The Scientific World Journal

Volume 2014 (2014), Article ID 164176, 8 pages

http://dx.doi.org/10.1155/2014/164176

Research Article

On Topological Structures of Fuzzy Parametrized Soft Sets

Serkan Atmaca and İdris Zorlutuna

Department of Mathematics, Cumhuriyet University, 58140 Sivas, Turkey

Received 10 February 2014; Accepted 28 March 2014; Published 28 April 2014

Academic Editor: Naim Cagman

Copyright © 2014 Serkan Atmaca and İdris Zorlutuna. 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 topological structure of fuzzy parametrized soft sets and fuzzy parametrized soft mappings. We define the notion of quasi-coincidence for fuzzy parametrized soft sets and investigated its basic properties. We study the closure, interior, base, continuity, and compactness and properties of these concepts in fuzzy parametrized soft topological spaces.

1. Introduction

In 1965, after Zadeh [1] generalized the usual notion of a set with the introduction of fuzzy set, the fuzzy set was carried out in the areas of general theories and applied to many real life problems in uncertain, ambiguous environment. In this manner, in 1968, Chang [2] gave the definition of fuzzy topology and introduced the many topological notions in fuzzy setting.

In 1999, Molodtsov [3] introduced the concept of soft set theory which is a completely new approach for modelling uncertainty and pointed out several directions for the applications of soft sets, such as game theory, perron integrations, and smoothness of functions. To improve this concept, many researchers applied this concept on topological spaces (e.g., [4–9]), group theory, ring theory (e.g., [10–14]), and also decision making problems (e.g., [15–18]).

Recently, researchers have combined fuzzy set and soft set to generalize the spaces and to solve more complicated problems. By this way, many interesting applications of soft set theory have been expanded. First combination of fuzzy set and soft set is fuzzy soft set and it was given by Maji et al. [19]. Then fuzzy soft set theory has been applied in several directions, such as topology (e.g., [20–23]), various algebraic structures (e.g., [24, 25]), and especially decision making (e.g., [26–29]). Another combination of fuzzy set and soft set was given by Çaman et al. [30] who called it fuzzy parametrized soft set (for short FP-soft set). In that paper, Çaman et al. defined operations on FP-soft sets and improved several results. After that, Çaman and Deli [31, 32] applied FP-soft sets to define some decision making methods and applied these methods to problems that contain uncertainties and fuzzy object.

In the present paper, we consider the topological structure of FP-soft sets. Firstly, we give some basic ideas of FP-soft sets and also studied results. We define FP-soft quasi-coincidence, as a generalization of quasi-coincidence in fuzzy manner [33], and use this notion to characterize concepts of FP-soft closure and FP-soft base in FP-soft topological spaces. We also introduce the notion of mapping on FP-soft classes and investigate the properties of FP-soft images and FP-soft inverse images of FP-soft sets. We define FP-soft topology in Chang’s sense. We study the FP-soft closure and FP-soft interior operators and properties of these concepts. Lastly we define FP-soft continuous mappings and we show that image of a FP-soft compact space is also FP-soft compact.

2. Preliminaries

Throughout this paper denotes initial universe, denotes the set of all possible parameters which are attributes, characteristic, or properties of the objects in , and the set of all subsets of will be denoted by .

Definition 1 (see [1]). A fuzzy set in is a function defined as follows:

where .

Here is called the membership function of , and the value is called the grade of membership of . This value represents the degree of belonging to the fuzzy set .

A fuzzy point in , whose value is at the support , is denoted by . A fuzzy point , where is fuzzy set in if .

Definition 2 (see [3]). A pair is called a soft set over if is a mapping defined by .

In other words, a soft set is a parametrized family of subsets of the set . Each set , , from this family may be considered as the set of -elements of the soft set .

Definition 3 (see [30]). Let be a fuzzy set over . A FP-soft set on the universe is defined as follows:

where the function is called approximate function such that if , and the function is called membership function of the set .

From now on, the set of all FP-soft sets over will be denoted by .

Definition 4 (see [30]). Let .

(1) is called the empty FP-soft set if for every , denoted by .(2) is called -universal FP-soft set if and for all , denoted by . If , then -universal FP-soft set is called universal FP-soft set, denoted by .

Definition 5 (see [30]). Let .

(1) is called a FP-soft subset of   if and for every and one writes .(2) and are said to be equal, denoted by if and .(3)The union of and , denoted by , is the FP-soft set, defined by the membership and approximate functions and for every , respectively.(4)The intersection of and , denoted by , is the FP-soft set, defined by the membership and approximate functions and for every , respectively.

Definition 6 (see [30]). Let . Then the complement of , denoted by , is the FP-soft set, defined by the membership and approximate functions and for every , respectively.

Clearly , , and .

Proposition 7 (see [30]). Let , , and . Then

(1);(2);(3), ;(4), ;(5), ;(6) , ;(7), .

3. Some Properties of FP-Soft Sets and FP-Soft Mappings

Definition 8. Let be an arbitrary index set and for all .

(1)The union of ’s, denoted by , is the FP-soft set, defined by the membership and approximate functions and for every , respectively.(2)The intersection of ’s, denoted by , is the FP-soft set, defined by the membership and approximate functions and for every , respectively.

Proposition 9. Let be an arbitrary index set and for all . Then

(1);(2).

Proof. Put and . Then for all ,

This completes the proof. The other can be proved similarly.

Definition 10. The FP-soft set is called FP-soft point if is fuzzy singleton and for . If , , then one denotes this FP-soft point by .

Definition 11. Let , . One says that read as belongs to the FP-soft set if and .

Proposition 12. Every nonempty FP-soft set can be expressed as the union of all the FP-soft points which belong to .

Proof. This follows from the fact that any fuzzy set is the union of fuzzy points which belong to it [33].

Definition 13. Let , . is said to be FP-soft quasi-coincident with , denoted by , if there exists such that or is not subset of . If is not FP-soft quasi-coincident with , then one writes .

Definition 14. Let , . is said to be FP-soft quasi-coincident with , denoted by , if or is not subset of . If is not FP-soft quasi-coincident with , then one writes .

Proposition 15. Let , . Then the following are true:    

(1);(2).(3);(4) there exists an such that ;(5);(6) if , then for all ;

Proof. (1) Consider

(2) Let . Then there exists an such that or is not subset of . If , and the proof is easy. If is not subset of , then . Hence .

(3) Suppose . Then there exists such that or is not subset of . But this is impossible.

(4) If , then there exists an such that or is not subset of . Put and . Then we have and .

Conversely, suppose for some . Then or is not subset of . Therefore, we have or is not subset of for . This shows .

(5) It is obvious from (1).

(6) Let , and . Then or is not subset of . Since , or is not subset of . Hence we have .

Proposition 16. Let be a family of FP-soft sets in , where is an index set. Then is FP-soft quasi-coincident with if and only if there exists some such that .

Proof. Obvious.

Definition 17. Let and be families of all FP-soft sets over and , respectively. Let and be two functions. Then a FP-soft mapping is defined as follows.

(1)For , the image of under the FP-soft mapping is the FP-soft set over defined by the approximate function, ,

where is fuzzy set in .

(2)For , then the preimage of under the FP-soft mapping is the FP-soft set over defined by the approximate function, ,

where is fuzzy set in .

If and are injective, then the FP-soft mapping is said to be injective. If and are surjective, then the FP-soft mapping is said to be surjective. The FP-soft mapping is called constant, if and are constant.

Theorem 18. Let and be crips sets , , , , where is an index set. Let be a FP-soft mapping. Then,

(1)if    then ;(2)if   then ;(3); the equality holds if is injective;(4); the equality holds if is surjective;(5);(6); the equality holds if is injective;(7);(8);(9);(10);(11);(12);(13); the equality holds if is surjective;(14).

Proof. We only prove (3), (5), (7), (9), (11), and (12). The others can be proved similarly.

(3) Put and . Since , it is sufficient to show that for all ,

This completes the proof.

(5) Put and . Then and for all ,

This completes the proof.

(7) Put and . Then and for all ,

This completes the proof.

(9) Put and . Then for all ,

where and are fuzzy sets over . This shows that the approximate functions of and are equal. This completes the proof.

(11) Put . Then for all ,

This shows that .

(12) Since is fuzzy empty set, that is, , the proof is clear.

4. FP-Soft Topological Spaces

Definition 19. A FP-soft topological space is a pair , where is a nonempty set and is a family of FP-soft sets over satisfying the following properties:

(T1);(T2)if , , then ;(T3)if , then .

is called a topology of FP-soft sets on . Every member of is called FP-soft open in . is called FP-soft closed in if .

Example 20. is a FP-soft topology on .

is a FP-soft topology on .

Example 21. Assume that is a universal set and is a set of parameters. If

then is a FP-soft topology on .

Theorem 22. Let be a FP-soft topological space and let denote family of all closed sets. Then,

(1);(2)if , , then ;(3)if , , then .

Proof. Straightforward.

Definition 23. Let be a FP-soft topological space and . The FP-soft closure of in , denoted by , is the intersection of all FP-soft closed supersets of .

Clearly, is the smallest FP-soft closed set over which contains , and is closed.

Theorem 24. Let be a FP-soft topological space and , . Then,

(1) and ;(2);(3);(4)if , then ;(5) is a FP-soft closed set if and only if ;(6).

Proof. (1), (2), (3), and (4) are obvious from the definition of FP-soft closure.

(5) Let be a FP-soft closed set. Since is the smallest FP-soft closed set which contains , then . Therefore, .

(6) Since and , then, by (4), , , and hence .

Conversely, since and are FP-soft closed sets, is a FP-soft closed set. Again since , by (4), then .

Definition 25. Let be a FP-soft topological space. A FP-soft set in is called FP-Q-neighborhood (briefly, FP-Q-nbd) of a FP-soft set if there exists a FP-soft open set in such that and .

Theorem 26. Let , . Then if and only if each FP-Q-nbd of is FP-soft quasi-coincident with .

Proof. Let . Suppose that is a FP-Q-nbd of and . Then there exists a FP-soft open set such that . Since , by Proposition 15(1), . Again since , does not belong to . This is a contradiction with .

Conversely, let each Q-nbd of be FP-soft quasi-coincident with . Suppose that does not belong to . Then there exists a FP-soft closed set which is containing such that does not belong to . By Proposition 15(5), we have . Then is a FP-Q-nbd of and, by Proposition 15(1), . This is a contradiction with the hypothesis.

Definition 27. Let be a FP-soft topological space and . The FP-soft interior of denoted by is the union of all FP-soft open subsets of .

Clearly, is the largest fuzzy soft open set contained in , and is FP-soft open.

Theorem 28. Let be a FP-soft topological space and , . Then,

(1) and ;(2);(3);(4)if , then ;(5) is a FP-soft open set if and only if ;(6).

Proof. Similar to that of Theorem 24.

Theorem 29. Let be a FP-soft topological space and . Then,

(1);(2).

Proof. We only prove (1). The other is similar. Consider

Theorem 30. Let be an operator satisfying the following:

(c1);(c2);(c3);(c4).

Then one can associate FP-soft topology in the following way:

Moreover with this FP-soft topology , for every .

Proof. (T1) By (c1), . By (c2) , so and .

(T2) Let , . By the definition of , and . By (c3), . So .

(T3) Let . Since is order preserving and ,, then . Then we have . Conversely, by (c2) we have . Hence, and .

Now we will show that with this FP-soft topology , for every . Let . Since , then . Since is order preserving, . Conversely, by (c4) we have . Then since and is the smallest FP-soft closed set over which contains , .

The operator is called the FP-soft closure operator.

Remark 31. By Theorem 24(1), (2), (3), and (6) and Theorem 30, we see that with a FP-soft closure operator we can associate a FP-soft topology and conversely with a given FP-soft topology we can associate a FP-soft closure operator.

Theorem 32. Let be an operator satisfying the following:

(i1);(i2), ;(i3), ;(i4), .

Then one can associate a FP-soft topology in the following way:

Moreover, with this fuzzy soft topology , for every .

Proof. Similar to that of Theorem 30.

The operator is called the FP-soft interior operator.

Remark 33. By Theorem 28(1), (2), (3), and (6) and Theorem 32, we see that with a FP-soft interior operator we can associate a FP-soft topology and conversely with a given FP-soft topology we can associate a FP-soft interior operator.

Definition 34. Let be a FP-soft topological space. A subcollection of is called a base for if every member of can be expressed as a union of members of .

Example 35. If we consider the FP-soft topology in Example 21, then one easily sees that the family is a basis for .

Proposition 36. Let be a FP-soft topological space and is subfamily of . is a base for if and only if for each in and for each FP-soft open Q-nbd of , there exists a such that .

Proof. Let be a base for , and let be a FP-soft open Q-nbd of . Then there exists a subfamily of such that . Suppose that for all . Then and for every . Therefore, we have and is not subset of since and . This is a contradiction.

Conversely, if is not a base for , then there exists a such that . Since , there exists such that or . Put and . Then in both cases, we obtain and . Therefore, we have and , that is, for all which is contained in . This is a contradiction.

Definition 37. Let and be two FP-soft topological spaces. A FP-soft mapping is called FP-soft continuous if , .

Example 38. Assume that , are two universal sets, , are two parameter sets, and is a FP-soft mapping, where , , , and and . If we take , , , and , then is a FP-soft continuous mapping.

The constant mapping is not continuous in general.

Example 39. Assume that , are two universal sets, , are two parameter sets, and is a constant FP-soft mapping, where and . If we take , , and , is not a FP-soft continuous since .

Let . We denote by the constant fuzzy set on ; that is, for all and .

Definition 40. Let . is called -universal FP-soft set if and for all , denoted by .

Definition 41 (see [21]). A FP-soft topology is called enriched if it satisfies and for all .

Theorem 42. Let be an enriched FP-soft topological space, a FP-soft topological space, and a constant FP-soft mapping. Then is FP-soft continuous.

Proof. Let . Put . Then , where and

for all . Hence or and so is FP-soft continuous.

Theorem 43. Let and be two FP-soft topological spaces and let be a FP-soft mapping. Then the following are equivalent:                      

(1) is FP-soft continuous;(2) is FP-soft closed for every FP-closed set   over ;(3), ;(4), ;(5), .

Proof. (1)(2) It is obvious from Theorem 18(9).

(2)(3) Let . Since , therefore we have . By Theorem 18(4), we get .

(3)(4) Let . If we choose instead of in (3), then . Hence by Theorem 18(3), .

(4)(5) These follow from Theorem 18(9) and Theorem 29.

(5)(1) Let . Since is a FP-soft open set, then . Consequently, is a FP-soft open and so is FP-soft continuous.

Theorem 44. Let be a FP-soft mapping and let be a base for . Then is FP-soft continuous if and only if , .

Proof. Straightforward.

Definition 45. A family of FP-soft sets is a cover of a FP-soft set if . It is a FP-soft open cover if each member of is a FP-soft open set. A subcover of is a subfamily of which is also a cover.

Definition 46. A family of FP-soft sets has the finite intersection property if the intersection of the members of each finite subfamily of is not empty FP-soft set.

Definition 47. A FP-soft topological space is FP-compact if each FP-soft open cover of has a finite subcover.

Example 48. Let , , and . Then is a FP-soft topology on , and is FP-compact.

Theorem 49. A FP-soft topological space is FP-soft compact if and only if each family of FP-soft closed sets with the finite intersection property has a nonempty FP-soft intersection.

Proof. If is a family of FP-soft sets in a FP-soft topological space , then is a cover of if and only if one of the following conditions hold:                   

(1) ;(2) ;(3).

Hence the FP-soft topological space is FP-soft compact if and only if each family of FP-soft open sets over such that no finite subfamily covers fails to be a cover, and this is true if and only if each family of FP-soft closed sets which has the finite intersection property has a nonempty FP-soft intersection.

Theorem 50. Let and be FP-soft topological spaces and let be a FP-soft mapping. If is FP-soft compact and is FP-soft continuous surjection, then is FP-soft compact.

Proof. Let be a cover of by FP-soft open sets. Then since is FP-soft continuous, the family of all FP-soft sets of the form , for , is a FP-soft open cover of which has a finite subcover. However, since is surjective, then for any FP-soft set over . Thus, the family of images of members of the subcover is a finite subfamily of which covers . Consequently, is FP-soft compact.

5. Conclusion

Topology is a branch of mathematics, whose concepts exist not only in almost all branches of mathematics, but also in many real life applications. In this paper, we introduce the topological structure of fuzzy parametrized soft sets and fuzzy parametrized soft mappings. We study some fundamental concepts in fuzzy parametrized soft topological spaces such as closures, interiors, bases, compactness, and continuity. Some basic properties of these concepts are also presented. This paper will form the basis for further applications of topology.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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