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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 835308, 10 pages
http://dx.doi.org/10.1155/2013/835308
Research Article

Some Results on Fuzzy Soft Topological Spaces

1Department of Mathematics, Kocaeli University, Kocaeli 41380, Turkey
2Department of Mathematics, Kafkas University, Kars 36100, Turkey

Received 24 February 2013; Accepted 22 April 2013

Academic Editor: Safa Bozkurt Coskun

Copyright © 2013 Cigdem Gunduz (Aras) and Sadi Bayramov. 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 some important properties of fuzzy soft topological spaces. Furthermore, fuzzy soft continuous mapping, fuzzy soft open and fuzzy soft closed mappings, and fuzzy soft homeomorphism for fuzzy soft topological spaces are given and structural characteristics are discussed and studied.

1. Introduction

In many complicated problems arising in the fields of economics, social sciences, engineering, medical science, and so forth. involving uncertainties, classical methods are found to be inadequate in latest year. A number of theories have been proposed for dealing with uncertainties in an efficient way like theory of fuzzy sets [1], theory of intuitionistic fuzzy sets [2], theory of vague sets, theory of interval mathematics [3, 4], and theory of rough sets [5]. However, these theories have their own difficulties.

In 1999, Molodtsov [6] introduced the concept of soft set which is completely new approach for modeling uncertainties. He applied several directions for the applications of soft sets, such as smoothness of functions, game theory, Riemann-integration, and theory of probability. Now, soft set theory and its applications are progressing rapidly in different fields. Maji et al. [7, 8] gave some new definitions on soft sets and presented an application of soft sets in decision making problems. Chen et al. [9], Feng et al. [10], Aktaş and Çağman [11], Sun et al. [12], and Ali et al. [13] improved the works of Maji et al. [7] that initiated the study involving both fuzzy sets and soft sets. Later some researchers studied the concept of fuzzy soft sets [1417]. Moreover, Shabir and Naz [18] presented soft topological spaces and defined some concepts of soft sets on this spaces and separation axioms. Later Tanay and Kandemir [19] initially gave the concept of fuzzy soft topology using the fuzzy soft sets and gave the basic notions of it by following Chang [20]. Pazar Varol and Aygun [21] defined fuzzy soft topology in Lowen’s sense.

The purpose of this paper is to discuss some important properties of fuzzy soft topological spaces. Firstly, we recall some basic definitions in fuzzy soft sets and the results from the literature. The newly introduced concept of parameters comes into play with the collection of parameterization fuzzy topologies on the initial universe. Later we investigate the properties of fuzzy soft interior and fuzzy soft closure of fuzzy sets. We can say that a fuzzy soft topological space gives a parameterized family of fuzzy topologies on the initial universe, but the converse is not true. Finally fuzzy soft continuous mapping, fuzzy soft open and fuzzy soft closed mappings, and fuzzy soft homeomorphism for fuzzy soft topological spaces are investigated, and some interesting results that may be of value for further research are obtained.

2. Preliminaries

In this section, we will introduce necessary definitions and theorems for fuzzy soft sets.

Definition 1 (see [6]). Let be an initial universe set and be a set of parameters. A pair is called a soft set over if and only if is a mapping from into the set of all subsets of the set , that is, , where is the power set of .

Definition 2 (see [15]). Let denote the set of all fuzzy sets on and . A pair is called a fuzzy soft set over , where is a mapping from into . That is, for each is a fuzzy set on .

Definition 3 (see [15]). For two fuzzy soft sets and over a common universe , we say that is a fuzzy soft subset of if(i),(ii)for each , ; that is, is a fuzzy subset of .This relationship is denoted by . Similarly, is said to be a fuzzy soft superset of if is a fuzzy soft subset of . This relationship is denoted by .

Definition 4 (see [15]). Two fuzzy soft sets and over a common universe are said to be fuzzy soft equal if is a fuzzy soft subset of and is a fuzzy soft subset of .

Definition 5 (see [15]). The union of two fuzzy soft sets and over a common universe is the fuzzy soft set , where , and, for all , This relationship is denoted by .

Definition 6 (see [15]). The intersection of two fuzzy soft sets and over a common universe is the fuzzy soft set , where , and, for all , . This is denoted by .

Definition 7 (see [14]). A fuzzy soft set over is said to be a null fuzzy soft set and is denoted by if and only if, for each , , where is the membership function of null fuzzy set over , which takes value 0 for all in .

Definition 8 (see [14]). A fuzzy soft set over is said to be an absolute fuzzy soft set and is denoted by if and only if, for each , , where is the membership function of absolute fuzzy set over , which takes value 1 for all in .

Definition 9 (see [15]). The complement of a fuzzy soft set is the fuzzy soft set , which is denoted by and where is a fuzzy set-valued function that is, for each , is a fuzzy set in , whose membership function for all . Here is the membership function of .

Let be a family of fuzzy soft sets over where, for each , is a fuzzy set-valued function, and each member of this family is denoted , that is, for each , . Then the union and the intersection of this family are defined as follows.

Definition 10 (see [10, 14]). Let . The union of nonempty family of fuzzy soft sets over a common universe , denoted by , is defined as the fuzzy soft set such that and for each , for all , where . Here is the membership function of the fuzzy set for each .

Definition 11 (see [10, 14]). Let . The intersection of nonempty family of fuzzy soft sets over a common universe , denoted by , is defined as the fuzzy soft set such that and for each , for all , where . Here is the membership function of the fuzzy set for each .

Definition 12 (see [18]). Let be the collection of soft sets over , then is said to be a soft topology on if (1) belong to ,(2)the union of any number of soft sets in belongs to ,(3)the intersection of any two soft sets in belongs to .The triplet is called a soft topological space over .
Let be the family of all fuzzy soft sets over .

Definition 13 (see [19]). A fuzzy soft topological space is a pair where is a nonempty set and is a family of fuzzy soft sets over satisfying the following properties:(i), (ii)if , then ,(iii)if , for all , then . is called a topology of fuzzy soft sets on . Every member of is called fuzzy soft open set. A fuzzy soft set is called -closed if and only if its complement is -open.

Definition 14 (see [21]). Let   be a fuzzy soft topological space and . The fuzzy soft closure of , denoted , is the intersection of all fuzzy soft closed supersets of .

Definition 15 (see [21]). Let be a fuzzy soft topological space and . The fuzzy soft interior of denoted by , is the union of all fuzzy soft open subsets of .

3. Fuzzy Soft Topology

Tanay and Kandemir [19] introduced the notion of fuzzy soft topology. Now we introduce some important properties of fuzzy soft topological spaces and define the fuzzy soft closure (fuzzy soft boundary) of a fuzzy soft set. From now on, let be an initial universe set and be the nonempty set of parameters be a fuzzy soft set. Let denote the set of all fuzzy sets on .

Proposition 16 (see [21]). Let be a fuzzy soft topological space over . Then the collection , for each , defines a fuzzy topology on .

Proposition 16 shows that, corresponding to each parameter , we have a fuzzy topology on . Thus a fuzzy soft topology on gives a parameterized family of fuzzy topologies on .

Example 17. Let , , and , where , and are fuzzy soft sets over , for , be defined as follows:Then defines a fuzzy soft topology on and hence is a fuzzy soft topological space over . Then it can be easily seen that are fuzzy topologies on .

Now we show that the converse of the above proposition does not hold.

Example 18. Let and . The fuzzy soft sets on , for , are defined as follows: Then are fuzzy topologies on . Here is not a fuzzy soft topology on because and so .

Proposition 19. Let and be two fuzzy soft topological spaces over the same universe , and then is a fuzzy soft topological spaces over .

Proof. (1) belong to .
(2) Let be a family of fuzzy soft sets in . Then and , for all . Thus and . This implies that .
(3) Let . Then and . Since and , so .

Notice that defines a fuzzy soft topology on and is a fuzzy soft topological space over .

Remark 20. The union of two fuzzy soft topologies on may not be a fuzzy soft topology on .

Example 21. Let , and , be two soft topologies defined on . The fuzzy soft sets on , for , are defined as follows:We define .
If we take then .

Definition 22. Let be a fuzzy soft topological space over and be a fuzzy soft set over . Then we associate with a fuzzy soft set over , denoted by and defined as , where is the closure of fuzzy set in for each .

Proposition 23. Let be a fuzzy soft topological space over and be a fuzzy soft set over  . Then .

Proof. For any is the smallest closed set in which contains . So and then is also a closed set in containing . This implies that . Thus .

Corollary 24. Let be a fuzzy soft topological space over   and be a fuzzy soft set over  . Then if and only if .

Proof. If , then is fuzzy soft closed set and so . Conversely if , then is a fuzzy soft closed set containing . By the definition of fuzzy soft closure of , . By proposition . Thus .

Example 25. Let , and , where and then is a fuzzy soft topological space over . If is defined as follows: then
From the simple calculation, are obtained. It is clear that

Definition 26. Let be a fuzzy soft topological space over and be a fuzzy soft set over . Then we associate with a fuzzy soft set over , denoted by and defined as , where is an interior of fuzzy set in , for each .

Proposition 27. Let be a fuzzy soft topological space over and be a fuzzy soft set over . Then .

Proof. For each is the biggest open fuzzy set in which is belonging . Since , then is satisfied.

Corollary 28. Let be a fuzzy soft topological space over and be a fuzzy soft set over . Then if and only if is a fuzzy soft closed set.

Proof. If , then is a fuzzy soft open set, and so is a fuzzy soft closed set. Conversely if is a fuzzy soft closed set, then is a fuzzy soft open set and . By the definition of fuzzy soft interior of is satisfied. Then by Proposition 27, since is obtained.

Example 29. Let and be a fuzzy soft topology over , and the fuzzy soft sets are defined as follows: Now we define fuzzy soft set as follows: Then Here Clearly, .

Definition 30. Let and be two fuzzy soft sets. The difference of two soft sets and over , denoted by , is defined as .

Definition 31. Let be a fuzzy soft topological space over and be a fuzzy soft set over . Then the fuzzy soft boundary of , denoted by , is defined as .

Theorem 32. Let be a fuzzy soft topological space over and be a fuzzy soft set over . Then(1),(2),(3),(4).

Proof. (1) Let be a fuzzy soft set over . Then we have
(2) From Definition 31, it follows that
(3) Suppose that . Then . Hence is obtained. Conversely, suppose that . From Definition 31, This implies that .
(4) Let . Then is a fuzzy soft closed set. Hence is obtained. Conversely, . From Definition 31, It is clear that . This implies that .

4. Fuzzy Soft Continuous Mappings

Definition 33 (see [21]). Let and be two fuzzy soft topological spaces and be a mapping. For each , if , then is said to be fuzzy soft continuous mapping of fuzzy soft topological spaces.
Let us investigate some properties of fuzzy soft continuous mappings.

Proposition 34. If mapping is a fuzzy soft continuous mapping, then, for each ,  is a fuzzy continuous mapping.

Proof. Let . Then there exists a fuzzy soft open set over such that . Since is a fuzzy soft continuous mapping, is a fuzzy soft open set over and is a fuzzy open set. This implies that is a fuzzy continuous mapping.

Now we give an example to show that the converse of above proposition does not hold.

Example 35. Let , and .

Here are fuzzy soft sets over and are fuzzy soft sets over , defined as follows: If we get the mapping defined as then is not a fuzzy soft continuous mapping. Since we have . Also, The mapping is a fuzzy continuous mapping. Because hence, is obtained.

Similarly, the mapping is a fuzzy continuous mapping. Because thus is obtained.

Definition 36. Let and be two fuzzy soft topological spaces and be a mapping.(a)If the image of each fuzzy soft open set over is a fuzzy soft open set in , then is said to be fuzzy soft open mapping. (b)If the image of each fuzzy soft closed set over is a fuzzy soft closed set in , then is said to be fuzzy soft closed mapping.

Proposition 37. If is a fuzzy soft open (closed), then, for each is fuzzy open (closed) mapping.

Proof. The proof is straightforward.

Theorem 38. Let and be two fuzzy soft topological spaces and be a mapping. (a) is a fuzzy soft open mapping if and only if, for each fuzzy soft set over , is satisfied.(b) is a fuzzy soft closed mapping if and only if for each fuzzy soft set over , is satisfied.

Proof. (a) Let be a fuzzy soft open mapping and be a fuzzy soft set over . is a fuzzy soft open set and . Since is a fuzzy soft open mapping, is a fuzzy soft open set in . Then . Thus is obtained.
Conversely, let be any fuzzy soft open set over . Then . From the condition of theorem, we have . Then . This implies that . This completes the proof.
(b) Let be a fuzzy soft closed mapping and be any fuzzy soft set over . Since is a fuzzy soft closed mapping, is a fuzzy soft closed set over and . Thus is obtained.
Conversely, let be any fuzzy soft closed set over . From the condition of theorem, . This means that . This completes the proof.

Note that the concepts of fuzzy soft continuous, fuzzy soft open, and fuzzy soft closed mappings are all independent of each other.

Example 39. Let , and . We define the mapping as.
Here the fuzzy soft sets are defined as follows: Here and is not a fuzzy soft closed set. Thus the mapping is not fuzzy soft continuous. Also it is not fuzzy soft closed mapping. But it is fuzzy soft open mapping.

Example 40. Let and . We define the mapping as .
Here the fuzzy soft sets are defined as follows: The mapping is a fuzzy soft closed mapping, but it is not both fuzzy soft open mapping and fuzzy soft continuous mapping.

Definition 41. Let and be two fuzzy soft topological spaces and be a mapping. If is a bijection, fuzzy soft continuous and is fuzzy soft continuous mapping, then is said to be fuzzy soft homeomorphism from to . When some homeomorphism exists, we say that is fuzzy soft homeomorphic to .

Theorem 42. Let and be two fuzzy soft topological spaces and be a bijective mapping. Then the following conditions are equivalent: (1)