- Email: [email protected]

On I-topology generated by fuzzy norm夡 Jin-xuan Fang Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210097, China Received 5 February 2005; received in revised form 25 March 2006; accepted 27 March 2006 Available online 27 April 2006

Abstract In this paper, we point out that an I-topology T· on the fuzzy normed linear space (X, · , min, max) constructed by Das and Das [Fuzzy topology generated by fuzzy norm, Fuzzy Sets and Systems 107 (1999) 349–354] is incompatible with the linear structure on X, that is, (X, · , min, max) is not an I-topological vector space with respect to the I-topology T· . Therefore, we construct a new I-topology T∗· on the fuzzy normed linear space (X, · , L, R) by using fuzzy norm · . We study some of its properties and prove that if R max, then (X, · , L, R) is a Hausdorff locally convex I-topological vector space with respect to the I-topology T∗· . In addition, we also study the relations among three I-topologies T∗· , T· and (), where () is the induced I-topology of the crisp vector topology determined by fuzzy norm · . © 2006 Elsevier B.V. All rights reserved. Keywords: Fuzzy normed linear space; I-topology; Induced I-topology; I-topological vector space; Locally convex I-topological vector space

1. Introduction Kaleva and Seikkala [9] introduced and studied fuzzy metric spaces. Inspired by this work, Felbin [6] introduced the concept of fuzzy normed linear space (FNS). Xiao and Zhu [14] studied the linear topological structure of FNSs. Das and Das [2] constructed a fuzzy topology T· on the FNS (X, · , min, max) and studied some basic properties of this fuzzy topology. According to the standardized terminology in [8], the fuzzy topology related in [2] should be called I-topology, where I = [0, 1]. It is well known that an ordinary normed linear space is necessarily a topological vector space for the topology determined by the norm. This leads naturally to the following question: is the FNS (X, · , min, max) an I-topological vector space with respect to the I-topology T· constructed by Das and Das? In this paper, we ﬁrst give a negative answer to this question, and so a shortcoming of the I-topology T· constructed ∗ on the FNS (X, · , L, R) and study some of by Das and Das is disclosed. Next, we introduce a new I-topology T· its properties. We prove that under the condition of R max, the FNS (X, · , L, R) is a Hausdorff locally convex ∗ . Finally, the relations among three I-topologies T ∗ , T I-topological vector space with respect to the I-topology T· · · and () on the FNS (X, · , L, R) are discussed, where () is the induced I-topology of the crisp vector topology ∗ ⊂ () ⊂ T , and give determined by fuzzy norm · (see [5,14]). We prove the following proper inclusions: T· · ∗ a necessary and sufﬁcient condition for the equality T· = () to hold. 夡

Project is supported by the Natural Science Foundation of Educational Department of Jiangsu Province of China (04KJB110061). E-mail address: [email protected]

0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2006.03.024

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2. Preliminaries Throughout this paper, let R = (−∞, +∞), R+ = [0, +∞) and I = [0, 1], and let N be the set of all positive integers. We denote the family of all fuzzy sets on X by I X . For r ∈ [0, 1], we denote a fuzzy set which takes the constant value r on X by r. A fuzzy point x on X (0 < 1) is a fuzzy set deﬁned by if y = x, x (y) = 0 otherwise. A crisp point x in X may regard as a fuzzy point x1 . Let A ∈ I X and x be a fuzzy point. We say x quasi-coincides with A, denoted by x ∈A, if A(x) > 1 − ; say x belongs to A, denoted by x ∈ A, if A(x) . We do not use the deﬁnition of a fuzzy real number in [7]. A mapping : R → I is called a fuzzy set of real numbers. A fuzzy set of real numbers is called non-negative if (t) = 0 for t < 0. The set of all non-negative upper semi-continuous normal convex fuzzy sets of real numbers is denoted by G+ . Obviously, every in G+ is a fuzzy + − interval [3], i.e., for every ∈ (0, 1], the -level set [] = {t ∈ R | (t) } is a closed interval [− , ] or [ , +∞). Deﬁnition 1 (Felbin [6]). Let X be a vector space over R, · : X → G+ and let the mappings L, R : I × I → I be symmetric, non-decreasing in both arguments and satisfy L(0, 0) = 0 and R(1, 1) = 1. Write [x] = [|x|]1 , |x|2 ]

for x ∈ X, ∈ (0, 1]

and suppose for all x ∈ X, x = , there exists 0 ∈ (0, 1] independent of x such that for all 0 , (A) |x|2 < +∞. The quadruple (X, · , L, R) is called a fuzzy normed linear space (FNS) and · a fuzzy norm, if the following conditions are satisﬁed: (FN-1) x = 0˜ if and only if x = ; (FN-2) rx = |r| x, x ∈ X, r ∈ R; (FN-3) for all x, y ∈ X, (a) whenever s |x|11 , t |y|11 and s + t |x + y|11 , x + y(s + t)L(x(s), y(t)), (b) whenever s |x|11 , t |y|11 and s + t |x + y|11 , x + y(s + t)R(x(s), y(t)). Remark 1. There is a little difference between the above deﬁnition and the original deﬁnition of FNS given in [6]. In [6], other than condition (A), the deﬁnition also requires that · : X → G+ to satisfy the condition (B) inf |x|1 > 0 for all x = .

Lemma 1 (Fang and Song [5]). Let (X, · , L, R) be a FNS, where R satisﬁes the condition lima→0 R(a, a) = 0. Then there exists a topology on X such that (X, ) is a metrizable Hausdorff topological vector space having U = {Uε, : ε > 0, ∈ (0, 1]} as a neighborhood base of , where Uε, = {x ∈ X : |x|2 < ε}.

(2.1)

is called the vector topology determined by fuzzy norm · . Lemma 2 (Xiao and Zhu [14])). Let (X, · , L, R) be a FNS. Then (1) R max implies that for every ∈ (0, 1], |x + y|2 |x|2 + |y|2 ,

∀ x, y ∈ X;

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(2) lima→0 R(a, a) = 0 implies that for every ∈ (0, 1], there exists = () ∈ (0, ] such that

|x + y|2 |x|2 + |y|2 ,

∀ x, y ∈ X.

Deﬁnition 2 (Das and Das [2]). Let (X, · , L, R) be a FNS. For ﬁxed x ∈ X, ∈ (0, 1] and ε > 0, the fuzzy set (x, ε) on X given by (x, ε)(y) =

if |y − x|2 < ε, 0 otherwise

(2.2)

is said to be an -sphere in (X, · , L, R). Remark 2. By (2.1) and (2.2), it is easy to verify (x, ε) = (x + Uε, ) ∩ = x + Uε, ∩ .

(2.3)

Deﬁnition 3 (Das and Das [2]). A fuzzy set on (X, · , L, R) is said to be · -open if for every x ∈ supp = {x ∈ X | (x) > 0}, there exist ε > 0 and ∈ (0, 1] such that (x, ε) ⊂ . Lemma 3 (Das and Das [2]). Let (X, · , L, R) be a FNS. Then the collection T· = { ∈ I X | is · -open} is an I-topology [1] on X. For convenience, we call it the Das’s I-topology. Remark 3. It is easy to see that T· is a stratiﬁed I-topology [11] (namely, an I-topology in the sense of Lowen [12]). Lemma 4 (Das and Das [2]). In the FNS (X, · , min, max), every -sphere (x, ε) is an I-open fuzzy set for T· , i.e., (x, ε) ∈ T· . Lemma 5. Let 0 < < 1 and (x0 , ε) be an -sphere in the FNS (X, · , min, max). Then for any given ∈ (, 1], the fuzzy set on X deﬁned by (x) =

if x = x0 , (x0 , ε)(x) if x = x0

is I-open for T· and (x0 , ε) ⊂ . Proof. By Lemma 4, (x0 , ε) ∈ T· . Besides, it is easy to see that (x0 , ε) ⊂ and supp = supp (x0 , ε). Thus, it follows from Deﬁnition 3 that ∈ T· . Let X be a vector space over the ﬁeld K (K = R or C), A, B ∈ I X and t ∈ K. Then A + B and tA are deﬁned by (A + B)(x) = sup (A(u) ∧ B(v)) u+v=x

and (tA)(x) = A(x/t), (0A)(x) =

t = 0,

sup A(y) if x = ,

y∈X

0

if x = ,

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respectively. In particular, for the fuzzy points x , y , we have (x + A)(y) = ∧ A(y − x), x + y = (x + y)∧ ,

(x + A)(y) = A(y − x);

tx = (tx) .

Deﬁnition 4 (Katsaras [10]). A stratiﬁed I-topology T on a vector space X is said to be an I-vector topology, if the following two mappings f : X × X → X, (x, y) → x + y, g : K × X → X,

(t, x) → tx,

are continuous, where K is equipped with the I-topology induced by the usual topology, and X × X and K × X are equipped with the corresponding product I-topologies. A vector space X with an I-vector topology T , denoted by (X, T ), is called an I-topological vector space (I-TVS). Deﬁnition 5 (Liu and Lao [11]). Let (X, ) be an I-topological space and x ∈ Pt(I X ). (1) A fuzzy set U on X is called Q-neighborhood of x iff there exists G ∈ such that x ∈ G ⊂ U. (2) A family Ux of Q-neighborhoods of x is called a Q-neighborhood base of x iff for every Q-neighborhood A of x , there exists U ∈ Ux such that U ⊂ A. Deﬁnition 6 (Wu and Fang [13]). An I-topological vector space (X, T ) is said to be of QL-type, if there exists a family U of fuzzy sets on X such that for each ∈ (0, 1], U = {U ∩ r | U ∈ U, r ∈ (1 − , 1]} is a Q-neighborhood base of in (X, T ). The family U is called a Q-prebase for T . Lemma 6 (Fang [4]). Let T be a stratiﬁed I-topology on a vector space X. Then (1) the mapping f (addition) is continuous iff for every fuzzy point (x, y) in X × X and every Q-neighborhood W of (x + y) , there exist a Q-neighborhood U of x and a Q-neighborhood V of y such that U + V ⊂ W ; (2) the mapping g (scalar multiplication) is continuous iff for every fuzzy point (t, x) in K × X and every Q-neighborhood W of (tx) , there exist a Q-neighborhood U of x and > 0 such that sU ⊂ W for all s ∈ K with |s − t| < . Lemma 7 (Fang [4]). Let (X, T ) be an I-TVS and U a Q-neighborhood base of in X, ∈ (0, 1]. Then the following conclusions hold: (1) If U ∈ U or U = r, where r ∈ (1 − , 1], then there exists 0 ∈ (0, ) such that for each ∈ [0 , 1] there is a V ∈ U such that V ⊂ U . (2) If U, V ∈ U , then there exists W ∈ U such that W ⊂ U ∩ V . (3) If U ∈ U , then there exists V ∈ U such that V + V ⊂ U . (4) If U ∈ U , then there exists V ∈ U such that tV ⊂ U for all t ∈ K with |t| 1. ˜ . (5) If U ∈ U and x ∈ X, there exists > 0 such that x ∈U Conversely, let X be a vector space over K such that every ∈ (0, 1] has a family U of fuzzy sets on X satisfying the conditions (1)–(5), then there exists a unique I-topology T on X such that (X, T ) is an I-TVS and U is a Q-neighborhood base of . Deﬁnition 7. An I-TVS (X, T ) is said to be locally convex, if for each ∈ (0, 1], there is a base of Q-neighborhoods of consisting of convex fuzzy sets.

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3. A shortcoming of Das’s I-topology Theorem 1. In the FNS (X, · , min, max), (1) the addition, i.e., the mapping f : (X, T· ) × (X, T· ), (x, y) → x + y, is continuous; however, (2) the scalar multiplication, i.e., the mapping g : R × (X, T· ), (t, x) → tx, is not continuous. Proof. (1) Suppose that W is a Q-neighborhood of (x + y) , then there exists ∈ T· such that (x + y) ∈ ⊂ W . By Deﬁnition 3, there exist ε > 0 and ∈ (0, 1] such that (x + y, ε) ⊂ ⊂ W . By conclusion (1) of Lemma 2 , it is not difﬁcult to verify that Uε/2, + Uε/2, ⊂ Uε, , and so we have (x + Uε/2, ) + (y + Uε/2, ) ⊂ (x + y) + Uε, , from which it follows that (x + Uε/2, ) ∩ + (y + Uε/2, ) ∩ ⊂ [(x + y) + Uε, ] ∩ . By Remark 2, we obtain ε ε (x, ) + (y, ) ⊂ (x + y, ε) ⊂ W. 2 2

(3.1)

If > 1 − , then it is easy to see that (x, ε/2), (y, ε/2) are the T· -open Q-neighborhoods of x and y , respectively. Thus, by (3.1) and Lemma 6 we know that the addition is continuous. If 1 − , we deﬁne the fuzzy sets A and B on X as follows: if z = x, A(z) = (x, 2ε )(z) if z = x and

B(z) =

if z = y, (y, 2ε )(z) if z = y,

where = (x + y) > 1 − . We can prove that A + B ⊂ W.

(3.2)

In fact, when z = x + y, we have (A + B)(z) = = (z); when z = x + y, we have (A + B)(z) = [A(x) ∧ B(z − x)] ∨ [A(z − y) ∧ B(y)] ∨ ε ε = (y, )(z − x) ∨ (x, )(z − y) ∨ 2 2

sup u+v=z

[A(u) ∧ B(v)]

u=x, v=y

sup u+v=z

u=x, v=y

ε ε [ (x, )(u) ∧ (y, )(v)], 2 2

and so if |z − (x + y)|2 < ε, then (A + B)(z) = (x + y, ε)(z)(z); if |z − (x + y)|2 ε, then we have |u−x|2 ε/2 or |v −y|2 ε/2, where z = u+v. Hence (A+B)(z) = 0 (z). This shows that A+B ⊂ ⊂ W , so (3.2) holds. By Lemma 5, we know that A and B are the T· -open Q-neighborhoods of x and y , respectively. By (3.2) and conclusion (1) of Lemma 6, we know that the addition is continuous. (2) Let (t0 , x) ∈ R × X, x = and ∈ (0, 1]. Taking , ∈ (0, 1] such that < 1 − < , we deﬁne a fuzzy set on X by if z = t0 x, (z) = if z = t0 x. Obviously, for every y ∈ supp and ε > 0, (y, ε) ⊂ and (t0 x) = > 1 − . Hence is a T· -open Q-neighborhood of t0 x . However, it is not difﬁcult to prove that for every Q-neighborhood U of x and every > 0, (t0 + )U ⊂ / .

(3.3)

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In fact, when z = (t0 + )x, we have [(t0 + )U ](z) = U (x) > 1 − ; but, (z) = < 1 − . So, (3.3) holds. From (2) of Lemma 6, we know that the scalar multiplication is not continuous. Remark 4. From conclusion (2) of Theorem 1, we know that the FNS (X, · , min, max) is not an I-topological vector space with respect to Das’s I-topology T· . This is a shortcoming of Das’s I-topology. 4. I -linear topological structure of a fuzzy linear normed space By modifying the deﬁnition of · -open fuzzy set, we give the following: Deﬁnition 8. A fuzzy set on (X, · , L, R) is said to be · -linearly open if for every x ∈ supp and ∈ (0, (x)), there exists ε > 0 such that (x, ε) ⊂ . Theorem 2. Let (X, · , L, R) be a FNS. Then the collection ∗ = { ∈ I X | is · -linearly open} T·

is a stratiﬁed I-topology on (X, · , L, R), and it is called the I-topology generated by the fuzzy norm. ∗ for Proof. (i) Let r ∈ (0, 1]. For every x ∈ supp r = X, ∈ (0, r) and any ε > 0, we have (x, ε) ⊂ r. So r ∈ T· all r ∈ (0, 1]. Again supp 0 = ∅, So 0 is also · -linearly open. ∗ and ( ∩ )(x) = > 0. Then for every ∈ (0, ), we have ∈ (0, (x)) (i = 1, 2). Since (ii) Let 1 , 2 ∈ T· 1 2 i ∗ i ∈ T· , there exists εi > 0 such that (x, εi ) ⊂ i (i = 1, 2). Let ε = min{ε1 , ε2 }, we have

(x, ε) ⊂ (x, ε1 ) ∩ (x, ε2 ) ⊂ 1 ∩ 2 . ∗ . Hence 1 ∩ 2 ∈ T· ∗ and ( (iii) Let {i }i∈ ⊂ T· i ∈ such that i∈ i )(x) = > 0. Then for every ∈ (0, ),there exists some 0 ∗ ∗ . ∈ (0, i0 (x)). Since i0 ∈ T· , there exists ε0 > 0 such that (x, ε0 ) ⊂ i0 ⊂ i∈ i . Hence i∈ i ∈ T· ∗ Therefore, T· is a stratiﬁed I-topology on (X, · , L, R).

Theorem 3. Let R satisfy the condition R max. Then every -sphere 0 (x0 , ε) in the FNS (X, · , L, R) is · ∗ . linearly open, i.e. 0 (x0 , ε) ∈ T· Proof. Let x ∈ supp 0 (x0 , ε), then |x0 − x|20 < ε and 0 (x0 , ε)(x) = 0 . For every ∈ (0, 0 ), let = ε − |x0 − x|20 . It is not difﬁcult to prove that (x, ) ⊂ 0 (x0 , ε). In fact, if |x − y|2 < , by Lemma 2 we have |x0 − y|20 |x0 − x|20 + |x − y|20 |x0 − x|20 + |x − y|2 < |x0 − x|20 + = ε.

Hence, (x, )(y) = < 0 = 0 (x0 , ε)(y). Again, if |x − y|2 , then (x, )(y) = 0 0 (x0 , ε)(y). Therefore, (x, ) ⊂ 0 (x0 , ε). This shows that 0 (x0 , ε) is · -linearly open. ∗ ) is a locally convex Theorem 4. Let (X, · , L, R) be a FNS, where R satisﬁes the condition R max. Then (X, T· I-TVS and for every ∈ (0, 1],

U = {Uε, ∩ | ε > 0, ∈ (1 − , 1]} = { (, ε) | ε > 0, ∈ (1 − , 1]} is a Q-neighborhood base of , where Uε, is deﬁned by (2.1). ∗ ) is a Hausdorff I-TVS. Also, if (X, · , L, R) satisﬁes condition (B), i.e., inf |x|1 > 0 for all x = , then (X, T· Proof. First, we verify that U ( ∈ (0, 1]) satisﬁes conditions (1)–(5) of Lemma 7. (1) Let U = Uε, ∩ ∈ U or U = with ∈ (1 − , 1], then there exists 0 ∈ (0, ) such that > 1 − 0 . So for each ∈ [0 , 1] we have > 1 − . This shows that Uε, ∩ ∈ U . Taking V = Uε, ∩ , we get V ⊂ U .

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(2) Let Uεi ,i ∩ i ∈ U (i = 1, 2). Taking ε = min{ε1 , ε2 } and = min{1 , 2 }, then we have Uε, ∈ U and Uε, ∩ ⊂ (Uε1 ,1 ∩ 1 ) ∩ (Uε2 ,2 ∩ 2 ). (3) Let Uε, ∩ ∈ U . Since R max, by (1) of Lemma 2 we have |x + y|2 |x|2 + |y|2 ,

∀ x, y ∈ X.

(4.1)

It follows from (4.1) that Uε/2, + Uε/2, ⊂ Uε, . Obviously, Uε/2, ∩ ∈ U and Uε/2, ∩ + Uε/2, ∩ ⊂ Uε, ∩ . (4) If Uε, ∩ ∈ U , then it is easy to see that Uε, is a balanced set, i.e., tUε, ⊂ Uε, for all t ∈ R with |t| 1 and then we have t (Uε, ∩ ) = tUε, ∩ ⊂ Uε, ∩ . (5) If Uε, ∩ ∈ U , then > 1 − . For every x ∈ X, taking t > |x|2 /ε, we have |(1/t)x|2 = (1/t)|x|2 < ε, ∼

which implies that (1/t)x ∈ Uε, , and so (Uε, ∩ )((1/t)x) = 1 ∧ = > 1 − . i.e., x ∈ t (Uε, ∩ ). Thus, from the second part of Lemma 7, there exists a unique I-topology T on X such that (X, T ) is an I-TVS and U is a Q-neighborhood base of . In addition, it is easy to verify that for each r ∈ (0, ], [ (, ε)]r = Uε, and Uε, is convex. Hence every (, ε) in U is a convex fuzzy set. Therefore, (X, T ) is locally convex. ∼ ∗ . If ∈ T ∗ and x ∈ , i.e., (x) > 1 − , then for every ∈ (1 − , (x)), there exists Now we prove T = T· · ε > 0 such that (x, ε) ⊂ . Notice that (x, ε) = x + Uε, ∩ is a Q-neighborhood of x for T , hence is also ∗ ⊂ T . On the other hand, if ∈ T and x ∈ supp , for every that of x for T . This shows that ∈ T , and so T· ∼

∈ (0, (x)), let = 1 − , then we have x ∈ . Thus there exists Uε, ∩ ∈ U such that x + Uε, ∩ ⊂ , i.e., ∗ , and so T ⊂ T ∗ . Therefore T ∗ = T . (x, ε) ⊂ . This shows that ∈ T· · · ∗ ) is Hausdorff. Finally, we prove that if (X, · , L, R) satisﬁes condition (B), then (X, T· Let x and y be two fuzzy points in X and x = y. By condition (B) of Deﬁnition 1, we know that inf |x − y|1 = ∗ and (y, /2) is a Q-neighborhood of y for T ∗ . > 0. Obviously, 1 (x, /2) is a Q-neighborhood of x for T· 1 · Now we prove that 1 (x, /2) ∩ 1 (y, /2) = ∅. If not, then there exists z ∈ X such that 1 (x, /2)(z) > 0 and 1 (y, /2)(z) > 0, i.e. |x − z|12 < /2 and |z − y|12 < /2. Thus, we have |x − y|11 |x − y|12 |x − z|12 + |z − y|12 < /2 + /2 = , ∗ ) is Hausdorff. which is a contradiction. Therefore, (X, T·

Using Theorem 4, we can obtain the following: (n)

Corollary 1. Let (X, · , L, R) be a FNS, where R satisﬁes the condition R max, and let {x } be a sequence n (n)

∗ T·

of fuzzy points in X and x be a fuzzy point in X. Then x −→ x if and only if limn→∞ |x (n) − x|2 = 0 for all n ∈ (1 − , 1] and limn→∞ n . Corollary 2. Let (X, · , L, R) be a FNS, where R satisﬁes the condition R max, and let {xn } be a sequence of ∗ T·

crisp points in X and x be a crisp point in X. Then xn −→ x if and only if limn→∞ |xn − x|2 = 0 for all ∈ (0, 1],

i.e., xn −→ x. 5. Relations among three I-topologies on fuzzy normed linear space ∗ and () on a FNS (X, · , L, R), where In this section, we discuss the relations among three I-topologies T· , T· ∗ T· is Das’s I-topology, T· is the I-topology generated by fuzzy norm and () is the induced I-topology by the crisp vector topology determined by fuzzy norm. ∗ ⊂ () ⊂ T . Theorem 5. Let (X, · , L, R) be a FNS, where R satisﬁes lima→0 R(a, a) = 0. Then T· ·

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∗ ⊂ (), we need only to prove that ∈ T ∗ implies that ∈ (), i.e., () = {x ∈ X | Proof. To prove T· · ∗ , there exists ε > 0 (x) > } ∈ for all ∈ [0, 1). Let x ∈ (), i.e., (x) > . Take ∈ (, (x)). By ∈ T· such that (x, ε) ⊂ . Since (x, ε) = (x + Uε, ) ∩ , we have x + Uε, ⊂ ( (x, ε)) ⊂ (). This shows that () is a neighborhood of x in (X, ), and so () ∈ . Hence ∈ (). Let ∈ () and (x) > 0. Take any r ∈ (0, (x)). We have x ∈ r () ∈ . So, by Lemma 1, there exist ε > 0 and ∈ (0, 1] such that x + Uε, ⊂ r (). Set = min{r, }, it follows that

(x, ε) = (x + Uε, ) ∩ ⊂ (x + Uε, ) ∩ r ⊂ . Hence ∈ T· , which shows that () ⊂ T· .

∗ = () if and only if Theorem 6. Let (X, · , L, R) be a FNS, where R satisﬁes the condition R max, then T·

for any , ∈ (0, 1), | · |2 is equivalent to | · |2 , i.e., there exist C1 , C2 > 0 such that

C1 |x|2 |x|2 C2 |x|2

∀ x ∈ X.

Proof. Note that the condition R max implies lima→0 R(a, a) = 0. By Lemma 1, (X, ) is a crisp topological vector space, where is the vector topology determined by the fuzzy norm · . Thus, we know that (X, ()) is a (QL)-type I-TVS and for each ∈ (0, 1], B = {Uε, ∩ r | ε > 0, ∈ (0, 1], r ∈ (1 − , 1]} is a Q-neighborhood base of in (X, ()). By Theorem 3, we know that for each ∈ (0, 1], U = {Uε, ∩ | ε > 0, ∈ (1 − , 1]} ∗ ). is a Q-neighborhood base of in (X, T· ∗ Necessity: Let T· = () and , ∈ (0, 1), > . We write = 1 − . Then for r ∈ (, 1], U1, ∩ r ∈ B . Since ∗ = (), it is easy to know that U ∗ T· 1, ∩ r ∈ B is also a Q-neighborhood of in (X, T· ). So, there exist ε > 0 and s ∈ (1 − , 1] such that Uε,s ∩ s ⊂ U1, ∩ r, from which it follows that

Uε,s = (Uε,s ∩ s) ⊂ (U1, ∩ r) = U1, .

(5.1)

Notice that (εx/|x|2 + 1/n) ∈ Uε, ⊂ Uε,s for all n ∈ N. It follows from (5.1) that |x|2 < (1/ε)(|x|2 +1/n),

and so |x|2 (1/ε)|x|2 . Besides, since > , we have

|x|2 |x|2 (1/ε)|x|2

for all x ∈ X.

Therefore | · |2 is equivalent to | · |2 . ∗ . Let ∈ () and x ∈ supp . Then for every Sufﬁciency: By Theorem 5, we need only to prove () ⊂ T· ∈ (0, (x)), we have x ∈ () ∈ . By Lemma 1, there exist ε > 0 and ∈ (0, 1] such that x + Uε, ⊂ (). In the following, we split two cases to discuss: (1) If , then we have x + Uε, ⊂ x + Uε, ⊂ (), and so (x, ε) = (x + Uε, ) ∩ ⊂ .

(5.2)

(2) If > , and notice that | · |2 is equivalent to | · |2 , then there exists m > 0 such that

m|x|2 |x|2 |x|2 for all x ∈ X.

(5.3)

It follows from (5.3) that Umε, ⊂ Uε, ⊂ Uε, . So, we have x + Umε, ⊂ x + Uε, ⊂ (), which implies that (x, mε) = (x + Umε, ) ∩ ⊂ . ∗ . This shows that () ⊂ T ∗ . From (5.2) and (5.4), we infer ∈ T· ·

(5.4)

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∗ is a proper subset of () and () is a proper In the following, we shall give two examples, which show that T· subset of T· . Let C(R) denote the set of all continuous real valued functions on R. For each ∈ (0, 1], we deﬁne a function | · | : C(R) → [0, ∞) by

|x| =

max

t∈[−1/,1/]

|x(t)|.

Obviously, for every x ∈ C(R), 0 < < 1 implies |x| |x| . Lemma 8. Let x ∈ C(R) with x = , t > 0 and ∈ (0, 1), we deﬁne x (t) by x (t) = sup{ ∈ (0, 1] | t |x| }. Then the following conclusions hold: (1) (2) (3) (4)

If t > |x| , then x (t). kx (t) = x (t/|k|) for all k = 0. x+y (s + t) max{ x (s), y (t)}. x (t) is upper semi-continuous for t.

Proof. (1) and (2) are obvious. (3) By deﬁnition of x+y (s+t), for any ε > 0, there exists 0 > x+y (s+t)−ε such that s+t |x+y| 0 |x| 0 +|y| 0 , which implies that s |x| 0 or t |y| 0 . So, we have x (s) 0 or y (t) 0 . It follows that max{ x (s), y (t)} 0 > x+y (s + t) − ε. By the arbitrariness of ε, (3) is proved. (4) We need only to prove that for each ∈ (0, 1], ( x ) = {t | x (t) } is a closed set. Let tn ∈ ( x ) with tn → t. Since x (tn ), there exists n > − /2n such that tn |x| n |x|−/2n . Setting n = − /2n, we have max

s∈[−1/ n ,1/ n ]

|x(s)|tn

(n = 1, 2, . . .).

So, there exists sn ∈ [−1/ n , 1/ n ] such that |x(sn )|tn (n = 1, 2, . . .). Since {sn } ⊂ [−2/, 2/], there exists a subsequence {sni } of {sn } such that sni → s0 . Notice that sni ∈ [−1/ ni , 1/ ni ] and ni → . Hence s0 ∈ [−1/, 1/]. It follows from |x(sni )| tni that |x(s0 )|t, i.e., |x| t, which implies x (t) , i.e., t ∈ ( x ) . Therefore ( x ) is a closed set. Example 1. Let , ∈ (0, 1) with > , n ∈ (0, ) with n 0. We deﬁne a mapping · : C(R) → G+ as follows: (i) If |x| = 0, then ⎧ 0 ⎪ ⎪ ⎨ 1 x(t) = ⎪ ⎪ ⎩ x (t) (ii) If there exists some n0 ⎧ ⎨1 x(t) = n ⎩ 0

if if if if

t < 0, 0 t |x| , |x| < t |x| , t > |x| .

∈ N such that |x|n

0 −1

= 0 and |x|n = 0 (0 = ), then 0

if t = 0, if t ∈ (|x|n−1 , |x|n ] (n = n0 , n0 + 1, . . .), otherwise.

(iii) If |x| = 0 for all ∈ (0, 1], i.e., x = , then 1 if t = 0, ˜ x(t) = i.e., = 0. 0 otherwise, ∗ is a proper subset of (). Then (C(R), · , min, max) is a FNS, and T·

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In fact, (FN-1) holds obviously. ˜ Hence, 0 · x = = 0˜ = |0| · x. (FN-2) It is easy to verify 0 · x = 0. Let k = 0. In the case of (i), by Lemma 7, we have ⎧ 0 if t/|k| < 0 ⎪ ⎪ ⎨ 1 if t/|k| ∈ [0, |x| ] (|k| · x)(t) = x(t/|k|) = if t/|k| ∈ (|x| , |x| ] ⎪ ⎪ ⎩ x (t/|k|) if t/|k| > |x| ⎧ 0 if t < 0 ⎪ ⎪ ⎨ 1 if t ∈ [0, |kx| ] = = k · x(t), if t ∈ (|kx| , |kx| ] ⎪ ⎪ ⎩ kx (t) if t > |kx| and so k · x = |k| · x. In the case of (ii), since |x|n = 0 ⇐⇒ |kx|n = 0 (k = 0), we have ⎧ ⎨ 1 if t/|k| = 0 (|k| · x)(t) = x(t/|k|) = n if t/|k| ∈ (|x|n−1 , |x|n ] (n = n0 , n0 + 1, . . .) ⎩ 0 otherwise ⎧ ⎨ 1 if t = 0 = n if t ∈ (|kx|n−1 , |kx|n ] (n = n0 , n0 + 1, . . .) ⎩ 0 otherwise = kx(t) and so k · x = |k| · x. In the case of (iii), clearly, k · = = 0˜ = |k| · . (FN-3) Obviously, |x|11 = 0 for every x ∈ C(R), and so (a) of (FN-3) holds. (b) Let s |x|11 , t |y|11 and s + t |x + y|11 , i.e., s, t 0. We shall prove that x + y(s + t) max{x(s), y(t)}. Clearly, we may assume that s, t > 0. (i) Let |x + y| = 0. We have ⎧ if s + t ∈ (0, |x| ], ⎨1 if s + t ∈ (|x + y| , |x + y| ], x + y(s + t) = ⎩ x+y (s + t) if s + t > |x + y| . Notice that 0 < s + t |x + y| ⇒ s |x| or t |y| ⇒ x(s) = 1 or y(t) = 1. Hence, when 0 < s + t |x + y| , we have x + y(s + t) = 1 = max{x(s), y(t)}. Notice that |x + y| < s + t |x + y| ⇒ s |x| or t |y| ⇒ x(s) or y(t) . Hence, when |x + y| < s + t |x + y| , we have x + y(s + t) = {x(s), y(t)}. When |x + y| < s + t, we consider the following two cases: Case 1: |x| < s and |y| < t. This implies x(s) x (s) and y(t) y (t). Thus, by Lemma 7, we have x + y(s + t) = x+y (s + t) max{ x (s), y (t)} max{x(s), y(t)}. Case 2: |x| s or |y| t. This implies x(s) and y(t) . Thus, we have x + y(s + t) = x+y (s + t) max{x(s), y(t)}. (ii) Let there exist some n0 ∈ N, such that |x + y|n

0 −1

= 0 and |x + y|n = 0. 0

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Notice that s + t |x + y|n ⇒ s |x|n or t |y|n ⇒ x(s) n or y(t) n (n = n0 , n0 + 1, . . .). Hence,

when s + t ∈ |x + y|n−1 , |x + y|n , we have x + y(s + t) = n max{x(s), y(t)},

n = n0 , n0 + 1, . . . .

(iii) Let x + y = . Obviously, x + y(s + t) max{x(s), y(t)}. (FN-3) is proved. Therefore (C(R), min, max) is a FNS. Since |x|2 = |x| and |x|2 = |x| for all x ∈ C(R) with |x| = 0, | · |2 is not equivalent to | · |2 . By ∗ Theorems 5 and 6, we can infer that T· is a proper subset of (). Example 2. Let us consider the FNS (R, · , L, max), where · : R → G+ is deﬁned by 1 if t = |x|, x(t) = 0 otherwise.

(5.5)

We deﬁne a fuzzy set on R by 1 if x ∈ [−1, 1], (x) = 1 2 otherwise. Then we can prove that ∈ T· but ∈ / (), and so () is a proper subset of T· by Theorem 5. In fact, for every ∈ (0, 1], it follows from (5.5) that |x|2 = |x|, and so we have Uε, = (−ε, ε) for each ε > 0. This shows that the topology determined by the fuzzy norm · on R is the ordinary topology on R. Taking 0 < 21 and ε > 0, we have (x, ε) ⊂ . Hence ∈ T· . On the other hand, since 1/2 () = [−1, 1] ∈ / , we have ∈ / (). 6. Conclusions It is well known that every normed linear space is a topological vector space for the norm topology. However, the fuzzy normed linear space (X, · , min, max) is not an I-topological vector space with respect to Das’s I-topology T· generated by fuzzy norm (see Theorem 1 and Remark 4). In order to get over this shortcoming, we have constructed a ∗ on the FNS (X, · , L, R) and have shown that under the condition of R max, (X, · , L, R) new I-topology T· ∗ (see Theorems 2 and 4). is a Hausdorff locally convex I-topological vector space with respect to the I-topology T· ∗ We also have proved the following proper inclusions: T· ⊂ () ⊂ T· , where () is the induced I-topology of the crisp vector topology determined by the fuzzy norm · (see Theorem 5, Examples 1 and 2). Up to now, most of the researches about fuzzy normed linear spaces are based upon the crisp vector topology determined by the fuzzy norm (see [5,6,14–16]). The present paper provides a new framework for studying theory of ∗ generated by the fuzzy norm. How to establish the corresponding theory FNSs, which is based on the I-topology T· of FNSs, under this framework, we will consider further later. References [1] [2] [3] [4] [5]

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[13] C.-x. Wu, J.-x. Fang, QL-type fuzzy topological vector space, Chinese Ann. Math. 6A (3) (1985) 355–364. [14] j.-z. Xiao, X.-h. Zhu, On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets and Systems 125 (2002) 153–161. [15] j.-z. Xiao, X.-h. Zhu, Fuzzy normed space of operators and its completeness, Fuzzy Sets and Systems 133 (2003) 389–399. [16] j.-z. Xiao, X.-h. Zhu, The uniform boundedness principles for fuzzy normed spaces, J. Fuzzy Math. 12 (2004) 543–560.

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