Soft filter

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.


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 Ç agman et al. [6] defined the concepts of soft open set, soft interior, soft closure, soft limit point, soft Hausdorff space. Aygünoglu 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 A E and F : A ! P X ð Þ is a set valued mapping. In other words, a soft set over X is a parameterized family of subsets of the universe X. For 8e 2 A, F e ð Þ may be considered as the set of e-approximate elements of the soft set (F, A). It is worth noting that F e ð Þ 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 X ¼ u 1 ; u 2 ; u 3 ; u 4 ; u 5 ; u 6 f gbe the wedding rooms under consideration, and E ¼ e 1 ¼ big; e 2 ¼ central; e 3 ¼ f cheap; e 4 ¼ quality; e 5 ¼ elegantg be the parameter set, The value of f A ðeÞ may be arbitrary.
Note that the set of all soft sets over X will be denoted by SðX; EÞ: Definition 3 [14] Let F A 2 SðX; EÞ: If f A ðeÞ ¼ ; for all e 2 E, then F A is called an empty soft set, denoted by F U or U: f A ðeÞ ¼ ; means that there is no element in X related to the parameter e 2 E: Therefore, we do not display such elements in the soft sets, as it is meaningless to consider such parameters.  [9] Let I be an arbitrary index set and fðF A Þ i g i2I be a subfamily of SðX; EÞ.
(a) The union of these soft sets is the soft set G C , where  is not only a soft neighbourhood of the soft point e F , it is also a soft neighbourhood for all soft points of the soft set H C by Definition 12.
The neighbourhood system of a soft point e F , denoted by N s $ ðe F Þ, is the family of all its soft neighbourhoods.

Remark 5
The union of soft filters over X is generally not a soft filter over X.
Now, we investigate the least upper bound of the family of soft filters over X.