Continuity of Fuzziﬁed Functions Using the Generalized Extension Principle

: To fuzzify the crisp functions, the extension principle has been widely used for performing this fuzziﬁcation. The purpose of this paper is to investigate the continuity of fuzziﬁed function using the more generalized extension principle. The Hausdorff metric will be invoked to study the continuity of fuzziﬁed function. We also apply the principle of continuity of fuzziﬁed function to the fuzzy topological vector space


Introduction
Let U be a universal set.A fuzzy subset Ã of U is defined as a set of ordered pairs: Ã = {(x, ξ Ã(x)) : x ∈ U} , where ξ Ã : U → [0, 1] is called the membership function of Ã.The set of all fuzzy subsets of U is denoted by F (U).
We consider an onto function f : U → V, where V is another universal set.This function is also called as a crisp function.The purpose is to fuzzify the crisp function f as a fuzzy function f : F (U) → F (V); that is, for any Ã ∈ F (U), it means f ( Ã) ∈ F (V).We remark that this fuzzy function is completely different from the concept of fuzzy function studied in Hajek [1], Demirci [2] and Höhle et al. [3] in which the fuzzy function is treated as a fuzzy relation.The principle for fuzzifying the crisp functions is called the extension principle, which was proposed by Zadeh [4][5][6].In particular, if V = R, then f : U → R is called a real-valued function, and f : F (U) → F (R) induced from f is called a fuzzy-valued function.
For any Ã ∈ F (U), the extension principle says that the membership function of B ≡ f ( Ã) is defined by the following supremum: ( Suppose that U and V are taken as the topological spaces such that the original crisp function f : (U, τ U ) → (V, τ V ) is continuous.In this paper, we shall investigate the continuity of fuzzified function f : F (U) → F (V). Román-Flores et al. [7] has studied the continuity of this fuzzified function when U and V are taken as the Euclidean space R n .In this paper, we are going to extend these results to the case of normed spaces and normable topological vector spaces using the generalized extension principle discussed in Wu [8].
Let U 1 and U 2 be two universal sets.We consider the onto crisp function f : U 1 × U 2 → V.The purpose is to fuzzify the crisp function f as a fuzzy function f : F (U 1 ) × F (U 2 ) → F (V).
If the universal set U is endowed with a topology, then the 0-level set of Ã is defined by: which is the closure of the support {x ∈ U : ξ Ã(x) > 0} of Ã.However, if U is not assumed to be a topological space, then the 0-level set is usually taken to be the whole set U.
Fullér and Keresztfalvi [10] generalized Theorem 1 by considering the t-norm.In this case, the extension principle presented in Equation ( 2) can be generalized in the following form: since min{x, y} is a t-norm.Therefore, Theorem 1 can be generalized as follows.
Fullér and Keresztfalvi [10] also obtained the following interesting result.
Theorem 3. (Fullér and Keresztfalvi [10]) Let U 1 , U 2 , V be locally compact topological spaces.Let Ã(i) ∈ F (U i ) for i = 1, 2 such that the membership functions ξ Ã(i) are upper semicontinuous and the 0-level sets Ã(i) 0 are compact subsets of U i for i = 1, 2. Let f : U 1 × U 2 → V be a continuous and onto crisp function and let f : F (U 1 ) × F (U 2 ) → F (V) be a fuzzy function induced from f via the extension principle defined in Equation (5).If the t-norm t is upper semicontinuous, then the following equality: α 2 (8) holds true for each α ∈ (0, 1].
Based on the Hausdorff space, Wu [8] generalizes Theorems 2 and 3 to the case of generalized t-norm T n : [0, 1] n → [0, 1] that is recursively defined by: Let U 1 , • • • , U n , V be universal sets and let f : for each y ∈ V.This definition extends the definition given in Equation (7).In the sequel, we are going to consider the extension principle using an operator called W n that is more general than T n .Let W n : [0, 1] n → [0, 1] be a function defined on [0, 1] n , which does not assume any extra conditions.Let U 1 , • • • , U n , V be universal sets and let f : for each y ∈ V. Of course, this definition extends the definition given in Equation (9).In this paper, we are going to investigate the continuity of this kind of fuzzy function.We also remark that the operator W n is a kind of aggregation operator studied in Calvo et al. [11] and Grabisch et al. [12].

Generalized Extension Principle on Normed Spaces
Let (X, • ) be a normed space.Then, we see that the norm • can induce a topology τ such that (X, τ) becomes a Hausdorff topological vector space in which τ is also called a norm topology.
the product vector space.Since each normed space (U i , • U i ) can induce a Hausdorff topological vector space (U i , τU i ), we can form a product topological vector space (U, is the product topology.On the other hand, we can define a product norm on the product vector space U using the norms which is shown in Kreyszig ([13], p. 71) or the p-norm: which is shown in Conway ([14], p. 72), where 1 ≤ p < ∞ and • U i are taken to be the same norm We also see that the product norms • (1) can induce the norm topologies τ1 and τ2 , respectively.We can show that τ However, in general, the norm topology generated by the product norm does not necessarily equal to the product topology τ U 1 ×•••×U n .Therefore, it is needed to investigate the generalized extension principle in the case of normed space separately.We can simply regard U as a vector space over R. Therefore, we can define a norm to make it as a normed space (U, • U ).In this case, we can induce a norm topology τU .Alternatively, we can consider the product norm for the product vector space U over R. Let h : R n → R be a real-valued function defined on R n .In general, the product norm on U can be defined as: Of course, the product norm is also a norm for U. Based on this product norm, we can also induce a product norm topology τU and, by referring to Equation (11), the open ball in the product normed space (U, We have the following interesting result. be the product vector space which is endowed with a norm • U .Given any > 0 and u = (u 1 , • • • , u n ) ∈ U, if there exist ˆ 1 , ˆ 2 > 0 such that the following inclusions hold true: and: where τU is the norm topology induced by the norm • U and τ U 1 ×•••×U n is the product topology.If the product vector space U is endowed with the product norm • U 1 ×•••×U n such that the inclusions Equations ( 12) and ( 13) are satisfied, then we also have By the assumption in Equation ( 12), it follows that By the assumption in Equation ( 13), we see that U n be the product vector space.Then, the following statements hold true.
(i) We consider the normed space (U, where the product norm is defined by Equation (11).Given any Proof.It suffices to prove the case of (ii).By definition, we have: The results follow immediately from Proposition 1 by taking ˆ 1 = ˆ 2 = .
Remark 1.Note that if the product norm is taken as the maximum norm • (1) defined above, then the assumption in part (ii) of Proposition 2 is satisfied automatically.

Let (X,
• X ) and (Y, • Y ) be two normed spaces.Recall that the function f : (X, • X ) → (Y, • Y ) is continuous at x 0 if and only if, given any > 0, there exists δ > 0 such that x − x 0 X < δ implies f (x) − f (x 0 ) Y < .The function f is continuous on X if and only if f is continuous at each point x 0 ∈ X.Then, we have the following easy observation.Remark 2. Let (X, • X ) and (Y, • Y ) be two normed spaces such that τX and τY are two norm topologies induced by the norms • X and • Y , respectively.It is well-known that the function f Then three kinds of continuity for the function f :

•
Suppose that the product vector space ) is continuous at x 0 if and only if, given any > 0, there exists Then the continuity of the function f : (U, similarly realized.We also see that f : (U, is continuous in the topological sense.Propositions 1 and 2 say that this kind of continuity will be equivalent to one of the above two continuities under some suitable conditions.
We say that W n is nondecreasing if and only if In the subsequent discussion, the function W n will satisfy some of the following conditions: Next, we are going to present the generalized extension principle on normed spaces.
be a fuzzy function induced from f via the extension principle defined in Equation (10).Assume that (U i , • U i ) and (V, • V ) are taken to be the normed spaces for i = 1, • • • , n, and that the product vector space U = U 1 × • • • × U n is endowed with a norm • U such that the inclusions Equations ( 12) and ( 13) are satisfied.We also assume that the following supremum: is attained for each y ∈ V.Then, the following equality: holds true for each α ∈ (0, 1].The results for the 0-level sets are given below.

•
We have: If we further assume that condition (a) for W n is satisfied, then: • If we further assume that the function f for W n is satisfied, then: If the product vector space U is endowed with the product norm • U 1 ×•••×U n such that the inclusions Equations ( 12) and ( 13) are satisfied, then we also have the same results.The assumptions satisfying the inclusions Equations ( 12) and ( 13) are not needed when we say that the function f : • U and the product norm Proof.Since a normed space can induce a Hausdorff topological space, the results follow immediately from Remarks 2 and 3, Proposition 1 and Wu ([8], Theorem 5.1).
By referring to Remark 1, from part (ii) of Proposition 2, we see that if the product norm is taken as the maximum norm • (1) be a fuzzy function induced from f via the extension principle defined in Equation (10).Suppose that the following supremum: is attained for each y ∈ V, and that that condition (b) for W n is satisfied.Then, for each α ∈ (0, 1], we have the following equalities. Let (U i , • U i ) and (V, • V ) be now taken to be the normed spaces for i = 1, • • • , n, and let the product vector space U = U 1 × • • • × U n be endowed with a norm • U such that the inclusions Equations ( 12) and ( 13) are satisfied.If we further assume that the function f : (U, • U ) → (V, • V ) is continuous, then we also have the following equality: If the product vector space U is endowed with the product norm • U 1 ×•••×U n such that the inclusions Equations ( 12) and ( 13) are satisfied, then we also have the same results.The assumptions satisfying the inclusions Equations ( 12) and ( 13) are not needed when we say that the function f : • U and the product norm Proof.Since a normed space can induce a Hausdorff topological space, the results follow immediately from Remarks 2 and 3, Proposition 1 and Wu ([8], Theorem 5.2).
endowed with a norm • U such that the inclusions Equations ( 12) and ( 13)

continuous and onto crisp function, and let
) be a fuzzy function induced from f via the extension principle defined in Equation (10).Suppose that conditions (c) and (d) for W n are satisfied.Then, the following statements hold true.
we have the following equality: For the 0-level sets, if we further assume that conditions (a) for W n is satisfied, then: Under this further assumption, the α-level sets ) are closed and compact subsets of V for all α ∈ [0, 1].(iii) If we further assume that conditions (b) for W n is satisfied, then: If the product vector space U is endowed with the product norm • U 1 ×•••×U n such that the inclusions Equations ( 12) and ( 13) are satisfied, then we also have the same results.The assumptions satisfying the inclusions Equations ( 12) and ( 13) are not needed when we say that the function f : • U and the product norm Proof.Since a normed space can induce a Hausdorff topological space, the results follow immediately from Remarks 2 and 3, Proposition 1 and Wu ([8], Theorem 5.3 and Corollary 5.1).
Since we are going to consider the concept of convexity, we need to impose the vector addition and scalar multiplication upon the universal set U. Therefore, the universal set U is taken as a vector space.For any fuzzy subset Ã of U, we say that Ã is convex if and only if each α-level set Ãα = {x : that is, the membership function ξ Ã is quasi-concave.Let U be a vector space over R which is endowed with a topology τ and let Ã be a fuzzy subset of U. Since Ã0 = cl( α>0 Ãα ), we see that if Ã is convex, then its 0-level set Ã0 is also a convex subset of U. Definition 1.Let U be a vector space over R which is endowed with a topology τ.We denote by F cc (U) the set of all fuzzy subsets of U such that each ã ∈ F cc (U) satisfies the following conditions: ) is called a fuzzy vector.Moreover, for any fuzzy number ã ∈ F cc (R), we see that each of its α-level sets ãα is a closed, bounded and convex subset of R, i.e., a closed interval in R.
and let the product vector space U = U 1 × • • • × U n be endowed with a norm • U such that the inclusions Equations ( 12) and ( 13) are satisfied.Let f : (U, • U ) → (V, • V ) be a continuous and onto crisp function.We also assume that f is linear in the sense of: ) be a fuzzy function induced from f via the extension principle defined in Equation (10).Suppose that conditions (a), (c), (d), (e) and (f) for W n are satisfied.For )) α are compact, convex and closed subsets of V for all α ∈ [0, 1].(ii) For each α ∈ (0, 1], we have the following equality: For the 0-level sets, we also have: (iii) If we further assume that conditions (b) for W n is satisfied, then: If the product vector space U is endowed with the product norm • U 1 ×•••×U n such that the inclusions Equations ( 12) and ( 13) are satisfied, then we also have the same results.The assumptions satisfying the inclusions Equations ( 12) and ( 13) are not needed when we say that the function f : • U and the product norm Proof.Since a normed space can induce a Hausdorff topological space, the results follow immediately from Remarks 2 and 3, Proposition 1 and Wu ([8], Theorem 6.1 and Corollary 6.1).
The linearity in Theorem 7 can be replaced by assuming the convexity.
and let the product vector space U = U 1 × • • • × U n be endowed with a norm • U such that the inclusions Equations ( 12) and ( 13) are satisfied.Let f : (U, • U ) → (V, • V ) be a continuous and onto crisp function and let f : be a fuzzy function induced from f via the extension principle defined in Equation (10).We further assume that f Suppose that conditions (b), (c), (d) and (e) for W n are satisfied.For we have the following equality: For the 0-level sets, we also have: If the product vector space U is endowed with the product norm • U 1 ×•••×U n such that the inclusions Equations ( 12) and ( 13) are satisfied, then we also have the same results.The assumptions satisfying the inclusions Equations ( 12) and ( 13) are not needed when we say that the function f : • U and the product norm Proof.Since a normed space can induce a Hausdorff topological space, the results follow immediately from Remarks 2 and 3, Proposition 1 and Wu ([8], Theorem 6.2).Remark 4. For n = 2, the following functions: satisfy the conditions of Theorem 8.
Let (X, • ) be a normed space.We recall that the norm • can induce a norm topology τ such that (X, τ) becomes a Hausdorff topological vector space.Conversely, if (X, τ) is a topological vector space, we say that X is normable if and only if there exists a norm • which can induce a norm topology τ such that τ = τ.A subset Y of a topological vector space (X, τ) is called bounded if and only if, for each neighborhood N of θ, where θ is the zero element of X, there is a real number r such that Y ⊆ rN.We have the following interesting results.(i) A topological Hausdorff vector space is normable if and only if there is a bounded convex neighborhood of 0. (ii) A finite product of normable spaces is normable.
From part (ii) of Proposition 3, we see that Theorems 4-8 are still valid if the normed spaces are replaced by the normable topological vector spaces.However, if we consider the continuity of the function f : , then the assumption satisfying the inclusions Equations ( 12) and ( 13) is not needed, since, in this case, we can just apply Wu ([8], Theorems 5.1, 5.2, 5.3, 6.1 and 6.2) by considering the product topology τ U 1 ×•••×U n instead of the norm topology τU or the product norm topology τU 1 ×•••×U n .

Continuity of Fuzzified Functions
Let (X, • ) be a normed space.We denote by K(X) the family of all compact subsets of X in the sense of norm topology τX induced by the norm • .Let A and B be any two compact subsets of X.We can define the Hausdorff metric d H between A and B as follows: If X is a normable topological vector space, then we can also define the Hausdorff metric d H . Now we have the following simple observation.Lemma 1.The function f : (X, • ) → R defined by f (x) = x − x 0 is continuous for any given x 0 ∈ X.
Proof.We have: The continuity follows immediately from the above inequality.
We need some useful properties from topological space.Proposition 4. The following statements hold true.
(i) (Royden ([16], p. 158)) If X is a Hausdorff space, then a compact subset of X is closed.(ii) (Royden ([16], p. 158)) If f is a continuous function from the topological space X to another topological space Y, then the image f (K) is a compact subset of Y when K is a compact subset of X. (iii) (Royden ([16], p. 161)) If f is an upper semicontinuous real-valued function defined on X, then f assumes its maximum on a compact subset of X.If f is a lower semicontinuous real-valued function defined on X, then f assumes its minimum on a compact subset of X. (iv) (Royden ([16], p. 158)) Let X be a topological space and let K be a compact subset of X.If A is a closed subset of X and is also a subset of K, then A is also a compact subset of X. (v) (Royden ( [16], p. 166)) (Tychonoff's Theorem) Let (X i , τ X i ) be n topological spaces and let K i be compact subsets of Proof.For any A, B ∈ K(X), i.e., A and B are compact subsets of (X, • X ), since f is continuous, by part (ii) of Proposition 4, we see that are also compact subsets of Y, i.e., C, D ∈ K(Y).We need to show that, for any > 0, there exists δ > 0 such that, for any By Lemma 1 and part (iii) of Proposition 4, we see that the above infimum are attained, i.e., a * − b 0 X < δ and a 0 − b * < δ for some a * ∈ A and b * ∈ B. Since f is continuous at a 0 and b 0 , we have f (a * ) − f (b 0 ) Y < /2 and f (a 0 ) − f (b * ) Y < /2.Since a 0 and b 0 are any elements of A and B, respectively, we have: and: Since C and D are compact subsets of Y, there exist c * ∈ C and d * ∈ D such that: From Equations ( 16) and (17), we obtain: Let (X, τ) be a normable topological vector space.Then, we can topologize K(X) by considering the Hausdorff metric d H .This topological space is denoted by (K(X), τ H ).
Proof.The result follows immediately from Proposition 5.
To study the continuity of fuzzified function, we are going to consider the space F c (U) instead of F (U), which is defined below.Let U be a topological space.We denote by F c (U) the set of all fuzzy subsets of U such that, for each Ã ∈ F c (U), its α-level sets Ãα are compact subsets of U for all α ∈ [0, 1].Let (U, • U ) be a normed space.Then, we can define the Hausdorff metric d H U on K(U).For any Ã ∈ F c (U), we see that Ãα ∈ K(U) for all α ∈ [0, 1].Therefore, for any Ã, B ∈ F c (U), we can define a distance d F c (U) between Ã and B as: Then, we can show that (F c (U), d F c (U) ) forms a metric space.Based on this metric d F c (U) , we can induce a topology τ F c (U) that is called a metric topology for F c (U).
Let (U i , • U i ) be normed spaces for i = 1, • • • , n and let U = U 1 × • • • × U n be the product vector space that is endowed with the norm • U or the product norm • U 1 ×•••×U n .In this case, we can define a Hausdorff metric d H U on K(U) based on the norm • U or the product norm Using the Tychonoff's theorem in part (v) of Proposition 4, we see that Ãα ∈ K(U) for all α ∈ [0, 1]; that is, the product set Ãα is a compact subset of the product space U.For any Ã, B ∈ F c (U), we define d F c (U) between Ã and B as follows: Then, we can also show that (F c (U), d F c (U) ) forms a metric space.Based on this metric d F c (U) , we can induce a metric topology τ F c (U) for F c (U).Now, if we assume that U 1 , • • • , U n are normable topological vector spaces and let U = U 1 × • • • × U n be the product vector space, then part (ii) of Proposition 3 says that U is also a normable topological vector space.Therefore, we still can define a Hausdorff metric d H U on K(U).In this case, we can also obtain the metric space (F c (U), the extension principle defined in Equation (10).However, even f is continuous, we cannot always induce a fuzzy function f : In fact, we can just induce a fuzzy function f : , where the range is the space F (V).In other words, given The purpose is to present the sufficient conditions to guarantee this desired result.First of all, we need a useful lemma.
Lemma 2. (Wu [8]) Let U 1 , • • • , U n be topological spaces and let V be a Hausdorff space.Let f : be a fuzzy function induced from f via the extension principle defined in Equation (10).Assume that Ã(i) ∈ F (U i ) such that its membership function ξ Ã(i) is upper semicontinuous and each 0-level set Ã(i) 0 of Ã(i) is a compact subset of U i for all i = 1, • • • , n. Suppose that conditions (c) and (d) for W n are satisfied.Then, the supremum given in Equation ( 14) is attained.
be a continuous and onto crisp function.Suppose that conditions (b), (c) and (d) for W n are satisfied.
Proof.Since the α-level sets Ã(i) α of Ã(i) are compact subsets of U i for all α ∈ [0, 1] and i = 1, • • • , n, part (i) of Proposition 4 says that Ã(i) α are also closed subsets of U i for all α ∈ [0, 1] and i = 1, • • • , n.In other words, the membership functions ξ Ã(i) are upper semicontinuous for all i = 1, • • • , n.Therefore, applying Lemma 2, we see that the supremum given in Equation ( 14) is attained for each y ∈ V.According to Wu ([8], Theorem 5.2), since f is continuous, we have: , which is a compact subset of V by part (ii) of Proposition 4 and the Tychonoff's theorem in part (v) of Proposition 4, i.e., f . This completes the proof.
Under the assumptions of Proposition 7, we can indeed induce a fuzzy function f : then we need to apply Theorem 5 to induce the fuzzy function f : . Now we have the following result.
Theorem 9. Let (U i , • U i ) and (V, • V ) be the normed spaces for i = 1, • • • , n, and let the product vector space U = U 1 × • • • × U n be endowed with a norm • U such that the inclusions Equations ( 12) and ( 13) are satisfied.We assume that f : (U, • U ) → (V, • V ) is a continuous and onto crisp function.Suppose that conditions (b), (c) and (d) for W n are satisfied.Let f : ) be a fuzzy function induced from f via the extension principle defined in Equation (10).Then, the fuzzy function f : ) is continuous.If the product vector space U is endowed with the product norm • U 1 ×•••×U n such that the inclusions Equations ( 12) and ( 13) are satisfied, then we also have the same results.The assumptions satisfying the inclusions Equations ( 12) and ( 13) are not needed when we say that the function f : • U and the product norm Proof.Since a normed space can induce a Hausdorff topological space, from Remarks 2 and 3, Proposition 1 and the arguments of Proposition 7, Lemma 2 says that the supremum given in Equation ( 14) is attained for each y ∈ V. Therefore, from Theorem 5, we have: From the arguments of Proposition 7, we can induce a fuzzy function f : We need to show: For A, B ∈ K(U), Proposition 6 says that, for any > 0, there exists δ > 0 such that: Since d F c (U) ( Ãm , Ã0 ) → 0 as m → 0, there exists m 0 ∈ N such that: for m > m 0 , where Ãmα and Ã0α denote the α-level sets of Ãm and Ã0 , respectively, which also says that d H U ( Ãmα , Ã0α ) < δ for all α ∈ [0, 1].Therefore, according to Equation ( 19), we have Applying Lemma 2, we see that the supremum given in Equation ( 14) is attained for each y ∈ V. From Theorem 5, we also see that, for each α ∈ [0, 1], ( f ( Ãm )) α = f ( Ãmα ) and ( f ( Ã0 )) α = f ( Ã0α ).Therefore, we obtain: for m > m 0 .This completes the proof.
Theorem 10.Let U 1 , • • • , U n , V be normable topological vector spaces.We also assume that V is a Hausdorff space.Let f : be a fuzzy function induced from f via the extension principle defined in Equation (10).Then, the fuzzy function f : Proof.From Lemma 2 and Wu ([8], Theorem 5.2), we see that, for each α ∈ . Since the normable topological vector spaces are also Hausdorff spaces, the result follows immediately from the same arguments of Theorem 9.
Example 1.To apply Theorem 10, we take V = R m and U i = R for i = 1, • • • , n.Let f : R n → R m be a continuous and onto function, where the continuity is based on the usual topologies τ R n and τ R m .Suppose that conditions (b), (c) and (d) for W n are satisfied.For convenient, we write: ) be a fuzzy function induced from f via the extension principle defined in (10).Then the fuzzy function f : In this case, the fuzzy function Let U be a topological space.We denote by F 0 (U) the set of all fuzzy subsets of U such that, for each Ã ∈ F 0 (U), its membership function ξ Ã is upper semicontinuous and the 0-level set Ã0 is a compact subset of U.
Remark 5. We have the following observations.

•
Let U be a topological space.For any Ã ∈ F 0 (U), by the upper semicontinuity of ξ Ã, we see that Ãα are closed subsets of U for all α ∈ (0, 1].Since Ãα ⊆ Ã0 for all α ∈ (0, 1] and Ã0 is a compact subset of U, part (iv) of Proposition 4 says that Ãα are also compact subsets of U for all α ∈ (0, 1].Therefore we conclude that F 0 (U) ⊆ F c (U).

•
Let U be a Hausdorff space.For any Ã ∈ F c (U), we see that Ãα are compact subsets of U for all α ∈ [0, 1].

•
If U is taken as a normable topological vector space or (U, • ) is taken as the normed space, then F 0 (U) = F c (U), since the normable topological vector space and normed space are also the Hausdorff spaces.
Let U 1 , • • • , U n be topological spaces and let V be a Hausdorff space.Let f : ) be a continuous and onto crisp function.In general, we cannot induce a fuzzy function f : via the extension principle defined in Equation (10).The following proposition presents the sufficient conditions to guarantee this property.
) be a continuous and onto crisp function.Suppose that conditions (c) and (d) for W n are satisfied.If Proof.The result follows immediately from Wu ([8], Theorem 5.3).
Under the assumptions of Proposition 8, we can induce a fuzzy function f : We also have the following interesting observations.Remark 6.We remark that Theorems 9 and 10 are still valid if F c (U i ) and F c (V) are replaced by F 0 (U i ) and F 0 (V) for all i = 1, • • • , n, since the third observation of Remark 5 says that Theorems 9 and 10 present the continuity of fuzzified functions based on the spaces F c (U i ) and F c (V) for i = 1, • • • , n.In the sequel, we are going to investigate the continuity of fuzzified functions based on the spaces F cc (U i ) and F cc (V) and let the product vector space U = U 1 × • • • × U n be endowed with a norm • U such that the inclusions Equations ( 12) and ( 13) are satisfied.We assume that f : (U, • U ) → (V, • V ) is a linear, continuous and onto crisp function.Suppose that conditions (b), (c), (d), (e) and (f) for W n are satisfied.Let f : be a fuzzy function induced from f via the extension principle defined in Equation (10).Then the fuzzy function f : is endowed with the product norm • U 1 ×•••×U n such that the inclusions Equations ( 12) and ( 13) are satisfied, then we also have the same results.The assumptions satisfying the inclusions Equations ( 12) and ( 13) are not needed when we say that the function f : Proof.From Remarks 2 and 3 and part (iii) of Theorem 7, we see that, for each α ∈ [0, 1], The result follows immediately from the same arguments of Theorem 9.
Remark 7. Suppose that W 2 satisfies the following conditions: for any a ∈ [0, 1].Then, we have If we take W 2 (a, b) = min{a, b}, then conditions in Equation ( 21) are satisfied automatically.
The following result will be used in the further discussion.
Proposition 9. (Wu [8]) Let U be a Hausdorff topological vector space over R. Suppose that conditions ( ) for W n are satisfied.Then, we have the following results.
(iii) If we take λ to be a nonnegative or nonpositive fuzzy number, then λ ⊗ ã ∈ F cc (U).
Definition 2. Let U be a vector space over R and let F be a subset of F (U).We say that F is a fuzzy vector space over R if and only if ã ⊕ b ∈ F and λ ã ∈ F for any ã, b ∈ F and λ ∈ R. In other words, F is closed under the fuzzy addition and scalar multiplication.
Proposition 10.Let U be a Hausdorff topological vector space over R.Then, the following statements hold true.
(i) Suppose that conditions (b), (c), (d) and (e) for W n are satisfied.Then F cc (U) is a fuzzy vector space over R. (ii) Suppose that conditions (c) and (d) for W n are satisfied.Then F 0 (U) = F c (U) is a fuzzy vector space over R.
Proof.Part (i) follows immediately from parts (i) and (iv) of Proposition 9. Part (ii) follows immediately from Proposition 8 and the arguments of Proposition 9.
If we consider F (R) instead of R, then we can introduce another concept of fuzzy vector space in which we consider the so-called fuzzy scalar multiplication instead of scalar multiplication.Definition 3. Let U be a vector space over R. Let F be a subset of F (U) and F R be a subset of F (R).We say that F is a fuzzy vector space over F R if and only if ã ⊕ b ∈ F and λ ã ∈ F for ã, b ∈ F and λ ∈ F R .
In other words, F is closed under the fuzzy addition and fuzzy scalar multiplication, where the fuzzy scalar multiplication should be defined in another way.Proposition 11.Let U be a Hausdorff topological vector space over R. Suppose that conditions (c) and (d) for W n are satisfied.If the fuzzy scalar multiplication is defined as λ ã = λ ⊗ ã, then F 0 (U) = F c (U) is a fuzzy vector space over F 0 (R) = F c (R).
Proof.The result follows immediately from Remark 5, Proposition 8 and the arguments of Proposition 9.
We say that the fuzzy number ã ∈ F cc (R) is nonnegative if and only if ξ ã(r) = 0 for all r < 0, and nonpositive if and only if ξ ã(r) = 0 for all r > 0. We denote by F + cc (R) the set of all nonnegative fuzzy numbers, and by F − cc (R) the set of all nonpositive fuzzy numbers.Let ã be a fuzzy number.We define the membership functions of ã+ and ã− as follows: if r = 0 and ξ ã(r ) < 1 for all r > 0 ξ ã(0) if r = 0 and there exists a r > 0 such that ξ ã(r ) = 1 0 otherwise and: if r = 0 and ξ ã(r ) < 1 for all r < 0 ξ ã(0) if r = 0 and there exists a r < 0 such that ξ ã(r ) = 1 0 otherwise.
If the fuzzy scalar multiplication is defined as λ ã = λ ⊗ ã, then F cc (U) is a fuzzy vector space over F ± cc (R).(ii) If the fuzzy scalar multiplication is defined as: where λ = λ+ ⊕ λ− , then F cc (U) is a fuzzy vector space over F cc (R).
In the sequel, we are going to introduce the concept of the fuzzy topological vector space.Recall that, for each normed space (U i , • U i ), we can induce a metric space (F c (U i ), where the metric d F c (U i ) is defined in (18).Given any > 0, the open ball with center Ã ∈ F c (U i ) is given by: Based on these open balls, we can induce a topological space (F c (U i ), τ F c (U i ) ), where τ F c (U i ) is the so-called metric topology.Based on the topological spaces is the product topology.On the other hand, we can induce another metric space (F c (U), d F c (U) ) by considering the norm • U or the product norm The open ball with center Ã ∈ F c (U) is given by: Based on these open balls, we can also induce the metric topology τ F c (U) for F c (U).To introduce the concept of the fuzzy topological vector space, we need to provide some suitable conditions to guarantee the equality τ ; that is, the product topology is equal to the metric topology.Proposition 13.Let (U i , • U i ) be normed spaces for i = 1, • • • , n and let U = U 1 × • • • × U n be the product vector space which is endowed with a norm • U .Given any > 0 and: if there exist ˆ 1 , ˆ 2 > 0 such that the following inclusions hold true: and: then the product topology τ F c (U 1 )×•••×F c (U n ) is equal to the metric topology τ F c (U) for F c (U).If the product vector space U is endowed with the product norm • U 1 ×•••×U n such that the inclusions Equations ( 22) and ( 23) are satisfied, then the product topology is also equal to the metric topology.
Proof.The results follow immediately from the same arguments of Proposition 1.
Lemma 3. Let (U i , • U i ) be normed spaces for i = 1, • • • , n and let U = U 1 × • • • × U n be the product vector space.Then, the following statements hold true.
(i) We consider the normed space (U, • U ).Given any > 0, assume that (u Then, we have the following inclusions: and B 1 ( (ii) We consider the product normed space (U, where the product norm is defined by (11).Given any > 0, assume that h(x 1 , • • • , x n ) < if and only if x i < for all i = 1, • • • , n.Then, we also have the inclusions in Equations ( 24) and (25).
Proof.To prove part (i), for B = ( B(1) , for all a i ∈ Ã(i) α and α ∈ [0, 1], which also says that: . By considering another case, we can similarly show that: for all α ∈ [0, 1].Then, we have: Therefore, we conclude the inclusion: On the other hand, for B(i) for all α ∈ [0, 1].We consider the case of: Therefore, we conclude the inclusion: The above same arguments can similarly apply to part (ii).This completes the proof.
be the product vector space.Then, the following statements hold true.
(i) We consider the normed space (U, • U ).Given any > 0, assume that (u where the product norm is defined by (11).
Given any > 0, assume that h Proof.The results follow immediately from Proposition 13 and Lemma 3.
We recall that if (U, τ) is a topological vector space, then the mappings of vector addition (x, y) → x + y and scalar multiplication (λ, x) → λx are continuous under this topology τ.Therefore, we can also introduce the concept of fuzzy topological vector space as follows.Definition 4. Let U be a vector space over R and let F be a subset of F (U).We say that (F , τ F ) is a fuzzy topological vector space over R if and only if the following conditions are satisfied: • F is a fuzzy vector space over R; • the mapping of fuzzy addition where τ R×F is the product topology for R × F .Remark 8. Let (U, • U ) be a normed space, which can also induce a norm topology τU .It is well-known that the mapping f 1 : (U × U, τ U×U ) → (U, τU ) defined by (x, y) → x + y and the mapping f 2 : (R × U, τ R×U ) → (U, τU ) defined by (λ, x) → λx are continuous, where τ U×U is the product topology for U × U formed by the norm topology τU and τ R×U is the product topology for R × U formed by the norm topology τU and the usual topology for R. In this case, the normed space (U, • U ) becomes a Hausdorff topological vector space (U, τU ).Let U = U × U be the product vector space such that U is endowed with the norm • U or the product norm • U×U , which can also induce the norm topology τU or the product norm topology τU×U .Let U R = R × U be the product vector space such that U R is endowed with the norm • U R or the product norm • R×U , which can also induce the norm topology τU R or the product norm topology τR×U .In this case, the mapping f 1 : (U × U, τU ) → (U, τU ) or f 1 : (U × U, τU×U ) → (U, τU ) defined by (x, y) → x + y and the ampping f 2 : (R × U, τU R ) → (U, τU ) or f 2 : (R × U, τR×U ) → (U, τU ) defined by (λ, x) → λx are not necessarily continuous unless the inclusions in Equations ( 12) and ( 13) are satisfied.
Theorem 15.Let (U, • U ) be a normed space and let the product vector space U = U × U be endowed with a norm • U such that the inclusions in Equations ( 12), ( 13), ( 22) and (23) are satisfied.Then, the following statements hold true.
(i) Suppose that conditions (b), (c) and (d) for W n are satisfied.Then, is a fuzzy topological vector space over R. (ii) Suppose that conditions (b), (c), (d) and (e) for W n are satisfied.Then, (F cc (U), τ F cc (U) ) is a fuzzy topological vector space over R.
If the product vector space U is endowed with the product norm • U×U such that the inclusions in Equations ( 12), ( 13), ( 22) and ( 23) are satisfied, then we also have the same results.The assumptions satisfying the inclusions Equations ( 12) and ( 13) are not needed when we say that the function f : U 1 × • • • × U n → V is continuous directly in topological sense without considering the norm • U and the product norm • U×U .
Proof.To prove part (i), using part (ii) of Proposition 10, it follows that F c (U) = F 0 (U) is a fuzzy vector space over R. From Remark 8, Proposition 13 and Theorem 9, we see that the mappings of fuzzy addition: (R × F c (U), τ R×F c (U) ) by (λ, ã) → λ ã are continuous, where λ ∈ R is regarded as the crisp number 1{λ} .
To prove part (ii), we consider the mapping f 1 : U × U → U defined by (x 1 , x 2 ) → x 1 + x 2 and the mapping f 2 : R × U → U defined by (λ, x) → λx.From the arguments of Proposition 9, we see that, for any convex subsets A 1 , A 2 of U and any convex subset A 3 of R, f 1 (A 1 , A 2 ) and f 2 (A 3 , A 1 ) are also convex subsets of U. Therefore, the result follows immediately from Remark 8, Proposition 13 and Theorem 13.This completes the proof.Corollary 1.Let (U, • U ) be a normed space and let the product vector space U = U × U be endowed with a norm • U such that, given any > 0, (u 1 , u 2 ) U < if and only if u i U i < for i = 1, 2. Then, the following statements hold true.
(i) Suppose that conditions (b), (c) and (d) for W n are satisfied.Then, (F c (U), τ F c (U) ) = (F 0 (U), τ F 0 (U) ) is a fuzzy topological vector space over R. (ii) Suppose that conditions (b), (c), (d) and (e) for W n are satisfied.Then (F cc (U), τ F cc (U) ) is a fuzzy topological vector space over R.
Proof.The results follow immediately from Remark 8, part (i) of Proposition 2 part (i) of Proposition 14 and Theorem 15.
Corollary 2. Let (U, • U ) be a normed space and let the product vector space U = U × U be endowed with a product norm • U×U that is defined by (11) such that, given any > 0, h(x 1 , x 2 ) < if and only if x i < for i = 1, 2. Then, the following statements hold true.
(i) Suppose that conditions (b), (c) and (d) for W n are satisfied.Then: (F c (U), τ F c (U) ) = (F 0 (U), τ F 0 (U) ) is a fuzzy topological vector space over R. (ii) Suppose that conditions (b), (c), (d) and (e) for W n are satisfied.Then, (F cc (U), τ F cc (U) ) is a fuzzy topological vector space over R.
Proof.The results follow immediately from Remark 8, part (ii) of Proposition 2 (ii), part (ii) of Proposition 14 and Theorem 15.
In the sequel, we consider the case of fuzzy scalar multiplication.
Definition 5. Let U be a vector space over R. Let F be a subset of F (U) and let F R be a subset of F (R).
We say that (F , τ F ) is a fuzzy topological vector space over F R if and only if the following conditions are satisfied: • F is a fuzzy vector space over F R ; • the mapping of fuzzy addition (F × F , τ F ×F ) → (F , τ F ) defined by ( ã, b) → ã ⊕ b is continuous, where τ F ×F is the product topology for F × F ; and • the mapping of fuzzy scalar multiplication (F R × F , τ F R ×F ) → (F , τ F ) defined by ( λ, ã) → λ ã is continuous, where τ F R ×F is the product topology for F R × F .Theorem 16.Let (U, • U ) be a normed space and let the product vector space U = U × U be endowed with a norm • U such that the inclusions in Equations ( 12), ( 13), ( 22) and (23) are satisfied.Suppose that conditions (b), (c) and (d) for W n are satisfied.If the fuzzy scalar multiplication is defined as λ ã = λ ⊗ ã, then (F c (U), τ F c (U) ) = (F 0 (U), τ F 0 (U) ) is a fuzzy topological vector space over F c (R) = F 0 (R).If the product vector space U is endowed with the product norm • U×U such that the inclusions in Equations ( 12), ( 13), ( 22) and ( 23) are satisfied, then we also have the same results.The assumptions satisfying the inclusions Equations ( 12) and ( 13) are not needed when the normed space (U, • U ) is directly regarded as a Hausdorff topological vector space over R without considering the norm • U and the product norm • U×U .
Proof.Proposition 11 says that F c (U) = F 0 (U) is a fuzzy vector space over F c (R) = F 0 (R).
Corollary 3. Let (U, • U ) be a normed space and let U = U × U be the product vector space such that one of the following conditions is satisfied: • the product vector space U is endowed with a norm • U such that, given any > 0, (u 1 , u 2 ) U < if and only if u i U i < for i = 1, 2; and
be a continuous and onto crisp function.Suppose that conditions (b), (c) and (d) for W n are satisfied.Let f :

(
F c (U) × F c (U), τ F c (U)×F c (U) ) → (F c (U), τ F c (U) ) by ( ã, b) → ã ⊕ b,and scalar multiplication: Therefore, we conclude that O ∈ τU .Conversely, for any open set O ∈ τU and any element u Let d H X and d H Y be the Hausdorff metrics defined by • X and • Y , respectively.Then the function F :