On Local Boundedness of I-Topological Vector Spaces

Iranian Journal of Fuzzy Systems Vol. 9, No. 5, (2012) pp. 93-104 93

ON LOCAL BOUNDEDNESS OF I-TOPOLOGICAL VECTOR SPACES

J. X. FANG AND H. ZHANG

Abstract. The notion of generalized locally bounded I-topological vector spaces is introduced. Some of their important properties are studied. The relationship between this kind of spaces and the locally bounded I-topological vector spaces introduced by Wu and Fang [Boundedness and locally bounded fuzzy topological vector spaces, Fuzzy Math. 5 (4) (1985) 87−94] is discussed. Moreover, we also use the family of generalized fuzzy quasi-norms to characterize the generalized locally bounded I-topological vector spaces, and give some applications of this characterization.

1. Introduction The concept of fuzzy topological vector space was introduced rationally by Kat- saras in 1981 [3]. According to the terminology of standardization in [2], it has now been renamed as I-topological vector space (briefly, I-tvs), where I = [0, 1]. It is well-known that, the local boundedness is an important concept in theory of classical topological vector spaces. So it is natural and necessary to find its coun- terpart in research of I-tvs. Out of this consideration, Wu and Fang [8] introduced the notion of locally bounded I-tvs and studied some of its properties. We know that the locally bounded I-topological vector spaces defined in [8] are a special subclass of (QL)- type I-topological vector spaces [7]. This has led to the following questions: Whether we can investigate the local boundedness of general I-tvs (Do not have to be (QL)-type)? This article will give a definite answer. This paper is organized as follows. In section 2, we briefly recall some basic concepts and lemmas that we will use in the sequel. In section 3, we introduce the notion of generalized locally bounded I-topological vector space, which contains that of locally bounded I-topological vector space introduced by Wu and Fang [8] as a special case. In section 4, we introduce the notion of a family of generalized fuzzy quasi-norms, and use it to characterize generalized locally bounded L-tvs. In section 5, as applications of the results obtained in section 4, we characterize the Hausdorff separation property, convergence of a net of fuzzy points in generalized locally bounded I-tvs by means of the family of generalized fuzzy quasi-norms.

Received: January 2011; Revised: September 2011; Accepted: October 2011 Key words and phrases: I-topological vector spaces, Generalized locally bounded I-topological vector spaces, Family of generalized fuzzy quasi-norms. This work was supported by the National Natural Science Foundation of China (No.10671094) and the Specialized Research Fund for the Doctor Program of Higher Education of China (No. 20060319001) . 94 J. X. Fang and H. Zhang

2. Basic Concepts and Lemmas Throughout this paper, IX denotes a family of all fuzzy subsets of X. A fuzzy subset which takes the constant value r on X (0 ≤ r ≤ 1) is denoted by r.A fuzzy subset of X is called a fuzzy point [4], denoted by xλ, if it takes value 0 at y ∈ X \{x} and its value at x is λ. The set of all fuzzy points on X is denoted X by Pt(I ). A fuzzy point xλ is said to be quasi-coincident with a fuzzy subset U, denoted by xλ∈e U, if U(x) > 1 − λ (see [4]). An I-topology on X [4] is a subset τ of IX which contains the constant fuzzy sets 0 and 1 and is closed with respect to finite infima and arbitrary suprema. The pair (X, τ) is called an I-topological space. An I-topology τ on X is called fully stratified [4] if r ∈ τ for each r ∈ [0, 1].

X Deﬁnition 2.1. [4] Let (X, τ) be an I-topological space and xλ ∈ Pt(I ). A fuzzy subset U of X is called Q-neighborhood of xλ iﬀ there exists G ∈ τ such that xλ∈e G ⊆ U.

The family consisting of all Q-neighborhoods of xλ, is called the system of Q- neighborhoods of xλ and is denoted by NQ(xλ).

X Definition 2.2. [4] Let (X, τ) be an I-topological space and xλ ∈ Pt(I ). A family Uxλ of Q-neighborhoods of xλ is called a Q-neighborhood base of xλ iff for each A ∈ NQ(xλ) there exists U ∈ Uxλ such that U ⊆ A. In the following, let X be a vector space over the field K (R or C) and θ the zero element of X.

Deﬁnition 2.3. [3] Let A, B ∈ IX and k ∈ K. Then A + B and kA are deﬁned respectively by (A + B)(x) = W{A(s) ∧ B(t): s + t = x}, (kA)(x) = A(x/k), whenever k 6= 0; W A(z), if x = θ, (0A)(x) = z∈X 0, if x 6= θ.

X In particular, for xλ, yµ ∈ Pt(I ), we have

xλ + yµ = (x + y)λ∧µ, kxλ = (kx)λ. Deﬁnition 2.4. [3] A stratiﬁed I-topology τ on X is said to be a I-vector topology, if the following two mappings are both continuous: f : X × X → X,(x, y) 7→ x + y,

g : K × X → X,(k, x) 7→ kx, where K is equipped with the I-topology induced by the usual topology, X × X and K × X are equipped with the corresponding product I-topologies. A vector space X with an I-vector topology τ, denoted by (X, τ), is called a I-topological vector space (for short, an I-tvs). On Local Boundedness of I-topological Vector Spaces 95

Deﬁnition 2.5. [7] An I-tvs (X, τ) is said to be (QL)-type of, if there exists a family U of fuzzy subsets on X such that for each λ ∈ (0, 1],

Uλ = {U ∩ r | U ∈ U, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ in (X, τ). The family U is called a Q-prebase for τ. Definition 2.6. [8] A fuzzy set B in (X, τ) is said to be λ-bounded (λ ∈ (0, 1] is given), if for each Q-neighborhood U of θλ in X, there exists t > 0 and r ∈ (1−λ, 1] such that B ∩ r ⊆ tU. B is said to be bounded, if it is λ-bounded for each λ ∈ (0, 1]. Definition 2.7. [8] An I-tvs (X, τ) is said to be locally bounded if it has a bounded neighborhood of θ. Remark 2.8. It is not difficult to prove that U ∈ IX is a bounded neighborhood of θ in an I-tvs (X, τ) iff {tU ∩ r | t > 0, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ for each λ ∈ (0, 1]. Therefore, each locally bounded I-tvs is necessarily (QL)-type I-tvs. X Definition 2.9. [1] Let U ⊂ I . A family {Uλ}λ∈(0,1] of nonempty subsets of U is called the stratified structure of U if the following two equalities hold: U = S S λ∈(0,1] Uλ and Uλ = µ∈(0,λ) Uµ. Definition 2.10. [1] Let (X, τ) be an I-tvs. A family U of fuzzy subsets on X is called a local base with the stratified structure {Uλ}λ∈(0,1] of (X, τ) if {Uλ}λ∈(0,1] is the stratified structure of U and Uλ is a Q-neighborhood base of θλ in (X, τ) for each λ ∈ (0, 1].

Lemma 2.11. [6] Each I-tvs (X, τ) has an Q-neighborhood base of θλ consisting of balanced fuzzy sets on X for each λ ∈ (0, 1]. Lemma 2.12. [1] Let (X, τ) be an I-tvs and U be a local base with stratified structure {Uλ}λ∈(0,1] of (X, τ). Then it has the following properties: (1) If U ∈ Uλ, V ∈ Uλ or V = r (1 − λ < r ≤ 1), then there exists W ∈ Uλ such that W ⊆ U ∩ V ; (2) If U ∈ Uλ, then there exists V ∈ Uλ such that V + V ⊆ U; (3) If U ∈ Uλ, then there exists V ∈ Uλ such that kV ⊆ U for all |k| ≤ 1; (4) If U ∈ Uλ, then for each x ∈ X, there is an α > 0 such that xλ∈e αU. Conversely, let X be a vector space over K and U be a family of fuzzy subsets on X with the stratified structure {Uλ}λ∈(0,1] which satisfies the above conditions (1)−(4). Then there exists a unique I-topology τ on X such that (X, τ) is an I-tvs and U is a local base with the stratified structure {Uλ}λ∈(0,1] of (X, τ). Lemma 2.13. [8] Let (X, τ) be an I-tvs and A, B ∈ IX . Then the following statements hold: (1) Every fuzzy point xα on X is bounded. (2) If B is λ-bounded and B0 ⊆ B, then B0 is also λ-bounded. (3) If A and B are λ-bounded, then A + B is also λ-bounded. (4) If B is λ-bounded, then kB is also λ-bounded for all k ∈ K. (5) If B is λ-bounded, then the balanced hull ba(B) of B is also λ-bounded. 96 J. X. Fang and H. Zhang

3. Generalized Locally Bounded I-tvs Deﬁnition 3.1. An I-tvs (X, τ) is said to be generalized locally bounded if for each λ ∈ (0, 1], there exists a λ-bounded Q-neighborhood Uλ of θλ, and 0 < µ < λ ≤ 1 implies Uλ ⊆ Uµ.

Lemma 3.2. Let (X, τ) be an I-tvs. If Uλ is a λ-bounded Q-neighborhood of θλ on X, then Uλ = {(1/n)Uλ ∩ r : n ∈ N, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ.

Proof. Since Uλ is a Q-neighborhood of θλ, it is easy to see that for each n ∈ N and r ∈ (1 − λ, 1], (1/n)Uλ ∩ r is also a Q-neighborhood of θλ. Let W be an arbitrary Q-neighborhood of θλ. By Lemma 2.11, we may assume that it is balanced. Since Uλ is λ-bounded, there exist r ∈ (1 − λ, 1] and t > 0 such that Uλ ∩ r ⊆ tW . 1 t Take n0 ∈ with n0 ≥ t. Then we have Uλ ∩ r ⊆ W ⊆ W . Hence Uλ is a N n0 n0 Q-neighborhood base of θλ. Lemma 3.3. Let (X, τ) be a generalized locally bounded I-tvs. Then for each λ ∈ (0, 1], there exists a balanced and λ-bounded Q-neighborhood Bλ of θλ, and 0 < µ < λ ≤ 1 implies Bµ ⊆ Bλ. Proof. Since (X, τ) is a generalized locally bounded I-tvs, for each λ ∈ (0, 1] there exists a λ-bounded Q-neighborhood Uλ of θλ, and 0 < µ < λ ≤ 1 implies Uλ ⊆ Uµ. Put Bλ = ba(Uλ). By Lemma 2.13, we know that Bλ is λ-bounded. Obviously, it is also a balanced Q-neighborhood of θλ, and 0 < µ < λ ≤ 1 implies Bλ ⊆ Bµ.

Theorem 3.4. Let (X, τ) be an I-tvs. Then (X, τ) is generalized locally bounded iff there exists a family {Bλ}λ∈(0,1] of fuzzy sets on X satisfying the following two conditions: (i) 0 < µ < λ ≤ 1 implies Bλ ⊆ Bµ; (ii) Bλ = {tBλ ∩ r | t > 0, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ for each λ ∈ (0, 1]. Proof. Necessity: Suppose that (X, τ) is a generalized locally bounded I-tvs. Then for each λ ∈ (0, 1], there exists a λ-bounded Q-neighborhood Bλ of θλ, and 0 < µ < λ ≤ 1 implies Bλ ⊆ Bµ. And then, by Lemma 3.2, we can conclude that Bλ = {tBλ ∩ r | t > 0, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ. Sufficiency: Suppose that {Bλ}λ∈(0,1] is a family of fuzzy sets on X satisfying conditions (i) and (ii). Then it is not difficult to see that Bλ is a Q-neighborhood of θλ, and for each W ∈ NQ(θλ), there exist t > 0 and r ∈ (1 − λ, 1] such that tBλ ∩ r ⊆ W . Hence Bλ is a λ-bounded Q-neighborhood of θλ. This shows that (X, τ) is generalized locally bounded.

Theorem 3.5. Every locally bounded I-tvs is necessarily generalized locally bounded. Proof. Let (X, τ) be a locally bounded I-tvs. Then it has a bounded neighborhood U of θ. By Remark 2.8, we know that Uλ = {tU ∩ r | t > 0, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ for each λ ∈ (0, 1]. Put Bλ = U for each λ ∈ (0, 1], then On Local Boundedness of I-topological Vector Spaces 97

Bλ = {tBλ ∩ r | t > 0, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ. Hence, by Theorem 3.4, we conclude that (X, τ) is generalized locally bounded.

The following example indicates that the converse of Theorem 3.5 is invalid in general. Example 3.6. Let (X,T ) be a classical nontrivial locally bounded topological vector space. Assume that U is a bounded neighborhood of θ and U 6= X (since T is nontrivial). Let c ∈ (0, 1) be given. For each λ ∈ (0, 1], we deﬁne Uλ as follows: {r | r ∈ (1 − λ, 1]} , λ ∈ (0, c], U = λ {tU ∧ r | t > 0, r ∈ (1 − λ, 1]} , λ ∈ (c, 1]. Then similar to Example 3.1 in [10], we can prove that there exists a unique I- topology Nc(T ) on X such that (X,Nc(T )) is an I-tvs and Uλ is a Q-neighborhood X base of θλ for each λ ∈ (0, 1]. We shall prove that (I ,Nc(T )) is generalized locally bounded. In fact, if λ ∈ (0, c], then 1 is a λ-bounded Q-neighborhood of θλ; If λ ∈ (c, 1], then U is a λ-bounded Q-neighborhood of θλ. However, we can show that (X,Nc(T )) is not of (QL)-type, and so it is not locally bounded. In fact, if (X,Nc(T )) is a (QL)-type I-tvs, then there exists a family of fuzzy sets B = {B} such that Bλ = {B ∩ r | B ∈ B, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ for each λ ∈ (0, 1]. Obviously, for each λ ∈ (0, 1], B ∈ B is also a Q- neighborhood of θλ. Notice that Uλ = {r : r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ for each λ ∈ (0, c]. Hence, for each B ∈ B, there exists r0 ∈ (1 − λ, 1] such that r0 ⊆ B, which implies that B(x) ≥ r0 > 1 − λ for all λ ∈ (0, c] and x ∈ X, and so B = X. Hence Bλ = {r : r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ for each λ ∈ (0, 1]. Since U is a neighborhood of θ, U is a Q-neighborhood of θλ for each λ ∈ (0, 1]. So, there exists r ∈ (1 − λ, 1] such that r ⊆ U, which implies that U = X. This contradicts U 6= X. Therefore, (X,Nc(T )) is not locally bounded.

4. Characterization of Generalized Locally Bounded I-tvs In this section, we first introduce the notion of a family of generalized fuzzy quasi-norms and investigate some of its properties, then characterize generalized locally bounded I-tvs in terms of a family of generalized fuzzy quasi-norms. X Definition 4.1. A family of mappings k · k(λ) : Pt(I ) → [0, ∞](λ ∈ (0, 1]) is called a family of generalized fuzzy quasi-norms on X, if it satisfies the following conditions: (GQN-1) for each λ ∈ (0, 1], kθλk(λ) = 0 and kxλk(λ) < +∞ for all x ∈ X; X (GQN-2) kkxαk(λ) = |k|kxαk(λ) for each xα ∈ Pt(I ) and k ∈ K with k 6= 0; (GQN-3) for each λ ∈ (0, 1], there exist s ∈ (1 − λ, 1] and k0 ≥ 1 such that kxα + yαk(λ) ≤ k0 kxαk(λ) + kyαk(λ) for all α ∈ (1 − s, 1], x, y ∈ X;

(GQN-4) kxαk(λ) = inf kxβk(λ), and 0<β<α X 0 < µ < λ ≤ 1 implies that kxαk(µ) ≤ kxαk(λ) for all xα ∈ Pt(I ); 98 J. X. Fang and H. Zhang

(GQN-5) for each λ ∈ (0, 1] there exist λ0 ∈ (0, λ), s ∈ (1 − λ0, 1] and n0 ≥ 1 such that

kxαk(λ) ≤ n0kxαk(λ0) for all α ∈ (1 − s, 1] and x ∈ X. Lemma 4.2. Let k · k(λ) λ∈(0,1] be a family of generalized fuzzy quasi-norms on X. For each λ ∈ (0, 1] and t > 0, deﬁne a fuzzy set Uλ,t on X by Uλ,t(x) = sup 1 − α | kxαk(λ) < t . (1)

Then Uλ,t has the following properties: (P-1) xα∈e Uλ,t if and only if kxαk(λ) < t; S (P-2) Uλ,t = 0 0;

(P-4) 0 < µ < λ ≤ 1 =⇒ Uλ,t ⊆ Uµ,t; (P-5) Uλ,t is semi-balanced, i.e., kUλ,t ⊆ Uλ,t for all k ∈ K with 0 < |k| ≤ 1. (P-6) for each λ ∈ (0, 1], there exist λ0 ∈ (0, λ), s ∈ (1 − λ0, 1] and n0 ≥ 1 such 1 that Uλ0, ∩ s ⊆ Uλ,1. n0

Proof. (P-1) If xα∈e Uλ,t, i.e., Uλ,t(x) > 1 − α, then by (1) there exists α0 ∈ (0, α) such that kxα0 k(λ) < t. By (GQN-4), we have kxαk(λ) ≤ kxα0 k(λ) < t. Conversely, if kxαk(λ) < t, by (GQN-4), there exists 0 < β < α such that kxβk(λ) < t, and so Uλ,t ≥ 1 − β > 1 − α, i.e., xα∈e Uλ,t.

(P-2) By the deﬁnition of Uλ,t, it is easy to see that Uλ,s ⊆ Uλ,t for each s ∈ (0, t). S Hence 0 0 and x ∈ X, we have n o Uλ,st(x) = sup 1 − α | kxαk(λ) < st = sup 1 − α | k((1/s)x)αk(λ) < t

= Uλ,t(x/s) = (sUλ,t)(x). Hence (P-3) holds.

(P-4) If 0 < µ < λ ≤ 1 and xα∈e Uλ,t, then by (GQN-4) and (P-1) we have kxαk(µ) ≤ kxαk(λ) < t, and so xα∈e Uµ,t. Hence (P-4) holds. (P-5) If 0 < |k| ≤ 1, then by (P-1) it is easy to know that the following implications hold: 1 x ∈ kU =⇒ x ∈ U =⇒ kx k < |k|t ≤ t =⇒ x ∈ U . α e λ,t k α e λ,t α (λ) α e λ,t Hence kUλ,t ⊆ Uλ,t. This shows that Uλ,t is semi-balanced.

(P-6) By (GQN-5), for each λ ∈ (0, 1], there exist λ0 ∈ (0, λ), s ∈ (1 − λ0, 1] and n0 ≥ 1 such that kxαk(λ) ≤ n0kxαk(λ0) for all α ∈ (1 − s, 1] and x ∈ X. Thus, 1 1 we can prove that Uλ0, ∩ s ⊆ Uλ,1. In fact, by (P-1), if xα∈ Uλ0, ∩ s, then n0 e n0 On Local Boundedness of I-topological Vector Spaces 99

1 kxαk < and α ∈ (1 − s, 1], and so kxαk < 1, i.e., xα∈ Uλ,1. This shows (λ0) n0 (λ) e that U 1 ∩ s ⊆ U . λ0, λ,1 n0 X Theorem 4.3. Let P = k · k(λ) : Pt(I ) → [0, ∞] λ∈(0,1] be a family of generalized fuzzy quasi-norms on X. Then there exists a unique I-topology τ on X such that (X, τ) is a generalized locally bounded I-tvs, and

Bλ = {Uλ,t ∩ r | t > 0, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ for each λ ∈ (0, 1], where Uλ,t is deﬁned by (1). The topology τ is said to be generated by the family of generalized fuzzy quasi-norms P. Proof. Deﬁne a family U of fuzzy sets on X as follows:

U = {Uα,t ∩ r | α ∈ (0, 1), r ∈ (0, 1], t > 0}.

We first prove that {Uλ}λ∈(0,1] is the stratified structure of U, where Uλ is defined by

Uλ = {Uα,t ∩ r | α ∈ (0, λ), r ∈ (1 − λ, 1], t > 0} . (2) S In fact, it is evident that Uλ ⊆ U. If Uα,t ∩ r ∈ U, then 0 < α < 1 and λ∈(0,1] 0 < r ≤ 1. Take µ ∈ (max{α, 1 − r}, 1], then α < µ and r ∈ (1 − µ, 1], and so S S Uα,t ∩ r ∈ Uµ. This shows that U ⊆ Uλ. Hence U = Uλ. In addition, λ∈(0,1] λ∈(0,1] S by (2), it is evident that Uµ ⊆ Uλ. If Uα,t ∩ r ∈ Uλ, then 0 < α < λ, µ∈(0,λ) 1 − λ < r ≤ 1. Take µ ∈ (max{α, 1 − r}, λ), then Uα,t ∩ r ∈ Uµ. This shows that S S Uλ ⊆ Uµ. Hence Uλ = Uµ. 0<µ<λ 0<µ<λ Next, we prove that {Uλ}λ∈(0,1] satisﬁes conditions (1)−(4) in Lemma 2.12

(1) Let Ui = Uαi,ti ∩ri ∈ Uλ, where 0 < αi < λ, ti > 0 and ri ∈ (1−λ, 1] (i = 1, 2). Put α = max{α1, α2}, r = min{r1, r2} and t = min{t1, t2}. Then V = Uα,t∩r ∈ Uλ, and by (P-2) and (P-4), it is evident that V ⊆ U1 ∩ U2. Let U = Uα,t ∩ r ∈ Uλ where 0 < α < λ, t > 0 and r ∈ (1 − λ, 1], and let V = s with s ∈ (1 − λ, 1]. Put q = min{r, s}. Then it is obvious that W = Uα,t ∩ q ∈ Uλ and W ⊆ U ∩ V .

(2) Let U = Uα,t ∩ r ∈ Uλ, where 0 < α < λ, t > 0 and r ∈ (1 − λ, 1]. For the above α, by (GQN-3), there exists s ∈ (1 − α, 1] and k0 ≥ 1 such that kxβ + yβk(α) ≤ k0 kxβk(α) + kyβk(α) (3) for all β ∈ (1 − s, 1] and x, y ∈ X. Put V = Uα, t ∩ q, where q = min{r, s}. 2k0 Obviously, V ∈ Uλ. We shall prove that V + V ⊆ U. In fact, if zβ∈ V +V , there exist x, y ∈ X with x+y = z such that xβ, yβ∈ Uα, t e e 2k0 and q > 1 − β. By (P-1), it follows that kxβk(α) < t/2k0 and kyβk(α) < t/2k0. Note that s ≥ q > 1 − β, i.e., β ∈ (1 − s, 1]. So, by (3), we get

kzβk(α) = kxβ + yβk(α) ≤ k0[kxβk(α) + kyβk(α)] < t, 100 J. X. Fang and H. Zhang i.e., zβ∈e Uα,t. Note that r ≥ q > 1 − β. Hence zβ∈e U. This shows that V + V ⊆ U. (3) Let U = Uα,t ∩ r ∈ Uλ, where 0 < α < λ, t > 0 and r ∈ (1 − λ, 1]. Put µ = min{Uα,t(θ), r} and V = Uα,t ∩ µ. It is evident that V ∈ Uλ. By (P-5), we know that Uα,t is semi-balanced, and so V is also semi-balanced. Note that (0 · V )(θ) = supz∈X Uα,t(z) ∧ µ ≥ Uα,t(θ) ∧ µ = µ. Hence 0 · V = θµ ⊆ V . This shows that V is balanced, and so kV ⊆ V ⊆ U for all k ∈ K with |k| ≤ 1. (4) Let U = Uα,t ∩ r ∈ Uλ, where 0 < α < λ, t > 0 and r ∈ (1 − λ, 1]. For x ∈ X, by (GQN-1) and (GQN-4), kxλk(α) ≤ kxαk(α) < +∞. Put s = kxλk(α) + 1 /t. By (GQN-2), we have

1 1 kxλk(α) xλ = kxλk(α) = · t < t. s (α) s kxλk(α) + 1

Hence, (1/s)xλ∈e Uα,t, Note that r > 1 − λ. So (1/s)xλ∈e Uα,t ∩ r, i.e., xλ∈e sU. Thus, by Lemma 2.12, there exists a unique I-topology τ on X such that (X, τ) is an I-tvs, and U is a local base with the stratiﬁed structure {Uλ}λ∈(0,1] of (X, τ), and so Uλ is a Q-neighborhood base of θλ for each λ ∈ (0, 1]. Finally, we prove that (X, τ) is generalized locally bounded. Note that 0 < µ < λ ≤ 1 implies Uλ,1 ⊆ Uµ,1 and tUλ,1 = Uλ,t (see (P-3) and (P-4)). By Theorem 3.1, we now need only to prove that for each λ ∈ (0, 1],

Bλ = {Uλ,t ∩ r | t > 0, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ in (X, τ). Note that Uλ = {Uα,t ∩ r | 0 < α < λ, t > 0, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ in (X, τ). Hence, by (P-6), it is easy to see that every member in Bλ is a Q-neighborhood of θλ in (X, τ). On the other hand, by (P-3) and (P-4), tUλ,1 = Uλ,t ⊆ Uα,t for each α ∈ (0, λ). Hence, Bλ is also a Q-neighborhood base of θλ in (X, τ). Thus, by Theorem 3.1, we conclude that (X, τ) is a generalized locally bounded L-tvs. Theorem 4.4. Let (X, τ) be a generalized locally bounded I-tvs. Then the topology τ can be generated by a family of generalized fuzzy quasi-norms k · k(λ) λ∈(0,1] on X, i.e., Bλ = {Uλ,t ∩ r | t > 0, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ in (X, τ) for each λ ∈ (0, 1], where Uλ,t is deﬁned by (1). Proof. Since (X, τ) is generalized locally bounded, by Lemma 3.3, there exists a balanced and λ-bounded Q-neighborhood Bλ of θλ for each λ ∈ (0, 1], and 0 < µ < X λ ≤ 1 implies Bλ ⊆ Bµ. For each λ ∈ (0, 1], deﬁne a mapping k · k(λ) : Pt(I ) → [0, ∞] by kxαk(λ) = inf t > 0 | xα∈e tBλ . (4)

We shall verify that {k · k(λ)}λ∈(0,1] is a family of generalized fuzzy quasi-norms on X. (GQN-1) Since Uλ is a Q-neighborhood of θλ, θλ∈e Bλ, which implies θλ = tθλ∈e tBλ for all t > 0, and so kθλk(λ) = 0. By (4) of Lemma 2.12 we easily On Local Boundedness of I-topological Vector Spaces 101 know that for each x ∈ X there exists s > 0, such that xλ∈e sBλ. So, by (4) we have kxλk(λ) ≤ s < +∞.

(GQN-2) Let λ ∈ (0, 1] and k ∈ K with k 6= 0. Since Bλ is balanced, xα∈e (t/k)Bλ ⇔ xα∈e (t/|k|)Bλ, and so kkxαk(λ) = inf{t > 0 | kxα∈e tBλ} = inf{t > 0 | xα∈e (t/k)Bλ} = inf{t > 0 | xα∈e (t/|k|)Bλ} = |k| inf{s > 0 | xα∈e sBλ} = |k|kxαk(λ).

(GQN-3) Note that Bλ is a λ-bounded Q-neighborhood of θλ for each λ ∈ (0, 1]. By Lemma 3.2, we know that that Uλ = {(1/n)Bλ ∩ r | n ∈ N, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ. So, by Lemma 2.12, there exist s ∈ (1 − λ, 1] and k0 ≥ 1 such that (1/k0)Bλ ∩ s + (1/k0)Bλ ∩ s ⊆ Bλ. (5) From (5), we can prove that

kxα + yαk(λ) ≤ k0(kxαk(λ) + kyαk(λ)), for all α ∈ (1 − s, 1], x, y ∈ X. (6) In fact, ∀ α ∈ (1 − s, 1] and x, y ∈ X, without loss of generality, we can assume that kxαk(λ) = a < ∞, kyαk(λ) = b < ∞. By (4), for each ε > 0, there exist t1, t2 > 0 such that t1 < a + ε/2, t2 < b + ε/2 and xα∈e t1Bλ, yα∈e t2Bλ. Note that s > 1 − α, it follows from (5) that 1 1 t1 t2 xα + yα ∈e Bλ ∩ s + Bλ ∩ s k0 k0 k0 k0 t1 t2 = (t1 + t2) Bλ ∩ s + Bλ ∩ s k0(t1 + t2) k0(t1 + t2) 1 1 ⊆ (t1 + t2) Bλ ∩ s + Bλ ∩ s ⊆ (t1 + t2)Bλ, k0 k0 which shows that xα + yα∈e k0(t1 + t2)Bλ. By (4), we get

kxα + yαk(λ) ≤ k0(t1 + t2) < k0(a + b + ε). By the arbitrariness of ε, (6) is proved. (GQN-4) Let 0 < β < α ≤ 1, then xβ∈e tBλ implies that xα∈e tBλ. So, by (4), we have kxαk(λ) ≤ kxβk(λ). Now, we prove that kxαk(λ) = inf0<β<α kxβk(λ). If kxαk(λ) = ∞, then the above inequality implies that kxβk(λ) = ∞ for all 0 < β < α. Hence kxαk(λ) = inf0<β<α kxβk(λ); If kxαk(λ) < ∞, by (4) we know that for each ε > 0, there exists t < kxαk(λ)+ε such that xα∈e tBλ, and so there exists µ ∈ (0, α), such that xµ∈e tBλ. By (4), we get kxµk(λ) ≤ t < kxαk(λ) + ε. Therefore kxαk(λ) = inf0<β<α kxβk(λ). Since 0 < µ < λ ≤ 1 implies Bλ ⊆ Bµ, it follows from (4) that kxαk(µ) ≤ kxαk(λ). (GQN-4) is proved.

(GQN-5) We know that Bλ is a λ-bounded Q-neighborhood of θλ for each λ ∈ (0, 1]. It is evident that for each λ ∈ (0, 1], there exists λ0 ∈ (0, λ) such that 102 J. X. Fang and H. Zhang

Bλ is a Q-neighborhood of θλ0 . Since Bλ0 is λ0-bounded, by Lemma 3.2 Uλ0 =

{(1/n)Bλ0 ∩ r | n ∈ N, r ∈ (1 − λ0, 1]} is a Q-neighborhood base of θλ0 , and so there exist s ∈ (1 − λ0, 1] and n0 ≥ 1 such that (1/n0)Bλ0 ∩ s ⊆ Bλ. From this we can prove that

kxαk(λ) ≤ n0kxαk(λ0) for all α ∈ (1 − s, 1] and x ∈ X. (7) In fact, for α ∈ (1−s, 1] and x ∈ X, without loss of generality, let us assume that kxαk(λ0) = a < ∞, then by (1) for each ε > 0 there exists 0 < t < c + ε such that xα∈e tBλ0 , which implies that (1/n0)xα∈e (1/n0)tBλ0 ∩ s ⊆ tBλ, i.e., xα∈e n0tBλ. By (4), we have kxαk(λ) ≤ n0t < n0(a + ε). By the arbitrariness of ε, we get kxαk(λ) ≤ n0kxαk(λ0), (7) is proved. This shows that {k · kλ}λ∈(0,1] is a family of generalized fuzzy quasi-norms on X. By Theorem 4.3, there exists a unique L-topologyτ ˜ such that (X, τ˜) is a generalized locally bounded I-tvs and Bλ = {Uλ,t ∩ r | r > 1 − λ, t > 0} is a Q-neighborhood of θλ in (X, τ˜) for each λ ∈ (0, 1], where Uλ,t is deﬁned by (1). We shall prove that τ˜ = τ. To this end, we need only to prove that t Bλ ⊆ Uλ,t ⊆ tBλ, for all λ ∈ (0, 1] and t > 0. (8) 2

In fact, by the deﬁnition of k · k(λ) and (P-1), it is easy to see that the following implications hold: t t xα∈e 2 Bλ ⇒ kxαk(λ) ≤ 2 < t ⇒ xα∈e Uλ,t. t Hence 2 Bλ ⊆ Uλ,t. Moreover, if xα∈e Uλ,t, then kxαk(λ) < t, and so there exists s ∈ (0, t) such that xα∈e sBλ ⊆ tBλ, which shows that Uλ,t ⊆ tBλ. Therefore (8) holds. Remark 4.5. From Theorem 4.4, we know that a generalized locally bounded I- tvs (X, τ) can be completely characterized in terms of a family of generalized fuzzy quasi-norms k · k(λ) λ∈(0,1]. So, for convenience, we also use X, k · k(λ) λ∈(0,1] to denote a generalized locally bounded I-tvs.

5. Applications In this section, as applications of Theorem 4.1 and Theorem 4.2, we shall use the family of generalized fuzzy quasi-norms to characterize Hausdorff separation property and convergence of a net of fuzzy points in generalized locally bounded I-tvs. We first recall some necessary concepts and results. Definition 5.1. [4] An I-topological space (X, τ) is said to be Hausdorff, if for X each xλ, yµ ∈ Pt(I ) with x 6= y, there exist P ∈ NQ(xλ) and Q ∈ NQ(yµ) such that P ∩ Q = 0. Lemma 5.2. [6] Let (X, τ) be an I-tvs. Then the following statements are mutually equivalent: (1) (LX , δ) is Hausdorff; (2) For each λ ∈ (0, 1], θλ is closed; On Local Boundedness of I-topological Vector Spaces 103

(3) For each λ ∈ (0, 1] and x ∈ X with x 6= θ, there exists U ∈ NQ(θλ) such that U(x) = 0.

Definition 5.3. [4] Let (X, τ) be an I-topological space and {x(n)} a net of λn n∈Λ fuzzy points in X. {x(n)} is said to be convergent to x ∈ Pt(IX ), if for each λn n∈Λ λ U ∈ N (x ), there exists n ∈ Λ such that x(n)∈ U whenever n ∈ Λ with n ≥ n . Q λ 0 λn e 0 Theorem 5.4. A generalized locally bounded I-tvs X, k · k(λ) λ∈(0,1] is Haus- dorff iff kx1k(λ) > 0 for each λ ∈ (0, 1] and each x ∈ X with x 6= θ.

Proof. Necessity: Assume that there exist some λ ∈ (0, 1] and x ∈ X with x 6= θ such that kx1k(λ) = 0, which implies that kx1k(λ) < t for each t > 0. By (P- 1), x1∈e Uλ,t for each t > 0. We know that Bλ = {Uλ,t ∩ r | t > 0, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θλ, where Uλ,t is defined by (1) (see Theorem 4.4). Hence, for each U ∈ NQ(θλ), there exist t > 0 and r ∈ (1 − λ, 1] such that Uλ,t ∩ r ⊆ U. Thus, it follows from x1∈e Uλ,t that U(x) > 0. Therefore, by Lemma 5.2, X, k · k(λ) λ∈(0,1] is not Hausdorff. Sufficiency: Suppose that for each λ ∈ (0, 1] and each x ∈ X with x 6= θ, kx1k(λ) > 0. Put t = kx1k(λ). By (P-1), x1 is not quasi-coincident with Uλ,t, i.e., Uλ,t(x) = 0. Note that Uλ,t ∈ NQ(θλ). Hence, by Lemma 5.2, X, k · k(λ) λ∈(0,1] is Hausdorff. Theorem 5.5. Let X, k · k(λ) λ∈(0,1] be a generalized locally bounded I-tvs and {x(n)} a net of fuzzy points in X, k · k . Then {x(n)} is λn n∈Λ (λ) λ∈(0,1] λn n∈Λ X (n) convergent to xλ ∈ Pt(I ) iff lim k(x − x)λn k(λ) = 0 and lim λn ≥ λ. n→∞ n→∞

Proof. Necessity: We know that for each t > 0 and r ∈ (1 − λ, 1], Uλ,t ∩ r is a Q-neighborhood of θλ (where Uλ,t is defined by (1)), and so x + Uλ,t ∩ r is a Q-neighborhood of x . Since {x(n)} → x , there exists n ∈ Λ such that λ λn n∈Λ λ 0 x(n)∈ x+U ∩r whenever n ∈ Λ with n ≥ n , i.e., (x(n) −x) ∈ U and r > 1−λ λn e λ,t 0 λn e λ,t n (n) whenever n ≥ n0, which implies that k(x −x)λn k(λ) < t and λn > 1−r whenever (n) n ≥ n0. This shows that lim k(x − x)λn k(λ) = 0 and lim λn ≥ λ. n→∞ n→∞ Sufficiency: Note that Bλ = {Uλ,t ∩ r | t > 0, r ∈ (1 − λ, 1]} is a Q-neighborhood base of θ . Hence, to prove that {x(n)} → x , it suffices to prove that for each λ λn n∈Λ λ W ∈ B there exists n ∈ Λ such that x(n)∈ x + W whenever n ∈ Λ with n ≥ n . λ 0 λn e 0 (n) Let W = Uλ,t ∩ r ∈ Bλ. Since lim k(x − x)λn k(λ) = 0 and lim λn ≥ λ, n→∞ n→∞ (n) for t > 0 and r > 1 − λ there exists n0 ∈ Λ such that k(x − x)λn k(λ) < t and λ > 1−r whenever n ∈ Λ with n ≥ n , which implies that x(n)∈ x+U ∩r = x+W n 0 λn e λ,t whenever n ≥ n0. This completes the proof. 104 J. X. Fang and H. Zhang

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Jin-Xuan Fang∗, School of Mathematical Science, Nanjing Normal University, Nan- jing, Jiangsu 210023, P. R. China E-mail address: [email protected]

Hui Zhang, Department of Mathematics, Anhui NormalUniversity, Wuhu, Anhui 241000, P. R. China E-mail address: [email protected]

*Corresponding author