Topological Structure of Vague Soft Sets

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.


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][15][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.

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 = Definition 2 (see [3]). Let , be two vague sets in the universe = { 1 , 2 , . . . , }; then the union, intersection, and complement of vague sets are defined as follows: (1) Definition 3 (see [3]). Let , be two vague sets in the 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 : → ( ).

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)0,̂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 19. Let ( , 1 , ) and ( , 2 , ) be two vague soft topological spaces over the same universe . If each vague soft set ( , ) ∈ 1 is in 2 , then 2 is called vague soft finer than 1 , or 1 is vague soft coarser than 2 .
Proposition 20. Let { | ∈ } be a family of all vague soft topologies on ; then ⋂ ∈ is the coarsest vague soft topology on .