Research Article | Open Access

A. A. Ramadan, M. El-Dardery, Hu Zhao, "On Convergence in -Valued Fuzzy Topological Spaces", *Abstract and Applied Analysis*, vol. 2015, Article ID 730940, 10 pages, 2015. https://doi.org/10.1155/2015/730940

# On Convergence in -Valued Fuzzy Topological Spaces

**Academic Editor:**Abdelghani Bellouquid

#### Abstract

We introduce the concept of *L*-fuzzy neighborhood systems using complete *MV*-algebras and present important links with the theory of *L*-fuzzy topological spaces. We investigate the relationships among the degrees of *L*-fuzzy *r*-adherent points (*r*-convergent, *r*-cluster, and *r*-limit, resp.) in an *L*-fuzzy topological spaces. Also, we investigate the concept of *LF*-continuous functions and their properties.

#### 1. Introduction

Å ostak [1â€“3] introduced a new definition of -fuzzy topology as the concept of the degree of the openness of fuzzy set. It is an extension ofâ€‰â€‰-fuzzy topology defined by Chang [4]. It has been developed in many directions [5â€“11]. The study of neighborhood systems and convergence of nets in Chang fuzzy topology was initiated by Pao-Ming and Ying-Ming [11] and Liu and Luo [12]. In [13] Ying introduced the degree to which a fuzzy point belongs to a fuzzy subset by and gave the idea of graded neighborhood on fuzzy topological spaces. This plays an important role in the theory of convergence in Chang fuzzy topology see also [14â€“18]. Following Ying [13], Demirci [5] introduced the idea of graded neighborhood systems in smooth toplogical spaces [19] (a smooth topology is similar to fuzzy topology as defined by Å ostak [1], Hazra and Samanta [6]) in a different approach but restricted himself to the -valued fuzzy sets.

In this paper, we study the concept of -fuzzy neighborhood systems and present important links with the theory of -fuzzy topological spaces and investigate some of their properties. We investigate the relationships among the degrees of -fuzzy* r*-adherent points (*r*-convergent,* r*-cluster, and* r*-limit, resp.) nets in an -fuzzy topological spaces. Also, we give some related examples to illustrate some of the introduced notions. In the end, we characterize -continuous functions in terms of some of the various notions introduced in this paper.

#### 2. Preliminaries

Throughout the text we consider as a completely distributive lattice with and , respectively, being the universal upper and lower bound and . A lattice is called order dense if for each such that , there exist such that . If is a completely distributive lattice and , then there must be such that , where means ,â€‰â€‰ such that . If and , we always assume [20] and some properties of can be found in [12].

A completely distributive lattice (or , in short) is called a residuated lattice [9, 21â€“23] if it satisfies the following conditions: for each ,(R1) is a commutative monoid,(R2)if , then ( is isotone operation),(R3)(Galois correspondence) .

In a residuated lattice , is called complement of .

A residuated lattice is called a -algebra [9, 21, 23] if it satisfies the following conditions: for each ,(B1), (B2), (B3).

A -algebra is called an -algebra if , for each .

Lemma 1 (see [9, 21, 23]). *Let be a complete MV-algebra. For each , , one has the following properties:*(1)

*,*(2)

*,*(3)

*If , , and ,*(4)

*,*(5)

*iff ,*(6)

*,*(7)

*,*(8)

*,*(9)

*,*(10)

*and ,*(11)

*,*(12)

*,*(13)

*,*(14)

*,*(15)

*and .*

In this paper, we always assume that is a complete* MV*-algebra. Let be a nonempty set, and the family denotes the set of all -fuzzy subsets of a given set . For , we denote , , and as , , and .

A fuzzy point for is an element of such that The set of all fuzzy points in is denoted by . For and if and only if .

Given a mapping , we write for the mapping defined by ; we write for the mapping defined by for all .

For a given set , define a binary mapping as For each , can be interpreted as the degree to which is fuzzy included in . It is called the -fuzzy inclusion order [24].

Lemma 2 (see [24]). *For each , and , the following properties hold:*(1)*,
*(2)* and , for any ,*(3)*, for any ,*(4)* if and only if and ,*(5)*,
*(6)*, for any ,*(7)*, for any .*

Lemma 3 (see [16]). *Let be a mapping. Then the following statement hold:*(1)*, for each *(2)*, for each .*

In particular, if the mapping is bijective, and then the equalities hold.

*Definition 4 (see [1, 9]). *A map is called an *-fuzzy topology* on if it satisfies the following conditions:(LO1),(LO2), for all ,(LO3), for any .The pair is called an *-fuzzy topological space*.

Let and be -fuzzy topologies on . We say that is* finer* than â€‰â€‰( is* coarser* than ), denoted by , if for all . Let and be -fuzzy topological space spaces. A map is -fuzzy continuous (-continuous, for short) if .

Theorem 5 (see [7, 9]). *Let be an -fuzzy topological space. For each and , one defines operators as follows:
*

For each and , one has the following properties:(I1),(I2),(I3)if and , then ,(I4),(I5), (I6).

*Definition 6 (see [12]). *Let be a directed set. A function is called a fuzzy net in . Let , and one says that is a fuzzy net in if for every .

*Definition 7 (see [12, 25]). *Let be a fuzzy net and .(1) is often in if for each , there exists such that and .(2) is finally in if there exists such that for each with , one has .

*Definition 8 (see [12, 25]). *Let and be two fuzzy nets. A fuzzy net is called a subnet of if there exists a function , called by a cofinal selection on , such that(1);
(2)for every , there exists such that , for .

#### 3. -Fuzzy Neighborhood Systems

*Definition 9. *Let and . Then the degree to which belongs to is

*Definition 10. *Let be an -fuzzy topological space, , , and . The degree to which is a* r*-neighborhood of is defined by
A mapping is called* the **-fuzzy neighborhood system* of .

Theorem 11. *Let be an -fuzzy topological space and let be the fuzzy neighborhood system of . For all and , the following properties hold:*(1)* and ,*(2)*,*(3)*, if ,*(4)*, if ,*(5)*,
*(6)(7)*.*

*Proof. *, , and are easily proved.

is proved from the following:

In if , then and , and there exists with and such that . Again, there exists with and such that . So, , , and . Hence,

In if , then , for each . It implies

is proved from

Theorem 12. *Let be a nonempty set. Let for each , and satisfying the above conditions . Define by
**Then one has the following:*(a)* is an -fuzzy topology on ;*(b)*if is the -fuzzy neighborhood system of induced by , then ;*(c)*if â€™s satisfy the conditions (6) and (7), then
*(d)*.*

*Proof. *(a) (LO1) It is easily proved from Theorem 11.

(LO2) It is proved from the following:

(LO3) If , then for each , and note that
so there exists , with such that . Put , and then . By Theorem 11, we have
It implies . Furthermore, by Lemma 2, we have
So . Hence, . Therefore, .

(b) If , then there exists with such that . Since
then, for each ,

Thus, . So . Hence, . We can easily obtain .

(c) We only show that

if and only if .

It is trivial.

From condition (7),

(d) From the proof of Theorem 11, we easily obtain .

If , there exists with ,â€‰â€‰ such that . Note that
and there exists with such that (thus ). So . Therefore, .

By Theorem 12, we have the following corollary.

Corollary 13. *The set of all -fuzzy topologies on and the set of all -fuzzy neighborhood systems on are in one to one correspondence.*

*Example 14. *Let ,â€‰â€‰ be a set, , and let be defined as follows:
We define an -fuzzy topology on as
From Definition 10, as follows:
From Theorem 12(c), we have

#### 4. R-Convergence

*Definition 15. *Let be an -fuzzy topological space, , and . The degree to which a fuzzy net in is* r*-convergent to and is* r*-cluster to are defined, respectively, as follows:

*Definition 16. *Let be be an -fuzzy topological space, , and . The degree to which is* r*-adherent point of is defined by

Proposition 17. *Let be an -fuzzy topological space. For each and , one has*(1)*,
*(2)*,
*(3)

*Proof. *(1) From Lemma 2, we have

(2) From Theorem 5, we have

(3) From Theorem 11, we have

Theorem 18. *Let be an -fuzzy topological space. Let be fuzzy net and let be a subnet of . For , the following properties hold:*(1)*if , , and ,*(2)*,
*(3)*,
*(4)*,
*(5)*, and *

*Proof. *(1) is easily proved.

In (2) if is finally in , is often in . Hence

In (3) if is finally in , is finally in . Hence

In (4) let be often in . We will show that is often in . Let . Since is a subnet of , there exists a cofinal selection . For each , there exists such that for . Since is often in , for , there exists such that for . Put . Then and . Thus, is often in . Hence

In (5) one has
The other case is the same.

Proposition 19. *Let be an -fuzzy topological space, let be a fuzzy net, , and . Then one has
*

*Proof. *Since is finally in , is often in . We easily show that
We only show that
Let . If , then . Put . Define a relation on by
For each , since by Theorem 11,
Hence, and . Thus, is a directed set. For each , that is, , we have ; that is, there exists such that . Thus, we can define a fuzzy net by where and .

We will show that if , then is not often in . Suppose that is often in . For , there exists such that such that
and . Since implies , it implies
It is contradiction for the definition of . Thus, if is often in , then ; that is, . Therefore,

Theorem 20. *Let be -fuzzy topological space and let be fuzzy nets such that for each . Define fuzzy nets by, for each ,
**
For each , the following properties hold:*(1)*ifâ€‰â€‰ for all , then
*(2)*,
*(3)*,
*(4)*,
*(5)*if is order dense, then .*

*Proof. *
In let be finally (often) in . Then let be finally (often) in , respectively. Thus it is trivial. , , and are easily proved.

In since and , by , we have

Suppose that . Since is order dense, then there exist and a fuzzy point such that
Since and , by the definition , there exist such that and are finally in and , respectively, with
Since is finally in , there exists such that for every with . Since is finally in , there exists such that for every with . Let such that and . For , we have
Thus, is finally in . It implies
It is a contradiction. Hence, we have

*Example 21. *Let be defined as Example 14. Let be a set and as follows:
We define -fuzzy topology as follows:

In general, .

Let be a natural numbers. Define fuzzy nets by
From Theorem 20, â€‰â€‰is a fuzzy net. Let . From Definition 15, we have for ,
Since or is finally in ,
Similarly, . For ,

In general, .

Define fuzzy nets by
From Theorem 20,â€‰â€‰â€‰â€‰is a fuzzy net. Let . For all ,
Since or is often in , for ,
Similarly, . For

#### 5. Fuzzy -Limit Nets and -Continuous Mappings

*Definition 22. *Let be an -fuzzy toplogical space. Let be fuzzy net in , , and . Then the degree to which is* r*-limit to is defined, denoted by , if .

Theorem 23. *Let be -fuzzy topological space and let be fuzzy nets such that for each . If is order dense, , and , then
*

*Proof. *From Theorem 20, is a fuzzy net. We easily proved it from the following:

Theorem 24. *Let be -fuzzy topological space. Let be a fuzzy net and . Then, if is an order dense, the following statements hold:*(1)*;*(2)*.*

*Proof. * For each , by Theorem 18, we have
Hence