JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 576237 10.1155/2013/576237 576237 Research Article On Fuzzy Modular Spaces Shen Yonghong 1, 2 Chen Wei 3 Herrera Luis Javier 1 School of Mathematics Beijing Institute of Technology Beijing 100081 China bit.edu.cn 2 School of Mathematics and Statistics Tianshui Normal University Tianshui 741001 China tsnc.edu.cn 3 School of Information Capital University of Economics and Business Beijing 100070 China cueb.edu.cn 2013 25 3 2013 2013 12 11 2012 30 01 2013 18 02 2013 2013 Copyright © 2013 Yonghong Shen and Wei Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 . Soon after, Musielak and Orlicz  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:

ρ(x)=0 if and only if x=θ (i.e., x is the null vector θ),

ρ(x)=ρ(-x),

ρ(αx+βy)ρ(x)+ρ(y) for all x,yX and α,β0 with α+β=1.

A modular space Xρ is defined by a corresponding modular ρ, that is, Xρ={xX:ρ(λx)0asλ0}.

Based on definition of the modular space, Kozłowski [3, 4] introduced the notion of modular function space. In the sequel, Kozłowski and Lewicki  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  studied the existence of best approximations in modular function spaces by elements of sublattices. In 1990, Khamsi et al.  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 .

In 2007, Nourouzi  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 xX, the distribution function ρ(x) is denoted by ρx, and ρx(t) is the value ρx at t) satisfying the following conditions:

ρx(0)=0,

ρx(t)=1 for all t>0 if and only if x=θ,

ρ-x(t)=ρx(t),

ραx+βy(s+t)ρx(s)ρy(t) for all x,yX and α,β,s,t0+, α+β=1.

Especially, for every xX,  t>0 and α{0}, if (1)ραx(t)=ρx(t|α|β),where  β(0,1], then we say that (X,ρ) is β-homogeneous.

Recently, further studies have been made on the probabilistic modular spaces. Nourouzi  extended the well-known Baire's theorem to probabilistic modular spaces by using a special condition. Fallahi and Nourouzi  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 , 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 [<xref ref-type="bibr" rid="B21">18</xref>]).

A binary operation *:[0,1]×[0,1][0,1] is called a continuous t-norm if it satisfies the following conditions:

* is commutative and associative;

* is continuous;

a*1=a for every a[0,1];

a*bc*d whenever ac, bd 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 .

Definition 2 (George and Veeramani [<xref ref-type="bibr" rid="B5">17</xref>]).

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,zX,  s,t>0:

M(x,y,t)>0,

M(x,y,t)=1 if and only if x=y,

M(x,y,t)=M(y,x,t),

M(x,y,t)*M(y,z,s)M(x,z,t+s),

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, -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,yX,  s,t>0 and α,β0 with α+β=1:

μ(x,t)>0,

μ(x,t)=1 for all t>0 if and only if x=θ,

μ(x,t)=μ(-x,t),

μ(αx+βy,s+t)μ(x,s)*μ(y,t),

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 xX,  t>0 and λ{0}, (2)μ(λx,t)=μ(x,t|λ|β),where  β(0,1].

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 xX. Then (X,μ,*) is a -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 (3)μ(x,t)=1eρ(x)/t for all xX. Then (X,μ,*) is a β-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 μ(αx+βy,s+t)μ(x,s)*μ(y,t), as α,β0 with α+β=1.

Since ρ is a modular on X, for all x,yX, we have (4)ρ(αx+βy)ρ(x)+ρ(y).

Then, we can obtain (5)ρ(αx+βy)t+stρ(x)+t+ssρ(y), that is, (6)1t+sρ(αx+βy)1tρ(x)+1sρ(y).

Therefore (7)eρ(αx+βy)/(t+s)eρ(x)/t·eρ(y)/s=eρ(x)/t*Peρ(y)/s.

Thus, we have μ(αx+βy,s+t)μ(x,s)*μ(y,t).

On the other hand, for all λ{0}, since ρ(λx)=|λx|β=|λ|β·|x|β=|λ|βρ(x), it follows that (8)μ(λx,t)=μ(x,t|λ|β).

Hence, we know that (X,μ,*) is a β-homogeneous -modular space.

Theorem 7.

If (X,μ,*) is a -modular space, then μ(x,·) is nondecreasing for all xX.

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 (9)μ(x,s)*μ(θ,t-s)=μ(x,s)*μ(y,t-s)μ(αx+βy,t)=μ(x,t)<μ(x,s).

It should be noted that, in general, a fuzzy modular and a fuzzy metric (in the sense of George and Veeramani ) 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=. Take t-norm a*b=a*Mb. For every t(0,), we define (10)μ(x,t)=kk+|x|, 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, and α,β0 with α+β=1. Without loss of generality, we assume that |x||y|. Since |αx+βy||y|, we then obtain (11)μ(αx+βy,t+s)=kk+|αx+βy|kk+|y|=min{kk+|x|,kk+|y|}=μ(x,t)*Mμ(y,s).

Hence (μ,*M) is a fuzzy modular on X. However, if we set (12)M(x,y,t)=μ(x-y,t)=kk+|x-y|, it is easy to verify that (M,*M) is not a fuzzy metric on X.

Example 9.

Let X=. Take t-norm a*b=a*Mb. For every x,yX and t(0,), we define (13)M(x,y,t)={1,x=y,12,xy,x,y,14,x,yor  x,y,14,xy,x,y.

It can easily be shown that (M,*M) is a fuzzy metric on X. Set (14)μ(x,t)=M(x,θ,t)={1,x=0,12,x{0},14,x.

If we take α=2/2, β=1-2/2, xy, and x,y, then we know that αx+βy. 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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M191"><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:math></inline-formula>-Homogeneous Fuzzy Modular

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

Definition 10.

Let (X,μ,*) be a -modular space. The μ-ball B(x,r,t) with center xX and radius r, 0<r<1,  t>0 is defined as (15)B(x,r,t)={yX:μ(x-y,t)>1-r}.

An element xE 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 [<xref ref-type="bibr" rid="B5">17</xref>]).

If the t-norm * is continuous, then

for every r1,r2(0,1) with r1>r2, there exists r3(0,1) such that r1*r3r2,

for every r4(0,1), there exists r5(0,1) such that r5*r5r4.

Theorem 12.

If (X,μ,*) is a β-homogeneous -modular space, then B(x,r,t/2β+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/2β+1), it is obvious that {yX:μ(x-y,t/2β+1>1-r)}{yX:μ(x-y,t/2)>1-r}.

Theorem 13.

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

Proof.

By Definition 10, for every yB(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 ϵy>0 such that μ(x-y,(t1-ϵ)/2β-1)>1-r for some ϵ>0 with (t1-ϵ)/2β-1>0 and ϵ/2β-1(0,ϵy). Set r0=μ(x-y,(t1-ϵ)/2β-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*r11-s.

Next, we show that B(y,1-r1,ϵ/2β-1)B(x,r,2t1). For every zB(y,1-r1,ϵ/2β-1), we have μ(y-z,ϵ/2β-1)>r1. Therefore, (16)μ(x-z,t)=μ(x-z,2t1)μ(2(x-y),2(t1-ϵ))*μ(2(y-z),2ϵ)=μ(x-y,t1-ϵ2β-1)*μ(y-z,ϵ2β-1)r0*r11-s>1-r. Thus zB(x,r,t) and hence B(y,1-r1,ϵ/2β-1)B(x,r,t).

Theorem 14.

Let (X,μ,*) be a β-homogeneous -modular space. Define (17)𝒯μ={AX:xA  ifandonlyifthereexist  t>0  andr(0,1)  suchthat  B(x,r,t)A{AX:xA}}. Then 𝒯μ is a topology on X.

Proof.

The proof will be divided into three parts.

Obviously, ,X𝒯μ.

Suppose that A,B𝒯μ. If xAB, then xA and xB.

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 yB(x,r,t), then we know that μ(x-y,t)>1-r. According to Theorem 7, we can obtain

(18)μ(x-y,t1)μ(x-y,t)>1-r1-r1.

Thus, yB(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)AB. That is to say, AB𝒯μ.

Suppose that 𝒯μ𝒯μ. If xA𝒯μA, then there exists U𝒯μ such that xU. Since U𝒯μ, there exist 0<r<1 and t>0 such that B(x,r,t)UA𝒯μA. Hence, A𝒯μA𝒯μ.

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 𝒯μ induced by a β-homogeneous -modular space is first countable.

Theorem 16.

Every β-homogeneous -modular space is Hausdorff.

Proof.

For the β-homogeneous -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*r1r0.

Next, we consider the μ-balls B(x,1-r1,t/2β+1) and B(y,1-r1,t/2β+1) and then show that B(x,1-r1,t/2β+1)B(y,1-r1,t/2β+1)= using reduction to absurdity. If there exists zB(x,1-r1,t/2β+1)B(y,1-r1,t/2β+1), then (19)r=μ(x-y,t)μ(2(x-z),t2)*μ(2(z-y),t2)=μ(x-z,t2β+1)*μ(z-y,t2β+1)r1*r1r0, 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 -modular space.

Definition 17.

Let (X,μ,*) be a -modular space.

A sequence {xn} in X is said to be μ-convergent to a point xX, denoted by xnμx, if for every r(0,1) and t>0, there exists n0 such that xnB(x,r,t) for all nn0.

A subset AX 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 xA.

A subset BX 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).

A subset CX is called a μ-closed if and only if for every sequence {xn}C, xnμx implies xC.

Theorem 18.

Every μ-compact subset A of a β-homogeneous -modular space (X,μ,*) is μ-bounded.

Proof.

Suppose that A is a μ-compact subset of the given β-homogeneous -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/2β+1):xA} is a μ-open cover of A. Since A is μ-compact, there exist x1,x2,,xnA such that Ai=1nB(xi,r,t/2β+1). For every xA, there exists i such that xB(xi,r,t/2β+1). Therefore, we have μ(x-xi,t/2β+1)>1-r. Set α=min{μ(xi,t/2β+1):1in}. Clearly, we know that α>0. Thus, we have (20)μ(x,t)=μ((x-xi)+xi,t)μ(2(x-xi),t2)*μ(2xi,t2)=μ(x-xi,t2β+1)*μ(xi,t2β+1)(1-r)*α>1-s for some s(0,1). This shows that A is μ-bounded.

Theorem 19.

Let (X,μ,*) be a β-homogeneous -modular space, and let 𝒯μ be the topology induced by the β-homogeneous modular. Then for a sequence {xn} in X, xnμx if and only if μ(x-xn,t)1 as n.

Proof.

Fix t>0. Suppose that xnμx. Then for every r(0,1), there exists n0 such that xnB(x,r,t) for all nn0. Namely, μ(xn-x,t)>1-r for all nn0. Thus, we have 1-μ(xn-x,t)<r for all nn0. Because r is arbitrary, we can verify that μ(xn-x,t)1 as n.

On the other hand, if for every t>0, μ(x-xn,t)1 as n, then for every r(0,1), there exists n0 such that 1-μ(x-xn,t)<r for all nn0. Therefore, we know that μ(x-xn,t)>1-r for all nn0. Thus xnB(x,r,t) for all nn0, and hence xnμx as n.

4. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M423"><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:math></inline-formula>-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 -modular space.

A sequence {xn} in X is a μ-Cauchy sequence if and only if for every ϵ(0,1) and t>0, there exists n0 such that μ(xm-xn,t)>1-ϵ for all m,nn0.

The -modular space (X,μ,*) is called μ-complete if every μ-Cauchy sequence is μ-convergent.

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

Theorem 21.

Let (X,μ,*M) be a β-homogeneous -modular space. Then every μ-convergent sequence {xn} in X is a μ-Cauchy sequence.

Proof.

Suppose that the sequence {xn}  μ-converges to xX. Therefore, for every ϵ(0,1) and t>0, there exists n0 such that μ(xn-x,t/2β+1)>1-ϵ for all nn0. For all m,nn0, we have (21)μ(xm-xn,t)μ(2(xm-x),t2)*Mμ(2(xn-x),t2)μ(xm-x,t2β+1)*Mμ(xn-x,t2β+1)>(1-ϵ)*M(1-ϵ)=1-ϵ.

Hence {xn} is a μ-Cauchy sequence in X.

Remark 22.

The proof of Theorem 21 shows that, in the -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 -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 -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 {xn} is a μ-Cauchy sequence of Y¯. Therefore, for every n and t>0, there exists ynY such that μ(xn-yn,t/4β+1)>1-1/(n+1). According to Theorem 7, we have μ(xn-yn,t/2β+1)>1-(1/(n+1)). In addition, for every r(0,1) and t>0, there exists an n0 such that μ(xn-xm,t/4β+1)>1-r for all m,nn0. That is to say, μ(xn-xm,t/4β+1)1 as m,n. Next, we will show that the sequence {yn} is a μ-Cauchy sequence of Y. For every m,nn0, we have (22)μ(yn-ym,t)μ(2(yn-xn),t2)*μ(2(xn-ym),t2)μ(2(yn-xn),t2)*μ(4(xn-xm),t4)*μ(4(xm-ym),t4)=μ(yn-xn,t2β+1)*μ(xn-xm,t4β+1)*μ(xm-ym,t4β+1)>(1-1n+1)*(1-r)*(1-1m+1).

Since t-norm * is continuous, it follows that μ(yn-ym,t)1 as m,n. Now, we assume that the sequence {yn}  μ-converges to xX. Thus, for every ϵ(0,1) and t>0, there exists an n1 such that μ(x-yn,t/2β+1)>1-ϵ for all nn1. Therefore, for all nn1, we can obtain (23)μ(xn-x,t)μ(2(xn-yn),t2)*μ(2(yn-xn),t2)=μ(xn-yn,t2β+1)*μ(yn-xn,t2β+1)>(1-ϵ)*(1-1n+1).

According to the arbitrary of ϵ and by letting n, it follows that limnμ(xn-x,t)=1. That is, an arbitrary μ-Cauchy sequence {xn} of Y¯μ-converges to xX. The proof of the theorem is now completed.

Theorem 25.

Let (X,μ,*) be a β-homogeneous -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 -modular space (X,μ,*) is μ-complete.

Proof.

It follows from Theorem 24.

5. Baire's Theorem and Uniform Limit Theorem

In , 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 Un  (n=1,2,) be a countable number of μ-dense and μ-open sets in the μ-complete β-homogeneous -modular space (X,μ,*M). Then n=1Un 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 ϵy>0 such that μ(x-y,(t-ϵ)/2β-1)>1-r for some ϵ>0 with (t-ϵ)/2β-1>0 and ϵ/2β-1(0,ϵy). Choose r(0,r), ϵ/2β-1(0,ϵy) and zB(y,r,ϵ/4β)¯, there exists a sequence {zn} in B(y,r,ϵ/4β)¯ such that znμz and hence we have (24)μ(z-y,ϵ2β-1)μ(2(z-zn),ϵ2β)*Mμ(2(zn-y),ϵ2β)=μ(z-zn,ϵ4β)*Mμ(zn-y,ϵ4β)>1-r for some n. Therefore, we can obtain (25)μ(x-z,2t)=μ(2(z-y),2ϵ)*Mμ(2(x-y),2(t-ϵ))=μ(z-y,ϵ2β-1)*Mμ(x-y,t-ϵ2β-1)>(1-r)*M(1-r)=1-r.

This shows that B(y,r,ϵ/4β)¯B(x,r,2t). It means that if A is a nonempty μ-open set of X, then AU1 is nonempty and μ-open. Now, let x1AU1, there exist r1(0,1) and t1>0 such that B(x1,r1,t1/2β-1)AU1. Choose r1<r1 and t1=min{t1,1} such that B(x1,r1,t1/2β-1)¯AU1. Since U2 is μ-dense in X, we can obtain B(x1,r1,t1/2β-1)U2. Let x2B(x1,r1,t1/2β-1)U2, there exist r2(0,1/2) and t2>0 such that B(x2,r2,t2/2β-1)B(x1,r1,t1/2β-1)U2. Choose r2<r2 and t2=min{t2,1/2} such that B(x2,r2,t2/2β-1)¯AU2. By induction, we can obtain a sequence {xn} in X and two sequence {rn},  {tn} such that 0<rn<1/n, 0<tn<1/n and B(xn,rn,tn/2β-1)¯AUn.

Next, we show that {xn} is a μ-Cauchy sequence. For given t>0 and r(0,1), we can choose k such that 2tk<t and rk<r. Then for m,nk, since xm,xnB(xk,rk,tk/2β-1), we have (26)μ(xm-xn,2t)μ(xm-xn,4tk)μ(2(xm-xk),2tk)*Mμ(2(xn-xk),2tk)=μ(xm-xk,tk2β-1)*Mμ(xn-xk,tk2β-1)1-rk>1-r.

According to the arbitrary of t, it follows that {xn} is a μ-Cauchy sequence. Since X is μ-complete, there exists xX such that xnμx. But xnB(xk,rk,tk/2β-1) for all nk, and therefore xB(xk,rk,tk/2β-1)  ¯AUk for all k. Thus A(n=1Un). Hence n=1Un is μ-dense in X.

Definition 27.

Let X be any nonempty set and let (Y,μ,*) be a -modular space. A sequence {fn} 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 such that μ(fn(x)-f(x),t)>1-r for all nn0 and for every xX.

Theorem 28 (Uniform limit theorem).

Let fn:XY be a sequence of continuous functions from a topological space X to a β-homogeneous -modular space (Y,μ,*). If {fn}  μ-converges uniformly to f:XY, then f is continuous.

Proof.

Let V be a μ-open set of Y and x0f-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 {fn}  μ-converges uniformly to f, given s(0,1) and t>0, there exists n0 such that μ(fn(x)-f(x),t/4β+1)>1-s for all nn0 and for every xX. Moreover, fn is continuous for every n, there exists a neighborhood U of x0 such that fn(U)B(fn(x0),s,t/4β+1). Therefore, we know that μ(fn(x)-fn(x0),t/4β+1)>1-s for every xU. Thus, we have (27)μ(f(x)-f(x0),t)μ(2(f(x)-fn(x)),t2)*μ(2(fn(x)-f(x0)),t2)=μ(f(x)-fn(x),t2β+1)*μ(fn(x)-f(x0),t2β+1)μ(f(x)-fn(x),t2β+1)*μ(2(fn(x)-fn(x0)),t2β+2)*μ(2(fn(x0)-f(x0)),t2β+2)=μ(f(x)-fn(x),t2β+1)*μ(fn(x)-fn(x0),t4β+1)*μ(fn(x0)-f(x0),t4β+1)(1-s)*(1-s)*(1-s)>1-r.

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, 2022], 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).

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