Abstract
We have studied some soft neighbourhood properties in a soft topological space and introduced soft filters which are defined over an initial universe with a fixed set of parameters. We have set up a soft topology with the help of a soft filter. We also have introduced the concepts of the greatest lower bound and the least upper bound of the family of soft filters over an initial universe, soft filter subbase and soft filter base. Also, we have explored some basic properties of these concepts.
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Introduction
Mathematics is based on exact concepts and there is not vagueness for mathematical concepts. Since in many other fields such as medicine, engineering, economics and sociology, the notions are vague, researchers need to define some new concepts for vagueness. To deal with the these problems in real life, researchers proposed several methods such as fuzzy set theory, rough set theory and soft set theory. Fuzzy set theory [1] proposed by Zadeh in 1965 provides an appropriate framework for representing and processing vague concepts. The basic idea of fuzzy set theory hinges on fuzzy membership function. By fuzzy membership function, we can determine the belonging of an element to set to a degree. Rough set theory [2] which is proposed by Pawlak in 1982 is another mathematical approach to vagueness to catch the granularity induced by vagueness in information systems. It based on equivalence relation. The advantage of rough set method is that it does not need any additional information about data, like membership in fuzzy set theory.
Theory of fuzzy sets and theory of rough sets can be considered as tools for dealing with vagueness but both of these theories have their own difficulties. The reason for these difficulties is, possibly, the inadequacy of the parametrization tool of the theory as mentioned by Molodtsov [3] in 1999. Soft set theory [3] was initiated by Molodtsov as a completely new approach for modeling vagueness and uncertainty. According to Molodtsov [3, 4], the soft set theory has been successfully applied to many fields, such as functions smoothness, game theory, Riemann integration, theory of measurement and so on. He also showed how soft set theory is free from the parametrization inadequacy syndrome of fuzzy set theory, rough set theory, probability theory and game theory. Soft systems provide a very general framework with the involvement of parameters. Research works on soft sets are progressing rapidly in recent years.
Topological structure of soft sets was studied by many authors: Shabir and Naz [5] introduced the soft topological spaces which are defined over an initial universe with a fixed set of parameters. They studied the concepts of soft open set, soft interior point, soft neighbourhood of a point, soft separation axioms and subspace of a soft topological space. As a different approach to soft topology Çağman et al. [6] defined the concepts of soft open set, soft interior, soft closure, soft limit point, soft Hausdorff space. Aygünoğlu and Aygün [7] defined soft continuity of soft mapping, soft product topology and studied properties of soft projection mappings, soft compactness and generalized Tychonoff theorem to the soft topological space. Min [8] gave some results on soft topological spaces. Zorlutuna et al. [9] also investigated soft interior point, soft neighbourhood and soft continuity. They introduced the relationships between soft topology and fuzzy topology. Hussain and Ahmad [10] continued and discussed the properties of soft interior, soft exterior and soft boundary on soft topology. Varol and Aygün [11] defined convergence of sequences in soft topological space, diagonal soft set and studied the properties of soft Hausdorff space.
The main purpose of this paper is to introduce soft filters which are defined over an initial universe with a fixed set of parameters. First, we have given some basic ideas about soft sets, soft topological spaces and the results already studied. Then, we have discussed some basic properties of soft neighbourhoods. We have defined soft filters and studied some basic properties of soft filters. We have set up a soft topology with the help of a soft filter. We have introduced some new concepts in soft filters such as the greatest lower bound and the least upper bound of the family of soft filters, soft filter subbase and soft filter base. Also, we have explored some basic properties of these concepts.
Preliminaries
In this section, we present the basic definitions and results of soft set theory which may be found in earlier studies.
Let X be an initial universe set and E be the set of all possible parameters with respect to X. Parameters are often attributes, characteristics or properties of the objects in X. Let P(X) denote the power set of X. Then, a soft set over X is defined as follows.
Definition 1
[3] A pair (F, A) is called a soft set over X where and is a set valued mapping. In other words, a soft set over X is a parameterized family of subsets of the universe X. For , may be considered as the set of -approximate elements of the soft set (F, A). It is worth noting that may be arbitrary. Some of them may be empty, and some may have nonempty intersection.
Example 1
[12] Miss Zeynep and Mr. Ahmet are going to marry and they want to hire a wedding room. The soft set (F, E) describes the "capacity of the wedding room". Let be the wedding rooms under consideration, and be the parameter set, The soft set (F,E) is as follows:
Definition 2
[13] A soft set on the universe X is defined by the set of ordered pairs , where such that if and if Here, is called an approximate function of the soft set The value of may be arbitrary.
Note that the set of all soft sets over X will be denoted by
Definition 3
[14] Let If for all , then is called an empty soft set, denoted by or means that there is no element in X related to the parameter Therefore, we do not display such elements in the soft sets, as it is meaningless to consider such parameters.
Definition 4
[14] Let If for all , then is called an A-universal soft set, denoted by If then the A-universal soft set is called a universal soft set, denoted by
Definition 5
[5] Let Y be a nonempty subset of X, then denotes the soft set over X for which for all In particular, will be denoted by .
Definition 6
[14] Let Then is a soft subset of denoted by if for all
Definition 7
[14] Let Then and are soft equal, denoted by , if and only if for all .
Definition 8
[14] Let Then, the soft union , the soft intersection , and the soft difference of and are defined by the approximate functions , , respectively, and the soft complement of is defined by the approximate function , where is the complement of the set ; that is, for all
Definition 9
[9] Let I be an arbitrary index set and be a subfamily of .
-
(a)
The union of these soft sets is the soft set , where for each . We write .
-
(b)
The intersection of these soft sets is the soft set , where for all . We write .
Definition 10
[9] The soft set is called a soft point in , denoted by , if for the element , and for all . The soft point is said to be in the soft set , denoted by , if for the element and .
Proposition 1
[9] Letand. Ifthen.
Definition 11
[5] Let . A soft topology on X, denoted by , is a collection of soft subsets of X having the following properties:
-
(1)
,
-
(2)
If , then
-
(3)
If , , then
The pair is called a soft topological space. Every member of is called a soft open set. A soft set is called soft closed in if
Definition 12
[9] A soft set in a soft topological space is called a soft neighbourhood of the soft point if there exists a soft open set such that .
Remark 1
A soft set in a soft topological space is not only a soft neighbourhood of the soft point , it is also a soft neighbourhood for all soft points of the soft set by Definition 12.
The neighbourhood system of a soft point , denoted by , is the family of all its soft neighbourhoods.
Theorem 1
[9] The neighbourhood systematin a soft topological spacehas the following properties:
-
If, then,
-
If and, then,
-
If, , then,
-
If, then there is a such that for each.
Definition 13
Let be a soft topological space and let be a family of some soft neighbourhoods of soft point . If, for each soft neighbourhood of , there exists a such that , then we say that is a soft neighbourhood base at .
Theorem 2
Let be a soft topological space and. is soft open if and only if it is a soft neighbourhood of each of its soft points.
Proof
If is a soft open set, then for each soft point , . Since , is a soft neighbourhood of each of its soft points by Definition 12.
For each soft point , there exists a soft set such that . For each , since we obtain and . Conversely, for each , since , we obtain. Consequently, we get . Hence, is a soft open set by Definition 11.
Theorem 3
If for each soft pointthere corresponds a familysuch that the properties, , and in Theorem 1 are satisfied, then there is a unique soft topological structure over such that for each, is the family of-soft neighbourhoods of.
Proof
Let It is clear that, is a soft topology over X. The family certainly satisfies axioms and in Definition 11: for , this follows immediately from in Theorem 1, and for , from in Theorem 1. The axiom in Definition 11 is a result of and in Theorem 1. It remains to show that, in the soft topology defined by , is the set of -soft neigbourhoods of for each . It follows from in Theorem 1 that every -soft neighbourhood of belongs to . Conversely, let be a soft set belonging to , and let be the soft set of soft points such that . If we can show that , and , then the proof will be complete. Since for every soft point belongs to by reason of in Theorem 1 and the hypothesis , we obtain . Since and , we have . It remains to show that , i.e. that for each . If then by in Theorem 1 there is a soft set such that for each we have . Since means that , it follows that and, therefore, by in Theorem 1, that .
Soft filter
Definition 14
Let , then is a called a soft filter on if satisfies the following properties:
-
-
-
and
Remark 2
satisfies the following property from and : “The all finite intersections of soft sets of are not soft empty”.
Remark 3
It is clear that by .
Example 2
The family is a soft filter over X.
Example 3
Let . Then, the family is a soft filter over X and it is called atomic soft filter.
Example 4
Let be the natural numbers and E be a finite set. Consider the family , then is a soft filter and it is called soft Frechet filter.
Example 5
Let X be an infinite set and E be a finite set. Consider the family , then is a soft filter and it is called soft cofinite filter.
Example 6
Let X be an uncountable set and E be a countable set. Consider the family , then is a soft filter and it is called soft cocountable filter.
Proposition 2
Let be a soft filter over X. Then the collection
for each , does not define a filter over X.
Proof
If for any then . Hence is not a filter over in general topology.
Example 7
Let , and be a soft filter defined over where . Since , then is not a filter over .
Theorem 4
Let be a soft topological space. The neighbourhood system is a soft filter for every Also, it is called soft neighbourhoods filter of the soft point
Proof
-
By in Theorem 1, since , we obtain .
-
This is clearly seen by in Theorem 1.
-
This is clearly seen by in Theorem 1.
Now, we set up a soft topology with the help of a soft filter.
Theorem 5
If, for every , there exists a soft filter which satisfies the following two properties, then there exists an unique soft topology such that consists of the -soft neighbourhoods of the soft point
-
Every soft set in the soft filter contains the soft point
-
For every there exists a such that for every
Proof
Since the axioms and are equivalent to the neighbourhood axioms , by Theorem 3, there exists a soft topology such that consists of the -soft neighbourhoods of the soft point.
Example 8
Let be a soft topological space and . Since cannot be an element of for every and , then the soft neighbourhood base is not a soft filter over X.
Definition 15
Let and be soft filters over X. If , then is soft finer than . If , then is soft strictly finer than . If either or , then is comparable with .
Proposition 3
Let be a family of soft filters. Then, the family is partially ordered according to “”.
Theorem 6
Let be a family of soft filters over X. Then is a soft filter over X.
Proof
-
Since for each , then does not belong to .
-
Let , . Then, , for each . Since for each , so we obtain .
-
Let and . Since for each and , we get for each . Hence .
Remark 4
The soft filter in Theorem 6 is called the greatest lower bound of the family .
Remark 5
The union of soft filters over is generally not a soft filter over .
Example 9
Let , and be two soft filters defined over where and Now, we define . If we take then and but . Hence is not a soft filter over X.
Now, we investigate the least upper bound of the family of soft filters over X.
Proposition 4
Let Then there exists a soft filter which contains the soft family if has the following property: "The all finite intersections of soft sets of are not soft empty”.
Proof
Let .
Then we give the family which consists of finite intersections of elements of ; .
Then, the family is a soft filter over .
-
Since , for every , and so .
-
Let . There exist soft sets such that and . From the definition of , . Since , we obtain .
-
Let and . Then, there exists a soft set such that . Since , we obtain .
Remark 6
The soft filter in Proposition 4 is said to be generated by and is said to be soft filter subbase of . It is clear that, .
Proposition 5
The soft filter which is generated by is the coarsest soft filter which contains
Proof
Suppose that . By Proposition 4, . By Remark 6, for every , there exists a such that . Since then . Since is a soft filter, by in Definition 14. Hence we obtain .
Theorem 7
The family of soft filters over X has a least upper bound if and only if for all finite subfamilies of and all , the intersection is not soft empty.
Proof
If there exists a least upper bound of the family , by and in Definition 14, for all finite subfamilies of and all , the intersection .
Let for all finite subfamilies of and all . Then, the soft filter generated by
is the least upper bound of the family by Proposition 5.
Definition 16
Let , then is said to be a soft filter base on if
-
and ,
-
.
Remark 7
which is in Proposition 4 is a soft filter base.
Remark 8
It is clear that, every soft filter is a soft filter base.
Example 10
Let . Then, the family is a soft filter base over X.
Example 11
Let be a soft topological space and . The soft neighbourhood base is a soft filter base over X.
-
Clearly, For every , . Then, . Hence, we obtain .
-
Let , . Since , , we get . By Definition 13, there exists a such that . Hence we get is a soft filter base of soft neighbourhoods filter by Definition 16.
Author contribution
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
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Yüksel, Ş., Tozlu, N. & Ergül, Z.G. Soft filter. Math Sci 8, 119 (2014). https://doi.org/10.1007/s40096-014-0119-4
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DOI: https://doi.org/10.1007/s40096-014-0119-4