On Fuzzy Modular Spaces

Abstract

The concept of fuzzy modular space is first proposed in this paper. Afterwards, a Hausdorff topology induced by a 𝛽-homogeneous fuzzy modular is defined and some related topological properties are also examined. And then, several theorems on 𝜇-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 𝜌 on an arbitrary vector space 𝑋 is said to be a modular if it satisfies the following conditions:(M-1)𝜌(𝑥)=0 if and only if 𝑥=𝜃 (i.e., 𝑥 is the null vector 𝜃),(M-2)𝜌(𝑥)=𝜌(−𝑥), (M-3)𝜌(𝛼𝑥+𝛽𝑦)≤𝜌(𝑥)+𝜌(𝑦) for all 𝑥,𝑦∈𝑋 and 𝛼,𝛽≥0 with 𝛼+𝛽=1.

A modular space 𝑋_𝜌 is defined by a corresponding modular 𝜌, that is, 𝑋_𝜌={𝑥∈𝑋∶𝜌(𝜆𝑥)→0as𝜆→0}.

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 (𝑋,𝜌) is called a probabilistic modular space if 𝑋 is a real vector space, 𝜌 is a mapping from 𝑋 into the set of all distribution functions (for 𝑥∈𝑋, the distribution function 𝜌(𝑥) is denoted by 𝜌_𝑥, and 𝜌_𝑥(𝑡) is the value 𝜌_𝑥 at 𝑡∈ℝ) satisfying the following conditions:(PM-1)𝜌_𝑥(0)=0, (PM-2)𝜌_𝑥(𝑡)=1 for all 𝑡>0 if and only if 𝑥=𝜃,(PM-3)𝜌_−𝑥(𝑡)=𝜌_𝑥(𝑡), (PM-4)𝜌_{𝛼𝑥+𝛽𝑦}(𝑠+𝑡)≥𝜌_𝑥(𝑠)∧𝜌_𝑦(𝑡) for all 𝑥,𝑦∈𝑋 and 𝛼,𝛽,𝑠,𝑡∈ℝ⁺₀, 𝛼+𝛽=1.

Especially, for every 𝑥∈𝑋,  𝑡>0 and 𝛼∈ℝ⧵{0}, if 𝜌_𝛼𝑥(𝑡)=𝜌_𝑥𝑡|𝛼|^𝛽,where𝛽∈(0,1], (1) then we say that (𝑋,𝜌) is 𝛽-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 𝑡-norm and present some related notions.

Definition 1 (Schweizer and Sklar [18]). A binary operation ∗∶[0,1]×[0,1]→[0,1] is called a continuous t-norm if it satisfies the following conditions:(TN-1)∗ is commutative and associative;(TN-2)∗ is continuous;(TN-3)𝑎∗1=𝑎 for every 𝑎∈[0,1];(TN-4)𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐, 𝑏≤𝑑 and 𝑎,𝑏,𝑐,𝑑∈[0,1].

Three common examples of the continuous 𝑡-norm are (1)  𝑎∗_𝑀𝑏=min{𝑎,𝑏};  (2)  𝑎∗_𝑃𝑏=𝑎⋅𝑏; (3)  𝑎∗_𝐿𝑏=max{𝑎+𝑏−1,0}. For more examples, the reader can be referred to [19].

Definition 2 (George and Veeramani [17]). A fuzzy metric space is an ordered triple (𝑋,𝑀,∗) such that 𝑋 is a nonempty set, ∗ is a continuous t-norm, and 𝑀 is a fuzzy set on 𝑋×𝑋×(0,∞) satisfying the following conditions, for all 𝑥,𝑦,𝑧∈𝑋,  𝑠,𝑡>0:(F-1)𝑀(𝑥,𝑦,𝑡)>0, (F-2)𝑀(𝑥,𝑦,𝑡)=1 if and only if 𝑥=𝑦,(F-3)𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡), (F-4)𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑦,𝑧,𝑠)≤𝑀(𝑥,𝑧,𝑡+𝑠), (F-5)𝑀(𝑥,𝑦,⋅)∶(0,∞)→(0,1] 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 (𝑋,𝜇,∗) is said to be a fuzzy modular space (shortly, ℱ-modular space) if 𝑋 is a real or complex vector space, ∗ is a continuous 𝑡-norm, and 𝜇 is a fuzzy set on 𝑋×(0,∞) satisfying the following conditions, for all 𝑥,𝑦∈𝑋,  𝑠,𝑡>0 and 𝛼,𝛽≥0 with 𝛼+𝛽=1:  (FM-1)𝜇(𝑥,𝑡)>0,  (FM-2)𝜇(𝑥,𝑡)=1 for all 𝑡>0 if and only if 𝑥=𝜃, (FM-3)𝜇(𝑥,𝑡)=𝜇(−𝑥,𝑡),  (FM-4)𝜇(𝛼𝑥+𝛽𝑦,𝑠+𝑡)≥𝜇(𝑥,𝑠)∗𝜇(𝑦,𝑡),  (FM-5)𝜇(𝑥,⋅)∶(0,∞)→(0,1] is continuous.

Generally, if (𝑋,𝜇,∗) is a fuzzy modular space, we say that (𝜇,∗) is a fuzzy modular on 𝑋. Moreover, the triple (𝑋,𝜇,∗) is called 𝛽-homogeneous if for every 𝑥∈𝑋,  𝑡>0 and 𝜆∈ℝ⧵{0}, 𝜇()𝑡𝜆𝑥,𝑡=𝜇𝑥,|𝜆|^𝛽,where𝛽∈(0,1]. (2)

Example 4. Let 𝑋 be a real or complex vector space and let 𝜌 be a modular on 𝑋. Take 𝑡-norm 𝑎∗𝑏=𝑎∗_𝑀𝑏. For every 𝑡∈(0,∞), define 𝜇(𝑥,𝑡)=𝑡/(𝑡+𝜌(𝑥)) for all 𝑥∈𝑋. Then (𝑋,𝜇,∗) is a ℱ-modular space.

Remark 5. Note that the above conclusion still holds even if the 𝑡-norm is replaced by 𝑎∗𝑏=𝑎∗_𝑃𝑏 and 𝑎∗𝑏=𝑎∗_𝐿𝑏, respectively.

Example 6. Let 𝑋=𝑅. 𝜌 is a modular on 𝑋, which is defined by 𝜌(𝑥)=|𝑥|^𝛽, where 𝛽∈(0,1]. Take 𝑡-norm 𝑎∗𝑏=𝑎∗_𝑃𝑏. For every 𝑡∈(0,∞), we define 𝜇()=1𝑥,𝑡𝑒^{𝜌(𝑥)/𝑡} (3) for all 𝑥∈𝑋. Then (𝑋,𝜇,∗) is a 𝛽-homogeneous ℱ-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 𝜇(𝛼𝑥+𝛽𝑦,𝑠+𝑡)≥𝜇(𝑥,𝑠)∗𝜇(𝑦,𝑡), as 𝛼,𝛽≥0 with 𝛼+𝛽=1.
Since 𝜌 is a modular on 𝑋, for all 𝑥,𝑦∈𝑋, we have 𝜌𝛼𝑥+𝛽𝑦≤𝜌(𝑥)+𝜌𝑦. (4)
Then, we can obtain 𝜌𝛼𝑥+𝛽𝑦≤𝑡+𝑠𝑡𝜌(𝑥)+𝑡+𝑠𝑠𝜌𝑦, (5) that is, 1𝜌≤1𝑡+𝑠𝛼𝑥+𝛽𝑦𝑡𝜌(𝑥)+1𝑠𝜌𝑦. (6)
Therefore 𝑒^{𝜌(𝛼𝑥+𝛽y)/(𝑡+𝑠)}≤𝑒^{𝜌(𝑥)/𝑡}⋅𝑒^{𝜌(𝑦)/𝑠}=𝑒^{𝜌(𝑥)/𝑡}∗_𝑃𝑒^{𝜌(𝑦)/𝑠}. (7)
Thus, we have 𝜇(𝛼𝑥+𝛽𝑦,𝑠+𝑡)≥𝜇(𝑥,𝑠)∗𝜇(𝑦,𝑡).
On the other hand, for all 𝜆∈ℝ⧵{0}, since 𝜌(𝜆𝑥)=|𝜆𝑥|^𝛽=|𝜆|^𝛽⋅|𝑥|^𝛽=|𝜆|^𝛽𝜌(𝑥), it follows that 𝜇()𝑡𝜆𝑥,𝑡=𝜇𝑥,|𝜆|^𝛽. (8)
Hence, we know that (𝑋,𝜇,∗) is a 𝛽-homogeneous ℱ-modular space.

Theorem 7. If (𝑋,𝜇,∗) is a ℱ-modular space, then 𝜇(𝑥,⋅) is nondecreasing for all 𝑥∈𝑋.

Proof. Suppose that 𝜇(𝑥,𝑡)<𝜇(𝑥,𝑠) for some 𝑡>𝑠>0. Without loss of generality, we can take 𝛼=1, 𝛽=0, and 𝑦=𝜃 is the null vector in 𝑋. By Definition 3, we can obtain 𝜇()()()()().𝑥,𝑠∗𝜇𝜃,𝑡−𝑠=𝜇𝑥,𝑠∗𝜇𝑦,𝑡−𝑠≤𝜇𝛼𝑥+𝛽𝑦,𝑡=𝜇𝑥,𝑡<𝜇𝑥,𝑠 (9)
Since 𝜇(𝜃,𝑡−𝑠)=1, we have 𝜇(𝑥,𝑠)<𝜇(𝑥,𝑠). 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 𝑋=ℝ. Take 𝑡-norm 𝑎∗𝑏=𝑎∗_𝑀𝑏. For every 𝑡∈(0,∞), we define 𝜇()=𝑘𝑥,𝑡|𝑥|,𝑘+ (10) where 𝑘>0 is a constant.
Here, we only show that 𝜇(𝑥,𝑡) satisfies the condition (FM-4) of Definition 3, since other conditions can be easily verified.
For every 𝑥,𝑦∈ℝ, and 𝛼,𝛽≥0 with 𝛼+𝛽=1. Without loss of generality, we assume that |𝑥|≤|𝑦|. Since |𝛼𝑥+𝛽𝑦|≤|𝑦|, we then obtain 𝜇=𝑘𝛼𝑥+𝛽𝑦,𝑡+𝑠≥𝑘𝑘+𝛼𝑥+𝛽𝑦𝑦𝑘𝑘+=min|𝑥|,𝑘𝑘+𝑦()∗𝑘+=𝜇𝑥,𝑡_𝑀𝜇𝑦,𝑠. (11)
Hence (𝜇,∗_𝑀) is a fuzzy modular on 𝑋. However, if we set 𝑀=𝑘𝑥,𝑦,𝑡=𝜇𝑥−𝑦,𝑡,𝑘+𝑥−𝑦 (12) it is easy to verify that (𝑀,∗_𝑀) is not a fuzzy metric on 𝑋.

Example 9. Let 𝑋=ℝ. Take 𝑡-norm 𝑎∗𝑏=𝑎∗_𝑀𝑏. For every 𝑥,𝑦∈𝑋 and 𝑡∈(0,∞), we define 𝑀=1𝑥,𝑦,𝑡1,𝑥=𝑦,21,𝑥≠𝑦,𝑥,𝑦∈ℤ,41,𝑥∈ℤ,𝑦∈ℝ⧵ℤor𝑥∈ℝ⧵ℤ,𝑦∈ℤ,4,𝑥≠𝑦,𝑥,𝑦∈ℝ⧵ℤ. (13)

It can easily be shown that (𝑀,∗_𝑀) is a fuzzy metric on 𝑋. Set 𝜇()()=1𝑥,𝑡=𝑀𝑥,𝜃,𝑡1,𝑥=0,2{0},1,𝑥∈ℤ⧵4,𝑥∈ℝ⧵ℤ. (14)

If we take 𝛼=√2/2, 𝛽=1−√2/2, 𝑥≠𝑦, and 𝑥,𝑦∈ℤ, then we know that 𝛼𝑥+𝛽𝑦∈ℝ⧵ℤ. Thus, for all 𝑡,𝑠>0, we have 𝜇(𝛼𝑥+𝛽𝑦,𝑡+𝑠)=1/4. But 𝜇(𝑥,𝑡)∗_𝑀𝜇(𝑦,𝑠)=min{𝜇(𝑥,𝑡),𝜇(𝑦,𝑠)}=1/2. Obviously, (𝜇,∗_𝑀) is not a fuzzy modular on 𝑋.

3. Topology Induced by a 𝛽-Homogeneous Fuzzy Modular

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

Definition 10. Let (𝑋,𝜇,∗) be a ℱ-modular space. The 𝜇-ball 𝐵(𝑥,𝑟,𝑡) with center 𝑥∈𝑋 and radius 𝑟, 0<𝑟<1,  𝑡>0 is defined as 𝐵(𝑥,𝑟,𝑡)=𝑦∈𝑋∶𝜇𝑥−𝑦,𝑡>1−𝑟. (15)
An element 𝑥∈𝐸 is called a 𝜇-interior point of 𝐸 if there exist 𝑟∈(0,1) and 𝑡>0 such that 𝐵(𝑥,𝑟,𝑡)⊆𝐸. Meantime, we say that 𝐸 is a 𝜇-open set in 𝑋 if and only if every element of 𝐸 is a 𝜇-interior point.

Lemma 11 (George and Veeramani [17]). If the 𝑡-norm ∗ is continuous, then(L1) for every 𝑟₁,𝑟₂∈(0,1) with 𝑟₁>𝑟₂, there exists 𝑟₃∈(0,1) such that 𝑟₁∗𝑟₃≥𝑟₂,(L2) for every 𝑟₄∈(0,1), there exists 𝑟₅∈(0,1) such that 𝑟₅∗𝑟₅≥𝑟₄.

Theorem 12. If (𝑋,𝜇,∗) is a 𝛽-homogeneous ℱ-modular space, then 𝐵(𝑥,𝑟,𝑡/2^𝛽+1)⊂𝐵(𝑥,𝑟,𝑡/2).

Proof. By Theorem 7, for every 𝑟∈(0,1) and 𝑡>0, since 𝜇(𝑥−𝑦,𝑡/2)≥𝜇(𝑥−𝑦,𝑡/2^𝛽+1), it is obvious that {𝑦∈𝑋∶𝜇(𝑥−𝑦,𝑡/2^𝛽+1>1−𝑟)}⊂{𝑦∈𝑋∶𝜇(𝑥−𝑦,𝑡/2)>1−𝑟}.

Theorem 13. Let (𝑋,𝜇,∗) be a 𝛽-homogeneous ℱ-modular space. Every 𝜇-ball 𝐵(𝑥,𝑟,𝑡) in (𝑋,𝜇,∗) is a 𝜇-open set.

Proof. By Definition 10, for every 𝑦∈𝐵(𝑥,𝑟,𝑡), we have 𝜇(𝑥−𝑦,𝑡)>1−𝑟. Without loss of generality, we may assume that 𝑡=2𝑡₁. Since 𝜇(𝑥−𝑦,⋅) is continuous, there exists an 𝜖_𝑦>0 such that 𝜇(𝑥−𝑦,(𝑡₁−𝜖)/2^𝛽−1)>1−𝑟 for some 𝜖>0 with (𝑡₁−𝜖)/2^𝛽−1>0 and 𝜖/2^𝛽−1∈(0,𝜖_𝑦). Set 𝑟₀=𝜇(𝑥−𝑦,(𝑡₁−𝜖)/2^𝛽−1). Since 𝑟₀>1−𝑟, there exists an 𝑠∈(0,1) such that 𝑟₀>1−𝑠>1−𝑟. According to Lemma 11, we can find an 𝑟₁∈(0,1) such that 𝑟₀∗𝑟₁≥1−𝑠.
Next, we show that 𝐵(𝑦,1−𝑟₁,𝜖/2^𝛽−1)⊂𝐵(𝑥,𝑟,2𝑡₁). For every 𝑧∈𝐵(𝑦,1−𝑟₁,𝜖/2^𝛽−1), we have 𝜇(𝑦−𝑧,𝜖/2^𝛽−1)>𝑟₁. Therefore, 𝜇(𝑥−𝑧,𝑡)=𝜇𝑥−𝑧,2𝑡₁≥𝜇2𝑥−𝑦,2𝑡₁2𝑡−𝜖∗𝜇𝑦−𝑧,2𝜖=𝜇𝑥−𝑦,¹−𝜖2^𝛽−1𝜖∗𝜇𝑦−𝑧,2^𝛽−1≥𝑟₀∗𝑟₁≥1−𝑠>1−𝑟. (16) Thus 𝑧∈𝐵(𝑥,𝑟,𝑡) and hence 𝐵(𝑦,1−𝑟₁,𝜖/2^𝛽−1)⊂𝐵(𝑥,𝑟,𝑡).

Theorem 14. Let (𝑋,𝜇,∗) be a 𝛽-homogeneous ℱ-modular space. Define 𝒯_𝜇=()()}.𝐴⊂𝑋∶𝑥∈𝐴𝑖𝑓𝑎𝑛𝑑𝑜𝑛𝑙𝑦𝑖𝑓𝑡ℎ𝑒𝑟𝑒𝑒𝑥𝑖𝑠𝑡𝑡>0𝑎𝑛𝑑𝑟∈0,1𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐵𝑥,𝑟,𝑡⊂𝐴 (17) Then 𝒯_𝜇 is a topology on 𝑋.

Proof. The proof will be divided into three parts.(i) Obviously, ∅,𝑋∈𝒯_𝜇.(ii) Suppose that 𝐴,𝐵∈𝒯_𝜇. If 𝑥∈𝐴∩𝐵, then 𝑥∈𝐴 and 𝑥∈𝐵.
Therefore, there exist 0<𝑟₁,  𝑟₂<1 and 𝑡₁,  𝑡₂>0 such that 𝐵(𝑥,𝑟₁,𝑡₁)⊂𝐴 and 𝐵(𝑥,𝑟₂,𝑡₂)⊂𝐵. Set 𝑟=min{𝑟₁,𝑟₂}, 𝑡=min{𝑡₁,𝑡₂}. Now, we claim that 𝐵(𝑥,𝑟,𝑡)⊂𝐵(𝑥,𝑟₁,𝑡₁).
If 𝑦∈𝐵(𝑥,𝑟,𝑡), then we know that 𝜇(𝑥−𝑦,𝑡)>1−𝑟. According to Theorem 7, we can obtain 𝜇𝑥−𝑦,𝑡₁≥𝜇𝑥−𝑦,𝑡>1−𝑟≥1−𝑟₁. (18)Thus, 𝑦∈𝐵(𝑥,𝑟₁,𝑡₁), that is, 𝐵(𝑥,𝑟,𝑡)⊂𝐵(𝑥,𝑟₁,𝑡₁).
Similarly, 𝐵(𝑥,𝑟,𝑡)⊂𝐵(𝑥,𝑟₂,𝑡₂).
Hence, 𝐵(𝑥,𝑟,𝑡)⊂𝐵(𝑥,𝑟₁,𝑡₁)∩𝐵(𝑥,𝑟₂,𝑡₂)⊂𝐴∩𝐵. That is to say, 𝐴∩𝐵∈𝒯_𝜇.(iii) Suppose that 𝒯^′_𝜇⊂𝒯_𝜇. If 𝑥∈⋃_{𝐴∈𝒯′𝜇}𝐴, then there exists 𝑈∈𝒯^′_𝜇 such that 𝑥∈𝑈. Since 𝑈∈𝒯_𝜇, there exist 0<𝑟<1 and 𝑡>0 such that 𝐵(𝑥,𝑟,𝑡)⊂𝑈⊂⋃_{𝐴∈𝒯′𝜇}𝐴. Hence, ⋃_{𝐴∈𝒯′𝜇}𝐴∈𝒯_𝜇.

Obviously, if we take 𝑟=𝑡=(1/𝑛)(𝑛=1,2,3,…), then the family of 𝜇-ball 𝐵(𝑥,1/𝑛,1/𝑛),  (𝑛=1,2,3,…) constitutes a countable local base at 𝑥. Therefore, we can obtain Theorem 15.

Theorem 15. The topology 𝒯_𝜇 induced by a 𝛽-homogeneous ℱ-modular space is first countable.

Theorem 16. Every 𝛽-homogeneous ℱ-modular space is Hausdorff.

Proof. For the 𝛽-homogeneous ℱ-modular space (𝑋,𝜇,∗), let 𝑥 and 𝑦 be two distinct points in 𝑋. By Definition 3, we can easily obtain 0<𝜇(𝑥−𝑦,𝑡)<1 for all 𝑡>0. Set 𝑟=𝜇(𝑥−𝑦,𝑡). According to Lemma 11, for every 𝑟₀∈(𝑟,1), there exists 𝑟₁∈(0,1) such that 𝑟₁∗𝑟₁≥𝑟₀.
Next, we consider the 𝜇-balls 𝐵(𝑥,1−𝑟₁,𝑡/2^𝛽+1) and 𝐵(𝑦,1−𝑟₁,𝑡/2^𝛽+1) and then show that 𝐵(𝑥,1−𝑟₁,𝑡/2^𝛽+1)∩𝐵(𝑦,1−𝑟₁,𝑡/2^𝛽+1)=∅ using reduction to absurdity. If there exists 𝑧∈𝐵(𝑥,1−𝑟₁,𝑡/2^𝛽+1)∩𝐵(𝑦,1−𝑟₁,𝑡/2^𝛽+1), then 2(),𝑡𝑟=𝜇𝑥−𝑦,𝑡≥𝜇𝑥−𝑧22,𝑡∗𝜇𝑧−𝑦2𝑡=𝜇𝑥−𝑧,2^𝛽+1𝑡∗𝜇𝑧−𝑦,2^𝛽+1≥𝑟₁∗𝑟₁≥𝑟₀, (19) which is a contradiction. Hence (𝑋,𝜇,∗) is Hausdorff.

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

Definition 17. Let (𝑋,𝜇,∗) be a ℱ-modular space.(i) A sequence {𝑥_𝑛} in 𝑋 is said to be 𝜇-convergent to a point 𝑥∈𝑋, denoted by 𝑥→−_𝑛^𝜇𝑥, if for every 𝑟∈(0,1) and 𝑡>0, there exists 𝑛₀∈ℕ such that 𝑥_𝑛∈𝐵(𝑥,𝑟,𝑡) for all 𝑛≥𝑛₀.(ii) A subset 𝐴⊂𝑋 is called 𝜇-bounded if and only if there exist 𝑡>0 and 𝑟∈(0,1) such that 𝜇(𝑥,𝑡)>1−𝑟 for all 𝑥∈𝐴.(iii) A subset 𝐵⊂𝑋 is called 𝜇-compact if and only if every 𝜇-open cover of 𝐵 has a finite subcover (or equivalently, every sequence in 𝐵 has a 𝜇-convergent subsequence in 𝐵).(iv) A subset 𝐶⊂𝑋 is called a 𝜇-closed if and only if for every sequence {𝑥_𝑛}⊂𝐶, 𝑥→−_𝑛^𝜇𝑥 implies 𝑥∈𝐶.

Theorem 18. Every 𝜇-compact subset 𝐴 of a 𝛽-homogeneous ℱ-modular space (𝑋,𝜇,∗) is 𝜇-bounded.

Proof. Suppose that 𝐴 is a 𝜇-compact subset of the given 𝛽-homogeneous ℱ-modular space (𝑋,𝜇,∗). Fix 𝑡>0 and 𝑟∈(0,1), it is easy to see that the family of 𝜇-ball {𝐵(𝑥,𝑟,𝑡/2^𝛽+1)∶𝑥∈𝐴} is a 𝜇-open cover of 𝐴. Since 𝐴 is 𝜇-compact, there exist 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝐴 such that 𝐴⊆⋃^𝑛_𝑖=1𝐵(𝑥_𝑖,𝑟,𝑡/2^𝛽+1). For every 𝑥∈𝐴, there exists 𝑖 such that 𝑥∈𝐵(𝑥_𝑖,𝑟,𝑡/2^𝛽+1). Therefore, we have 𝜇(𝑥−𝑥_𝑖,𝑡/2^𝛽+1)>1−𝑟. Set 𝛼=min{𝜇(𝑥_𝑖,𝑡/2^𝛽+1)∶1≤𝑖≤𝑛}. Clearly, we know that 𝛼>0. Thus, we have 𝜇(𝑥,𝑡)=𝜇𝑥−𝑥_𝑖+𝑥_𝑖,𝑡≥𝜇2𝑥−𝑥_𝑖,𝑡2∗𝜇2𝑥_𝑖,𝑡2=𝜇𝑥−𝑥_𝑖,𝑡2^𝛽+1∗𝜇𝑥_𝑖,𝑡2^𝛽+1≥(1−𝑟)∗𝛼>1−𝑠 (20) for some 𝑠∈(0,1). This shows that 𝐴 is 𝜇-bounded.

Theorem 19. Let (𝑋,𝜇,∗) be a 𝛽-homogeneous ℱ-modular space, and let 𝒯_𝜇 be the topology induced by the 𝛽-homogeneous modular. Then for a sequence {𝑥_𝑛} in 𝑋, 𝑥→−_𝑛^𝜇𝑥 if and only if 𝜇(𝑥−𝑥_𝑛,𝑡)→1 as 𝑛→∞.

Proof. Fix 𝑡>0. Suppose that 𝑥→−_𝑛^𝜇𝑥. Then for every 𝑟∈(0,1), there exists 𝑛₀∈ℕ such that 𝑥_𝑛∈𝐵(𝑥,𝑟,𝑡) for all 𝑛≥𝑛₀. Namely, 𝜇(𝑥_𝑛−𝑥,𝑡)>1−𝑟 for all 𝑛≥𝑛₀. Thus, we have 1−𝜇(𝑥_𝑛−𝑥,𝑡)<𝑟 for all 𝑛≥𝑛₀. Because 𝑟 is arbitrary, we can verify that 𝜇(𝑥_𝑛−𝑥,𝑡)→1 as 𝑛→∞.
On the other hand, if for every 𝑡>0, 𝜇(𝑥−𝑥_𝑛,𝑡)→1 as 𝑛→∞, then for every 𝑟∈(0,1), there exists 𝑛₀∈ℕ such that 1−𝜇(𝑥−𝑥_𝑛,𝑡)<𝑟 for all 𝑛≥𝑛₀. Therefore, we know that 𝜇(𝑥−𝑥_𝑛,𝑡)>1−𝑟 for all 𝑛≥𝑛₀. Thus 𝑥_𝑛∈𝐵(𝑥,𝑟,𝑡) for all 𝑛≥𝑛₀, and hence 𝑥→−_𝑛^𝜇𝑥 as 𝑛→∞.

4. 𝜇-Completeness of a Fuzzy Modular Space

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

Definition 20. Let (𝑋,𝜇,∗) be a ℱ-modular space.(i) A sequence {𝑥_𝑛} in 𝑋 is a 𝜇-Cauchy sequence if and only if for every 𝜖∈(0,1) and 𝑡>0, there exists 𝑛₀∈ℕ such that 𝜇(𝑥_𝑚−𝑥_𝑛,𝑡)>1−𝜖 for all 𝑚,𝑛≥𝑛₀.(ii) The ℱ-modular space (𝑋,𝜇,∗) is called 𝜇-complete if every 𝜇-Cauchy sequence is 𝜇-convergent.

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

Theorem 21. Let (𝑋,𝜇,∗_𝑀) be a 𝛽-homogeneous ℱ-modular space. Then every 𝜇-convergent sequence {𝑥_𝑛} in 𝑋 is a 𝜇-Cauchy sequence.

Proof. Suppose that the sequence {𝑥_𝑛}  𝜇-converges to 𝑥∈𝑋. Therefore, for every 𝜖∈(0,1) and 𝑡>0, there exists 𝑛₀∈ℕ such that 𝜇(𝑥_𝑛−𝑥,𝑡/2^𝛽+1)>1−𝜖 for all 𝑛≥𝑛₀. For all 𝑚,𝑛≥𝑛₀, we have 𝜇𝑥_𝑚−𝑥_𝑛,𝑡≥𝜇2𝑥_𝑚,𝑡−𝑥2∗_𝑀𝜇2𝑥_𝑛,𝑡−𝑥2≥𝜇𝑥_𝑚𝑡−𝑥,2^𝛽+1∗_𝑀𝜇𝑥_𝑛𝑡−𝑥,2^𝛽+1>(1−𝜖)∗_𝑀(1−𝜖)=1−𝜖. (21)
Hence {𝑥_𝑛} is a 𝜇-Cauchy sequence in 𝑋.

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

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

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

Theorem 24. Let (𝑋,𝜇,∗) be a 𝛽-homogeneous ℱ-modular space, and let 𝑌 be a subset of 𝑋. If every 𝜇-Cauchy sequence of 𝑌 is 𝜇-convergent in 𝑋, then every 𝜇-Cauchy sequence of 𝑌 is also 𝜇-convergent in 𝑋, where 𝑌 denotes the 𝜇-closure of 𝑌.

Proof. Suppose that the sequence {𝑥_𝑛} is a 𝜇-Cauchy sequence of 𝑌. Therefore, for every 𝑛∈ℕ and 𝑡>0, there exists 𝑦_𝑛∈𝑌 such that 𝜇(𝑥_𝑛−𝑦_𝑛,𝑡/4^𝛽+1)>1−1/(𝑛+1). According to Theorem 7, we have 𝜇(𝑥_𝑛−𝑦_𝑛,𝑡/2^𝛽+1)>1−(1/(𝑛+1)). In addition, for every 𝑟∈(0,1) and 𝑡>0, there exists an 𝑛₀∈ℕ such that 𝜇(𝑥_𝑛−𝑥_𝑚,𝑡/4^𝛽+1)>1−𝑟 for all 𝑚,𝑛≥𝑛₀. That is to say, 𝜇(𝑥_𝑛−𝑥_𝑚,𝑡/4^𝛽+1)→1 as 𝑚,𝑛→∞. Next, we will show that the sequence {𝑦_𝑛} is a 𝜇-Cauchy sequence of 𝑌. For every 𝑚,𝑛≥𝑛₀, we have 𝜇𝑦_𝑛−𝑦_𝑚,𝑡≥𝜇2𝑦_𝑛−𝑥_𝑛,𝑡2∗𝜇2𝑥_𝑛−𝑦_𝑚,𝑡2≥𝜇2𝑦_𝑛−𝑥_𝑛,𝑡2∗𝜇4𝑥_𝑛−𝑥_𝑚,𝑡4∗𝜇4𝑥_𝑚−𝑦_𝑚,𝑡4=𝜇𝑦_𝑛−𝑥_𝑛,𝑡2^𝛽+1∗𝜇𝑥_𝑛−𝑥_𝑚,𝑡4^𝛽+1∗𝜇𝑥_𝑚−𝑦_𝑚,𝑡4^𝛽+1>11−∗()∗1𝑛+11−𝑟1−.𝑚+1 (22)
Since 𝑡-norm ∗ is continuous, it follows that 𝜇(𝑦_𝑛−𝑦_𝑚,𝑡)→1 as 𝑚,𝑛→∞. Now, we assume that the sequence {𝑦_𝑛}  𝜇-converges to 𝑥∈𝑋. Thus, for every 𝜖∈(0,1) and 𝑡>0, there exists an 𝑛₁∈ℕ such that 𝜇(𝑥−𝑦_𝑛,𝑡/2^𝛽+1)>1−𝜖 for all 𝑛≥𝑛₁. Therefore, for all 𝑛≥𝑛₁, we can obtain 𝜇𝑥_𝑛−𝑥,𝑡≥𝜇2𝑥_𝑛−𝑦_𝑛,𝑡2∗𝜇2𝑦_𝑛−𝑥_𝑛,𝑡2=𝜇𝑥_𝑛−𝑦_𝑛,𝑡2^𝛽+1∗𝜇𝑦_𝑛−𝑥_𝑛,𝑡2^𝛽+1>()∗11−𝜖1−.𝑛+1 (23)
According to the arbitrary of 𝜖 and by letting 𝑛→∞, it follows that lim_𝑛→∞𝜇(𝑥_𝑛−𝑥,𝑡)=1. That is, an arbitrary 𝜇-Cauchy sequence {𝑥_𝑛} of 𝑌𝜇-converges to 𝑥∈𝑋. The proof of the theorem is now completed.

Theorem 25. Let (𝑋,𝜇,∗) be a 𝛽-homogeneous ℱ-modular space, and let 𝑌 be a dense subset of 𝑋. If every 𝜇-Cauchy sequence of 𝑌 is 𝜇-convergent in 𝑋, then the 𝛽-homogeneous ℱ-modular space (𝑋,𝜇,∗) is 𝜇-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 𝑈_𝑛(𝑛=1,2,…) be a countable number of 𝜇-dense and 𝜇-open sets in the 𝜇-complete 𝛽-homogeneous ℱ-modular space (𝑋,𝜇,∗_𝑀). Then ⋂^∞_𝑛=1𝑈_𝑛 is 𝜇-dense in 𝑋.

Proof. First of all, if 𝐵(𝑥,𝑟,2𝑡) is a 𝜇-ball in 𝑋 and 𝑦 is an arbitrary element of it, then we know that 𝜇(𝑥−𝑦,2𝑡)>1−𝑟. Since 𝜇(𝑥−𝑦,⋅) is continuous, there exists an 𝜖_𝑦>0 such that 𝜇(𝑥−𝑦,(𝑡−𝜖)/2^𝛽−1)>1−𝑟 for some 𝜖>0 with (𝑡−𝜖)/2^𝛽−1>0 and 𝜖/2^𝛽−1∈(0,𝜖_𝑦). Choose 𝑟^′∈(0,𝑟), 𝜖/2^𝛽−1∈(0,𝜖_𝑦) and 𝑧∈𝐵(𝑦,𝑟^′,𝜖/4^𝛽), there exists a sequence {𝑧_𝑛} in 𝐵(𝑦,𝑟^′,𝜖/4^𝛽) such that 𝑧→−_𝑛^𝜇𝑧 and hence we have 𝜇𝜖𝑧−𝑦,2^𝛽−1≥𝜇2𝑧−𝑧_𝑛,𝜖2^𝛽∗_𝑀𝜇2𝑧_𝑛,𝜖−𝑦2^𝛽=𝜇𝑧−𝑧_𝑛,𝜖4^𝛽∗_𝑀𝜇𝑧_𝑛𝜖−𝑦,4^𝛽>1−𝑟 (24) for some 𝑛∈ℕ. Therefore, we can obtain 𝜇(𝑥−𝑧,2𝑡)=𝜇2𝑧−𝑦,2𝜖∗_𝑀𝜇2()𝜖𝑥−𝑦,2𝑡−𝜖=𝜇𝑧−𝑦,2^𝛽−1∗_𝑀𝜇𝑥−𝑦,𝑡−𝜖2^𝛽−1>(1−𝑟)∗_𝑀(1−𝑟)=1−𝑟. (25)
This shows that 𝐵(𝑦,𝑟^′,𝜖/4^𝛽)⊆𝐵(𝑥,𝑟,2𝑡). It means that if 𝐴 is a nonempty 𝜇-open set of 𝑋, then 𝐴∩𝑈₁ is nonempty and 𝜇-open. Now, let 𝑥₁∈𝐴∩𝑈₁, there exist 𝑟₁∈(0,1) and 𝑡₁>0 such that 𝐵(𝑥₁,𝑟₁,𝑡₁/2^𝛽−1)⊆𝐴∩𝑈₁. Choose 𝑟^′₁<𝑟₁ and 𝑡^′₁=min{𝑡₁,1} such that 𝐵(𝑥₁,𝑟^′₁,𝑡^′₁/2^𝛽−1)⊆𝐴∩𝑈₁. Since 𝑈₂ is 𝜇-dense in 𝑋, we can obtain 𝐵(𝑥₁,𝑟^′₁,𝑡^′₁/2^𝛽−1)∩𝑈₂≠∅. Let 𝑥₂∈𝐵(𝑥₁,𝑟^′₁,𝑡^′₁/2^𝛽−1)∩𝑈₂, there exist 𝑟₂∈(0,1/2) and 𝑡₂>0 such that 𝐵(𝑥₂,𝑟₂,𝑡₂/2^𝛽−1)⊆𝐵(𝑥₁,𝑟^′₁,𝑡^′₁/2^𝛽−1)∩𝑈₂. Choose 𝑟^′₂<𝑟₂ and 𝑡^′₂=min{𝑡₂,1/2} such that 𝐵(𝑥₂,𝑟^′₂,𝑡^′₂/2^𝛽−1)⊆𝐴∩𝑈₂. By induction, we can obtain a sequence {𝑥_𝑛} in 𝑋 and two sequence {𝑟^′_𝑛},  {𝑡^′_𝑛} such that 0<𝑟^′_𝑛<1/𝑛, 0<𝑡^′_𝑛<1/𝑛 and 𝐵(𝑥_𝑛,𝑟^′_𝑛,𝑡^′_𝑛/2^𝛽−1)⊆𝐴∩𝑈_𝑛.
Next, we show that {𝑥_𝑛} is a 𝜇-Cauchy sequence. For given 𝑡>0 and 𝑟∈(0,1), we can choose 𝑘∈ℕ such that 2𝑡^′_𝑘<𝑡 and 𝑟^′_𝑘<𝑟. Then for 𝑚,𝑛≥𝑘, since 𝑥_𝑚,𝑥_𝑛∈𝐵(𝑥_𝑘,𝑟^′_𝑘,𝑡^′_𝑘/2^𝛽−1), we have 𝜇𝑥_𝑚−𝑥_𝑛,2𝑡≥𝜇𝑥_𝑚−𝑥_𝑛,4𝑡^′_𝑘≥𝜇2𝑥_𝑚−𝑥_𝑘,2𝑡^′_𝑘∗_𝑀𝜇2𝑥_𝑛−𝑥_𝑘,2𝑡^′_𝑘=𝜇𝑥_𝑚−𝑥_𝑘,𝑡^′_𝑘2^𝛽−1∗_𝑀𝜇𝑥_𝑛−𝑥_𝑘,𝑡^′_𝑘2^𝛽−1≥1−𝑟_𝑘>1−𝑟. (26)
According to the arbitrary of 𝑡, it follows that {𝑥_𝑛} is a 𝜇-Cauchy sequence. Since 𝑋 is 𝜇-complete, there exists 𝑥∈𝑋 such that 𝑥→−_𝑛^𝜇𝑥. But 𝑥_𝑛∈𝐵(𝑥_𝑘,𝑟^′_𝑘,𝑡^′_𝑘/2^𝛽−1) for all 𝑛≥𝑘, and therefore 𝑥∈𝐵(𝑥_𝑘,𝑟^′_𝑘,𝑡^′_𝑘/2^𝛽−1)⊆𝐴∩𝑈_𝑘 for all 𝑘. Thus 𝐴∩(⋂^∞_𝑛=1𝑈_𝑛)≠∅. Hence ⋂^∞_𝑛=1𝑈_𝑛 is 𝜇-dense in 𝑋.

Definition 27. Let 𝑋 be any nonempty set and let (𝑌,𝜇,∗) be a ℱ-modular space. A sequence {𝑓_𝑛} of functions from 𝑋 to 𝑌 is said to 𝜇-converge uniformly to a function 𝑓 from 𝑋 to 𝑌 if given 𝑡>0 and 𝑟∈(0,1); there exists 𝑛₀∈ℕ such that 𝜇(𝑓_𝑛(𝑥)−𝑓(𝑥),𝑡)>1−𝑟 for all 𝑛≥𝑛₀ and for every 𝑥∈𝑋.

Theorem 28 (Uniform limit theorem). Let 𝑓_𝑛∶𝑋→𝑌 be a sequence of continuous functions from a topological space 𝑋 to a 𝛽-homogeneous ℱ-modular space (𝑌,𝜇,∗). If {𝑓_𝑛}  𝜇-converges uniformly to 𝑓∶𝑋→𝑌, then 𝑓 is continuous.

Proof. Let 𝑉 be a 𝜇-open set of 𝑌 and 𝑥₀∈𝑓⁻¹(𝑉). Since 𝑉 is 𝜇-open, there exist 𝑟∈(0,1) and 𝑡>0 such that 𝐵(𝑓(𝑥₀),𝑟,𝑡)⊂𝑉. Owing to 𝑟∈(0,1), we can choose 𝑠∈(0,1) such that (1−𝑠)∗(1−𝑠)∗(1−𝑠)>1−𝑟. Since {𝑓_𝑛}  𝜇-converges uniformly to 𝑓, given 𝑠∈(0,1) and 𝑡>0, there exists 𝑛₀∈ℕ such that 𝜇(𝑓_𝑛(𝑥)−𝑓(𝑥),𝑡/4^𝛽+1)>1−𝑠 for all 𝑛≥𝑛₀ and for every 𝑥∈𝑋. Moreover, 𝑓_𝑛 is continuous for every 𝑛∈ℕ, there exists a neighborhood 𝑈 of 𝑥₀ such that 𝑓_𝑛(𝑈)⊂𝐵(𝑓_𝑛(𝑥₀),𝑠,𝑡/4^𝛽+1). Therefore, we know that 𝜇(𝑓_𝑛(𝑥)−𝑓_𝑛(𝑥₀),𝑡/4^𝛽+1)>1−𝑠 for every 𝑥∈𝑈. Thus, we have 𝜇𝑓(𝑥)−𝑓𝑥₀,𝑡≥𝜇2𝑓(𝑥)−𝑓_𝑛(𝑥),𝑡2∗𝜇2𝑓_𝑛(𝑥)−𝑓𝑥₀,𝑡2=𝜇𝑓(𝑥)−𝑓_𝑛(𝑥),𝑡2^𝛽+1∗𝜇𝑓_𝑛(𝑥)−𝑓𝑥₀,𝑡2^𝛽+1≥𝜇𝑓(𝑥)−𝑓_𝑛(𝑥),𝑡2^𝛽+1∗𝜇2𝑓_𝑛(𝑥)−𝑓_𝑛𝑥₀,𝑡2^𝛽+2∗𝜇2𝑓_𝑛𝑥₀−𝑓𝑥₀,𝑡2^𝛽+2=𝜇𝑓(𝑥)−𝑓_𝑛(𝑥),𝑡2^𝛽+1∗𝜇𝑓_𝑛(𝑥)−𝑓_𝑛𝑥₀,𝑡4^𝛽+1∗𝜇𝑓_𝑛𝑥₀−𝑓𝑥₀,𝑡4^𝛽+1≥(1−𝑠)∗(1−𝑠)∗(1−𝑠)>1−𝑟. (27)
This shows that 𝑓(𝑥)∈𝐵(𝑓(𝑥₀),𝑟,𝑡)⊂𝑉. Hence 𝑓(𝑈)⊂𝑉; that is, 𝑓 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 𝑡-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 𝛽-homogeneous fuzzy modular and examined some related topological properties. It should be pointed out that the 𝛽-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 𝛽-homogeneous fuzzy modular spaces.

Further research will focus on the following problems. (1) 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. (2) It has important theoretical values to explore what conditions a fuzzy modular and a fuzzy metric can induce mutually. (3) 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. (4) 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).