On the Fuzzy Number Space with the Level Convergence Topology

We characterize compact sets of E1 endowed with the level convergence topology τ . We also describe the completion ̂ E1, ̂ U of E1 with respect to its natural uniformity, that is, the pointwise uniformity U, and show other topological properties of ̂ E1, as separability. We apply these results to give an Arzela-Ascoli theorem for the space of E1, τ -valued continuous functions on a locally compact topological space equipped with the compact-open topology.


Introduction
Not long after Chang and Zadeh 1 introduced the concept of fuzzy numbers with the consideration of the properties of probability functions, this topic became an active and important area of research because of the development of various theories of fuzzy numbers and their applications in fuzzy optimization, fuzzy decision making, and so forth.In this framework, the study of several types of convergence in fuzzy number spaces plays a central role see, e.g., 2, 3 .This paper is a contribution to the study of one of these types of convergence, the so-called level convergence.
Let F R denote the family of all fuzzy subsets on the real numbers R, that is, the set of functions u : R → 0, 1 .For u ∈ F R and λ ∈ 0, 1 , the λ-level set of u is defined by The fuzzy number space E 1 is the set of elements u of F R satisfying the following properties: 1 u is normal, that is, there exists an x 0 ∈ R with u x 0 1; 2 u is convex, that is, u λx 1 − λ y ≥ min{u x , u y } for all x, y ∈ R, λ ∈ 0, 1 ; 3 u is upper-semicontinuous; 4 u 0 is a compact set in R.
The λ-level set u λ of u ∈ E 1 is a compact interval for each λ ∈ 0, 1 .We denote u λ u − λ , u λ .Notice that every r ∈ R can be considered a fuzzy number since r can be identified with the fuzzy number r defined as r t : Recall also that Goetschel and Voxman provided the following representation theorem of fuzzy numbers see 4 .
Theorem 1.1.Let u ∈ E 1 and u λ u − λ , u λ , λ ∈ 0, 1 .Then the pair of functions u − λ and u λ has the following properties: i u − λ is a bounded left continuous nondecreasing function on 0, 1 ; ii u λ is a bounded left continuous nonincreasing function on 0, 1 ; iii u − λ and u λ are right continuous at λ 0; Conversely, if a pair of functions α λ and β λ satisfies the above conditions (i)-(iv), then there exists a unique u ∈ E 1 such that u λ α λ , β λ for each λ ∈ 0, 1 .
Actually, the function u − λ is a nondecreasing function on 0, 1 , and the function u λ is a nonincreasing function on 0, 1 .This is an easy consequence of condition iii of the Goetschel and Voxman representation theorem.
Since u λ is a compact interval of R for all λ ∈ 0, 1 , given two elements u, v ∈ E 1 , u λ u − λ , u λ and v λ v − λ , v λ , we can consider the Hausdorff distance d between u λ and v λ .It is well known that for compact intervals of the reals, the Hausdorff distance is given by The Hausdorff distance allows us to endow E 1 with the following metric.
is a metric on E 1 .It is called the supremum metric on E 1 , and E 1 , d ∞ is a complete metric space.
Definition 1.3.We say that the net An easy characterization of level convergence is as follows.
Proposition 1.4.The net Notice that a net {u k } k∈D ⊂ E 1 d ∞ -converges to u ∈ E 1 if and only if lim k u k converges uniformly to u and lim k u − k converges uniformly to u − .Thus, the d ∞ -convergence implies the τ -convergence.The converse fails to be true see Example 2.1 in 6 .In 7 , Fang and Huang described the topology τ associated with this level convergence in E 1 .They showed that E 1 , τ is a Hausdorff, first countable topological space.
In Section 2, we will first prove that the subset of all elements u of E 1 such that u − and u are continuous is dense in E 1 , and we will also study compact sets of E 1 , τ , giving a characterization to be compared with the one provided in 7 .In Section 3, we will provide a description of the completion E 1 , U of E 1 with the pointwise uniformity U, and we will show that its underlying topological space is separable, Lindel öf thus, strongly paracompact and a Baire space.In the last section, an Arzela-Ascoli theorem for the space of E 1 , τ -valued continuous functions will be stated.

Two Useful Results
Our first result provides a dense subset of E 1 , τ .We will consider the subset C 1 of E 1 given by where u 1 is defined as and linearly at intermediate values, and u 2 is defined as and linearly at intermediate values.It is an easy matter to check that u 1 , u 2 ∈ C 1 and that By 8, Theorem 6.1 , we know that the topology of pointwise convergence and the topology of uniform convergence coincide in the set of all monotone continuous functions defined on the unit interval.Taking into account these facts and properties i and ii of Theorem where, as usual, τ d ∞ stands for the topology induced by the metric d ∞ .
There are several characterizations of compact subsets of spaces of functions F depending on the topology we endow the space with.For instance, if we deal with the topology of uniform convergence τ u on some spaces of continuous functions, the Ascoli-Arzela's Theorem asserts that K ⊂ F, τ u is compact if and only if K is pointwise bounded, closed and equicontinuous.Dealing with the pointwise convergence topology τ p in more general spaces of functions F, it is well known that K ⊂ F, τ p is compact if and only if K is closed and pointwise bounded.Hence, it is clear that closed and pointwise bounded subsets A ⊂ F satisfying τ p τ u are also equicontinuous.

Definition 2.2 see 5 . It is said that
As it is customary, we say that a subset A of E 1 is pointwise closed, if A is a closed subset of the product space R 0,1 × R 0,1 or equivalently, of the product space R × R 0,1 . in E 1 , τ that does not converge see, e.g., 9 .Since E 1 , τ is first countable, the set K {u n } ∞ n 1 is uniformly support bounded and τ -closed in E 1 , τ but it is not compact.
We also notice that there exist compact metric subsets extended linearly at intermediate values, and v n 0 for all n ≥ 1.It is clear that the sequence

Topological and Uniform Properties of E 1
First we prove the following.
Proof.We will denote by CMI resp., CMD, the set of all nondecreasing monotone continuous functions resp., the set of all nonincreasing monotone continuous functions on 0, 1 .It is well known that the space C 0, 1 of all continuous real-valued functions on 0, 1 endowed with the topology of the uniform convergence is separable by the classical Stone-Weierstrass theorem.Separability is a hereditary property in the realm of metric spaces, so that the spaces CMI and CMD equipped with the τ u -topology are separable spaces.Since the topologies of uniform convergence and pointwise convergence coincide on CMI × CMD, the product space CMI × CMD, τ p is a separable metric space and, consequently, the space C 1 is separable.The result now follows from Theorem 2.1.
The correspondence u ∈ E 1 i → u − , u is a uniform isomorphism when we consider the natural admissible structure U on E 1 , τ and the pointwise uniform structure or pointwise uniformity uniform structure V on R 0,1 × R 0,1 , τ p which has as a base the sets U λ 1 , λ 2 , . . ., λ n , ε n ≥ 1, ε > 0 of the following form: where, for all n ≥ 1, λ 1 , λ 2 , . . ., λ n runs over 0, 1 n .Throughout, we shall freely identify, without explicit mention, the uniformity U with the restriction of the uniformity It is a well-known fact that the uniform space E 1 , U is not complete 9 .Hence, our next goal is to describe its completion E 1 , U .Remark that a consequence of the previous theorem is the following.
Corollary 3.2.The underlying topological space of E 1 , U is separable.
Theorem 3.3.The completion E 1 , U of the uniform space E 1 , U is the subspace of R 0,1 × R 0,1 , V whose elements u, v verify the following properties: i u is a nonincreasing function on 0, 1 ; ii v is a nondecreasing function on 0, 1 ; Proof.The uniform space R 0,1 × R 0,1 , V is a product of complete uniform spaces, so that it is complete.Hence cl R 0,1 ×R 0,1 E 1 is a complete uniform space containing E 1 as dense subset.By 10, Theorem 8.3.12 , cl R 0,1 ×R 0,1 E 1 can be identified with the completion of E 1 , U .We move on to the description of cl R 0,1 ×R 0,1 E 1 .First notice that if u, v ∈ cl R 0,1 ×R 0,1 E 1 , then u is nonincreasing and v is nondecreasing since a pointwise limit of a net of nonincreasing resp., nondecreasing, functions is a nonincreasing resp., a nondecreasing, function.Moreover, since every u − , u ∈ E 1 satisfies u − 1 ≤ u 1 , condition iii is an easy consequence of the definition of the limit of a net.Thus, every element in cl R 0,1 ×R 0,1 E 1 verifies conditions i -iii .The converse follows from an argument similar to the one used in Theorem 2.1.
Corollary 3.4.The uniformity induced by the metric d ∞ and the uniformity U induce the same topology on the set C 1 , but the first is finer than the second.
As we have seen above, the topology τ d ∞ induced by the metric d ∞ and the level topology τ coincide in C 1 .This question dealing with E 1 seems to have received almost no attention in the literature.We borrow from 11 the necessary techniques in order to obtain a characterization of subsets Recall that a function f between two first countable spaces X and Y is continuous if and only if f lim n x n lim n f x n for every sequence {x n } in the space X.For every u ∈ E 1 and λ ∈ 0, 1 , we will denote by u λ the limit of u t when t → λ .First we need the following helpful result.
Proof.The proof for {u } ∞ n 1 proceeds along the same lines as the proof for {u − } ∞ n 1 so that we only show our lemma for the sequence {u − } ∞ n 1 .Let u u − , u be the limit point of {u n } ∞ n 1 .Since u − is left-continuous on 0, 1 and bearing in mind that u − is nondecreasing, there exists Then, for all n ≥ n 0 and all λ ∈ λ 0 − δ, λ 0 , we obtain

3.2
The proof is completed by invoking the fact that u − n is left continuous at λ 0 for each n < n 0 .
Theorem 3.6.For a subset A of E 1 the following conditions are equivalent: ii for each sequence {u n } ∞ n 1 { u − n , u n } n≥1 ⊂ A which levelly converges to an element u − , u in A, we have lim n u − n λ u − λ and lim n u n λ u λ for all λ ∈ 0, 1 ; iii every τ -convergent sequence {u n } ∞ n 1 ⊂ A is right equicontinuous.
Proof.i ⇒ ii easily follows from the definition of uniform convergence and ii ⇒ iii runs along similar lines to the ones used in Lemma 3.5.To see iii ⇒ i , we need to prove that the identity map from Then we can clearly assume that {u − n } n≥1 does not converge uniformly to u − .Under this assumption, we can find ε > 0, an infinite sequence of natural numbers Let us suppose, with no loss of generality that the sequence {λ n k } k∈N converges to λ 0 ∈ 0, 1 .Notice that {λ n k } k∈N has infinite many different elements because {u − } ≥1 pointwise converges to u − .Assume that there exists an infinite subsequence of {λ n k } k∈N whose elements are greater than λ 0 .For simplicity, we shall denote this subsequence again by {λ n k } k∈N .Our hypothesis and the fact that u − and the elements of {u − n } ∞ n 1 are right continuous at λ 0 allow us to find a sequence {n k } k such that which leads us to a contradiction.On the other hand, if there is an infinite subsequence of {λ n k } k∈N whose elements are less than λ 0 , by means of Lemma 3.5, an argument similar to the previous one also permits us to obtain a contradiction.Thus, the proof is complete.
We finish this section by giving some topological properties of E 1 , τ and the underlying topological space E 1 , τ p of the uniform space E 1 , U .Recall that a topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set in the sequence.The space is said to be Lindelöf, if every open cover has a countable subcover.Theorem 3.7.E 1 , τ p is a hemicompact space.
Proof.Let K n denote the set of all u ∈ E 1 such that |u − 0 − u 0 | ≤ n.It is clear that these sets are pointwise closed and pointwise bounded so that they are compact subsets of R × R 0,1 .Since E 1 is a closed subset of R × R 0,1 , they are compact subsets of E 1 , τ p .Now, given a Hemicompact spaces are Lindel öf, and every Lindel öf space is strongly paracompact 10, Theorem 5.3.11so that we conclude the following.
Corollary 3.8.E 1 , τ p is a Lindelöf space.In particular, it is strongly paracompact and, consequently, a normal space.
Recall that a strongly paracompact space X is Dieudonné complete, that is, the finest uniformity on X is complete.Thus, the natural uniformity on E 1 is not complete but the finest uniformity is.

Journal of Function Spaces and Applications
We close the section by showing that both E 1 , τ and E 1 , τ p are Baire spaces, that is, the Baire category theorem holds in E 1 , τ and in E 1 , τ p : for each sequence G 1 , G 2 , . . . of open dense subsets of E 1 , τ resp., of E 1 , τ p the intersection n≥1 G n is a dense set.Theorem 3.9.E 1 , τ and E 1 , τ p are Baire spaces.
Proof.Since a topological space X that contains a dense subspace which is a Baire space is a Baire space itself 10, 3.9.J b , it suffices to prove that C 1 , τ is a Baire space.To see this, first recall that the topologies τ d ∞ and τ coincide in C 1 .Next, notice that the space C 1 , d ∞ is a metric complete space and apply that metric complete spaces are Baire spaces.

An Arzela-Ascoli Theorem for C X, E 1 , τ l
In this section, X will denote a locally compact topological space.We shall assume that the space C X, E 1 , τ l of E 1 -valued continuous functions is endowed with the compact-open topology and given a subset F of C X, E 1 , τ l , we shall write F x : {f x : f ∈ F} for any x ∈ X.
As in 12 , we have the following.
ii For any x ∈ X, F x is pointwise closed and uniformly support-bounded; iii F is evenly equicontinuous.
When we deal with C X, E 1 , τ l , the definition of evenly continuous can be translated as follows.
Definition 4.2.A subset F of C X, E 1 , τ l is said to be evenly equicontinuous if for any x 0 ∈ X, u ∈ E 1 , {α 1 , . . ., α n } ⊂ 0, 1 and > 0, there exists an open neighborhood V of x 0 and {β For the sake of completeness, we give the proof of Theorem 4.1 following the same pattern as in 12 .
Proof of Theorem 4.1.We have the following.

Necessity. Let us suppose that
ii It is apparent that, for any x ∈ X, the map L x : C X, E 1 , τ l → E 1 , τ l defined to be L x f : f x is continuous.Hence, since F is compact, then L x F F x is compact in E 1 , τ l for each x ∈ K. Therefore, by Theorem 2.3, F x is, for any x ∈ K, pointwise closed and uniformly support bounded.

4.5
Sufficiency.Let us first check that F, the closure of F in the pointwise convergence topology, is also evenly equicontinuous.Fix x 0 ∈ K, u ∈ E 1 , {α 1 , . . ., α n } ⊂ 0, 1 and > 0.Then, by the evenly equicontinuity of F, there exists a neighborhood V of x 0 and {β 1 , . . ., β m } ⊂ 0, 1 such that if f ∈ F and for all x ∈ V .Let us take g ∈ F such that max 1≤i≤m g x 0 β i − u β i , g x 0 − β i − u − β i < .

4.8
We must check max 1≤i≤n g x α i − u α i , g x − α i − u − α i < 4.9 checked that both topologies coincide on F cl F .To this end, we can consider F as a closed subset of the product x∈K {F x }.By Tychonoff's theorem, x∈K {F x } is compact in the topology of pointwise convergence since, by Theorem 2.3, F x is compact for every x ∈ K. Thus, F is compact and, since by hypothesis, F is closed, then F is compact, as was proved.

Example 2 . 4 .
There are examples of τ -closed and uniformly support bounded subsets of E 1 which are not τ -compact.Consider a uniformly support bounded Cauchy sequence {u n } ∞ n 1