassov [Intuitionistic fuzzy sets. in: Sgurev V. (Ed.), VII ITKR’s Session, Sofia June, 1983; Fuzzy Sets Syst. 20 (1986) 87], we define the notion of intuitionistic fuzzy metric spaces as a natural generalization of fuzzy metric spaces due to George and Veeramani [Fuzzy Sets Syst. 64 (1994) 395] and prove some known results of metric spaces including Baire’s theorem and the Uniform limit theorem for intuitionistic fuzzy metric spaces. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction The concept of fuzzy topology may have very important applications in quantum particle physics particularly in connections with both string and ð1Þ theory which were given and studied by Elnaschie [8,9]. One of the most important problems in fuzzy topology is to obtain an appropriate concept of fuzzy metric space. This problem has been investigated by many authors [5,10,12,15,17] from different points of views. In particular, George and Veeramani [12] have introduced and studied a notion of fuzzy metric space with the help of continuous tnorms, which constitutes a slight but appealing modification of the one due to Kramosil and Michalek [17]. On the other hand, Atanassov [1] introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets [24], and later there has been much progress in the study of intuitionistic fuzzy sets by many authors [1–7,13,15]. In this paper, using the idea of intuitionistic fuzzy sets, we define the notion of intuitionistic fuzzy metric spaces with the help of continuous t-norms and continuous t-conorms as a generalization of fuzzy metric space due to George and Veeramani [12]. In Section 3, we define a Hausdorff topology on this intuitionistic fuzzy metric space and show that every metric induces an intuitionistic fuzzy metric. Further we introduce the notion of Cauchy sequences in an intuitionistic fuzzy metric space and prove the Baire’s theorem [21] for intuitionistic fuzzy metric spaces. In Section 4, we are finding a necessary and sufficient condition for an intuitionistic fuzzy metric space to be complete and show that every separable intuitionistic fuzzy metric space is second countable and that every subspace of an intuitionistic fuzzy metric space is separable. Finally, we prove the Uniform limit theorem [20] for intuitionistic fuzzy metric spaces.
2. Intuitionistic fuzzy metric spaces Definition 2.1 [22]. A binary operation : ½0:1 ½0; 1 ! ½0; 1 is continuous t-norm if is satisfying the following conditions:
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Tel.: +82-51-752-7457; fax: +82-51-628-7432. E-mail address: [email protected] (J.H. Park).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.02.051
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(a) (b) (c) (d)
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is commutative and associative; is continuous; a 1 ¼ a for all a 2 ½0; 1; a b 6 c d whenever a 6 c and c 6 d, and a; b; c; d 2 ½0; 1.
Definition 2.2 [22]. A binary operation } : ½0:1 ½0; 1 ! ½0; 1 is continuous t-conorm if } is satisfying the following conditions: (a) (b) (c) (d)
} is commutative and associative; } is continuous; a}0 ¼ a for all a 2 ½0; 1; a}b 6 c}d whenever a 6 c and c 6 d, and a; b; c; d 2 ½0; 1.
Note 2.3. The concepts of triangular norms (t-norms) and triangular conorms (t-conorms) are known as the axiomatic skeletons that we use for characterizing fuzzy intersections and unions, respectively. These concepts were originally introduced by Menger [19] in his study of statistical metric spaces. Several examples for these concepts were proposed by many authors (see [6,7,11,14,16,22,23]). Remark 2.4 (a) For any r1 ; r2 2 ð0; 1Þ with r1 > r2 , there exist r3 ; r4 2 ð0; 1Þ such that r1 r3 P r2 and r1 P r4 }r2 . (b) For any r5 2 ð0; 1Þ, there exist r6 ; r7 2 ð0; 1Þ such that r6 r6 P r5 and r7 }r7 6 r5 . Definition 2.5. A 5-tuple ðX ; M; N ; ; }Þ is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, is a continuous t-norm, } is a continuous t-conorm and M, N are fuzzy sets on X 2 ð0; 1Þ satisfying the following conditions: for all x; y; z 2 X , s; t > 0, (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)
Mðx; y; tÞ þ N ðx; y; tÞ 6 1; Mðx; y; tÞ > 0; Mðx; y; tÞ ¼ 1 if and only if x ¼ y; Mðx; y; tÞ ¼ Mðy; x; tÞ; Mðx; y; tÞ Mðy; z; sÞ 6 Mðx; z; t þ sÞ; Mðx; y; Þ : ð0; 1Þ ! ð0; 1 is continuous; N ðx; y; tÞ > 0; N ðx; y; tÞ ¼ 0 if and only if x ¼ y; N ðx; y; tÞ ¼ Nðy; x; tÞ; N ðx; y; tÞ}N ðy; z; sÞ P N ðx; z; t þ sÞ; N ðx; y; Þ : ð0; 1Þ ! ð0; 1 is continuous.
Then ðM; N Þ is called an intuitionistic fuzzy metric on X . The functions Mðx; y; tÞ and N ðx; y; tÞ denote the degree of nearness and the degree of non-nearness between x and y with respect to t, respectively. Remark 2.6. Every fuzzy metric space ðX ; M; Þ is an intuitionistic fuzzy metric space of the form ðX ; M; 1 M; ; }Þ such that t-norm and t-conorm } are associated [18], i.e. x}y ¼ 1 ðð1 xÞ ð1 yÞÞ for any x; y 2 X . Remark 2.7. In intuitionistic fuzzy metric space X , Mðx; y; Þ is non-decreasing and N ðx; y; Þ is non-increasing for all x; y 2 X . Example 2.8 (Induced intuitionistic fuzzy metric). Let ðX ; dÞ be a metric space. Denote a b ¼ ab and a}b ¼ minf1; a þ bg for all a; b 2 ½0; 1 and let Md and Nd be fuzzy sets on X 2 ð0; 1Þ defined as follows: Md ðx; y; tÞ ¼
htn
htn ; þ mdðx; yÞ
Nd ðx; y; tÞ ¼
dðx; yÞ ktn þ mdðx; yÞ
for all h; k; m; n 2 Rþ . Then ðX ; Md ; Nd ; ; }Þ is an intuitionistic fuzzy metric space.
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Remark 2.9. Note the above example holds even with the t-norm a b ¼ minfa; bg and the t-conorm a}b ¼ maxfa; bg and hence ðM; NÞ is an intuitionistic fuzzy metric with respect to any continuous t-norm and continuous t-conorm. In the above example by taking h ¼ k ¼ m ¼ n ¼ 1, we get t dðx; yÞ Md ðx; y; tÞ ¼ ; Nd ðx; y; tÞ ¼ : t þ dðx; yÞ t þ dðx; yÞ We call this intuitionistic fuzzy metric induced by a metric d the standard intuitionistic fuzzy metric. Example 2.10. Let X ¼ N. Define a b ¼ maxf0; a þ b 1g and a}b ¼ a þ b ab for all a; b 2 ½0; 1 and let M and N be fuzzy sets on X 2 ð0; 1Þ as follows: 8x < Mðx; y; tÞ ¼ y :y x
if x 6 y; if y 6 x;
8
if x 6 y; if y 6 x
for all x; y 2 X and t > 0. Then ðX ; M; N ; ; }Þ is an intuitionistic fuzzy metric space. Remark 2.11. Note that, in the above example, t-norm and t-conorm } are not associated. And there exists no metric d on X satisfying Mðx; y; tÞ ¼
t ; t þ dðx; yÞ
N ðx; y; tÞ ¼
dðx; yÞ ; t þ dðx; yÞ
where Mðx; y; tÞ and N ðx; y; tÞ are as defined in above example. Also note that the above functions ðM; N Þ is not an intuitionistic fuzzy metric with the t-norm and t-conorm defined as a b ¼ minfa; bg and a}b ¼ maxfa; bg. 3. Topology induced by an intuitionistic fuzzy metric Definition 3.1. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space, and let r 2 ð0; 1Þ, t > 0 and x 2 X . The set Bðx; r; tÞ ¼ fy 2 X : Mðx; y; tÞ > 1 r; N ðx; y; tÞ < rg is called the open ball with center x and radius r with respect to t. Theorem 3.2. Every open ball Bðx; r; tÞ is an open set. Proof. Let Bðx; r; tÞ be an open ball with center x and radius r with respect to t. Let y 2 Bðx; r; tÞ. Then Mðx; y; tÞ > 1 r and Nðx; y; tÞ < r. Since Mðx; y; tÞ > 1 r, there exists t0 2 ð0; tÞ such that Mðx; y; t0 Þ > 1 r and Nðx; y; t0 Þ < r. Put r0 ¼ Mðx; y; t0 Þ. Since r0 > 1 r, there exists s 2 ð0; 1Þ such that r0 > 1 s > 1 r. Now for given r0 and s such that r0 > 1 s, there exist r1 ; r2 2 ð0; 1Þ such that r0 r1 > 1 s and ð1 r0 Þ}ð1 r2 Þ 6 s. Put r3 ¼ maxfr1 ; r2 g and consider the open ball Bðy; 1 r3 ; t t0 Þ. We claim Bðy; 1 r3 ; t t0 Þ Bðx; r; tÞ. Now, let z 2 Bðy; 1 r3 ; t t0 Þ. Then Mðy; z; t t0 Þ > r3 and N ðy; z; t t0 Þ < r3 . Therefore Mðx; z; tÞ P Mðx; y; t0 Þ Mðy; z; t t0 Þ P r0 r3 P r0 r1 P 1 s > 1 r and Nðx; z; tÞ 6 N ðx; y; t0 Þ}N ðy; z; t t0 Þ 6 ð1 r0 Þ}ð1 r3 Þ 6 ð1 r0 Þ}ð1 r2 Þ 6 s < r: Thus z 2 Bðx; r; tÞ and hence Bðy; 1 r3 ; t t0 Þ Bðx; r; tÞ.
h
Remark 3.3. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space. Define sðM;N Þ ¼ fA X : for each x 2 A, there exist t > 0 and r 2 ð0; 1Þ such that Bðx; y; tÞ Ag. Then sðM;N Þ is a topology on X . Remark 3.4 (a) From Theorem 3.2 and Remark 3.3, every intuitionistic fuzzy metric ðN ; MÞ on X generates a topology sðM ;N Þ on X which has as a base the family of open sets of the form fBðx; r; tÞ : x 2 X ; r 2 ð0; 1Þ; t > 0g. (b) Since fBðx; 1n ; 1nÞ : n ¼ 1; 2; . . .g is a local base at x, the topology sðM;N Þ is first countable.
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Theorem 3.5. Every intuitionistic fuzzy metric space is Hausdorff. Proof. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space. Let x and y be two distinct points in X . Then 0 < Mðx; y; tÞ < 1 and 0 < N ðx; y; tÞ < 1. Put r1 ¼ Mðx; y; tÞ, r2 ¼ N ðx; y; tÞ and r ¼ maxfr1 ; 1 r2 g. For each r0 2 ðr; 1Þ, there exist r3 and r4 such that r3 r3 P r0 and ð1 r4 Þ}ð1 r4 Þ 6 1 r0 . Put r5 ¼ maxfr3 ; r4 g and consider the open balls Bðx; 1 r5 ; 2t Þ and Bðy; 1 r5 ; 2t Þ. Then clearly Bðx; 1 r5 ; 2t Þ \ Bðy; 1 r5 ; 2t Þ ¼ /. For if there exists z 2 Bðx; 1 r5 ; 2t Þ \ Bðy; 1 r5 ; 2t Þ, then t t M z; y; P r5 r5 P r3 r3 P r0 > r1 r1 ¼ Mðx; y; tÞ P M x; z; 2 2 and t t 6 ð1 r5 Þ}ð1 r5 Þ 6 ð1 r4 Þ}ð1 r4 Þ 6 1 r0 < r2 ; r2 ¼ N ðx; y; tÞ 6 N x; z; }N z; y; 2 2 which is a contradiction. Hence ðX ; M; N ; ; }Þ is Hausdorff.
h
Remark 3.6. Let ðX ; dÞ be a metric space. Let Mðx; y; tÞ ¼
t ; t þ dðx; yÞ
N ðx; y; tÞ ¼
dðx; yÞ ; kt þ dðx; yÞ
k 2 Rþ
be the intuitionistic fuzzy metric defined on X . Then the topology sd induced by the metric d and the topology sðM ;N Þ induced by the intuitionistic fuzzy metric ðM; N Þ are the same. Definition 3.7. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space. A subset A of X is said to be IF-bounded if there exist t > 0 and r 2 ð0; 1Þ such that Mðx; y; tÞ > 1 r and Nðx; y; tÞ < r for all x; y 2 A. Remark 3.8. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space induced by a metric d on X . Then A X is IF-bounded if and only if it is bounded. Theorem 3.9. Every compact subset A of an intuitionistic fuzzy metric space ðX ; M; N ; ; }Þ is IF-bounded. Proof. Let A be a compact subset of an intuitionistic fuzzy metric space X . Fix t > 0 and 0 < S r < 1. Consider an open cover fBðx; r; tÞ : x 2 Ag of A. Since A is compact, there exist x1 ; x2 ; . . . ; xn 2 A such that A ni¼1 Bðxi ; r; tÞ. Let x; y 2 A. Then x 2 Bðxi ; r; tÞ and y 2 Bðxj ; r; tÞ for some i; j. Thus we have Mðx; xi ; tÞ > 1 r, N ðx; xi ; tÞ < r, Mðy; xj ; tÞ > 1 r and Nðy; xj ; tÞ < r. Now, let a ¼ minfMðxi ; xj ; tÞ : 1 6 i; j 6 ng and b ¼ maxfN ðxi ; xj ; tÞ : i 6 i; j 6 ng. Then a > 0 and b > 0. Now, we have Mðx; y; 3tÞ P Mðx; xi ; tÞ Mðxi ; xj ; tÞ Mðxj ; y; tÞ P ð1 rÞ ð1 rÞ a > 1 s1
for some 0 < s1 < 1
and N ðx; y; 3tÞ 6 N ðx; xi ; tÞ}N ðxi ; xj ; tÞ}N ðxj ; y; tÞ 6 r}r}b < s2
for some 0 < s2 < 1:
Taking s ¼ maxfs1 ; s2 g and t0 ¼ 3t, we have Mðx; y; t0 Þ > 1 s and N ðx; y; t0 Þ < s for all x; y 2 A. Hence A is IF-bounded. h From Remark 3.8 and Theorems 3.5 and 3.9 we have the following: Remark 3.10. In an intuitionistic fuzzy metric space every compact set is closed and bounded. Theorem 3.11. Let ðX ; M; N ; ; }Þ be a fuzzy metric space and sðM;N Þ be the topology on X induced by the fuzzy metric. Then for a sequence fxn g in X , xn ! x if and only if Mðxn ; x; tÞ ! 1 and Nðxn ; x; tÞ ! 0 as n ! 1. Proof. Fix t > 0. Suppose xn ! x. Then for r 2 ð0; 1Þ, there exists n0 2 N such that xn 2 Bðx; r; tÞ for all n P n0 . Then 1 Mðxn ; x; tÞ < r and N ðxn ; x; tÞ < r and hence Mðxn ; x; tÞ ! 1 and Nðxn ; x; tÞ ! 0 as n ! 1.
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Conversely, if for each t > 0, Mðxn ; x; tÞ ! 1 and N ðxn ; x; tÞ ! 0 as n ! 1, then for r 2 ð0; 1Þ, there exists n0 2 N such that 1 Mðxn ; x; tÞ < r and N ðxn ; x; tÞ < r for all n P n0 . It follows that Mðxn ; x; tÞ > 1 r and N ðxn ; x; tÞ < r for all n P n0 . Thus xn 2 Bðx; r; tÞ for all n P n0 and hence xn ! x. h Definition 3.12. Let ðX ; M; N; ; }Þ be an intuitionistic fuzzy metric space. Then (a) a sequence fxn g in X is said to be Cauchy if for each > 0 and each t > 0, there exists n0 2 N such that Mðxn ; xm ; tÞ > 1 and N ðxn ; xm ; tÞ < for all n; m P n0 . (b) ðX ; M; N ; ; }Þ is called complete if every Cauchy sequence is convergent with respect to sðM;N Þ . Theorem 3.13. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space such that every Cauchy sequence in X has a convergent subsequence. Then ðX ; M; N ; ; }Þ is complete. Proof. Let fxn g be a Cauchy sequence and let fxin g be a subsequence of fxn g that converges to x. We prove that xn ! x. Let t > 0 and 2 ð0; 1Þ. Choose r 2 ð0; 1Þ such that ð1 rÞ ð1 rÞP 1 and r}r 6 . Since fxn g is Cauchy sequence, there is n0 2 N such that M xm ; xn ; 2t > 1 r and N xm ; xn ; 2t < r for all m; n P n0 . Since xni ! x, there is positive integer ip such that ip > n0 , M xip ; x; 2t > 1 r and N ðxip ; x; 2t Þ < r. Then, if n P n0 , t t M xip ; x; > ð1 rÞ ð1 rÞ P 1 Mðxn ; x; tÞ P M xn ; xip ; 2 2 and t t Nðxn ; x; tÞ 6 N xn ; xip ; }N xip ; x; < r}r 6 : 2 2 Therefore xn ! x and hence ðX ; M; N; ; }Þ is complete.
h
Theorem 3.14. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space and let A be a subset of X with the subspace intuitionistic fuzzy metric ðMA ; NA Þ ¼ ðMjA2 ð0;1Þ ; N jA2 ð0;1Þ Þ. Then ðA; MA ; NA ; ; }Þ is complete if and only if A is a closed subset of X . Proof. Suppose that A is a closed subset of X and let fxn g be a Cauchy sequence in ðA; MA ; NA ; ; }Þ. Then fxn g is a Cauchy sequence in X and hence there is a point X in X such that xn ! x. Then x 2 A ¼ A and thus fxn g converges in A. Hence ðA; MA ; NA ; ; }Þ is complete. Conversely, suppose that ðA; MA ; NA ; ; }Þ is complete and A is not closed. Let x 2 A n A. Then there is a sequence fxn g of points in A that converges to x and thus fxn g is a Cauchy sequence. Thus for each 0 < < 1 and each t > 0, there is n0 2 N such that Mðxn ; xm ; tÞ > 1 and N ðxn ; xm ; tÞ < for all n; m P n0 . Since fxn g is a sequence in A, Mðxn ; xm ; tÞ ¼ MA ðxn ; xm ; tÞ and Nðxn ; xm ; tÞ ¼ NA ðxn ; xm ; tÞ. Therefore fxn g is a Cauchy sequence in A. Since ðA; MA ; NA ; ; }Þ is complete, there is a y 2 A such that xn ! y. That is, for each 0 < < 1 and each t > 0, there is n0 2 N such that MA ðy; xn ; tÞ > 1 and NA ðy; xn ; tÞ < for all n P n0 . But since fxn g is a sequence in A and y 2 A, Mðy; xn ; tÞ ¼ MA ðy; xn ; tÞ and Nðy; xn ; tÞ ¼ NA ðy; xn ; tÞ. Hence fxn g converges in ðX ; M; N ; ; }Þ to both x and y. Since x 62 A and y 2 A, x 6¼ y. This is a contradiction. h Lemma 3.15. Let ðX ; M; N ; ; }Þ be an intuitionistic fuzzy metric space. If t > 0 and r; s 2 ð0; 1Þ such that ð1 sÞ ð1 sÞ P ð1 rÞ and s}s 6 r, then Bðx; s; 2t Þ Bðx; r; tÞ. Proof. Let y 2 Bðx; s; 2t Þ and let Bðy; s; 2t Þ be an open ball with center x and radius s. Since Bðy; s; 2t Þ \ Bðx; s; 2t Þ 6¼ /, there is a z 2 Bðy; s; 2t Þ \ Bðx; s; 2t Þ. Then we have t t M y; z; > ð1 sÞ ð1 sÞ P 1 r Mðx; y; tÞ P M x; z; 2 2 and t t < s}s 6 r: Nðx; y; tÞ 6 N x; z; }N y; z; 2 2 Hence z 2 Bðx; r; tÞ and thus Bðx; s; 2t Þ Bðx; r; tÞ.
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Theorem 3.16. A subset A of an intuitionistic fuzzy metric space ðX ; M; N; ; }Þ is nowhere dense if and only if every nonempty open set in X contains an open ball whose closure is disjoint from A. Proof. Let U be a nonempty open subset of X . Then there exists a nonempty open set V such that V U and V \ A 6¼ /. Let x 2 V . Then there exist r 2 ð0; 1Þ and t > 0 such that Bðx; r; tÞ V . Choose s 2 ð0; 1Þ such that ð1 sÞ ð1 sÞ P 1 r and s}s 6 r. By Lemma 3.15, Bðx; s; 2t Þ Bðx; r; tÞ. Thus Bðx; s; 2t Þ U and Bðx; s; 2t Þ \ A ¼ /. Conversely, suppose A is not nowhere dense. Then intðAÞ 6¼ /, so there exists a nonempty open set U such that U A. Let Bðx; r; tÞ be an open ball such that Bðx; r; tÞ U . Then Bðx; r; tÞ \ A 6¼ /. This is a contradiction. h Now, we shall prove Baire’s theorem for intuitionistic fuzzy metric space. Theorem 3.17 (Baire’s theorem). Let T fUn : n 2 Ng be a sequence of dense open subsets of a complete intuitionistic fuzzy metric space ðX ; M; N ; ; }Þ. Then n2N Un is also dense in X . Proof. Let V be a nonempty open set of X . Since U1 is dense in X , V \ U1 6¼ /. Let x1 2 V \ U1 . Since V \ U1 is open, there exist r1 2 ð0; 1Þ and t1 > 0 such that Bðx1 ; r1 ; t1 Þ V \ U1 . Choose r10 < r1 and t10 ¼ minðt1 ; 1Þ such that Bðx1 ; r10 ; t10 Þ V \ U1 . Since U2 is dense in X , Bðx1 ; r10 ; t10 Þ \ U2 6¼ /. Let x2 2 Bðx1 ; r10 ; t10 Þ \ U2 . Since Bðx1 ; r10 ; t10 Þ \ U2 is open, there exist r2 2 ð0; 12Þ and t2 > 0 such that Bðx2 ; r2 ; t2 Þ Bðx1 ; r10 ; t10 Þ \ U2 . Choose r20 < r2 and t20 ¼ minft2 ; 12g such that Bðx2 ; r20 ; t20 Þ Bðx1 ; r10 ; t10 Þ \ U2 . Continuing in this manner, we obtain a sequence fxn g in X and a sequence ftn0 g such that 0 < tn0 < 1n and 0 0 Bðxn ; rn0 ; tn0 Þ Bðxn1 ; rn1 ; tn1 Þ \ Un :
Now we claim that fxn g is a Cauchy sequence. For a given t > 0 and > 0, choose n0 2 N such that Then for n P n0 and m P n,
1 1 1 1 Mðxn ; xm ; tÞ P M xn ; xm ; P 1 > 1 ; N ðxn ; xm ; tÞ 6 N xn ; xm ; 6 < : n n n n
1 n0
< t and
1 n0
< .
0 0 Therefore fxn g is a Cauchy sequence. Since X is complete, there exists x 2 X such that xn ! x. Since xk 2 Bðx T n ; rn ; tn Þ for 0 0 0 ; t 0 Þ. Hence x 2 Bðx ; r 0 ; t 0 Þ Bðx k P n, we obtain x 2 Bðx ; r ; r ; t Þ \ U for all n. Therefore V \ ð U n n n1 n n1 n1 n2N n Þ 6¼ /. n n n n T Hence n2N Un is dense in X . h
Note 3.18. Since any complete intuitionistic fuzzy metric space cannot be represented as the union of a sequence of nowhere dense sets, it is not of the first category. Hence every complete intuitionistic fuzzy metric space is of the second category. Remark 3.19. Since every metric induces an intuitionistic fuzzy metric and intuitionistic fuzzy metric is a generalization of fuzzy metric, Baire’s theorem for complete metric space [21] and Baire’s theorem for complete fuzzy metric space [12] are particular cases of the above theorem. 4. Some properties of complete intuitionistic fuzzy metric spaces Definition 4.1. Let ðX ; N ; M; ; }Þ be an intuitionistic fuzzy metric space. A collection fFn gn2N is said to have intuitionistic fuzzy diameter zero if for each r 2 ð0; 1Þ and each t > 0, there exists n0 2 N such that Mðx; y; tÞ > 1 r and Nðx; y; tÞ < r for all x; y 2 Fn0 . Remark 4.2. A nonempty subset F of an intuitionistic fuzzy metric space X has intuitionistic fuzzy diameter zero if and only if F is a singleton set. Theorem 4.3. An intuitionistic fuzzy metric space ðX ; M; N; ; }Þ is complete if and only if every nested sequence fFn gn2N of nonempty closed sets with intuitionistic fuzzy diameter zero have nonempty intersection. Proof. First suppose that the given condition is satisfied. We claim that ðX ; M; N ; ; }Þ is complete. Let fxn g be a Cauchy sequence in X . Set Bn ¼ fxk : k P ng and Fn ¼ Bn , then we claim that fFn g has intuitionistic fuzzy diameter zero. For given s 2 ð0; 1Þ and t > 0, we choose r 2 ð0; 1Þ such thatð1 rÞ ð1 rÞ ð1 rÞ > 1 s and r}r}r < s. Since 1 1 fxn g is Cauchy there exists sequence, n0 2 N such that M xn ; xm ; 3 t > 1 r and N xn ; xm ; 3 t < r for all m; n P n0 . Therefore, M x; y; 13 t > 1 r and N x; y; 13 t < r for all x; y 2 Bn0 .
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yn0
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Let x; y 2 Fn0 . Then there exist sequences fx0n g and fyn0 g in Bn0 such that x0n ! x and yn0 ! y. Hence x0n 2 Bðx; r; 3t Þ and 2 Bðy; r; 3t Þ for sufficiently large n. Now we have t t t Mðx; y; tÞ P M x; x0n ; M x0n ; yn0 ; M yn0 ; y; > ð1 rÞ ð1 rÞ ð1 rÞ > 1 s 3 3 3
and t t t < r}r}r < s: Nðx; y; tÞ 6 N x; x0n ; }N x0n ; yn0 ; }N yn0 ; y; 3 3 3 Therefore, Mðx; y; tÞ > T 1 s and N ðx; y; tÞ < s for all x; y 2 Fn0 . Thus fFn g has intuitionistic fuzzy diameter zero and hence by hypothesis n2N Fn is nonempty. T Take x 2 n2N Fn . We show that xn ! x. Then, for r 2 ð0; 1Þ and t > 0, there exists n1 2 N such that Mðxn ; x; tÞ > 1 r and Nðxn ; x; tÞ < r for all n P n1 . Therefore, for each t > 0, Mðxn ; x; tÞ ! 1 and N ðxn ; x; tÞ ! 0 as n ! 1 and hence xn ! x. Therefore, ðX ; M; N ; ; }Þ is complete. Conversely, suppose that ðX ; M; N ; ; }Þ is complete and fFn gn2N is nested sequence of nonempty closed sets with intuitionistic fuzzy diameter zero. For each n 2 N, choose a point xn 2 Fn . We claim that fxn g is a Cauchy sequence. Since fFn g has intuitionistic fuzzy diameter zero, for t > 0 and r 2 ð0; 1Þ, there exists n0 2 N such that Mðx; y; tÞ > 1 r and Nðx; y; tÞ < r for all x; y 2 Fn0 . Since fFn g is nested sequence, Mðxn ; xm ; tÞ > 1 r and Nðxn ; xm ; tÞ < r for all n; m P n0 . Hence fxn g is a Cauchy sequence. Since ðX ; M; N; ; }Þ is complete, xn ! x for some x 2 X . Therefore, T x 2 Fn ¼ Fn for every n, and hence x 2 n2N Fn . This completes our proof. h T T Remark 4.4. The element x 2 n2N Fn is unique. For if there are two elements x; y 2 n2N Fn , since fFn g has intuitionistic fuzzy diameter zero, for each fixed t > 0, Mðx; y; tÞ > 1 1n and N ðx; y; tÞ < 1n for each n 2 N. This implies Mðx; y; tÞ ¼ 1 and N ðx; y; tÞ ¼ 0 and hence x ¼ y. Note that the topologies induced by the standard intuitionistic fuzzy metric and the corresponding metric are the same. So we have the following: Corollary 4.5. A metric space ðX ; dÞ is complete if and only if every nested sequence fFn gn2N of nonempty closed sets with diameter tending to zero have nonempty intersection. Theorem 4.6. Every separable intuitionistic fuzzy metric space is second countable. Proof. Let ðX ; M; N ; ; }Þ be the given separable intuitionistic fuzzy metric space. Let A ¼ fxn : n 2 Ng be a countable dense subset of X . Consider the family B ¼ fBðxj ; 1k ; 1k Þ : j; k 2 Ng. Then B is countable. We claim that B is a base for the family of all open sets in X . Let U be any open set in X and let x 2 U . Then there exist t > 0 and r 2 ð0; 1Þ such that Bðx; r; tÞ U . Since r 2 ð0; 1Þ, we can choose a s 2 ð0; 1Þ such that ð1 sÞ ð1 sÞ > 1 r and s}s < r. Take m 2 N such that m1 < minfs; 2t g. Since A is dense in X , there exists xj 2 A such that xj 2 Bðx; m1 ; m1 Þ. Now, if y 2 Bðxj ; m1 ; m1 Þ, then
t t 1 1 1 1 M y; xj ; P M x; xj ; M y; xj ; P 1 1 Mðx; y; tÞ P M x; xj ; 2 2 m m m m P ð1 sÞ ð1 sÞ > 1 r and
t t 1 1 1 1 Nðx; y; tÞ 6 N x; xj ; }N y; xj ; 6 N x; xj ; }N y; xj ; 6 } 6 s}s < r: 2 2 m m m m
Thus, y 2 Bðx; r; tÞ U and hence B is a base. h Remark 4.7. Since second countability is hereditary property and second countability implies separability, we obtain the following: Every subspace of a separable intuitionistic fuzzy metric space is separable. Definition 4.8. Let X be any nonempty set and ðY ; M; N ; ; }Þ be an intuitionistic fuzzy metric space. Then a sequence ffn g of functions from X to Y is said to converge uniformly to a function f from X to Y if given t > 0 and r 2 ð0; 1Þ, there exists n0 2 N such that Mðfn ðxÞ; f ðxÞ; tÞ > 1 r and Nðfn ðxÞ; f ðxÞ; tÞ < r for all n P n0 and for all x 2 X .
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Theorem 4.9 (Uniform limit theorem). Let fn : X ! Y be a sequence of continuous functions from a topological space X to an intuitionistic fuzzy metric space ðY ; M; N ; ; }Þ. If ffn g converges uniformly to f : X ! Y , then f is continuous. Proof. Let V be open set of Y and let x0 2 f 1 ðV Þ. We wish to find a neighborhood U of x0 such that f ðU Þ V . Since V is open, there exist t > 0 and r 2 ð0; 1Þ such that Bðf ðx0 Þ; r; tÞ V . Since r 2 ð0; 1Þ, we choose a s 2 ð0; 1Þ such that ð1 sÞ ð1 sÞ ð1 sÞ > 1 r and s}s}s < r. Since ff n g converges uniformly to f , given t > 0 and s 2 ð0; 1Þ, there exists n0 2 N such that M fn ðxÞ; f ðxÞ; 3t > 1 s and N fn ðxÞ; f ðxÞ; 3t < s for all n P n0 and for all x 2 X . Since fn continuous for all n 2 N, there exists a neighborhood U of x0 such that fn ðU Þ Bðfn ðx0 Þ; s; 3t Þ. Hence t M fn ðxÞ; fn ðx0 Þ; 3 > 1 s and N fn ðxÞ; fn ðx0 Þ; 3t < s for all x 2 U . Now
t t t M fn ðxÞ; fn ðx0 Þ; M fn ðx0 Þ; f ðx0 Þ; Mðf ðxÞ; f ðx0 Þ; tÞ P M f ðxÞ; fn ðxÞ; 3 3 3 P ð1 sÞ ð1 sÞ ð1 sÞ > 1 r and t t t N ðf ðxÞ; f ðx0 Þ; tÞ 6 N f ðxÞ; fn ðxÞ; }N fn ðxÞ; fn ðx0 Þ; }N fn ðx0 Þ; f ðx0 Þ; 6 s}s}s < r: 3 3 3 Thus, f ðxÞ 2 Bðf ðx0 Þ; r; tÞ V for all x 2 U . Hence f ðU Þ V and so f is continuous.
h
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