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On I(L) -topological vector spaces generated by a co-tower of L-topological vector spaces
Fuzzy Sets and Systems 160 (2009) 2926 – 2936 www.elsevier.com/locate/fss
On I (L)-topological vector spaces generated by a co-tower of L-topological vector spaces夡 Hua-Peng Zhang, Jin-Xuan Fang∗ Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210097, China Received 4 August 2007; received in revised form 16 February 2009; accepted 16 February 2009 Available online 28 February 2009
Abstract In this paper, a characterization for an I (L)-topological space to be generated by a given co-tower of L-topological spaces is obtained. Moreover, the relationship between some properties of an I (L)-topological vector space generated by a co-tower of Ltopological vector spaces and the corresponding properties of the given co-tower of L-topological vector spaces is investigated. Our results show that if an I (L)-topological vector space generated by a co-tower of L-topological vector spaces has some properties, such as local convexity and local boundedness, then all L-topological vector spaces in the co-tower also have the same properties. But the converse is incorrect even in the case of I-topological vector space generated by a co-tower of classical topological vector spaces. Finally, we supply a necessary and sufficient condition for an I (L)-topological vector space generated by a co-tower of L-topological vector spaces with some properties, such as local convexity and local boundedness, to have such properties too. © 2009 Elsevier B.V. All rights reserved. Keywords: L-fuzzy unit interval; L-topological vector space; Locally convex L-topological vector space; Locally bounded L-topological vector space
1. Introduction Kubiak [7] introduced the notion of I (L)-valued lower semi-continuous mappings, where I is the real unit interval [0, 1], L denotes a Hutton algebra [5], and I (L) is the L-fuzzy unit interval [6]. Using the notion of I (L)-valued lower semi-continuous mappings, Wang [11] introduced and studied the concept of induced I (L)-topological spaces. Moreover, Wang [12] obtained a necessary and sufficient condition for a chain of L-topological structures on a set X such that there exists an (not necessarily unique in general) I (L)-topological structure of which the level L-topological structures are exactly the given chain of L-topological structures. Inspired by Wang’s work, Fang and Yan [4,18] introduced and studied induced I (L)-topological vector spaces and supplied a new method to generate I (L)-topological vector spaces by a co-tower of L-topological vector spaces. In addition, they proved that an I (L)-topological vector space generated by a co-tower of L-topological vector spaces is Hausdorff [10] if and only if all L-topological vector spaces in the co-tower are Hausdorff. Thus, a natural question arises: whether an I (L)-topological vector space generated by a 夡 This work is supported by the National Natural Science Foundation of China (No. 10671094) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20060319001). ∗ Corresponding author. E-mail addresses: [email protected] (H.-P. Zhang), [email protected] (J.-X. Fang). 0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.02.020
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co-tower of L-topological vector spaces has some property if and only if all L-topological vector spaces in the co-tower have the same property? This paper is a continuation of the work in [18], where we will focus on study of the question raised above. In view of that locally convex L-topological vector spaces [15], locally bounded L-topological vector spaces [17] and L-fuzzy normable L-topological vector spaces [16] are three important subclasses of L-topological vector spaces. Hence, in this paper, we will answer the above question in detail: whether an I (L)-topological vector space generated by a co-tower of L-topological vector spaces is locally convex (locally bounded, I (L)-fuzzy normable) if and only if all L-topological vector spaces in the co-tower are locally convex (locally bounded, L-fuzzy normable)? If not, does there exist any necessary and sufficient condition such that the answer to the above question is positive? This paper is organized as follows. In Section 2, we recall some basic concepts and lemmas needed in the sequel. In Section 3, we give a characterization to an I (L)-topological space generated by a given co-tower of L-topological spaces. In particular, we obtain some corollaries, one of which will be needed in the counterexample given in the next section. In Section 4, we prove that an I (L)-topological vector space generated by a co-tower of L-topological vector spaces has some properties, such as local convexity and local boundedness, then all L-topological vector spaces in the co-tower also have the same properties. At the same time, we give a counterexample to show that in general, the converse is incorrect. Finally, we provide a necessary and sufficient condition for an I (L)-topological vector space generated by a co-tower of L-topological vector spaces with some properties, such as local convexity and local boundedness, to have such properties too. 2. Basic concepts and lemmas Throughout this paper, L denotes a Hutton algebra [5], i.e., L is a complete and completely distributive lattice equipped with an order-reversing involution : L → L. 0 and 1 are its bottom and top elements, respectively. M(L) denotes the set of all non-zero union-irreducible elements in L. The elements of M(L) are also called molecules [8] in L. L X denotes the family of all L-fuzzy sets on X. An L-fuzzy set which takes the constant value ∈ L on X is denoted by . An L-topology on X is called stratified, if it contains all constant L-fuzzy sets on X. We always assume that the L-topologies referred to in the present paper are all stratified and the lattice L is regular (i.e., the intersection of each pair of non-zero elements in L is not zero, or equivalently 1 ∈ M(L)). In this paper, the notations I, I0 and I1 stand for the intervals [0, 1], (0, 1] and [0, 1), respectively. The L-fuzzy unit interval I (L) is the set of all equivalence classes [], where : R → L is a monotone decreasing mapping satisfying (t) = 1 for t < 0 and (t) = 0 for t > 1, and ∈ [] iff (t−) = (t−) (equivalently, (t+) = (t+)) for all t ∈ I . The natural L-topology on I (L) is generated by the subbase {L t , Rt |t ∈ I }, where L t [] = ((t−)) and Rt [] = (t+). The partial order on I (L) is naturally defined by [][] iff (t−) (t−) (equivalently, (t+) (t+)) for all t ∈ I . Let [] ∈ I (L), we define [] = [ ], where ∈ L R is defined by (t) = ((1 − t)) for all t ∈ R. Thus, (I (L), , ) is a Hutton algebra. To simplify notation, we shall identify equivalence classes with their representatives in the sequel. For other symbols which are not mentioned, we refer to [3,8]. Definition 2.1 (Liu and Luo [8], Wang [10]). Let (L X , ) be an L-topological space and x ∈ M(L X ). P ∈ L X is called a closed R-neighborhood of x , if P ∈ and x P. The set of all closed R-neighborhoods of x is denoted by − (x ). A ∈ L X is called an R-neighborhood of x , if there exists P ∈ − (x ) such that A P. The set of all R-neighborhoods of x is denoted by (x ). U ⊆ (x ) is said to be an R-neighborhood base of x if for each P ∈ (x ), there exists Q ∈ U such that P Q. Remark 2.1. In the research of I-topological space, the widely used neighborhood structure of a fuzzy point is Qneighborhood introduced by Pu and Liu [9]. Recall that U ∈ I X is called a Q-neighborhood of x in the I-topological space (I X , ) if there exists G ∈ such that G U and G(x) > 1 − . A family U of Q-neighborhoods of x is called a Q-neighborhood base of x if for each Q-neighborhood V of x , there exists U ∈ U such that U V . It is easy to see that, in the case of L = I , U is a Q-neighborhood of x iff U is an R-neighborhood of x and U is a Q-neighborhood base of x iff U = {U |U ∈ U} is an R-neighborhood base of x .
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Definition 2.2 (Liu and Luo [8]). Let (L X , ) be an L-topological space. A ∈ L X is called a neighborhood of x , if there exists U ∈ such that x U A. Definition 2.3 (Liu and Luo [8], Wang [10]). Let (L X , ) be an L-topological space. (L X , ) is called T2 (or Hausdorff), if for every two molecules x and y with different support points x y, there exist U ∈ (x ) and V ∈ (y ) such that U ∨ V = 1. Definition 2.4 (Fang and Yan [3]). Let X be a vector space over K (K = R or C) and be an L-topology on X. The pair (L X , ) is called an L-topological vector space (briefly, L-tvs) if the following two mappings (the addition and the scalar multiplication on X): (1) f : X × X → X , (x, y)x + y and (2) g : K × X → X , (k, x)kx are both continuous, where X × X and K × X are equipped with the corresponding product L-topologies × and JK × , respectively, and JK denotes the usual topology on K. Definition 2.5 (Fang [2]). Let (L X , ) be an L-tvs and be the zero element in X. An L-fuzzy set A on X is said to be -bounded ( ∈ M(L)), if for each R-neighborhood Q of in (L X , ), there exist t > 0 and ∈ L with such that A ∧ t Q . A is said to be bounded if it is -bounded for each ∈ M(L). Definition 2.6 (Yan and Fang [17]). An L-tvs (L X , ) is said to be locally bounded if it has a bounded neighborhood of (i.e., 1 ). Remark 2.2. From Theorem 2 in [17] and Definition 2.5, it is not difficult to prove that U ∈ L X is a bounded neighborhood of in the L-tvs (L X , ) iff for each ∈ M(L), {tU ∨ |t > 0, } is an R-neighborhood base of . Recall that an I-tvs (I X , ) is said to be of (QL)-type if there exists a family U of fuzzy sets on X such that U = {U ∧ r |U ∈ U, r ∈ (1 − , 1]} is a Q-neighborhood base of for each ∈ I0 [13]. Remark 2.3. In the case of L = I , by Remark 2.2, it is easy to see that a locally bounded I-tvs is necessarily an I-tvs of (QL)-type. Definition 2.7 (Yan and Fang [15]). Let (L X , ) be an L-tvs. A family B of L-fuzzy sets on X is called a prebase of if for each ∈ M(L), B = {B ∨ |B ∈ B, } is an R-neighborhood base of . An L-tvs (L X , ) is said to be locally convex, if there exists a family B of absolutely convex L-fuzzy sets on X such that B is a prebase of . Obviously, in the case of L = I , an I-tvs has a prebase iff it is of (QL)-type; a locally convex I-tvs in the sense of Definition 2.7 is exactly a locally convex I-tvs introduced by Wu and Li [14], which is necessarily an I-tvs of (QL)-type (see [19] for detail). Remark 2.4. (1) By Remark 2.2, a locally bounded L-tvs necessarily has a prebase. (2) If B is a prebase of the L-tvs (L X , ), then each B in B is a neighborhood of and equivalently, for each B ∈ B and ∈ M(L), B is an R-neighborhood of (see Lemma 2 in [16]). Definition 2.8 (Kubiak [7]). For each t ∈ I , define mappings t , t : I (L) X → L X as follows: t () = ← (Rt ), t () = ← (L t ) for all ∈ I (L) X . Obviously t ()(x) = (x)(t+), t ()(x) = (x)(t−).
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Definition 2.9 (Kubiak [7]). Let (L X , ) be an L-topological space. A mapping : X → I (L) is called I (L)-valued lower (resp. upper) semi-continuous if for each t ∈ I , t () ∈ (resp. (t ()) ∈ ). We denote the set of all I (L)-valued lower semi-continuous mappings on X by I (L) (). It is easy to verify that I (L) () is an I (L)-topology on X, called the induced I (L)-topology of , and (I (L) X , I (L) ()) is called an induced I (L)-topological space. Definition 2.10 (Wang [11]). Define the mapping ∗ : L X → I (L) X satisfying ⎧ t < 0, ⎨ 1, P ∗ (x)(t+) = P(x), 0 t < 1, ⎩ 0, t 1 for all P ∈ L X and x ∈ X . Moreover, for each t ∈ I , define a mapping t X∗ : X → I (L) (for short, t ∗ ) by letting 1, s < t, t ∗ (x)(s+) = 0, s t for all x ∈ X . It is easy to see that ⎧ t 0, ⎨ 1, P ∗ (x)(t−) = P(x), 0 < t 1, ⎩ 0, t > 1,
t ∗ (x)(s−) =
1, s t, 0, s > t
for all x ∈ X . Definition 2.11 (Yan and Fang [18]). A co-tower of L-topologies on a set X (indexed by I1 ) is a family of L-topologies = {s | s ∈ I1 } such that s is generated by s
(t ()) = 1−t ( ); (t ()) = 1−t ( ); t (P ∗ ) = P for all t ∈ I1 ; t (P ∗ ) = P for all t ∈ I0 ; (P ∗ ) = (P )∗ ; (t ∗ ) = (1 − t)∗ [4].
Lemma 2.2 (Wang [11]). Let ∈ I (L) X be an I (L)-fuzzy set. Then the following equality holds: (t ∗ ∧ (t ())∗ ), = t∈I
where t ∗ and (t ())∗ are defined as in Definition 2.10. Remark 2.5. By Lemmas 2.1 and 2.2, for each ∈ I (L) X and s ∈ I , we have (s ())∗ ∨ s ∗ . Define ∈ I (L) as follows: 1, s < 0,
(s+) = 0, s 0.
(2.1)
It is straightforward by Definition 2.10 that is the bottom element in I (L) and 0∗X is an I (L)-fuzzy set which takes the constant value on X.
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For each ∈ L and t ∈ I0 , we define ,t ∈ I (L) as follows: ⎧ ⎨ 1, s < 0, ,t (s+) = , 0 s < t, ⎩ 0, s t.
(2.2)
Remark 2.6. It is not difficult to see that 1,1 is the top element in I (L) and 1∗X is an I (L)-fuzzy set which takes the constant value 1,1 on X. In addition, ⎧ ⎨ 1, s < 1 − t, ( ,t ) (s+) = , 1 − t s < 1, (2.3) ⎩ 0, s 1. Lemma 2.3. Let ∈ I (L) and P ∈ I (L) X . Then (1) is a molecule in I (L), i.e., ∈ M(I (L)) iff there exist ∈ M(L) and t ∈ I0 such that = ,t [11]; (2) x ,t P iff x t (P). Proof. (2) It is clear that x ,t P ⇐⇒ ,t P(x) ⇐⇒ ,t (t−)P(x)(t−). Note that ,t (t−) = and P(x)(t−) = t (P)(x). Hence (2) holds. Lemma 2.4 (Yan and Fang [18]). Let = {s }s∈I1 be a co-tower of L-topologies on X, then the operator ◦ : I (L) X → I (L) X defined by (s ∗ ∧ (ints (s ()))∗ ) ◦ = s∈I1
is a fuzzy interior operator on X, where ints denotes the interior operator with respect to s . Thus, it determines an I (L)-topology on X, denoted by ( ), called the I (L)-topology generated by . Moreover, ∈ ( ) iff s () ∈ s for all s ∈ I1 and s ( ( )) = s for each s ∈ I1 , where s ( ( )) = {s (U )|U ∈ ( )}. Particularly, (I (L) X , ( )) is an I (L)-tvs iff (L X , s ) is an L-tvs for each s ∈ I1 . Lemma 2.5 (Yan and Fang [18]). Suppose that (I (L) X , ( )) is an I (L)-topological space generated by a co-tower of L-topologies = {s }s∈I1 on X. If s ∈ I1 and G s ∈ s , then s ∗ ∧ (G s )∗ ∈ ( ). 3. A characterization for an I (L)-topological space to be generated by a given co-tower of L-topological spaces In this section, = {s }s∈I1 denotes a co-tower of L-topologies on X and Us (x ) denotes an R-neighborhood base of x in (L X , s ) for each s ∈ I1 and x ∈ M(L X ). Lemma 3.1. Let (I (L) X , ( )) be an I (L)-topological space generated by and s ∈ (0, 1). If P ∈ s (x ), then P ∗ ∨ (1 − s)∗ ∈ (x ,t ) for each t ∈ (1 − s, 1], where s (x ) and (x ,t ) denote the set of all R-neighborhoods of x in (L X , s ) and that of x ,t in (I (L) X , ( )), respectively. Proof. Since P ∈ s (x ), there exists Q ∈ s such that P Q and x Q. Hence P ∗ ∨ (1 − s)∗ Q ∗ ∨ (1 − s)∗ for each s ∈ (0, 1). It suffices to show that Q ∗ ∨ (1 − s)∗ is a closed R-neighborhood of x ,t for each t ∈ (1 − s, 1]. In fact, since Q ∈ s and (Q ∗ ∨ (1 − s)∗ ) = (Q )∗ ∧ s ∗ , we have Q ∗ ∨ (1 − s)∗ ∈ ( ( )) by Lemma 2.5. Note that Q ∗ (x)((1 − s)+) = Q(x) , (1 − s)∗ (x)((1 − s)+) = 0 and ,t ((1 − s)+) = . Hence we conclude that x ,t Q ∗ ∨ (1 − s)∗ . This completes the proof. Theorem 3.1. Let (I (L) X , ( )) be an I (L)-topological space generated by . Then U(x ,t ) = s∈(1−t,1) U∗s (x ) is an R-neighborhood base of x ,t for each x ,t ∈ M(I (L) X ), where U∗s (x ) = {P ∗ ∨ (1 − s)∗ |P ∈ Us (x )}.
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Proof. By Lemma 3.1, each member of U(x ,t ) is an R-neighborhood of x ,t . It remains to prove that for each ˜ Q ∈ − (x ,t ), there exists Q˜ ∈ U(x ,t ) such that Q Q. − In fact, since Q ∈ (x ,t ), we have ,t Q(x), and so there exists s0 ∈ (0, t) such that = ,t (s0 −)Q(x)(s0 −) = s0 (Q)(x), i.e., x s0 (Q). On the other hand, (s0 (Q)) = 1−s0 (Q ) ∈ 1−s0 by Lemmas 2.1 and 2.4. So s0 (Q) is a closed R-neighborhood of x in (L X , 1−s0 ). Therefore, there exists P0 ∈ U1−s0 (x ) such that s0 (Q) P0 , and so = P ∗ ∨ s ∗ . It remains to show that Q ∈ U(x ). (s0 (Q))∗ ∨ s0∗ P0∗ ∨ s0∗ . Hence Q P0∗ ∨ s0∗ by Remark 2.5. Put Q ,t 0 0 Note that s0 ∈ (0, t), which implies 1 − s0 ∈ (1 − t, 1), and so Q ∈ U(x ,t ) by the construction of U(x ,t ). This completes the proof. Theorem 3.2. Let (I (L) X , ) be an I (L)-topological space and = {s }s∈I1 be a co-tower of L-topologies on X. If U(x ,t ) = s∈(1−t,1) U∗s (x ) is an R-neighborhood base of x ,t for each x ,t ∈ M(I (L) X ), where U∗s (x ) = {P ∗ ∨ (1 − s)∗ |P ∈ Us (x )}, then is generated by , i.e., = ( ). Proof. By Lemma 2.4, it suffices to show that Q ∈ iff t (Q) ∈ t for each t ∈ I1 . Let Q ∈ and t ∈ I1 . To prove t (Q) ∈ t , it suffices to show that for each x ∈ M(L X ) with x (t (Q)) , (t (Q)) ∈ t (x ), where t (x ) is the set of all R-neighborhoods of x in (L X , t ). Note that (t (Q)) = 1−t (Q ). By Lemma 2.3, x (t (Q))
⇐⇒
x 1−t (Q )
⇐⇒
x ,1−t Q ,
and so Q is a closed R-neighborhood of x ,1−t in (I (L) X , ). Since U(x ,1−t ) = s∈(t,1) U∗s (x ) is an R-neighborhood base of x ,1−t , there exists s ∈ (t, 1) and P ∈ Us (x ) such that Q P ∗ ∨ (1 − s)∗ , and so 1−t (Q ) 1−t (P ∗ ∨ (1 − s)∗ ) = 1−t (P ∗ ) ∨ 1−t ((1 − s)∗ ) = P ∨ 0 = P. Hence 1−t (Q ) ∈ s (x ). On the other hand, since {s }s∈I1 is a co-tower, s > t ⇒ s ⊆ t , and so s (x ) ⊆ t (x ), which implies that 1−t (Q ) ∈ t (x ). Conversely, suppose that Q ∈ I (L) X and t (Q) ∈ t for all t ∈ I1 . To prove Q ∈ , it suffices to show that for each x ,r ∈ M(I (L) X ), x ,r Q implies that Q ∈ (x ,r ). In fact, x ,r Q implies that there exists s ∈ (0, r ) such that x s (Q ) = (1−s (Q)) ∈ 1−s , and so s (Q ) ∈ 1−s (x ). Hence there exists P ∈ U1−s (x ) such that s (Q ) P, and so Q (s (Q ))∗ ∨ s ∗ P ∗ ∨ s ∗ . On the other hand, we have P ∗ ∨ s ∗ ∈ U(x ,r ) since 1 − s ∈ (1 − r, 1). Therefore, Q ∈ (x ,r ). This completes the proof. By Theorems 3.1 and 3.2, we can obtain the following conclusion, which is a characterization for an I (L)-topological space to be generated by a given co-tower of L-topological spaces. Theorem 3.3. Let (I (L) X , ) be an I (L)-topological space. Then is generated by the co-tower of L-topologies = {s }s∈I1 on X iff U(x ,t ) = s∈(1−t,1) U∗s (x ) is an R-neighborhood base of x ,t for each x ,t ∈ M(I (L) X ), where U∗s (x ) = {P ∗ ∨ (1 − s)∗ |P ∈ Us (x )}.
(3.1)
In the case of L = {0, 1} (hence I (L) = I ), by Theorem 3.3, we have the following corollary. Corollary 3.1. Let { s }s∈I1 be a co-tower of crisp topologies on X and (I X , ) an I-topological space. Then is generated by { s }s∈I1 iff B(x ) = s∈(1−,1) B∗s (x) is a Q-neighborhood base of x for each x ∈ X and ∈ I0 , where B∗s (x) = {U ∧ s|U ∈ Bs (x)} and Bs (x) denotes a neighborhood base of x in (X, s ) for each s ∈ I1 . If for each s ∈ I1 , (L X , s ) is an L-tvs, then, by Theorem 3.3, we have the following conclusion. X Theorem 3.4. Let (I (L) , ) be∗ an I (L)-tvs. Then is generated by the co-tower of vector L-topologies = {s }s∈I1 on X iff U( ,t ) = s∈(1−t,1) Us ( ) is an R-neighborhood base of ,t for each ,t ∈ M(I (L)), where
U∗s ( ) = {P ∗ ∨ (1 − s)∗ |P ∈ Us ( )}.
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Proof. Necessity: It follows directly from Theorem 3.3. Sufficiency: Since (L X , s ) is an L-tvs and Us ( ) is an R-neighborhood base of , by Corollary 2.3 in [3], Us (x ) = X {x s ( )} is an R-neighborhood base of x for each x ∈ X . Since (I (L) , ) is an I (L)-tvs and U( ,t ) = + P|P ∈ U ∗ s∈(1−t,1) Us ( ) is an R-neighborhood base of ,t , we know that U(x ,t ) = {x + Q|Q ∈ U( ,t )} =
(x + U∗s ( ))
s∈(1−t,1)
is an R-neighborhood base of x ,t . By (3.1), it is easy to see that x + U∗s ( ) = {x + (P ∗ ∨ (1 − s)∗ )|P ∈ Us ( )} = {(x + P)∗ ∨ (1 − s)∗ |x + P ∈ Us (x )} = U∗s (x ). Therefore, is generated by by Theorem 3.3. In the case of L = {0, 1}, by Theorem 3.4, we have the following corollary. X Corollary 3.2. Let { s }s∈I1 be a co-tower of crisp vector topologies on X and (I , ) an I-tvs. Then is generated by ∗ { s }s∈I1 iff B( ) = s∈(1−,1) Bs () is a Q-neighborhood base of for each ∈ I0 , where B∗s () = {U ∧ s|U ∈ Bs ()} and Bs () denotes a neighborhood base of in (X, s ) for each s ∈ I1 .
4. Local convexity and local boundedness of I (L)-tvs generated by a co-tower of L-topological vector spaces In this section, we assume that (I (L) X , ) denotes an I (L)-tvs without special instruction and s ( ) = {s (U )|U ∈ } for each s ∈ I1 . By Theorem 2.1 in [18], we know that (L X , s ( )) is an L-tvs for each s ∈ I1 . Lemma 4.1. Let (I (L) X , ) be an I (L)-topological space, s ∈ I1 and U(x ,1−s ) be an R-neighborhood base of x ,1−s . Then B(x ) = {1−s (P)|P ∈ U(x ,1−s )} is an R-neighborhood base of x in (L X , s ), where s = s ( ). Proof. We first verify that each member of B(x ) is an R-neighborhood of x in (L X , s ). In fact, for each P ∈ U(x ,1−s ), there exists Q ∈ such that x ,1−s Q and P Q , and so x 1−s (Q ) and 1−s (P) 1−s (Q ) = (s (Q)) ∈ s . Hence 1−s (P) is an R-neighborhood of x in (L X , s ). It remains to show that for each R-neighborhood W of x in (L X , s ), there exists P ∈ U(x ,1−s ) such that W 1−s (P). Without loss of generality, we may assume that W ∈ s , and so there exists U ∈ such that W = (s (U )) = 1−s (U ). Note that x W , i.e., x 1−s (U ), which implies that x ,1−s U , and so U is a closed R-neighborhood of x ,1−s in (I (L) X , ). Hence there exists P ∈ U(x ,1−s ) such that U P, which implies W = 1−s (U ) 1−s (P). This completes the proof. In particular, in the case of L = {0, 1}, we obtain the following corollary. Corollary 4.1. Let (I X , ) be an I-topological space, s ∈ I1 and U(x1−s ) is a Q-neighborhood base of x1−s . Then B(x) = {s (U )|U ∈ U(x1−s )} is a neighborhood base of x in (X, s ()), where s () = {s (G)|G ∈ }. Lemma 4.2. If (I (L) X , ) has a prebase, then (L X , s ( )) also has a prebase for each s ∈ I1 . Moreover, for each s, t ∈ I1 with s < t, we have s ( ) ⊆ t ( ). Proof. Let U be a prebase in (I (L) X , ). We shall show that {s (U )|U ∈ U} is a prebase of (L X , s ( )) for each s ∈ I1 . It suffices to show that for each ∈ M(L), the family B = {(s (U )) ∨ |U ∈ U, } is an R-neighborhood base of in (L X , s ( )). Since U is a prebase in (I (L) X , ), by Remark 2.4, we have U () = for each U ∈ U. So (s (U )) () = 1−s (U )() = U ()((1 − s)−) = 0. Thus, it is not difficult to see that for each U ∈ U and ∈ L with , (s (U )) ∨ is an R-neighborhood of in (L X , s ( )).
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Let Q ∈ − ( ) in (L X , s ( )), then there exists G ∈ such that Q = (s (G)) = 1−s (G ). From Q we have ,1−s G , which implies that G ∈ − ( ,1−s ) in (I (L) X , ). Note that {U ∨ |U ∈ U, ,1−s } is an R-neighborhood base of ,1−s . So there exist U ∈ U and ∈ I (L) with ,1−s such that G U ∨ , and so Q = 1−s (G ) 1−s (U ) ∨ 1−s () = (s (U )) ∨ ((1 − s)−). On the other hand, from ,1−s we know that ((1 − s)−), which implies that (s (U )) ∨ ((1 − s)−) ∈ B . Hence B = {(s (U )) ∨ |U ∈ U, } is an R-neighborhood base of in (L X , s ( )). Finally, we prove that for each s, t ∈ I1 with s < t, we have s ( ) ⊆ t ( ). In fact, since B = {(s (U )) ∨ |U ∈ U, } is an R-neighborhood base of in (L X , s ( )) for each s ∈ I1 and note that s < t implies (s (U )) (t (U )) , we have s ( ) ⊆ t ( ), where s ( ) and t ( ) denote the sets of all R-neighborhoods of in (L X , s ( )) and (L X , t ( )), respectively. Since (L X , s ( )) and (L X , t ( )) are both L-topological vector spaces, s < t implies that s (x ) ⊆ t (x ) for each x ∈ M(L X ). Therefore, s ( ) ⊆ t ( ). This completes the proof. In particular, in the case of L = {0, 1}, we obtain the following corollary. Corollary 4.2. If (I X , ) is an I-tvs of (QL)-type, then for each s, t ∈ I1 with s < t, we have s () ⊆ t (). Theorem 4.1. If (I (L) X , ) is locally convex, then (L X , s ( )) is also locally convex for each s ∈ I1 . Proof. Let U be an absolutely convex prebase in (I (L) X , ). In the following, we shall show that {s (U )|U ∈ U} is an absolutely convex prebase of (L X , s ( )) for each s ∈ I1 . From Theorem 3.1 in [20], we know that s (U ) is an absolutely convex L-fuzzy set for each U ∈ U. On the other hand, by the proof of Lemma 4.2, {s (U )|U ∈ U} is a prebase of (L X , s ( )). Hence (L X , s ( )) is locally convex. Lemma 4.3. If B ∈ I (L) X is bounded in (I (L) X , ), then s (B) is bounded in (L X , s ( )) for each s ∈ I1 . Proof. Assume that B is bounded in (I (L) X , ), and let ∈ M(L) and Q ∈ − ( ) in (L X , s ( )), then there exists U ∈ such that Q = (s (U )) = 1−s (U ). From Q, we know that ,1−s U , which implies that U ∈ − ( ,1−s ). Since B is bounded in (I (L) X , ), there exist t > 0 and ∈ I (L) with ( ,1−s ) such that B ∧ t(U ) = tU.
(4.1)
It follows from (4.1) that s (B) ∧ (s+) = s (B) ∧ s () s (tU ) = ts (U ) = t Q .
(4.2)
Moreover, from (2.3) and ( ,1−s ) , it follows that there exists r ∈ [s, 1) such that (r +) . On the other hand, it follows from (4.2) and r s that s (B)∧(r +)t Q . Therefore, s (B) is -bounded. By the arbitrariness of ∈ M(L), we know that s (B) is bounded in (L X , s ( )). Lemma 4.4. If Q is a neighborhood of in (I (L) X , ), then s (Q) is a neighborhood of in (L X , s ( )) for each s ∈ I1 . Proof. Since Q is a neighborhood of in (I (L) X , ), there exists U ∈ such that 1,1 U Q. Hence 1 = s (1,1 ) s (U ) s (Q) for each s ∈ I1 . On the other hand, we have s (U ) ∈ s ( ), and so s (Q) is a neighborhood of in (L X , s ( )). Theorem 4.2. If (I (L) X , ) is locally bounded, then (L X , s ( )) is also locally bounded for each s ∈ I1 . Proof. Since (I (L) X , ) is locally bounded, there exists a bounded neighborhood Q of in (I (L) X , ). By Lemma 4.3, we know that s (Q) is bounded in (L X , s ( )). In addition, s (Q) is a neighborhood of in (L X , s ( )) by Lemma 4.4. Hence (L X , s ( )) is locally bounded.
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Theorem 4.3. If (I (L) X , ) is I (L)-fuzzy normable, then (L X , s ( )) is also L-fuzzy normable for each s ∈ I1 . Proof. An L-tvs (L X , ) is L-fuzzy normable iff it is Hausdorff and has a convex and bounded neighborhood of (see Theorems 2 and 3 in [16]). So (I (L) X , ) is Hausdorff and has a convex and bounded neighborhood Q of . By Theorem 2.1 in [18], (L X , s ( )) is Hausdorff, and s (Q) is convex by Theorem 3.1 in [20]. Moreover, s (Q) is bounded and a neighborhood of in (L X , s ( )) by Lemmas 4.3 and 4.4, respectively. Hence (L X , s ( )) is L-fuzzy normable. By Theorems 4.1–4.3 and Lemma 2.4, we get the following three corresponding corollaries. Corollary 4.3. Let (I (L) X , ( )) be an I (L)-tvs generated by the co-tower of vector L-topologies {s }s∈I1 on X. If (I (L) X , ( )) is locally convex, then (L X , s ) is also locally convex for each s ∈ I1 . Corollary 4.4. Let (I (L) X , ( )) be an I (L)-tvs generated by the co-tower of vector L-topologies {s }s∈I1 on X. If (I (L) X , ( )) is locally bounded, then (L X , s ) is also locally bounded for each s ∈ I1 . Corollary 4.5. Let (I (L) X , ( )) be an I (L)-tvs generated by the co-tower of vector L-topologies {s }s∈I1 on X. If (I (L) X , ( )) is I (L)-fuzzy normable, then (L X , s ) is L-fuzzy normable for each s ∈ I1 . By Lemma 3.5 in [21] and Theorems 4.1–4.3, we obtain the following three corresponding corollaries, which are main results in [20]. Corollary 4.6. Let (L X , ) be an L-tvs and (I (L) X , I (L) ()) be its induced I (L)-tvs. If (I (L) X , I (L) ()) is locally convex, then (L X , ) is also locally convex. Corollary 4.7. Let (L X , ) be an L-tvs and (I (L) X , I (L) ()) be its induced I (L)-tvs. If (I (L) X , I (L) ()) is locally bounded, then (L X , ) is also locally bounded. Corollary 4.8. Let (L X , ) be an L-tvs and (I (L) X , I (L) ()) be its induced I (L)-tvs. If (I (L) X , I (L) ()) is I (L)-fuzzy normable, then (L X , ) is L-fuzzy normable. Remark 4.1. Generally, the converse of Theorems 4.1, 4.2, and 4.3 are all incorrect even in the case that L = {0, 1} and is generated by a co-tower of crisp vector topologies. Please see the following example.
∞ ∞ ⊆ K satisfying Example 4.1. Let X = 1 , i.e., X is the set of all sequences {i }i=1 i=1 |i | < +∞. If we define addition and scalar multiplication on X pointwisely, then it is clear that X is a vector space
∞ over K. |i | and x2 = supi |i | Define · 1 : X → [0, +∞) and · 2 : X → [0, +∞) respectively by x1 = i=1 ∞ ∈ X . Then from the theory of classical functional analysis about p spaces, we know that · for each x = {i }i=1 1 and · 2 are both norms on X, T·2 T·1 , and it is obvious that (X, T·1 ) and (X, T·2 ) are both normable topological vector spaces, where T·1 and T·2 denote the topologies induced by · 1 and · 2 , respectively. In addition, {U·1 (t)|t > 0} and {U·2 (t)|t > 0} are neighborhood bases of in (X, T·1 ) and (X, T·2 ), respectively, where U·i (t) = {x ∈ X |xi < t}, i = 1, 2. For each ∈ I0 , we define a family U of fuzzy sets on X as follows: {U·2 (t) ∧ r |t > 0, r ∈ (1 − , 1)}, ∈ (0, 1/2], U = {U·1 (t) ∧ r |t > 0, r ∈ (1 − , 1)}, ∈ (1/2, 1]. It is not difficult to check that {U }∈I0 satisfies conditions (1)–(5) of Theorem 4.1 in [1]. So there exists a unique I-topology on X, denoted by , such that (I X , ) is an I-tvs and U is a Q-neighborhood base of for each ∈ I0 . Hence (X, s ()) is a topological vector space for each s ∈ I1 , where s () = {s (G)|G ∈ }. By Corollary 4.1, s (U1−s ) = {s (U )|U ∈ U1−s } is a neighborhood base of in (X, s ()). Note that {U·1 (t)|t > 0}, s ∈ [0, 1/2), s (U1−s ) = {U·2 (t)|t > 0}, s ∈ [1/2, 1),
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which implies T·1 , s ∈ [0, 1/2), s () = T·2 , s ∈ [1/2, 1). It is not difficult to see that {s ()}s∈I1 is a co-tower of vector topologies on X and for each s ∈ I1 (X, s ()) is normable, hence it is locally convex and locally bounded. Moreover, by Corollary 3.2, we can prove that is generated by {s ()}s∈I1 . In fact, for each s ∈ I1 , we put Bs () = s (U1−s ), B∗s () = {U ∧ s|U ∈ s (U1−s )} and B( ) = s∈(1−,1) B∗s (). Then when ∈ (0, 1/2] we have B( ) = {U·2 (t) ∧ s|t > 0, s ∈ (1 − , 1)} = U , which is a Q-neighborhood base of ; When ∈ (1/2, 1], we have B( ) = {U·1 (t) ∧ s|t > 0, s ∈ (1 − , 1/2)} ∪ {U·2 (t) ∧ s|t > 0, s ∈ [1/2, 1)}. Note that U = {U·1 (t) ∧r|t > 0, r ∈ (1 − , 1]} ( ∈ (1/2, 1]) is a Q-neighborhood base of in (I X , ). It is easy to see that {U·1 (t) ∧ s|t > 0, s ∈ (1 − , 1/2)} is also a Q-neighborhood base of . In addition, since U·1 (t) ⊆ U·2 (t) when ∈ (1/2, 1], for each t > 0, U·2 (t) ∧ s is a Q-neighborhood of for each t > 0 and s ∈ [1/2, 1). Therefore, B( ) is also a Q-neighborhood base of . This shows that for each ∈ I0 , B( ) = s∈(1−,1) B∗s () is a Qneighborhood base of . So, by Corollary 3.2, the I-topology is generated by {s ()}s∈I1 . Now, we verify that (I X , ) is not of (QL)-type, hence not locally convex (locally bounded, fuzzy normable). In fact, if (I X , ) is of (QL)-type, then, by Corollary 4.2, we have 1/4 () ⊆ 3/4 (), and so T·1 ⊆ T·2 , which contradicts with T·1 T·2 . Finally, we supply a necessary and sufficient condition for an I (L)-topological vector space generated by a co-tower of L-topological vector spaces with some properties, such as local convexity and local boundedness, to have such properties too. Definition 4.1. An L-tvs is said to have Property (A) if it is locally convex or locally bounded or L-fuzzy normable. Remark 4.2. From the discussion above, we know that if an L-tvs has Property (A), then it must have a prebase. Theorem 4.4. Let (I (L) X , ( )) be an I (L)-tvs generated by the co-tower of vector L-topologies {s }s∈I1 on X and all the L-topological vector spaces {(L X , s )}s∈I1 have Property (A). Then (I (L) X , ( )) also has Property (A) iff s = 0 for all s ∈ I1 . Proof. Necessity: Let (I (L) X , ( )) have Property (A). Then, by Remark 4.2 and Lemmas 2.4 and 4.2, s ⊆ t for each s, t ∈ I1 with s < t. On the other hand, we have t ⊆ s for each s, t ∈ I1 with s < t since {s }s∈I1 is a co-tower. Hence s = 0 for all s ∈ I1 . Sufficiency: Let s = 0 for all s ∈ I1 . Then, by Corollary 2.7 in [18], (I (L) X , ( )) is the induced I (L)-tvs of (L X , 0 ). Hence (I (L) X , ( )) has Property (A) by the main results in [20]. Acknowledgments The authors would like to thank the Editors-in-Chief and the anonymous referees for their valuable suggestions in improvement of the original manuscript. References [1] [2] [3] [4]
J.-X. Fang, On local bases of fuzzy topological vector spaces, Fuzzy Sets and Systems 87 (1997) 341–347. J.-X. Fang, The continuity of fuzzy linear order-homomorphism, J. Fuzzy Math. 5 (4) (1997) 829–838. J.-X. Fang, C.-H. Yan, L-fuzzy topological vector spaces, J. Fuzzy Math. 5 (1) (1997) 133–144. J.-X. Fang, C.-H. Yan, Induced I (L)-fuzzy topological vector spaces, Fuzzy Sets and Systems 121 (2001) 293–299.
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[5] U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Dordrecht, 1999. [6] B. Hutton, Normality in fuzzy topological spaces, J. Math. Anal. Appl. 50 (1975) 74–79. [7] T. Kubiak, L-fuzzy normal spaces and Tietze extension theorem, J. Math. Anal. Appl. 125 (1987) 141–153. [8] Y.-M. Liu, M.-K. Luo, Fuzzy Topology, World Scientific, Singapore, 1997. [9] P.-M. Pu, Y.-M. Liu, Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore–Smith convergence, J. Math. Anal. Appl. 76 (1980) 571–599. [10] G.-J. Wang, Theory of L-Fuzzy Topological Spaces, Shaanxi Normal University Press, Xi’an, 1988 (in Chinese). [11] G.-P. Wang, Induced I (L)-fuzzy topological spaces, Fuzzy Sets and Systems 43 (1991) 69–80. [12] G.-P. Wang, I (L)-fuzzy topologies determined by their level topologies, J. Math. (PRC) 17 (1) (1997) 29–32 (in Chinese). [13] C.-X. Wu, J.-X. Fang, (QL)-type fuzzy topological vector spaces, Chinese Ann. Math. 6A (3) (1985) 355–364 (in Chinese) or 6B (3) (1985) 376 (English summary). [14] C.-X. Wu, J.-H. Li, Convexity and fuzzy topological vector spaces, Science Exploration (China) 4 (1) (1984) 1–4. [15] C.-H. Yan, J.-X. Fang, L-fuzzy locally convex topological vector spaces, J. Fuzzy Math. 7 (3) (1999) 765–772. [16] C.-H. Yan, J.-X. Fang, Generalization of Kolmogoroff’s theorem to L-topological vector spaces, Fuzzy Sets and Systems 125 (2002) 177–183. [17] C.-H. Yan, J.-X. Fang, Locally bounded L-topological vector spaces, Inform. Sci. 159 (2004) 273–281. [18] C.-H. Yan, J.-X. Fang, I (L)-topological vector spaces and its level topological spaces, Fuzzy Sets and Systems 149 (2005) 485–492. [19] H. Zhang, J.-X. Fang, On locally convex I-topological vector spaces, Fuzzy Sets and Systems 157 (2006) 1995–2002. [20] H.-P. Zhang, J.-X. Fang, Local convexity and local boundedness of induced I (L)-topological vector spaces, Fuzzy Sets and Systems 158 (2007) 1496–1503. [21] H.-P. Zhang, J.-X. Fang, I (L)-topological groups and its level L-topological groups, Fuzzy Sets and Systems 158 (2007) 1504–1510.
On I (L)-topological vector spaces generated by a co-tower of L-topological vector spaces夡 Hua-Peng Zhang, Jin-Xuan Fang∗ Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210097, China Received 4 August 2007; received in revised form 16 February 2009; accepted 16 February 2009 Available online 28 February 2009
Abstract In this paper, a characterization for an I (L)-topological space to be generated by a given co-tower of L-topological spaces is obtained. Moreover, the relationship between some properties of an I (L)-topological vector space generated by a co-tower of Ltopological vector spaces and the corresponding properties of the given co-tower of L-topological vector spaces is investigated. Our results show that if an I (L)-topological vector space generated by a co-tower of L-topological vector spaces has some properties, such as local convexity and local boundedness, then all L-topological vector spaces in the co-tower also have the same properties. But the converse is incorrect even in the case of I-topological vector space generated by a co-tower of classical topological vector spaces. Finally, we supply a necessary and sufficient condition for an I (L)-topological vector space generated by a co-tower of L-topological vector spaces with some properties, such as local convexity and local boundedness, to have such properties too. © 2009 Elsevier B.V. All rights reserved. Keywords: L-fuzzy unit interval; L-topological vector space; Locally convex L-topological vector space; Locally bounded L-topological vector space
1. Introduction Kubiak [7] introduced the notion of I (L)-valued lower semi-continuous mappings, where I is the real unit interval [0, 1], L denotes a Hutton algebra [5], and I (L) is the L-fuzzy unit interval [6]. Using the notion of I (L)-valued lower semi-continuous mappings, Wang [11] introduced and studied the concept of induced I (L)-topological spaces. Moreover, Wang [12] obtained a necessary and sufficient condition for a chain of L-topological structures on a set X such that there exists an (not necessarily unique in general) I (L)-topological structure of which the level L-topological structures are exactly the given chain of L-topological structures. Inspired by Wang’s work, Fang and Yan [4,18] introduced and studied induced I (L)-topological vector spaces and supplied a new method to generate I (L)-topological vector spaces by a co-tower of L-topological vector spaces. In addition, they proved that an I (L)-topological vector space generated by a co-tower of L-topological vector spaces is Hausdorff [10] if and only if all L-topological vector spaces in the co-tower are Hausdorff. Thus, a natural question arises: whether an I (L)-topological vector space generated by a 夡 This work is supported by the National Natural Science Foundation of China (No. 10671094) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20060319001). ∗ Corresponding author. E-mail addresses: [email protected] (H.-P. Zhang), [email protected] (J.-X. Fang). 0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.02.020
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co-tower of L-topological vector spaces has some property if and only if all L-topological vector spaces in the co-tower have the same property? This paper is a continuation of the work in [18], where we will focus on study of the question raised above. In view of that locally convex L-topological vector spaces [15], locally bounded L-topological vector spaces [17] and L-fuzzy normable L-topological vector spaces [16] are three important subclasses of L-topological vector spaces. Hence, in this paper, we will answer the above question in detail: whether an I (L)-topological vector space generated by a co-tower of L-topological vector spaces is locally convex (locally bounded, I (L)-fuzzy normable) if and only if all L-topological vector spaces in the co-tower are locally convex (locally bounded, L-fuzzy normable)? If not, does there exist any necessary and sufficient condition such that the answer to the above question is positive? This paper is organized as follows. In Section 2, we recall some basic concepts and lemmas needed in the sequel. In Section 3, we give a characterization to an I (L)-topological space generated by a given co-tower of L-topological spaces. In particular, we obtain some corollaries, one of which will be needed in the counterexample given in the next section. In Section 4, we prove that an I (L)-topological vector space generated by a co-tower of L-topological vector spaces has some properties, such as local convexity and local boundedness, then all L-topological vector spaces in the co-tower also have the same properties. At the same time, we give a counterexample to show that in general, the converse is incorrect. Finally, we provide a necessary and sufficient condition for an I (L)-topological vector space generated by a co-tower of L-topological vector spaces with some properties, such as local convexity and local boundedness, to have such properties too. 2. Basic concepts and lemmas Throughout this paper, L denotes a Hutton algebra [5], i.e., L is a complete and completely distributive lattice equipped with an order-reversing involution : L → L. 0 and 1 are its bottom and top elements, respectively. M(L) denotes the set of all non-zero union-irreducible elements in L. The elements of M(L) are also called molecules [8] in L. L X denotes the family of all L-fuzzy sets on X. An L-fuzzy set which takes the constant value ∈ L on X is denoted by . An L-topology on X is called stratified, if it contains all constant L-fuzzy sets on X. We always assume that the L-topologies referred to in the present paper are all stratified and the lattice L is regular (i.e., the intersection of each pair of non-zero elements in L is not zero, or equivalently 1 ∈ M(L)). In this paper, the notations I, I0 and I1 stand for the intervals [0, 1], (0, 1] and [0, 1), respectively. The L-fuzzy unit interval I (L) is the set of all equivalence classes [], where : R → L is a monotone decreasing mapping satisfying (t) = 1 for t < 0 and (t) = 0 for t > 1, and ∈ [] iff (t−) = (t−) (equivalently, (t+) = (t+)) for all t ∈ I . The natural L-topology on I (L) is generated by the subbase {L t , Rt |t ∈ I }, where L t [] = ((t−)) and Rt [] = (t+). The partial order on I (L) is naturally defined by [][] iff (t−) (t−) (equivalently, (t+) (t+)) for all t ∈ I . Let [] ∈ I (L), we define [] = [ ], where ∈ L R is defined by (t) = ((1 − t)) for all t ∈ R. Thus, (I (L), , ) is a Hutton algebra. To simplify notation, we shall identify equivalence classes with their representatives in the sequel. For other symbols which are not mentioned, we refer to [3,8]. Definition 2.1 (Liu and Luo [8], Wang [10]). Let (L X , ) be an L-topological space and x ∈ M(L X ). P ∈ L X is called a closed R-neighborhood of x , if P ∈ and x P. The set of all closed R-neighborhoods of x is denoted by − (x ). A ∈ L X is called an R-neighborhood of x , if there exists P ∈ − (x ) such that A P. The set of all R-neighborhoods of x is denoted by (x ). U ⊆ (x ) is said to be an R-neighborhood base of x if for each P ∈ (x ), there exists Q ∈ U such that P Q. Remark 2.1. In the research of I-topological space, the widely used neighborhood structure of a fuzzy point is Qneighborhood introduced by Pu and Liu [9]. Recall that U ∈ I X is called a Q-neighborhood of x in the I-topological space (I X , ) if there exists G ∈ such that G U and G(x) > 1 − . A family U of Q-neighborhoods of x is called a Q-neighborhood base of x if for each Q-neighborhood V of x , there exists U ∈ U such that U V . It is easy to see that, in the case of L = I , U is a Q-neighborhood of x iff U is an R-neighborhood of x and U is a Q-neighborhood base of x iff U = {U |U ∈ U} is an R-neighborhood base of x .
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Definition 2.2 (Liu and Luo [8]). Let (L X , ) be an L-topological space. A ∈ L X is called a neighborhood of x , if there exists U ∈ such that x U A. Definition 2.3 (Liu and Luo [8], Wang [10]). Let (L X , ) be an L-topological space. (L X , ) is called T2 (or Hausdorff), if for every two molecules x and y with different support points x y, there exist U ∈ (x ) and V ∈ (y ) such that U ∨ V = 1. Definition 2.4 (Fang and Yan [3]). Let X be a vector space over K (K = R or C) and be an L-topology on X. The pair (L X , ) is called an L-topological vector space (briefly, L-tvs) if the following two mappings (the addition and the scalar multiplication on X): (1) f : X × X → X , (x, y)x + y and (2) g : K × X → X , (k, x)kx are both continuous, where X × X and K × X are equipped with the corresponding product L-topologies × and JK × , respectively, and JK denotes the usual topology on K. Definition 2.5 (Fang [2]). Let (L X , ) be an L-tvs and be the zero element in X. An L-fuzzy set A on X is said to be -bounded ( ∈ M(L)), if for each R-neighborhood Q of in (L X , ), there exist t > 0 and ∈ L with such that A ∧ t Q . A is said to be bounded if it is -bounded for each ∈ M(L). Definition 2.6 (Yan and Fang [17]). An L-tvs (L X , ) is said to be locally bounded if it has a bounded neighborhood of (i.e., 1 ). Remark 2.2. From Theorem 2 in [17] and Definition 2.5, it is not difficult to prove that U ∈ L X is a bounded neighborhood of in the L-tvs (L X , ) iff for each ∈ M(L), {tU ∨ |t > 0, } is an R-neighborhood base of . Recall that an I-tvs (I X , ) is said to be of (QL)-type if there exists a family U of fuzzy sets on X such that U = {U ∧ r |U ∈ U, r ∈ (1 − , 1]} is a Q-neighborhood base of for each ∈ I0 [13]. Remark 2.3. In the case of L = I , by Remark 2.2, it is easy to see that a locally bounded I-tvs is necessarily an I-tvs of (QL)-type. Definition 2.7 (Yan and Fang [15]). Let (L X , ) be an L-tvs. A family B of L-fuzzy sets on X is called a prebase of if for each ∈ M(L), B = {B ∨ |B ∈ B, } is an R-neighborhood base of . An L-tvs (L X , ) is said to be locally convex, if there exists a family B of absolutely convex L-fuzzy sets on X such that B is a prebase of . Obviously, in the case of L = I , an I-tvs has a prebase iff it is of (QL)-type; a locally convex I-tvs in the sense of Definition 2.7 is exactly a locally convex I-tvs introduced by Wu and Li [14], which is necessarily an I-tvs of (QL)-type (see [19] for detail). Remark 2.4. (1) By Remark 2.2, a locally bounded L-tvs necessarily has a prebase. (2) If B is a prebase of the L-tvs (L X , ), then each B in B is a neighborhood of and equivalently, for each B ∈ B and ∈ M(L), B is an R-neighborhood of (see Lemma 2 in [16]). Definition 2.8 (Kubiak [7]). For each t ∈ I , define mappings t , t : I (L) X → L X as follows: t () = ← (Rt ), t () = ← (L t ) for all ∈ I (L) X . Obviously t ()(x) = (x)(t+), t ()(x) = (x)(t−).
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Definition 2.9 (Kubiak [7]). Let (L X , ) be an L-topological space. A mapping : X → I (L) is called I (L)-valued lower (resp. upper) semi-continuous if for each t ∈ I , t () ∈ (resp. (t ()) ∈ ). We denote the set of all I (L)-valued lower semi-continuous mappings on X by I (L) (). It is easy to verify that I (L) () is an I (L)-topology on X, called the induced I (L)-topology of , and (I (L) X , I (L) ()) is called an induced I (L)-topological space. Definition 2.10 (Wang [11]). Define the mapping ∗ : L X → I (L) X satisfying ⎧ t < 0, ⎨ 1, P ∗ (x)(t+) = P(x), 0 t < 1, ⎩ 0, t 1 for all P ∈ L X and x ∈ X . Moreover, for each t ∈ I , define a mapping t X∗ : X → I (L) (for short, t ∗ ) by letting 1, s < t, t ∗ (x)(s+) = 0, s t for all x ∈ X . It is easy to see that ⎧ t 0, ⎨ 1, P ∗ (x)(t−) = P(x), 0 < t 1, ⎩ 0, t > 1,
t ∗ (x)(s−) =
1, s t, 0, s > t
for all x ∈ X . Definition 2.11 (Yan and Fang [18]). A co-tower of L-topologies on a set X (indexed by I1 ) is a family of L-topologies = {s | s ∈ I1 } such that s is generated by s
(t ()) = 1−t ( ); (t ()) = 1−t ( ); t (P ∗ ) = P for all t ∈ I1 ; t (P ∗ ) = P for all t ∈ I0 ; (P ∗ ) = (P )∗ ; (t ∗ ) = (1 − t)∗ [4].
Lemma 2.2 (Wang [11]). Let ∈ I (L) X be an I (L)-fuzzy set. Then the following equality holds: (t ∗ ∧ (t ())∗ ), = t∈I
where t ∗ and (t ())∗ are defined as in Definition 2.10. Remark 2.5. By Lemmas 2.1 and 2.2, for each ∈ I (L) X and s ∈ I , we have (s ())∗ ∨ s ∗ . Define ∈ I (L) as follows: 1, s < 0,
(s+) = 0, s 0.
(2.1)
It is straightforward by Definition 2.10 that is the bottom element in I (L) and 0∗X is an I (L)-fuzzy set which takes the constant value on X.
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For each ∈ L and t ∈ I0 , we define ,t ∈ I (L) as follows: ⎧ ⎨ 1, s < 0, ,t (s+) = , 0 s < t, ⎩ 0, s t.
(2.2)
Remark 2.6. It is not difficult to see that 1,1 is the top element in I (L) and 1∗X is an I (L)-fuzzy set which takes the constant value 1,1 on X. In addition, ⎧ ⎨ 1, s < 1 − t, ( ,t ) (s+) = , 1 − t s < 1, (2.3) ⎩ 0, s 1. Lemma 2.3. Let ∈ I (L) and P ∈ I (L) X . Then (1) is a molecule in I (L), i.e., ∈ M(I (L)) iff there exist ∈ M(L) and t ∈ I0 such that = ,t [11]; (2) x ,t P iff x t (P). Proof. (2) It is clear that x ,t P ⇐⇒ ,t P(x) ⇐⇒ ,t (t−)P(x)(t−). Note that ,t (t−) = and P(x)(t−) = t (P)(x). Hence (2) holds. Lemma 2.4 (Yan and Fang [18]). Let = {s }s∈I1 be a co-tower of L-topologies on X, then the operator ◦ : I (L) X → I (L) X defined by (s ∗ ∧ (ints (s ()))∗ ) ◦ = s∈I1
is a fuzzy interior operator on X, where ints denotes the interior operator with respect to s . Thus, it determines an I (L)-topology on X, denoted by ( ), called the I (L)-topology generated by . Moreover, ∈ ( ) iff s () ∈ s for all s ∈ I1 and s ( ( )) = s for each s ∈ I1 , where s ( ( )) = {s (U )|U ∈ ( )}. Particularly, (I (L) X , ( )) is an I (L)-tvs iff (L X , s ) is an L-tvs for each s ∈ I1 . Lemma 2.5 (Yan and Fang [18]). Suppose that (I (L) X , ( )) is an I (L)-topological space generated by a co-tower of L-topologies = {s }s∈I1 on X. If s ∈ I1 and G s ∈ s , then s ∗ ∧ (G s )∗ ∈ ( ). 3. A characterization for an I (L)-topological space to be generated by a given co-tower of L-topological spaces In this section, = {s }s∈I1 denotes a co-tower of L-topologies on X and Us (x ) denotes an R-neighborhood base of x in (L X , s ) for each s ∈ I1 and x ∈ M(L X ). Lemma 3.1. Let (I (L) X , ( )) be an I (L)-topological space generated by and s ∈ (0, 1). If P ∈ s (x ), then P ∗ ∨ (1 − s)∗ ∈ (x ,t ) for each t ∈ (1 − s, 1], where s (x ) and (x ,t ) denote the set of all R-neighborhoods of x in (L X , s ) and that of x ,t in (I (L) X , ( )), respectively. Proof. Since P ∈ s (x ), there exists Q ∈ s such that P Q and x Q. Hence P ∗ ∨ (1 − s)∗ Q ∗ ∨ (1 − s)∗ for each s ∈ (0, 1). It suffices to show that Q ∗ ∨ (1 − s)∗ is a closed R-neighborhood of x ,t for each t ∈ (1 − s, 1]. In fact, since Q ∈ s and (Q ∗ ∨ (1 − s)∗ ) = (Q )∗ ∧ s ∗ , we have Q ∗ ∨ (1 − s)∗ ∈ ( ( )) by Lemma 2.5. Note that Q ∗ (x)((1 − s)+) = Q(x) , (1 − s)∗ (x)((1 − s)+) = 0 and ,t ((1 − s)+) = . Hence we conclude that x ,t Q ∗ ∨ (1 − s)∗ . This completes the proof. Theorem 3.1. Let (I (L) X , ( )) be an I (L)-topological space generated by . Then U(x ,t ) = s∈(1−t,1) U∗s (x ) is an R-neighborhood base of x ,t for each x ,t ∈ M(I (L) X ), where U∗s (x ) = {P ∗ ∨ (1 − s)∗ |P ∈ Us (x )}.
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Proof. By Lemma 3.1, each member of U(x ,t ) is an R-neighborhood of x ,t . It remains to prove that for each ˜ Q ∈ − (x ,t ), there exists Q˜ ∈ U(x ,t ) such that Q Q. − In fact, since Q ∈ (x ,t ), we have ,t Q(x), and so there exists s0 ∈ (0, t) such that = ,t (s0 −)Q(x)(s0 −) = s0 (Q)(x), i.e., x s0 (Q). On the other hand, (s0 (Q)) = 1−s0 (Q ) ∈ 1−s0 by Lemmas 2.1 and 2.4. So s0 (Q) is a closed R-neighborhood of x in (L X , 1−s0 ). Therefore, there exists P0 ∈ U1−s0 (x ) such that s0 (Q) P0 , and so = P ∗ ∨ s ∗ . It remains to show that Q ∈ U(x ). (s0 (Q))∗ ∨ s0∗ P0∗ ∨ s0∗ . Hence Q P0∗ ∨ s0∗ by Remark 2.5. Put Q ,t 0 0 Note that s0 ∈ (0, t), which implies 1 − s0 ∈ (1 − t, 1), and so Q ∈ U(x ,t ) by the construction of U(x ,t ). This completes the proof. Theorem 3.2. Let (I (L) X , ) be an I (L)-topological space and = {s }s∈I1 be a co-tower of L-topologies on X. If U(x ,t ) = s∈(1−t,1) U∗s (x ) is an R-neighborhood base of x ,t for each x ,t ∈ M(I (L) X ), where U∗s (x ) = {P ∗ ∨ (1 − s)∗ |P ∈ Us (x )}, then is generated by , i.e., = ( ). Proof. By Lemma 2.4, it suffices to show that Q ∈ iff t (Q) ∈ t for each t ∈ I1 . Let Q ∈ and t ∈ I1 . To prove t (Q) ∈ t , it suffices to show that for each x ∈ M(L X ) with x (t (Q)) , (t (Q)) ∈ t (x ), where t (x ) is the set of all R-neighborhoods of x in (L X , t ). Note that (t (Q)) = 1−t (Q ). By Lemma 2.3, x (t (Q))
⇐⇒
x 1−t (Q )
⇐⇒
x ,1−t Q ,
and so Q is a closed R-neighborhood of x ,1−t in (I (L) X , ). Since U(x ,1−t ) = s∈(t,1) U∗s (x ) is an R-neighborhood base of x ,1−t , there exists s ∈ (t, 1) and P ∈ Us (x ) such that Q P ∗ ∨ (1 − s)∗ , and so 1−t (Q ) 1−t (P ∗ ∨ (1 − s)∗ ) = 1−t (P ∗ ) ∨ 1−t ((1 − s)∗ ) = P ∨ 0 = P. Hence 1−t (Q ) ∈ s (x ). On the other hand, since {s }s∈I1 is a co-tower, s > t ⇒ s ⊆ t , and so s (x ) ⊆ t (x ), which implies that 1−t (Q ) ∈ t (x ). Conversely, suppose that Q ∈ I (L) X and t (Q) ∈ t for all t ∈ I1 . To prove Q ∈ , it suffices to show that for each x ,r ∈ M(I (L) X ), x ,r Q implies that Q ∈ (x ,r ). In fact, x ,r Q implies that there exists s ∈ (0, r ) such that x s (Q ) = (1−s (Q)) ∈ 1−s , and so s (Q ) ∈ 1−s (x ). Hence there exists P ∈ U1−s (x ) such that s (Q ) P, and so Q (s (Q ))∗ ∨ s ∗ P ∗ ∨ s ∗ . On the other hand, we have P ∗ ∨ s ∗ ∈ U(x ,r ) since 1 − s ∈ (1 − r, 1). Therefore, Q ∈ (x ,r ). This completes the proof. By Theorems 3.1 and 3.2, we can obtain the following conclusion, which is a characterization for an I (L)-topological space to be generated by a given co-tower of L-topological spaces. Theorem 3.3. Let (I (L) X , ) be an I (L)-topological space. Then is generated by the co-tower of L-topologies = {s }s∈I1 on X iff U(x ,t ) = s∈(1−t,1) U∗s (x ) is an R-neighborhood base of x ,t for each x ,t ∈ M(I (L) X ), where U∗s (x ) = {P ∗ ∨ (1 − s)∗ |P ∈ Us (x )}.
(3.1)
In the case of L = {0, 1} (hence I (L) = I ), by Theorem 3.3, we have the following corollary. Corollary 3.1. Let { s }s∈I1 be a co-tower of crisp topologies on X and (I X , ) an I-topological space. Then is generated by { s }s∈I1 iff B(x ) = s∈(1−,1) B∗s (x) is a Q-neighborhood base of x for each x ∈ X and ∈ I0 , where B∗s (x) = {U ∧ s|U ∈ Bs (x)} and Bs (x) denotes a neighborhood base of x in (X, s ) for each s ∈ I1 . If for each s ∈ I1 , (L X , s ) is an L-tvs, then, by Theorem 3.3, we have the following conclusion. X Theorem 3.4. Let (I (L) , ) be∗ an I (L)-tvs. Then is generated by the co-tower of vector L-topologies = {s }s∈I1 on X iff U( ,t ) = s∈(1−t,1) Us ( ) is an R-neighborhood base of ,t for each ,t ∈ M(I (L)), where
U∗s ( ) = {P ∗ ∨ (1 − s)∗ |P ∈ Us ( )}.
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Proof. Necessity: It follows directly from Theorem 3.3. Sufficiency: Since (L X , s ) is an L-tvs and Us ( ) is an R-neighborhood base of , by Corollary 2.3 in [3], Us (x ) = X {x s ( )} is an R-neighborhood base of x for each x ∈ X . Since (I (L) , ) is an I (L)-tvs and U( ,t ) = + P|P ∈ U ∗ s∈(1−t,1) Us ( ) is an R-neighborhood base of ,t , we know that U(x ,t ) = {x + Q|Q ∈ U( ,t )} =
(x + U∗s ( ))
s∈(1−t,1)
is an R-neighborhood base of x ,t . By (3.1), it is easy to see that x + U∗s ( ) = {x + (P ∗ ∨ (1 − s)∗ )|P ∈ Us ( )} = {(x + P)∗ ∨ (1 − s)∗ |x + P ∈ Us (x )} = U∗s (x ). Therefore, is generated by by Theorem 3.3. In the case of L = {0, 1}, by Theorem 3.4, we have the following corollary. X Corollary 3.2. Let { s }s∈I1 be a co-tower of crisp vector topologies on X and (I , ) an I-tvs. Then is generated by ∗ { s }s∈I1 iff B( ) = s∈(1−,1) Bs () is a Q-neighborhood base of for each ∈ I0 , where B∗s () = {U ∧ s|U ∈ Bs ()} and Bs () denotes a neighborhood base of in (X, s ) for each s ∈ I1 .
4. Local convexity and local boundedness of I (L)-tvs generated by a co-tower of L-topological vector spaces In this section, we assume that (I (L) X , ) denotes an I (L)-tvs without special instruction and s ( ) = {s (U )|U ∈ } for each s ∈ I1 . By Theorem 2.1 in [18], we know that (L X , s ( )) is an L-tvs for each s ∈ I1 . Lemma 4.1. Let (I (L) X , ) be an I (L)-topological space, s ∈ I1 and U(x ,1−s ) be an R-neighborhood base of x ,1−s . Then B(x ) = {1−s (P)|P ∈ U(x ,1−s )} is an R-neighborhood base of x in (L X , s ), where s = s ( ). Proof. We first verify that each member of B(x ) is an R-neighborhood of x in (L X , s ). In fact, for each P ∈ U(x ,1−s ), there exists Q ∈ such that x ,1−s Q and P Q , and so x 1−s (Q ) and 1−s (P) 1−s (Q ) = (s (Q)) ∈ s . Hence 1−s (P) is an R-neighborhood of x in (L X , s ). It remains to show that for each R-neighborhood W of x in (L X , s ), there exists P ∈ U(x ,1−s ) such that W 1−s (P). Without loss of generality, we may assume that W ∈ s , and so there exists U ∈ such that W = (s (U )) = 1−s (U ). Note that x W , i.e., x 1−s (U ), which implies that x ,1−s U , and so U is a closed R-neighborhood of x ,1−s in (I (L) X , ). Hence there exists P ∈ U(x ,1−s ) such that U P, which implies W = 1−s (U ) 1−s (P). This completes the proof. In particular, in the case of L = {0, 1}, we obtain the following corollary. Corollary 4.1. Let (I X , ) be an I-topological space, s ∈ I1 and U(x1−s ) is a Q-neighborhood base of x1−s . Then B(x) = {s (U )|U ∈ U(x1−s )} is a neighborhood base of x in (X, s ()), where s () = {s (G)|G ∈ }. Lemma 4.2. If (I (L) X , ) has a prebase, then (L X , s ( )) also has a prebase for each s ∈ I1 . Moreover, for each s, t ∈ I1 with s < t, we have s ( ) ⊆ t ( ). Proof. Let U be a prebase in (I (L) X , ). We shall show that {s (U )|U ∈ U} is a prebase of (L X , s ( )) for each s ∈ I1 . It suffices to show that for each ∈ M(L), the family B = {(s (U )) ∨ |U ∈ U, } is an R-neighborhood base of in (L X , s ( )). Since U is a prebase in (I (L) X , ), by Remark 2.4, we have U () = for each U ∈ U. So (s (U )) () = 1−s (U )() = U ()((1 − s)−) = 0. Thus, it is not difficult to see that for each U ∈ U and ∈ L with , (s (U )) ∨ is an R-neighborhood of in (L X , s ( )).
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Let Q ∈ − ( ) in (L X , s ( )), then there exists G ∈ such that Q = (s (G)) = 1−s (G ). From Q we have ,1−s G , which implies that G ∈ − ( ,1−s ) in (I (L) X , ). Note that {U ∨ |U ∈ U, ,1−s } is an R-neighborhood base of ,1−s . So there exist U ∈ U and ∈ I (L) with ,1−s such that G U ∨ , and so Q = 1−s (G ) 1−s (U ) ∨ 1−s () = (s (U )) ∨ ((1 − s)−). On the other hand, from ,1−s we know that ((1 − s)−), which implies that (s (U )) ∨ ((1 − s)−) ∈ B . Hence B = {(s (U )) ∨ |U ∈ U, } is an R-neighborhood base of in (L X , s ( )). Finally, we prove that for each s, t ∈ I1 with s < t, we have s ( ) ⊆ t ( ). In fact, since B = {(s (U )) ∨ |U ∈ U, } is an R-neighborhood base of in (L X , s ( )) for each s ∈ I1 and note that s < t implies (s (U )) (t (U )) , we have s ( ) ⊆ t ( ), where s ( ) and t ( ) denote the sets of all R-neighborhoods of in (L X , s ( )) and (L X , t ( )), respectively. Since (L X , s ( )) and (L X , t ( )) are both L-topological vector spaces, s < t implies that s (x ) ⊆ t (x ) for each x ∈ M(L X ). Therefore, s ( ) ⊆ t ( ). This completes the proof. In particular, in the case of L = {0, 1}, we obtain the following corollary. Corollary 4.2. If (I X , ) is an I-tvs of (QL)-type, then for each s, t ∈ I1 with s < t, we have s () ⊆ t (). Theorem 4.1. If (I (L) X , ) is locally convex, then (L X , s ( )) is also locally convex for each s ∈ I1 . Proof. Let U be an absolutely convex prebase in (I (L) X , ). In the following, we shall show that {s (U )|U ∈ U} is an absolutely convex prebase of (L X , s ( )) for each s ∈ I1 . From Theorem 3.1 in [20], we know that s (U ) is an absolutely convex L-fuzzy set for each U ∈ U. On the other hand, by the proof of Lemma 4.2, {s (U )|U ∈ U} is a prebase of (L X , s ( )). Hence (L X , s ( )) is locally convex. Lemma 4.3. If B ∈ I (L) X is bounded in (I (L) X , ), then s (B) is bounded in (L X , s ( )) for each s ∈ I1 . Proof. Assume that B is bounded in (I (L) X , ), and let ∈ M(L) and Q ∈ − ( ) in (L X , s ( )), then there exists U ∈ such that Q = (s (U )) = 1−s (U ). From Q, we know that ,1−s U , which implies that U ∈ − ( ,1−s ). Since B is bounded in (I (L) X , ), there exist t > 0 and ∈ I (L) with ( ,1−s ) such that B ∧ t(U ) = tU.
(4.1)
It follows from (4.1) that s (B) ∧ (s+) = s (B) ∧ s () s (tU ) = ts (U ) = t Q .
(4.2)
Moreover, from (2.3) and ( ,1−s ) , it follows that there exists r ∈ [s, 1) such that (r +) . On the other hand, it follows from (4.2) and r s that s (B)∧(r +)t Q . Therefore, s (B) is -bounded. By the arbitrariness of ∈ M(L), we know that s (B) is bounded in (L X , s ( )). Lemma 4.4. If Q is a neighborhood of in (I (L) X , ), then s (Q) is a neighborhood of in (L X , s ( )) for each s ∈ I1 . Proof. Since Q is a neighborhood of in (I (L) X , ), there exists U ∈ such that 1,1 U Q. Hence 1 = s (1,1 ) s (U ) s (Q) for each s ∈ I1 . On the other hand, we have s (U ) ∈ s ( ), and so s (Q) is a neighborhood of in (L X , s ( )). Theorem 4.2. If (I (L) X , ) is locally bounded, then (L X , s ( )) is also locally bounded for each s ∈ I1 . Proof. Since (I (L) X , ) is locally bounded, there exists a bounded neighborhood Q of in (I (L) X , ). By Lemma 4.3, we know that s (Q) is bounded in (L X , s ( )). In addition, s (Q) is a neighborhood of in (L X , s ( )) by Lemma 4.4. Hence (L X , s ( )) is locally bounded.
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Theorem 4.3. If (I (L) X , ) is I (L)-fuzzy normable, then (L X , s ( )) is also L-fuzzy normable for each s ∈ I1 . Proof. An L-tvs (L X , ) is L-fuzzy normable iff it is Hausdorff and has a convex and bounded neighborhood of (see Theorems 2 and 3 in [16]). So (I (L) X , ) is Hausdorff and has a convex and bounded neighborhood Q of . By Theorem 2.1 in [18], (L X , s ( )) is Hausdorff, and s (Q) is convex by Theorem 3.1 in [20]. Moreover, s (Q) is bounded and a neighborhood of in (L X , s ( )) by Lemmas 4.3 and 4.4, respectively. Hence (L X , s ( )) is L-fuzzy normable. By Theorems 4.1–4.3 and Lemma 2.4, we get the following three corresponding corollaries. Corollary 4.3. Let (I (L) X , ( )) be an I (L)-tvs generated by the co-tower of vector L-topologies {s }s∈I1 on X. If (I (L) X , ( )) is locally convex, then (L X , s ) is also locally convex for each s ∈ I1 . Corollary 4.4. Let (I (L) X , ( )) be an I (L)-tvs generated by the co-tower of vector L-topologies {s }s∈I1 on X. If (I (L) X , ( )) is locally bounded, then (L X , s ) is also locally bounded for each s ∈ I1 . Corollary 4.5. Let (I (L) X , ( )) be an I (L)-tvs generated by the co-tower of vector L-topologies {s }s∈I1 on X. If (I (L) X , ( )) is I (L)-fuzzy normable, then (L X , s ) is L-fuzzy normable for each s ∈ I1 . By Lemma 3.5 in [21] and Theorems 4.1–4.3, we obtain the following three corresponding corollaries, which are main results in [20]. Corollary 4.6. Let (L X , ) be an L-tvs and (I (L) X , I (L) ()) be its induced I (L)-tvs. If (I (L) X , I (L) ()) is locally convex, then (L X , ) is also locally convex. Corollary 4.7. Let (L X , ) be an L-tvs and (I (L) X , I (L) ()) be its induced I (L)-tvs. If (I (L) X , I (L) ()) is locally bounded, then (L X , ) is also locally bounded. Corollary 4.8. Let (L X , ) be an L-tvs and (I (L) X , I (L) ()) be its induced I (L)-tvs. If (I (L) X , I (L) ()) is I (L)-fuzzy normable, then (L X , ) is L-fuzzy normable. Remark 4.1. Generally, the converse of Theorems 4.1, 4.2, and 4.3 are all incorrect even in the case that L = {0, 1} and is generated by a co-tower of crisp vector topologies. Please see the following example.
∞ ∞ ⊆ K satisfying Example 4.1. Let X = 1 , i.e., X is the set of all sequences {i }i=1 i=1 |i | < +∞. If we define addition and scalar multiplication on X pointwisely, then it is clear that X is a vector space
∞ over K. |i | and x2 = supi |i | Define · 1 : X → [0, +∞) and · 2 : X → [0, +∞) respectively by x1 = i=1 ∞ ∈ X . Then from the theory of classical functional analysis about p spaces, we know that · for each x = {i }i=1 1 and · 2 are both norms on X, T·2 T·1 , and it is obvious that (X, T·1 ) and (X, T·2 ) are both normable topological vector spaces, where T·1 and T·2 denote the topologies induced by · 1 and · 2 , respectively. In addition, {U·1 (t)|t > 0} and {U·2 (t)|t > 0} are neighborhood bases of in (X, T·1 ) and (X, T·2 ), respectively, where U·i (t) = {x ∈ X |xi < t}, i = 1, 2. For each ∈ I0 , we define a family U of fuzzy sets on X as follows: {U·2 (t) ∧ r |t > 0, r ∈ (1 − , 1)}, ∈ (0, 1/2], U = {U·1 (t) ∧ r |t > 0, r ∈ (1 − , 1)}, ∈ (1/2, 1]. It is not difficult to check that {U }∈I0 satisfies conditions (1)–(5) of Theorem 4.1 in [1]. So there exists a unique I-topology on X, denoted by , such that (I X , ) is an I-tvs and U is a Q-neighborhood base of for each ∈ I0 . Hence (X, s ()) is a topological vector space for each s ∈ I1 , where s () = {s (G)|G ∈ }. By Corollary 4.1, s (U1−s ) = {s (U )|U ∈ U1−s } is a neighborhood base of in (X, s ()). Note that {U·1 (t)|t > 0}, s ∈ [0, 1/2), s (U1−s ) = {U·2 (t)|t > 0}, s ∈ [1/2, 1),
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which implies T·1 , s ∈ [0, 1/2), s () = T·2 , s ∈ [1/2, 1). It is not difficult to see that {s ()}s∈I1 is a co-tower of vector topologies on X and for each s ∈ I1 (X, s ()) is normable, hence it is locally convex and locally bounded. Moreover, by Corollary 3.2, we can prove that is generated by {s ()}s∈I1 . In fact, for each s ∈ I1 , we put Bs () = s (U1−s ), B∗s () = {U ∧ s|U ∈ s (U1−s )} and B( ) = s∈(1−,1) B∗s (). Then when ∈ (0, 1/2] we have B( ) = {U·2 (t) ∧ s|t > 0, s ∈ (1 − , 1)} = U , which is a Q-neighborhood base of ; When ∈ (1/2, 1], we have B( ) = {U·1 (t) ∧ s|t > 0, s ∈ (1 − , 1/2)} ∪ {U·2 (t) ∧ s|t > 0, s ∈ [1/2, 1)}. Note that U = {U·1 (t) ∧r|t > 0, r ∈ (1 − , 1]} ( ∈ (1/2, 1]) is a Q-neighborhood base of in (I X , ). It is easy to see that {U·1 (t) ∧ s|t > 0, s ∈ (1 − , 1/2)} is also a Q-neighborhood base of . In addition, since U·1 (t) ⊆ U·2 (t) when ∈ (1/2, 1], for each t > 0, U·2 (t) ∧ s is a Q-neighborhood of for each t > 0 and s ∈ [1/2, 1). Therefore, B( ) is also a Q-neighborhood base of . This shows that for each ∈ I0 , B( ) = s∈(1−,1) B∗s () is a Qneighborhood base of . So, by Corollary 3.2, the I-topology is generated by {s ()}s∈I1 . Now, we verify that (I X , ) is not of (QL)-type, hence not locally convex (locally bounded, fuzzy normable). In fact, if (I X , ) is of (QL)-type, then, by Corollary 4.2, we have 1/4 () ⊆ 3/4 (), and so T·1 ⊆ T·2 , which contradicts with T·1 T·2 . Finally, we supply a necessary and sufficient condition for an I (L)-topological vector space generated by a co-tower of L-topological vector spaces with some properties, such as local convexity and local boundedness, to have such properties too. Definition 4.1. An L-tvs is said to have Property (A) if it is locally convex or locally bounded or L-fuzzy normable. Remark 4.2. From the discussion above, we know that if an L-tvs has Property (A), then it must have a prebase. Theorem 4.4. Let (I (L) X , ( )) be an I (L)-tvs generated by the co-tower of vector L-topologies {s }s∈I1 on X and all the L-topological vector spaces {(L X , s )}s∈I1 have Property (A). Then (I (L) X , ( )) also has Property (A) iff s = 0 for all s ∈ I1 . Proof. Necessity: Let (I (L) X , ( )) have Property (A). Then, by Remark 4.2 and Lemmas 2.4 and 4.2, s ⊆ t for each s, t ∈ I1 with s < t. On the other hand, we have t ⊆ s for each s, t ∈ I1 with s < t since {s }s∈I1 is a co-tower. Hence s = 0 for all s ∈ I1 . Sufficiency: Let s = 0 for all s ∈ I1 . Then, by Corollary 2.7 in [18], (I (L) X , ( )) is the induced I (L)-tvs of (L X , 0 ). Hence (I (L) X , ( )) has Property (A) by the main results in [20]. Acknowledgments The authors would like to thank the Editors-in-Chief and the anonymous referees for their valuable suggestions in improvement of the original manuscript. References [1] [2] [3] [4]
J.-X. Fang, On local bases of fuzzy topological vector spaces, Fuzzy Sets and Systems 87 (1997) 341–347. J.-X. Fang, The continuity of fuzzy linear order-homomorphism, J. Fuzzy Math. 5 (4) (1997) 829–838. J.-X. Fang, C.-H. Yan, L-fuzzy topological vector spaces, J. Fuzzy Math. 5 (1) (1997) 133–144. J.-X. Fang, C.-H. Yan, Induced I (L)-fuzzy topological vector spaces, Fuzzy Sets and Systems 121 (2001) 293–299.
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[5] U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Dordrecht, 1999. [6] B. Hutton, Normality in fuzzy topological spaces, J. Math. Anal. Appl. 50 (1975) 74–79. [7] T. Kubiak, L-fuzzy normal spaces and Tietze extension theorem, J. Math. Anal. Appl. 125 (1987) 141–153. [8] Y.-M. Liu, M.-K. Luo, Fuzzy Topology, World Scientific, Singapore, 1997. [9] P.-M. Pu, Y.-M. Liu, Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore–Smith convergence, J. Math. Anal. Appl. 76 (1980) 571–599. [10] G.-J. Wang, Theory of L-Fuzzy Topological Spaces, Shaanxi Normal University Press, Xi’an, 1988 (in Chinese). [11] G.-P. Wang, Induced I (L)-fuzzy topological spaces, Fuzzy Sets and Systems 43 (1991) 69–80. [12] G.-P. Wang, I (L)-fuzzy topologies determined by their level topologies, J. Math. (PRC) 17 (1) (1997) 29–32 (in Chinese). [13] C.-X. Wu, J.-X. Fang, (QL)-type fuzzy topological vector spaces, Chinese Ann. Math. 6A (3) (1985) 355–364 (in Chinese) or 6B (3) (1985) 376 (English summary). [14] C.-X. Wu, J.-H. Li, Convexity and fuzzy topological vector spaces, Science Exploration (China) 4 (1) (1984) 1–4. [15] C.-H. Yan, J.-X. Fang, L-fuzzy locally convex topological vector spaces, J. Fuzzy Math. 7 (3) (1999) 765–772. [16] C.-H. Yan, J.-X. Fang, Generalization of Kolmogoroff’s theorem to L-topological vector spaces, Fuzzy Sets and Systems 125 (2002) 177–183. [17] C.-H. Yan, J.-X. Fang, Locally bounded L-topological vector spaces, Inform. Sci. 159 (2004) 273–281. [18] C.-H. Yan, J.-X. Fang, I (L)-topological vector spaces and its level topological spaces, Fuzzy Sets and Systems 149 (2005) 485–492. [19] H. Zhang, J.-X. Fang, On locally convex I-topological vector spaces, Fuzzy Sets and Systems 157 (2006) 1995–2002. [20] H.-P. Zhang, J.-X. Fang, Local convexity and local boundedness of induced I (L)-topological vector spaces, Fuzzy Sets and Systems 158 (2007) 1496–1503. [21] H.-P. Zhang, J.-X. Fang, I (L)-topological groups and its level L-topological groups, Fuzzy Sets and Systems 158 (2007) 1504–1510.