{'fraction_non_alphanumeric': 0.05498959360280425, 'fraction_numerical': 0.012706758681126081, 'mean_word_length': 3.7552083333333335, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Topological Approach to Soft Covering Approximation Space

Theories of rough sets and soft sets are powerful mathematical tools for modelling various types of vagueness. Hybrid model combining a rough set with a soft set which is called soft rough set proposed by Feng et al. [3] in 2010. In this paper, we study soft covering based rough sets from the topological view. We present under which conditions soft covering lower approximation operation become interior operator and the soft covering upper approximation become closure operator. Also some new methods for generating topologies are obtained. Finally, we study the relationship between concepts of topology and soft covering lower and soft covering upper approximations.


Introduction
Mathematics is based on exact concepts and there is not vagueness for mathematical concepts. For this reason researchers need to define some new concepts for vagueness. The most successful approach is exactly Zadeh's fuzzy set [16] which is based on membership function. This theorical approach is used in several areas as engineering, medicine, economics and etc. Pawlak [11] initiated rough set theory in 1982 as a tool for uncertainty and imprecise data. The theory is based on partition or equivalence relation, which is rather strict. Covering based rough set [2,13] is an important extension of rough sets. Compared with rough sets, it often gives a more reasonable description to a subset of the universe. In recent years, covering based rough set theory has attracted more attentions. The studies of Zhu et al. [17,18,19,20] are fundamental and significant. In 1999 Molodtsov [10] gave soft set theory as a new tool for vagueness and showed in his paper that soft set theory can be applied to several areas. The hybrid models like fuzzy soft set [14], rough soft set [3], soft rough set [3] took attention from researchers. Feng et al. investigated the concept of soft rough set [3] which is a combination of soft set and rough set. It is known that the equivalence relation is used to form the granulation structure of the universe in the rough set model and also the soft set is used to form the granulation structure of the universe in the soft rough set model.
Topology is a branch of mathematics, whose concepts exist not only in almost all branches of mathematics, but also in many real life applications. Topology is also a mathematical tool to study rough sets [5,6,7,8,9]. It should be noted that the generation of topology by relation and the representation of topological concepts via relation will narrow the gap between topology and its applications.
The remaining part of this paper is arranged as follows: In section 3, we give a new concept called as soft covering based rough sets and its basic properties. Also we investigate the conditions under which the soft covering lower and upper approximation operations are also interior and closure operators, respectively. In section 4, we discuss methods of setting up topology in soft covering approximation space. The relationship between concepts of topology and soft covering lower and upper approximations are studied in section 5. and the special condition of soft covering approximation space is investigated in section 6.

Preliminaries
In this section, we introduce the fundamental ideas behind rough sets, soft sets and topological spaces.
First, we recall some concepts and properties of the Pawlak's rough sets.
which are called the lower and the upper approximation of X, respectively. In addition, are called the positive, negative and boundary regions of X, respectively. Now, we are ready to give the definition of rough sets: Let U be an initial universe set and E be the set of all possible parameters with respect to U . Usually, parameters are attributes, characteristics or properties of the objects in U . The notion of a soft set is defined as follows:

Theorem 4 [1] Every rough set may be considered as a soft set.
The following result indicates that soft sets and binary relations are closely related.
Theorem 5 [3] Let G = (F, A) be a soft set over U . Then G induces a binary relation R G ⊆ A × U , which is defined by where x ∈ A, y ∈ U . Conversely, assume that R is a binary relation from A to U . Define a set valued mapping F R : A −→ P (U ) by

Soft Covering Based Rough Sets
We know that a soft set is determined by the set-valued mapping from a set of parameters to the powerset of the universe. In this section, we will use a special soft set and by using this soft set, we will establish a soft covering approximation space.

Definition 7 [3] A soft set
We discussed some properties of soft covering upper and lower approximations in our previous work [15]. Following definitions are given in this paper.
Definition 10 Let S = (U, C G ) be a soft covering approximation space, for any x ∈ U , the soft minimal description of x is defined as following: Definition 11 Let S = (U, C G ) be a soft covering approximation space. For a set X ⊆ U , the soft covering lower and upper approximations are respectively defined as In addition, are called the soft covering positive, negative and boundary regions of X, respectively.
We give following two theorems in our previous work [15].
Theorem 14 Let G = (F, E) be a soft set over U , S = (U, C G ) be a soft covering approximation space and X, Y ⊆ U . Then the soft covering lower and upper approximations have the following properties: Theorem 15 Let G = (F, E) be a soft set over U , S = (U, C G ) be a soft covering approximation space and X, Y ⊆ U . Then the soft covering lower and upper approximations do not have the following properties: The symbol "-" denotes the complement of the set. The following examples show that the equalities mentioned above do not hold. The continuation of the paper, the parameter set E is supposed to be finite.  Proof. =⇒: S − (F (e 1 )∩F (e 2 )) ⊆ S − (F (e 1 )) = F (e 1 ) and S − (F (e 1 )∩F (e 2 )) ⊆ S − (F (e 2 )) = F (e 2 ), so S − (F (e 1 ) ∩ F (e 2 )) ⊆ F (e 1 ) ∩ F (e 2 ). By property 3 of Theorem 14, F (e 1 )∩F (e 2 ) ⊆ S − (F (e 1 )∩F (e 2 )), so F (e 1 )∩F (e 2 ) = S − (F (e 1 )∩ F (e 2 )). Hence, F (e 1 ) ∩ F (e 2 ) is a finite union of elements of C G .
⇐=: By the definition of soft covering upper approximation, S − (X) can be expressed as S − (X) = S − (X) ∪ F (e 1 ) ∪ ... ∪ F (e m ) where y i ∈ F (e i ) X and F (e i ) ∈ M d S (y i ) for some y i ∈ X − S − (X), 1 ≤ i ≤ m. It is obvious that Theorem 22 Let S = (U, C G ) be a soft covering approximation space and Proof. =⇒: By the assumption of this Theorem, O 2 ) Let for each i ∈ I, A i ∈ τ , i.e., S − (A i ) = A i . Then there exists a j ∈ Since this property is satisfied for each j ∈ I, we get Also by Theorem 14, we know that From (1) and (2), we get S − ( i∈I A i ) = i∈I A i . And so we conclude that Theorem 25 Let U be a nonempty universe set and S = (U, C G ) be a soft covering approximation space.For each e 1 , e 2 ∈ E, F (e 1 ) ∩ F (e 2 ) is a finite union of elements of C G .
be a collection of subsets of U . Then K is called a topology over U . Proof.
property is satisfied for each j ∈ I, we get Also by Theorem 14, we know that From (3) and (4), we obtain that S − ( Remark 26 Let U be a nonempty universe set and S = (U, C G ) be a soft covering approximation space. We can set up a topology over U when we consider the soft covering of the universe as a subbase.

Relationship between concepts of topology and soft covering lower and upper approximations
In soft covering based rough set theory the reference space is the soft covering approximation space. We will consider the soft covering of the universe as a subbase for topology and we will obtain the closure, the interior and the boundary of a set with respect to this topology, then we will compare these concepts with the soft covering upper approximation, the soft covering lower approximation and the soft covering boundary region of a set.
Proposition 28 Let S = (U, C G ) be a soft covering approximation space and X ⊆ U . The soft covering lower approximation is contained in the interior of a set defined by taking this soft covering as a subbase for topology.
Proof. Let C G be a soft covering of the universe U , X ⊆ U and x ∈ S − (X). Then, ∃F (e) ∈ S − (X) such that x ∈ F (e). Since F (e) is an element of subbase for the topology defined on U then every F (e) ∈ C G is open hence x ∈ ∪ {F (e) ⊆ U : F (e) ⊆ X open}. Thus x ∈ int(X) and S − (X) ⊆ int(X).
Proposition 29 Let S = (U, C G ) be a soft covering approximation space and X ⊆ U . The soft covering upper approximation of X can not be compared with the closure of X with respect to the topology induced by soft covering.
Corollary 30 Let S = (U, C G ) be a soft covering approximation space and X ⊆ U . The soft covering boundary region of X can not be compared with the boundary of X with respect to the topology induced by soft covering.
Example 31 Let S = (U, C G ) be a soft covering approximation space, where  Proof. Let G = (F, E) be a partition soft set then R − (X) = S − (X) and R − (X) = S − (X). And we know that R − (X) = int(X) and R − (X) = cl(X).