#### Abstract

The concept of fuzzy modular space is first proposed in this paper. Afterwards, a Hausdorff topology induced by a -homogeneous fuzzy modular is defined and some related topological properties are also examined. And then, several theorems on -completeness of the fuzzy modular space are given. Finally, the well-known Baire’s theorem and uniform limit theorem are extended to fuzzy modular spaces.

#### 1. Introduction and Preliminaries

In the 1960s, the concept of modular space was introduced by Nakano [1]. Soon after, Musielak and Orlicz [2] redefined and generalized the notion of modular space. A real function on an arbitrary vector space is said to be a *modular* if it satisfies the following conditions:(M-1) if and only if (i.e., is the null vector ),(M-2),
(M-3) for all and with .

A *modular space * is defined by a corresponding modular , that is, .

Based on definition of the modular space, Kozłowski [3, 4] introduced the notion of modular function space. In the sequel, Kozłowski and Lewicki [5] considered the problem of analytic extension of measurable functions in modular function spaces and discussed some extension properties by means of polynomial approximation. Afterwards, Kilmer and Kozłowski [6] studied the existence of best approximations in modular function spaces by elements of sublattices. In 1990, Khamsi et al. [7] initiated the study of fixed point theory for nonexpansive mappings defined on some subsets of modular function spaces. More researches on fixed point theory in modular function spaces can be found in [8–13].

In 2007, Nourouzi [14] proposed probabilistic modular spaces based on the theory of modular spaces and some researches on the Menger's probabilistic metric spaces. A pair is called a *probabilistic modular space* if is a real vector space, is a mapping from into the set of all distribution functions (for , the distribution function is denoted by , and is the value at ) satisfying the following conditions:(PM-1),
(PM-2) for all if and only if ,(PM-3),
(PM-4) for all and , .

Especially, for every , and , if
then we say that is *-homogeneous*.

Recently, further studies have been made on the probabilistic modular spaces. Nourouzi [15] extended the well-known Baire's theorem to probabilistic modular spaces by using a special condition. Fallahi and Nourouzi [16] investigated the continuity and boundedness of linear operators defined between probabilistic modular spaces in the probabilistic sense.

In this paper, following the idea of probabilistic modular space and the definition of fuzzy metric space in the sense of George and Veeramani [17], we apply fuzzy concept to the classical notions of modular and modular spaces and propose a novel concept named fuzzy modular spaces.

#### 2. Fuzzy Modular Spaces

In this section, following the idea of probabilistic modular space, we will introduce the concept of fuzzy modular space by using continuous -norm and present some related notions.

*Definition 1 (Schweizer and Sklar [18]). *A binary operation is called a *continuous t-norm* if it satisfies the following conditions:(TN-1) is commutative and associative;(TN-2) is continuous;(TN-3) for every ;(TN-4) whenever , and .

Three common examples of the continuous -norm are ; ; (3) . For more examples, the reader can be referred to [19].

*Definition 2 (George and Veeramani [17]). *A *fuzzy metric space* is an ordered triple such that is a nonempty set, is a continuous *t*-norm, and is a fuzzy set on satisfying the following conditions, for all , :(F-1),
(F-2) if and only if ,(F-3),
(F-4),
(F-5) is continuous.

Based on the notion of probabilistic modular space and Definition 2, we will propose a novel concept named fuzzy modular spaces.

*Definition 3. *The triple is said to be a *fuzzy modular space* (shortly, -modular space) if is a real or complex vector space, is a continuous -norm, and is a fuzzy set on satisfying the following conditions, for all , and with : (FM-1),
(FM-2) for all if and only if , (FM-3),
(FM-4),
(FM-5) is continuous.

Generally, if is a fuzzy modular space, we say that is a *fuzzy modular* on . Moreover, the triple is called *-homogeneous* if for every , and ,

*Example 4. * Let be a real or complex vector space and let be a modular on . Take -norm . For every , define for all . Then is a -modular space.

*Remark 5. *Note that the above conclusion still holds even if the -norm is replaced by and , respectively.

*Example 6. * Let . is a modular on , which is defined by , where . Take -norm . For every , we define
for all . Then is a -homogeneous -modular space.

*Proof. *We just need to prove the condition (FM-4) of Definition 3 and formula (2), because other conditions hold obviously. In the following, we first verify , as with .

Since is a modular on , for all , we have

Then, we can obtain
that is,

Therefore

Thus, we have .

On the other hand, for all , since , it follows that

Hence, we know that is a -homogeneous -modular space.

Theorem 7. *If is a -modular space, then is nondecreasing for all . *

* Proof. *Suppose that for some . Without loss of generality, we can take , , and is the null vector in . By Definition 3, we can obtain

Since , we have . Obviously, this leads to a contradiction.

It should be noted that, in general, a fuzzy modular and a fuzzy metric (in the sense of George and Veeramani [17]) do not necessarily induce mutually when the triangular norm is the same one. In essence, the fuzzy modular and fuzzy metric can be viewed as two different characterizations for the same set. The former is regarded as a kind of fuzzy quantization on the classical vector modular, while the latter is regarded as a fuzzy measure on the distance between two points. Next, we construct two examples to show that there does not exist direct relationship between a fuzzy modular and a fuzzy metric.

*Example 8. * Let . Take -norm . For every , we define
where is a constant.

Here, we only show that satisfies the condition (FM-4) of Definition 3, since other conditions can be easily verified.

For every , and with . Without loss of generality, we assume that . Since , we then obtain

Hence is a fuzzy modular on . However, if we set
it is easy to verify that is not a fuzzy metric on .

*Example 9. * Let . Take -norm . For every and , we define

It can easily be shown that is a fuzzy metric on . Set

If we take , , , and , then we know that . Thus, for all , we have . But . Obviously, is not a fuzzy modular on .

#### 3. Topology Induced by a -Homogeneous Fuzzy Modular

In this section, we will define a topology induced by a -homogeneous fuzzy modular and examine some topological properties. Let denote the set of all positive integers.

*Definition 10. *Let be a -modular space. The *-ball * with center and radius , , is defined as

An element is called a *-interior point* of if there exist and such that . Meantime, we say that is a *-open set* in if and only if every element of is a -interior point.

Lemma 11 (George and Veeramani [17]). *If the -norm is continuous, then*(L1)* for every with , there exists such that ,*(L2)* for every , there exists such that . *

Theorem 12. *If is a -homogeneous -modular space, then . *

*Proof. *By Theorem 7, for every and , since , it is obvious that .

Theorem 13. *Let be a -homogeneous -modular space. Every -ball in is a -open set. *

* Proof. *By Definition 10, for every , we have . Without loss of generality, we may assume that . Since is continuous, there exists an such that for some with and . Set . Since , there exists an such that . According to Lemma 11, we can find an such that .

Next, we show that . For every , we have . Therefore,
Thus and hence .

Theorem 14. *Let be a -homogeneous -modular space. Define
**
Then is a topology on . *

*Proof. *The proof will be divided into three parts.(i) Obviously, .(ii) Suppose that . If , then and .

Therefore, there exist , and , such that and . Set , . Now, we claim that .

If , then we know that . According to Theorem 7, we can obtain Thus, , that is, .

Similarly, .

Hence, . That is to say, .(iii) Suppose that . If , then there exists such that . Since , there exist and such that . Hence, .

Obviously, if we take , then the family of -ball , constitutes a countable local base at . Therefore, we can obtain Theorem 15.

Theorem 15. *The topology induced by a -homogeneous -modular space is first countable. *

Theorem 16. *Every -homogeneous -modular space is Hausdorff. *

*Proof. *For the -homogeneous -modular space , let and be two distinct points in . By Definition 3, we can easily obtain for all . Set . According to Lemma 11, for every , there exists such that .

Next, we consider the -balls and and then show that using reduction to absurdity. If there exists , then
which is a contradiction. Hence is Hausdorff.

In order to obtain some further properties, several basic notions derived from general topology are introduced in the -modular space.

*Definition 17. *Let be a -modular space.(i) A sequence in is said to be -*convergent* to a point , denoted by , if for every and , there exists such that for all .(ii) A subset is called -*bounded* if and only if there exist and such that for all .(iii) A subset is called -*compact* if and only if every -open cover of has a finite subcover (or equivalently, every sequence in has a -convergent subsequence in ).(iv) A subset is called a -*closed* if and only if for every sequence , implies .

Theorem 18. *Every -compact subset of a -homogeneous -modular space is -bounded. *

* Proof. *Suppose that is a -compact subset of the given -homogeneous -modular space . Fix and , it is easy to see that the family of -ball is a -open cover of . Since is -compact, there exist such that . For every , there exists such that . Therefore, we have . Set . Clearly, we know that . Thus, we have
for some . This shows that is -bounded.

Theorem 19. *Let be a -homogeneous -modular space, and let be the topology induced by the -homogeneous modular. Then for a sequence in , if and only if as . *

* Proof. *Fix . Suppose that . Then for every , there exists such that for all . Namely, for all . Thus, we have for all . Because is arbitrary, we can verify that as .

On the other hand, if for every , as , then for every , there exists such that for all . Therefore, we know that for all . Thus for all , and hence as .

#### 4. -Completeness of a Fuzzy Modular Space

In this section, we will establish some related theorems of -completeness of a fuzzy modular space.

*Definition 20. *Let be a -modular space.(i) A sequence in is a -*Cauchy sequence* if and only if for every and , there exists such that for all .(ii) The -modular space is called -*complete* if every -Cauchy sequence is -convergent.

In [16], Fallahi and Nourouzi proved that every -convergent sequence is a -Cauchy sequence in the -homogeneous -modular space. Here we will propose a similar result in -modular space. Noticing that the following theorem shows that a -convergent sequence is not necessarily a -Cauchy sequence in a general -modular space.

Theorem 21. *Let be a -homogeneous -modular space. Then every -convergent sequence in is a -Cauchy sequence. *

* Proof. *Suppose that the sequence -converges to . Therefore, for every and , there exists such that for all . For all , we have

Hence is a -Cauchy sequence in .

*Remark 22. * The proof of Theorem 21 shows that, in the -modular space, a -convergent sequence is not necessarily a -Cauchy sequence. However, the -homogeneity and the choice of triangular norms are essential to guarantee the establishment of theorem.

Theorem 23. *Every -closed subspace of -complete -modular space is -complete. *

*Proof. *From Definition 20, it is evident to see that the theorem holds.

Theorem 24. *Let be a -homogeneous -modular space, and let be a subset of . If every -Cauchy sequence of is -convergent in , then every -Cauchy sequence of is also -convergent in , where denotes the -closure of . *

*Proof. *Suppose that the sequence is a -Cauchy sequence of . Therefore, for every and , there exists such that . According to Theorem 7, we have . In addition, for every and , there exists an such that for all . That is to say, as . Next, we will show that the sequence is a -Cauchy sequence of . For every , we have

Since -norm is continuous, it follows that as . Now, we assume that the sequence -converges to . Thus, for every and , there exists an such that for all . Therefore, for all , we can obtain

According to the arbitrary of and by letting , it follows that . That is, an arbitrary -Cauchy sequence of -converges to . The proof of the theorem is now completed.

Theorem 25. *Let be a -homogeneous -modular space, and let be a dense subset of . If every -Cauchy sequence of is -convergent in , then the -homogeneous -modular space is -complete. *

*Proof. *It follows from Theorem 24.

#### 5. Baire's Theorem and Uniform Limit Theorem

In [15], Nourouzi extended the well-know Baire's theorem to probabilistic modular spaces. In this section, we will extend the Baire's theorem to fuzzy modular spaces in an analogous way. Moreover, the uniform limit theorem also can be extended to this type of spaces.

Theorem 26 (Baire's theorem). *Let be a countable number of -dense and -open sets in the -complete -homogeneous -modular space . Then is -dense in . *

*Proof. *First of all, if is a -ball in and is an arbitrary element of it, then we know that . Since is continuous, there exists an such that for some with and . Choose , and , there exists a sequence in such that and hence we have
for some . Therefore, we can obtain

This shows that . It means that if is a nonempty -open set of , then is nonempty and -open. Now, let , there exist and such that . Choose and such that . Since is -dense in , we can obtain . Let , there exist and such that . Choose and such that . By induction, we can obtain a sequence in and two sequence , such that , and .

Next, we show that is a -Cauchy sequence. For given and , we can choose such that and . Then for , since , we have

According to the arbitrary of , it follows that is a -Cauchy sequence. Since is -complete, there exists such that . But for all , and therefore for all . Thus . Hence is -dense in .

*Definition 27. * Let be any nonempty set and let be a -modular space. A sequence of functions from to is said to *-converge uniformly* to a function from to if given and ; there exists such that for all and for every .

Theorem 28 (Uniform limit theorem). *Let be a sequence of continuous functions from a topological space to a -homogeneous -modular space . If -converges uniformly to , then is continuous. *

* Proof. *Let be a -open set of and . Since is -open, there exist and such that . Owing to , we can choose such that . Since -converges uniformly to , given and , there exists such that for all and for every . Moreover, is continuous for every , there exists a neighborhood of such that . Therefore, we know that for every . Thus, we have

This shows that . Hence ; that is, is continuous.

*Remark 29. * All the results in this paper are still valid if the condition (FM-5) in Definition 3 is replaced by left continuity.

#### 6. Conclusions

In this paper, we have proposed the concept of fuzzy modular space based on the (probabilistic) modular space and continuous -norm, which can be regarded as a generalization of (probabilistic) modular space in the fuzzy sense. Meantime, two examples are given to show that a fuzzy modular and a fuzzy metric do not necessarily induce mutually when the triangular norm is the same one. In the sequel, we have defined a Hausdorff topology induced by a -homogeneous fuzzy modular and examined some related topological properties. It should be pointed out that the -homogeneity is essential to ensure the establishment of most important conclusions, and some properties also depend on the choice of triangular norms. Finally, we have extended the well-known Baire's theorem and uniform limit theorem to -homogeneous fuzzy modular spaces.

Further research will focus on the following problems. We first address the problem whether there is a relationship between a fuzzy modular and a fuzzy metric. If the aforementioned relationship exists, then the following issue should be simultaneously considered. It has important theoretical values to explore what conditions a fuzzy modular and a fuzzy metric can induce mutually. Similar to the fixed point theory in probabilistic or fuzzy metric spaces, it is an interesting and valuable research direction to construct fixed point theorems in fuzzy modular spaces. Inspired by [3, 4, 20–22], a problem worthy to be considered is extending the modular sequence (function) space and the Orlicz sequence space to fuzzy setting by the method used in this paper.

#### Acknowledgments

This work was supported by “Qing Lan” Talent Engineering Funds by Tianshui Normal University. The second author acknowledge the support of the Beijing Municipal Education Commission Foundation of China (no. KM201210038001), the National Natural Science Foundation (no. 71240002) and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (no. PHR201108333).