Some topological properties of fuzzy cone metric spaces

Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 2016, 799 805 Research Article Some topological properties of fuzzy cone metric spaces Tarkan Öner Department of Mathematics, Faculty of Sciences, Muğla Sıtkı Koçman University, 48000 Mugla, Turkey. Communicated by R. Saadati Abstract We prove Baire s theorem for fuzzy cone metric spaces in the sense of Öner et al. [T. Öner, M. B. Kemir, B. Tanay, J. Nonlinear Sci. Appl., 8 2015, 610 616]. A necessary sufficient condition for a fuzzy cone metric space to be precompact is given. We also show that every separable fuzzy cone metric space is second countable that a subspace of a separable fuzzy cone metric space is separable. c 2016 All rights reserved. Keywords: Fuzzy cone metric space, Baire s theorem, separable, second countable. 2010 MSC: 54A40, 54E35, 54E15, 54H25. 1. Introduction After Zadeh [13] introduced the theory of fuzzy sets, many authors have introduced studied several notions of metric fuzziness [2], [3], [4], [8], [9] metric cone fuzziness [1], [10] from different points of view. By modifying the concept of metric fuzziness introduced by George Veeramani [4], Öner et al. [10] studied the notion of fuzzy cone metric spaces. In particular, they proved that every fuzzy cone metric space generates a Hausdorff first-countable topology. Here we study further topological properties of these spaces whose fuzzy metric version can be found in [4], [5] [6]. We show that every closed ball is a closed set prove Baire s theorem for fuzzy cone metric spaces. Moreover, we prove that a fuzzy cone metric space is precompact if only if every sequence in it has a Cauchy subsequence. Further, we show that X 1 X 2 is a complete fuzzy cone metric space if only if X 1 X 2 are complete fuzzy cone metric spaces. Finally it is proven that every separable fuzzy cone metric space is second countable a subspace of a separable fuzzy cone metric space is separable. Email address: tarkanoner@mu.edu.tr Tarkan Öner Received 2015-07-26

T. Öner, J. Nonlinear Sci. Appl. 9 2016, 799 805 800 2. Preliminaries Let E be a real Banach space, θ the zero of E P a subset of E. Then P is called a cone [7] if only if 1 P is closed, nonempty, P {θ; 2 if a, b R, a, b 0 x, y P, then ax + by P ; 3 if both x P x P, then x = θ. Given a cone P, a partial ordering on E with respect to P is defined by x y if only if y x P. The notation x y will st for x y x y, while x y will st for y x int P [7]. Throughout this paper, we assume that all the cones have nonempty interiors. There are two kinds of cones: normal nonnormal ones. A cone P is called normal if there exists a constant K 1 such that for all t, s E, θ t s implies t K s, the least positive number K having this property is called normal constant of P [7]. It is clear that K 1 [11]. According to [12], a binary operation : [0, 1] [0, 1] [0, 1] is a continuous t-norm if it satisfies: 1 is associative commutative; 2 is continuous; 3 a 1 = a for all a [0, 1]; 4 a b c d whenever a c b d, a, b, c, d [0, 1]. In [10], we generalized the concept of fuzzy metric space of George Veeramani by replacing the 0, interval by intp where P is a cone as follows: A fuzzy cone metric space is a 3-tuple X, M, such that P is a cone of E, X is nonempty set, is a continuous t-norm M is a fuzzy set on X 2 int P satisfying the following conditions, for all x, y, z X t, s int P that is t θ, s θ FCM1 M x, y, t > 0; FCM2 M x, y, t = 1 if only if x = y; FCM3 M x, y, t = M y, x, t; FCM4 M x, y, t M y, z, s M x, z, t + s; FCM5 M x, y,. : int P [0, 1] is continuous. If X, M, is a fuzzy cone metric space, we will say that M is a fuzzy cone metric on X. In [10], it was proven that every fuzzy cone metric space X, M, induces a Hausdorff first-countable topology τ fc on X which has as a base the family of sets of the form {Bx, r, t : x X, 0 < r < 1, t θ, where Bx, r, t = {y X : Mx, y, t > 1 r for every r with 0 < r < 1 t θ. A fuzzy cone metric space X, M, is called complete if every Cauchy sequence in it is convergent, where a sequence {x n is said to be a Cauchy sequence if for any ε 0, 1 any t θ there exists a natural number n 0 such that M x n, x m, t > 1 ε for all n, m n 0, a sequence {x n is said to converge to x if for any t θ any r 0, 1 there exists a natural number n 0 such that M x n, x, t > 1 r for all n n 0 [10]. A sequence {x n converges to x if only if lim n M x n, x, t 1 for each t θ [10]. 3. Results Definition 3.1. Let X, M, be a fuzzy cone metric space. For t θ, the closed ball B[x, r, t] with center x radius r 0, 1 is defined by B[x, r, t] = {y X : Mx, y, t 1 r. Lemma 3.2. Every closed ball in a fuzzy cone metric space X, M, is a closed set. Proof. Let y B[x, r, t]. Since X is first countable, there exits a sequence {y n in B[x, r, t] converging to y. Therefore My n, y, t converges to 1 for all t θ. For a given ɛ 0, we have Mx, y, t + ɛ Mx, y n, t My n, y, ɛ.

T. Öner, J. Nonlinear Sci. Appl. 9 2016, 799 805 801 Hence Mx, y, t + ɛ lim n Mx, y n, t lim n My n, y, ɛ 1 r 1 = 1 r. If Mx, y n, t is bounded, then the sequence {y n has a subsequence, which we again denote by {y n, for which lim n Mx, y n, t exists. In particular for n N, take ɛ = t n. Then Hence Thus y B[x, r, t]. Therefore B[x, r, t] is a closed set. Mx, y, t + t 1 r. n Mx, y, t lim n Mx, y, t + t n 1 r. Theorem 3.3 Baire s theorem. Let X, M, be a complete fuzzy cone metric space. Then the intersection of a countable number of dense open sets is dense. Proof. Let X be the given complete fuzzy cone metric space, B 0 a nonempty open set, D 1, D 2, D 3,... dense open sets in X. Since D 1 is dense in X, we have B 0 D 1. Let x 1 B 0 D 1. Since B 0 D 1 is open, there exist 0 < r 1 < 1, θ such that B x 1, r 1, B 0 D 1. Choose r 1 < r 1 t 1 = min {, / such that B [x 1, r 1, t 1 ] B 0 D 1. Let B 1 = B x 1, r 1, t 1. Since D 2 is dense in X, we have B 1 D 2. Let x 2 B 1 D 2. Since B 1 D 2 is open, there exist 0 < r 2 < 1/2 t 2 θ such that B x 2, r 2, t 2 B 1 D 2. Choose r 2 < r 2 t 2 = min {t 2, t 2 /2 t 2 such that B [x 2, r 2, t 2 ] B 1 D 2. Let B 2 = B x 2, r 2, t 2. Similarly proceeding by induction, we can find an x n B n 1 D n. Since B n 1 D n is open, there exist 0 < r n < 1/n, t n θ such that B x n, r n, t n B n 1 D n. Choose an r n < r n t n = min {t n, t n /n t n such that B [x n, r n, t n] B n 1 D n. Let B n = B x n, r n, t n. Now we claim that {x n is a Cauchy sequence. For a given t θ, 0 < ε < 1, choose an n 0 such that t/n 0 t t, 1/n 0 < ε. Then for n n 0, m n, we have t M x n, x m, t M x n, x m, 1 1 n 0 t n 1 ε. Therefore {x n is a Cauchy sequence. Since X is complete, x n x in X. But x k B [x n, r n, t n] for all k n. Since B [x n, r n, t n] is closed, x B [x n, r n, t n] B n 1 D n for all n. Therefore B 0 n=1 D n. Hence n=1 D n is dense in X. Definition 3.4. A fuzzy cone metric space X, M, is called precompact if for each r, with 0 < r < 1, each t θ, there is a finite subset A of X, such that X = a A B a, r, t. In this case, we say that M is a precompact fuzzy cone metric on X. Lemma 3.5. A fuzzy cone metric space is precompact if only if every sequence has a Cauchy subsequence. Proof. Suppose that X, M, is a precompact fuzzy cone metric space. Let {x n be a sequence in X. For each m N there is a finite subset A m of X such that X = a A m B a, 1/m, t 0 /m t 0 where t 0 θ is a constant. Hence, for m = 1, there exists an a 1 A 1 a subsequence { x 1n of {xn such that x 1n B a 1, 1, t 0 / t 0 for every n N. Similarly, there exist an a 2 A 2 a subsequence { x2n of { x1n such that x2n B a 2, 1/2, t 0 /2 t 0 for every n N. By continuing this process, we get that for m N, m > 1, there is an a m A m a subsequence { x mn of { xm 1n such that x mn B a m, 1/m, t 0 /m t 0 for every n N. Now, consider the subsequence { x nn of {xn. Given r with 0 < r < 1 t θ there is an n 0 N such tha 1/n 0 1 1/n 0 > 1 r 2t 0 /n 0 t 0 t. Then, for every k, m n 0, we have

T. Öner, J. Nonlinear Sci. Appl. 9 2016, 799 805 802 M x kk, x mm, t 2t 0 M x kk, x mm, n 0 t 0 t 0 M x kk, a n0, M n 0 t 0 1 1n0 1 1n0 > 1 r. a n0, x mm, t 0 n 0 t 0 Hence x nn is a Cauchy sequence in X, M,. Conversely, suppose that X, M, is a nonprecompact fuzzy cone metric space. Then there exist an r with 0 < r < 1 t θ such that for each finite subset A of X, we have X a A B a, r, t. Fix x 1 X. There is an x 2 X B x 1, r, t. Moreover, there is an x 3 X 2 k=1 B x k, r, t. By continuing this process, we construct a sequence {x n of distinct points in X such that x n+1 / n k=1 B x k, r, t for every n N. Therefore {x n has no Cauchy subsequence. This completes the proof. Lemma 3.6. Let X, M, be a fuzzy cone metric space. If a Cauchy sequence clusters around a point x X, then the sequence converges to x. Proof. Let {x n be a Cauchy sequence in X, M, having a cluster point x X. Then, there is a subsequence { x kn of {xn that converges to x with respect to τ fc. Thus, given r with 0 < r < 1 t θ, there is an n 0 N such that for each n n 0, M x, x kn, t/2 > 1 s where s > 0 satisfies 1 s 1 s > 1 r. On the other h, there is n 1 kn 0 such that for each n, m n 1, we have Mx n, x m, t/2 > 1 s. Therefore, for each n n 1, we have Mx, x n, t Mx, x kn, t 2 Mx kn, x n, t 2 1 s 1 s > 1 r. We conclude that the Cauchy sequence {x n converges to x. Proposition 3.7. Let X 1, M 1, X 2, M 2, be fuzzy cone metric spaces. X 1 X 2, let Mx 1, x 2, y 1, y 2, t = M 1 x 1, y 1, t M 2 x 2, y 2, t. Then M is a fuzzy cone metric on X 1 X 2. Proof. FCM1. Since M 1 x 1, y 1, t > 0 M 2 x 2, y 2, t > 0, this implies that Therefore M 1 x 1, y 1, t M 2 x 2, y 2, t > 0. Mx 1, x 2, y 1, y 2, t > 0. For x 1, x 2, y 1, y 2 FCM2. Suppose that for all t θ, x 1, y 1, t = x 2, y 2, t. This implies that x 1 = y 1 x 2 = y 2 for all t θ. Hence M 1 x 1, y 1, t = 1 It follows that M 2 x 2, y 2, t = 1. Mx 1, x 2, y 1, y 2, t = 1. Conversely, suppose that Mx 1, x 2, y 1, y 2, t = 1. This implies that Since M 1 x 1, y 1, t M 2 x 2, y 2, t = 1.

T. Öner, J. Nonlinear Sci. Appl. 9 2016, 799 805 803 it follows that 0 < M 1 x 1, y 1, t 1 0 < M 2 x 2, y 2, t 1, M 1 x 1, y 1, t = 1 M 2 x 2, y 2, t = 1. Thus x 1 = y 1 x 2 = y 2. Therefore x 1, x 2 = y 1, y 2. FCM3. To prove that Mx 1, x 2, y 1, y 2, t = My 1, y 2, x 1, x 2, t we observe that M 1 x 1, y 1, t = M 1 y 1, x 1, t M 2 x 2, y 2, t = M 2 y 2, x 2, t. It follows that for all x 1, x 2, y 1, y 2 X 1 X 2 t θ Mx 1, x 2, y 1, y 2, t = My 1, y 2, x 1, x 2, t. FCM4. Since X 1, M 1, X 2, M 2, are fuzzy cone metric spaces, we have that M 1 x 1, z 1, t + s M 1 x 1, y 1, t M 1 y 1, z 1, s M 2 x 2, z 2, t + s M 2 x 2, y 2, t M 2 y 2, z 2, s for all x 1, x 2, y 1, y 2, z 1, z 2 X 1 X 2 t, s θ. Therefore Mx 1, x 2, z 1, z 2, t + s = M 1 x 1, z 1, t + s M 2 x 2, z 2, t + s M 1 x 1, y 1, t M 1 y 1, z 1, s M 2 x 2, y 2, t M 2 y 2, z 2, s M 1 x 1, y 1, t M 2 x 2, y 2, t M 1 y 1, z 1, s M 2 y 2, z 2, s Mx 1, x 2, y 1, y 2, t My 1, y 2, z 1, y 2, s. FCM5. Note that M 1 x 1, y 1, t M 2 x 2, y 2, t are continuous with respect to t is continuous too. It follows that Mx 1, x 2, y 1, y 2, t = M 1 x 1, y 1, t M 2 x 2, y 2, t is also continuous. Proposition 3.8. Let X 1, M 1, X 2, M 2, be fuzzy cone metric spaces. We define Mx 1, x 2, y 1, y 2, t = M 1 x 1, y 1, t M 2 x 2, y 2, t. Then M is a complete fuzzy cone metric on X 1 X 2 if only if X 1, M 1, X 2, M 2, are complete. Proof. Suppose that X 1, M 1, X 2, M 2, are complete fuzzy cone metric spaces. Cauchy sequence in X 1 X 2. Note that a n = x n 1, x n 2 a m = x m 1, x m 2. Also, Ma n, a m, t converges to 1. This implies that Let {a n be a Mx n 1, x n 2, x m 1, x m 2, t

T. Öner, J. Nonlinear Sci. Appl. 9 2016, 799 805 804 converges to 1 for each t θ. It follows that M 1 x n 1, x m 1, t M 2 x n 2, x m 2, t converges to 1 for each t θ. Thus M 1 x n 1, xm 1, t converges to 1 also M 2x n 2, xm 2, t converges to 1. Therefore {x n 1 is a Cauchy sequence in X 1, M 1, {x n 2 is a Cauchy sequence in X 2, M 2,. Since X 1, M 1, X 2, M 2, are complete fuzzy cone metric spaces, there exist x 1 X 1 x 2 X 2 such that M 1 x n 1, x 1, t converges to 1 M 2 x n 2, x 2, t converges to 1 for each t θ. Let a = x 1, x 2. Then a X 1 X 2. It follows that Ma n, a, t converges to 1 for each t θ. This shows that X 1 X 2, M, is complete. Conversely, suppose that X 1 X 2, M, is complete. We shall show that X 1, M 1, X 2, M 2, are complete. Let {x n 1 {xn 2 be Cauchy sequences in X 1, M 1, X 2, M 2, respectively. Thus M 1 x n 1, xm 1, t converges to 1 M 2x n 2, xm 2, t converges to 1 for each t θ. It follows that Mx n 1, x n 2, x m 1, x m 2, t = M 1 x n 1, x m 1, t M 2 x n 2, x m 2, t converges to 1. Then x n 1, xn 2 is a Cauchy sequence in X 1 X 2. Since X 1 X 2, M, is complete, there exists a pair x 1, x 2 X 1 X 2 such that Mx n 1, xn 2, x 1, x 2, t converges to 1. Clearly, M 1 x n 1, x 1, t converges to 1 M 2 x n 1, x 2, t converges to 1. Hence X 1, M 1, X 2, M 2, are complete. This completes the proof. Theorem 3.9. Every separable fuzzy cone metric space is second countable. Proof. Let X, M, be the given separable fuzzy cone metric space. Let A = {a n : n N be a countable dense subset of X. Consider { B = B a j, 1 k, : j, k N k where θ is constant. Then B is countable. We claim that B is a base for the family of all open sets in X. Let G be an open set in X. Let x G; then there exists r with 0 < r < 1 t θ such that B x, r, t G. Since r 0, 1, we can find an s 0, 1 such tha s 1 s > 1 r. Choose m N such tha/m < s /m t 2. Since A is dense in X, there exists an a j A such that a j B x, 1/m, /m. Now if y B a j, 1/m, /m, then M x, y, t M M x, a j, t y, a j, t 2 2 M x, a j, M m 1 1 1 1 m m 1 s 1 s > 1 r. Thus y B x, y, t hence B is a basis. y, a j, m Proposition 3.10. A subspace of a separable fuzzy cone metric space is separable. Proof. Let X be a separable fuzzy cone metric space Y a subspace of X. Let A = {x n : n N be a countable dense subset of X. For arbitrary but fixed n, k N, if there are points x X such that M x n, x, /k > 1 1/k, where θ is constant, choose one of them denote it by x nk. Let B = {x nk : n, k N; then B is countable. Now we claim that Y B. Let y Y. Given r with 0 < r < 1 t θ we can find a k N such tha 1/k 1 1/k > 1 r /k t 2. Since A is dense

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