Abstract and Applied Analysis

Volume 2015 (2015), Article ID 730940, 10 pages

http://dx.doi.org/10.1155/2015/730940

## On Convergence in -Valued Fuzzy Topological Spaces

^{1}Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt^{2}Department of Mathematics, Faculty of Science, Fayoum University, Fayoum, Egypt^{3}School of Science, Xian Polytechnic University, Xian 710048, China

Received 30 November 2014; Revised 28 December 2014; Accepted 29 December 2014

Academic Editor: Abdelghani Bellouquid

Copyright © 2015 A. A. Ramadan et al. 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.

#### 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

Šostak [1–3] introduced a new definition of -fuzzy topology as the concept of the degree of the openness of fuzzy set. It is an extension of -fuzzy topology defined by Chang [4]. It has been developed in many directions [5–11]. 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 [14–18]. 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 Šostak [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 , 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, 21–23] 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 ( 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*

*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 ’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 , 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 , 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*

*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)if 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, 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, is a fuzzy net. Let . For all ,
Since or is often in , for ,
Similarly, . For

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

*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
Suppose
Then there exist and such that
Since , there exists with is often in such that
Since is often in , for each there exists with and . Hence there exists a cofinal selection such that . Thus is a subnet of and is finally in . It is a contradiction.

From , we have
Conversely, let . Then , for is finally in . Let . Define a relation on by
Then is a directed set. If , then is not finally in . For each , there exists with such that . So, we can define . For each and , there exists with such that . Hence for every , since , we have . Therefore is a cofinal selection on . So is a fuzzy subnet of and is finally to every member of . If is often in , then is not finally of ; that is, . Thus
Since is arbitrary, we complete the proof.

*Theorem 25. Let be an order dense, let be -fuzzy topological space, and let be a fuzzy net. If every subnet of has a subnet of such that , then .*

*Proof. *Let . For each , since has a subnet with , by Theorem 18, we have
Hence, by Theorem 24,
Conversely, by Theorem 18,
Hence, by Theorem 24,
Hence, . Since from Theorem 18, ; that is, .

*Example 26. *Let ( be defined as in Example 21. Let be a natural number set. Define a fuzzy net by
Let . Since is often in , for ,
Since is finally in , for each ,
Thus, since for , does not exists.

Since for , .

*Theorem 27. Let and be -fuzzy topological spaces. For every fuzzy net in , , , and , the following statements are equivalent:(1)*