Topological Structure of Vague Soft Sets

Abstract

We introduce vague soft topological spaces which are defined over an initial universe with a fixed set of parameters. The notions of vague soft open sets, vague soft closed sets, vague soft interior, vague soft closure, and vague soft boundary are introduced and their basic properties and relations are investigated. Furthermore, with the help of examples they established that some properties of topological spaces and soft topological spaces do not hold in vague soft topological spaces. Vague soft connectedness and vague soft compactness are also studied.

1. Introduction

Researchers in economics, environmental science, social science, medical science, business, and many other fields deal daily with the complexities of modeling uncertain data. Classical methods are not always successful because the uncertainties appearing in these domains may be of various types. Fuzzy set theory [1], intuitionistic fuzzy set theory [2], vague set theory [3], interval mathematics [4], and other mathematical tools are well-known and often useful approaches to describing uncertainty. However, all of these theories have their own difficulties which have been pointed out in [5]. Molodtsov suggested that one reason for these difficulties may be due to the inadequacy of the parameterization tools of these theories. To overcome these difficulties, Molodtsov [5] introduced the concept of soft sets as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Since then, many researches have investigated soft sets and have established some significant conclusions. For example, Jun and Park [6] proposed the notion of soft ideals and idealistic soft BCK/BCI-algebras and constructed several examples. Majumdar and Samanta [7] further generalized the concept of fuzzy soft sets and some of their properties were studied, and relations on generalized fuzzy soft sets were also discussed by them. Maji et al. [8] introduced the concept of fuzzy soft sets by combining fuzzy sets and soft sets. By combining the vague set and the soft set, Xu et al. [9] introduced the notion of vague soft sets, derived its basic properties, and illustrated its potential applications. Wang and Qu [10] introduced the definitions of entropy, similarity measure, and distance measure of vague soft sets, and the relations between these measures are discussed in detail. About soft topology, Shabir and Naz [11] defined several basic notions on soft topology and studied many properties. Hussain and Ahmad [12] continued investigating the properties of soft topological spaces and strengthened the foundations of the theory of soft topological spaces. Tanay and Kandemir [13] introduced the concept of fuzzy soft topology and some of its structural properties are studied.

By definition, a soft set is a parameterized family of subsets of the universal set. In other words, a soft set is a mapping from a set of parameters to the power set of an initial universe set. In the real world, the difficulty is that the objects in the universal set may not precisely satisfy the problem’s parameters, which usually represent some attributes, characteristics, or properties of the objects in the universal set. The concept of fuzzy soft sets proposed in [8] partially resolves this difficulty but falls short in dealing with additional complexity; that is, the mapping may be too vague. It is, therefore, desirable to extend soft set theory and fuzzy soft set theory using the concept of vague set theory. Vague set theory is actually an extension of fuzzy set theory and vague sets are regarded as a special case of context-dependent fuzzy sets. The basic concepts of vague set theory and its extensions, as well as some interesting applications, can be found in [14–16]. Vague soft set theory makes descriptions of the object world more realistic, practical, and accurate, at least in some cases, making it a very promising tool. Since vague sets are equivalent to intuitionistic fuzzy sets [17], so vague soft sets are equivalent to intuitionistic fuzzy soft sets. Some scholars have studied intuitionistic fuzzy soft sets from different aspects. For example, Gunduz and Bayramov [18] introduced the concept of an intuitionistic fuzzy soft module and some operations on intuitionistic fuzzy soft sets were given; they also studied some of its basic properties. Jiang et al. [19] proposed the notion of the interval-valued intuitionistic fuzzy soft set, the complement, and/or union, intersection, and necessity, and possibility operations were defined on interval-valued intuitionistic fuzzy soft sets, and the basic properties of interval-valued intuitionistic fuzzy soft sets were discussed. They [20] also presented an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and gave some illustrative examples; the weighted intuitionistic fuzzy soft sets were introduced and its application to decision making was investigated. Zhang [21] proposed a novel approach to intuitionistic fuzzy soft set based decision making problems using rough set theory. However, there has been rather little work completed for topological structure in the context of intuitionistic fuzzy soft sets. The purpose of this paper is to further extend the concept of vague soft set theory proposed by Xu et al. in [9]. In this paper, the concept of vague soft topology is introduced and some of its structural properties are studied. Some results about vague soft connectedness and vague soft compactness are also investigated.

The rest of this paper is organized as follows. Section 2 recalls some basic concepts of vague sets, soft sets, and vague soft sets. In Section 3, we introduce the definitions of vague soft topology and some of its structural properties such as vague soft open sets, vague soft closed sets, vague soft interior, vague soft closure, and vague soft boundary are studied. The vague soft connectedness and vague soft compactness are also investigated. In the final section, some concluding comments are presented.

2. Preliminaries

In this section, we will recall several definitions and results which are necessary for our paper. They are stated as follows.

Definition 1 (see [3]). A vague set in the universe can be expressed by the following notion: ; that is, , and the condition should hold for any , where is called the membership degree (true membership) of element to the vague set , while is the degree of nonmembership (false membership) of the element to the vague set .

Definition 2 (see [3]). Let , be two vague sets in the universe ; then the union, intersection, and complement of vague sets are defined as follows:

Definition 3 (see [3]). Let , be two vague sets in the universe . If , then is called a vague subset of , denoted by , where .

Definition 4 (see [5]). Let be an initial universe set, the power set of , a set of parameters, and . A pair is called a soft set over , where is a mapping given by .

Definition 5 (see [9]). Let be an initial universe set, the set of all vague sets on , a set of parameters, and . A pair is called a vague soft set over , where is a mapping given by .

Definition 6 (see [9]). Let and be two vague soft sets over a universe . If and for all is a vague subset of , then is called a vague soft subset of . This relation is denoted by .

Definition 7 (see [9]). Two vague soft sets and over a universe are said to be vague soft equal if is a vague soft subset of and is a vague soft subset of . This relation is denoted by .

Definition 8 (see [9]). Let be a parameter set. The not set of denoted by is defined by , where .

Definition 9 (see [9]). The complement of vague soft set is denoted by and is defined by , where is a mapping given by .

Clearly is the same as and .

Definition 10 (see [9]). A vague soft set over is said to be a null vague soft set denoted by , if .

Definition 11 (see [9]). A vague soft set over is said to be an absolute vague soft set denoted by , if .

Definition 12 (see [9]). The union of two vague soft sets and over a universe is a vague soft set , where and ,

We denote it by .

Definition 13 (see [9]). The intersection of two vague soft sets and over a universe is a vague soft set , where and ,

We denote it by .

Proposition 14 (see [9]). If and are two vague soft sets over a universe , then(1),(2).

3. Vague Soft Topological Spaces

Let be an initial universe set, and let be the nonempty set of parameters.

Definition 15. Let be the collection of vague soft sets over ; then is said to be a vague soft topology on if(1) belong to ,(2)the union of any number of vague soft sets in belongs to ,(3)the intersection of any two vague soft sets in belongs to .

The triplet is called a vague soft topological space over .

Definition 16. Let be a vague soft topological space over ; then the members of are said to be vague soft open sets in .

Definition 17. Let be a vague soft topological space over . A vague soft set over is said to be a vague soft closed set in , if its complement belongs to .

Proposition 18. Let be a family of vague soft topologies on ; then is a vague soft topology on .

Proof. (1) and belong to .
(2) Let . Then for all , , so . Thus .
(3) Let . Then for all , . Since , so .

Definition 19. Let and be two vague soft topological spaces over the same universe . If each vague soft set is in , then is called vague soft finer than , or is vague soft coarser than .

Proposition 20. Let be a family of all vague soft topologies on ; then is the coarsest vague soft topology on .

Proof. is a vague soft topology on by Proposition 18. If each vague soft set , then for all . Hence is the coarsest vague soft topology on .

Remark 21. Let and be two vague soft topological spaces over the same universe ; then may not be a vague soft topological space over .

Example 22. Let , and , be two vague soft topologies defined on where , are vague soft sets over , defined as follows: Now, .
If we take , , then , , so is not a vague soft topological space on .

Definition 23. Let be a vague soft topological space, and let be a vague soft set over . Then vague soft interior of , denoted by , is defined as the union of all vague soft open sets contained in .

Clearly, is the largest vague soft open set contained in .

Theorem 24. Let be a vague soft topological space over , and let and be two vague soft sets over . Then the following properties hold:(1) and ;(2);(3) is a vague soft open set if and only if ;(4);(5) implies ;(6).

Proof. (1) and (2) are obvious.
(3) If is a vague soft open set, then . Since by part (2), so .
Suppose that . Since is a vague soft open set, so is a vague soft open set.
(4) Since is a vague soft open set, we have by part (3).
(5) If , then . is a vague soft open set contained in , so by Definition 23.
(6) Since , , so , by part (5). Thus . Since , so by part (5). is a vague soft open set; then . Hence .

Definition 25. Let be a vague soft topological space, and let be a vague soft set over . Then vague soft closure of , denoted by , is defined as the intersection of all vague soft closed sets containing .

Clearly, is the smallest vague soft closed set containing .

Theorem 26. Let be a vague soft topological space over , and let and be two vague soft sets over . Then the following properties hold:(1) and ;(2);(3) is a vague soft closed set if and only if ;(4);(5) implies ;(6).

Proof. The proof is similar to the proof of Theorem 24.

Theorem 27. Let be a vague soft topological space over , and let be a vague soft set over . Then the following properties hold:(1);(2).

Proof. (1) Suppose that is the family of all vague soft open sets contained in ; that is, . Then is the family of all vague soft closed sets containing ; that is, . Since by Proposition 14 (1), so .
(2) Suppose that is the family of all vague soft closed sets containing ; that is, . Then is the family of all vague soft open sets contained in ; that is, . Since by Proposition 14 (2), so .

Definition 28. Let be a vague soft topological space, and let be a vague soft set over . Then vague soft boundary of , denoted by , is defined as .

Clearly, the vague soft sets and have same vague soft boundary; that is, .

Theorem 29. Let be a vague soft topological space over , and let be a vague soft set over . Then the following properties hold:(1);(2); (3).

Proof. (1) is obvious.
(2) is as follows:
(3) is as follows:

Remark 30. Let be a vague soft topological space over , and let be a vague soft set over . Then the following properties may not hold:(1);(2).

Example 31. Let , , and where are vague soft sets over , defined as follows: Then defines a vague soft topology on and is a vague soft topological space over .
Let us take vague soft set , defined as follows: Then , Hence , .

Theorem 32. Let be a vague soft topological space over , and let be a vague soft set over .(1)If , then is a vague soft open set over .(2)If is a vague soft closed set over , then .

Proof. (1) Let . Then , or , or , which implies is a vague soft closed set. Hence is a vague soft open set.
(2) If is a vague soft closed set over , then , since ; that is, .

Remark 33. Let be a vague soft topological space over , and let be a vague soft set over .(1)If is a vague soft open set over , then may not hold.(2)If , then may not be a vague soft closed set over .

Example 34. Let us consider the vague soft topological space over in Example 31. We take is a vague soft open set over . Then Hence . We have , but is not a vague soft closed set over .

Theorem 35. Let be a vague soft topological space over , and let be a vague soft set over . If , then is a vague soft open set and vague soft closed set over .

Proof. First we prove that is a vague soft open set over . Consider . This implies that is a vague soft open set.
We now prove that is a vague soft closed set over . Consider . This implies that is a vague soft closed set.

Remark 36. If is a vague soft open set and vague soft closed set over , then may not be .

Example 37. Let , , and where , are vague soft sets over , defined as follows: Then defines a vague soft topology on and is a vague soft topological space over . We have is a vague soft open set and vague soft closed set over , but .

Definition 38. A vague soft topological space is called strongly disconnected, if there exist vague soft open sets and such that and .

Definition 39. A vague soft topological space is called strongly connected, if it is not vague soft strongly disconnected.

Definition 40. A vague soft topological space is called weakly disconnected, if there exists vague soft set which is both vague soft open set and vague soft closed set such that .

Definition 41. A vague soft topological space is called weakly connected, if it is not vague soft weakly disconnected.

Theorem 42. Let be a vague soft topological space over . If is strongly disconnected, then it is also weakly disconnected.

Proof. If is strongly disconnected, then there exist vague soft open sets and such that and . For any , we have , and , . When , then and , . When , then and . Hence ; that is, is a vague soft set which is both vague soft open set and vague soft closed set such that . Hence is weakly disconnected.

Remark 43. If is weakly disconnected, then it may not be strongly disconnected.

Example 44. Let , , and where and are two vague soft sets over , defined as follows: Then defines a vague soft topology on and is a vague soft topological space over . Because is a vague soft set which is both vague soft open set and vague soft closed set such that , is weakly disconnected, but it is not strongly disconnected.

Theorem 45. is strongly disconnected if and only if there exist vague soft closed sets and such that and .

Proof. If is strongly disconnected, then there exist vague soft open sets and such that and . Then and are vague soft closed sets such that and .
The proof is similar to the above.

Theorem 46. is weakly disconnected if and only if there exist vague soft open sets and such that .

Proof. If is weakly disconnected, then there exists vague soft set which is both vague soft open set and vague soft closed set such that . Take ; thus is a vague soft open set and .
If there exist vague soft open sets and such that , then is both vague soft open set and vague soft closed set. Since , , so is weakly disconnected.

Theorem 47. is weakly disconnected if and only if there exist vague soft sets and such that , , .

Proof. If is weakly disconnected, then there exists vague soft set which is both vague soft open set and vague soft closed set such that . Take ; thus is both vague soft open set and vague soft closed set. Since , are vague soft open sets, , . Hence , .
Suppose there exist vague soft set and such that and and . Since and are vague soft closed sets, and are vague soft closed sets. Since , so is a vague soft open set. Thus, vague soft set is both vague soft open set and vague soft closed set such that . Hence, is weakly disconnected.

Theorem 48. is weakly disconnected if and only if there exist vague soft sets and such that .

Proof. The proof is similar to the proof of Theorem 47.

Definition 49. Let is a vague soft topological space over . If a family of vague soft open sets satisfies the condition , then it is called a vague soft open cover of . If a finite subfamily of a vague soft open cover is also a vague soft open cover of , then it is called a finite subcover of .

Definition 50. A vague soft topological space is called vague soft compact if every vague soft open cover of has a finite subcover.

Example 51. Let , , and where is vague soft sets over , defined as follows: Then defines a vague soft topology on and is a vague soft topological space over .
is not vague soft compact, because the vague soft open cover has no finite subcover.

Definition 52. Let be a vague soft topological space, and let be a family of vague soft sets over . is said to satisfy the finite intersection property, if every finite subfamily satisfies the condition .

Theorem 53. A vague soft topological space is vague soft compact if and only if every family of vague soft closed sets which satisfies the finite intersection property has a nonempty intersection.

Proof. Let be a family of vague soft closed sets which satisfies the finite intersection property. Suppose that ; then , so is a vague soft open cover of . Since is vague soft compact, has a finite subcover, denoted by . Thus, , which is a contradiction. Hence, has a nonempty intersection.
Let is a vague soft open cover of . Then is a family of vague soft closed sets and . Since every family of vague soft closed sets which satisfies the finite intersection property has a nonempty intersection, so does not have the finite intersection property; that is, a subfamily of satisfies . Thus, ; that is, is a finite subcover of . Hence, is vague soft compact.

Definition 54. Let be a vague soft topological space, and let be a vague soft set over . If a family of vague soft open sets satisfies the condition , then it is called a vague soft open cover of . If a finite subfamily of a vague soft open cover is also a vague soft open cover of , then it is called a finite subcover of .

Definition 55. A vague soft set is called vague soft compact if every vague soft open cover of has a finite subcover.

Theorem 56. A vague soft set is vague soft compact if and only if every family of vague soft open sets with properties that, for any , and , has a finite subfamily such that and .

Proof. It follows from Definitions 54 and 55.

4. Conclusion

Topology is an important and major area of mathematics. In this paper, we have introduced vague soft topological spaces which are defined over an initial universe with a fixed set of parameters. The notions and properties of vague soft open set, vague soft closed set, vague soft interior, vague soft closure, vague soft boundary, vague soft connectedness, and vague soft compactness are introduced and investigated. The findings in this paper will enhance and promote the further study on vague soft topological space to establish a general framework for practical application.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous referees for their constructive comments as well as helpful suggestions which helped in improving this paper significantly. The works described in this paper are supported by the National Natural Science Foundation of China under Grant nos. 11171271, 11326048; the Postdoctoral Science Foundation of China under Grant no. 2013M532079; and the Science Research Foundation of Northwest University under Grant no. 12NW04.