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
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.
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 . 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  and Liu and Luo . In  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 , Demirci  introduced the idea of graded neighborhood systems in smooth toplogical spaces  (a smooth topology is similar to fuzzy topology as defined by Šostak , Hazra and Samanta ) 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.
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  and some properties of can be found in .
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 -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 .
Lemma 2 (see ). 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 ). 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.
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 .
For each and , one has the following properties:(I1),(I2),(I3)if and , then ,(I4),(I5), (I6).
Definition 6 (see ). 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 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 ’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.
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)
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 .
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
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