#### Abstract

There are many approaches to deal with vagueness and ambiguity including soft sets and rough sets. Feng et al. initiated the concept of possible hybridization of soft sets and rough sets. They introduced the concept of soft rough sets, in which parameterized subsets of a universe set serve as the building blocks for lower and upper approximations of a subset. Topological notions play a vital role in rough sets and soft rough sets. So, the basic objectives of the current work are as follows: first, we find answers to some very important questions, such as how to determine the probability that a subset of the universe is definable. Some more similar questions are answered in rough sets and their extensions. Secondly, we enhance soft rough sets from topological perspective and introduce topological soft rough sets. We explore some of their properties to improve existing techniques. A comparison has been made with some existing studies to show that accuracy measure of proposed technique shows an improvement. Proposed technique has been employed in decision-making problem for diagnosing heart failure. For this two algorithms have been given.

#### 1. Introduction

Mathematical modeling for the vagueness and uncertainty of data has many different methods, for instance, rough set theory [1], fuzzy set theory [2], soft set theory [3], and topology [4]. Pawlak [1] introduced the classical rough sets model in the early eighties to study vagueness of data, which originate from daily life situations. The key of this methodology is an equivalence relation which is constructed from the data of an information system. In general, it is very difficult to find an equivalence relation in such data. Therefore application of this technique is very limited. Therefore authors relaxed the condition of equivalence relation by some more general relations such as similarity relations (reflexive and symmetric) [5, 6], preorder relations (reflexive and transitive) [7], reflexive relations [8], general binary relations [8–14], topological approaches [15, 16], and coverings [17–19].

Soft set theory is another mathematical model to deal with uncertainty, when data is collected from real-life situations. This concept was introduced by Molodtsove [3]. This theory has applications in many fields, for instance, game theory, operations research, integration of Riemann, and measurement theory [3]. Recently, scientists and researchers have shown their inclination to the idea of soft sets to apply it in numerous areas. For more information about this theory and its applications, we refer the reader to the references (soft set theoretical concepts [20, 21], soft sets and soft topological spaces [22–24], soft rough sets and their applications [25–28], and medical applications of soft sets and their extensions [29–35]).

In rough set theory [1], basic requirement is to have an equivalence relation among the elements of the set under consideration. But in daily life situations it is not easy to find such an equivalence relation. Perhaps this limitation is associated with rough set theory due to the lack of parameterization tools. The idea of soft rough sets was initiated and studied by Feng et al. in [24] which are very useful in intelligent systems. The concept of the lower (resp., upper) approximation of this theory is particularly useful to extract knowledge hidden in an information system. Decision-making has a crucial part in our daily life, and this method produces the best alternate among dissimilar selections. Chen et al. [34] proposed the choice values of objects in a soft set and considered how to use this notion to address decision-making problems. In [35], Roy and Maji generalized this method for new decision-making problems. There are several subsequent advances after Maji et al.’s work, such as the uni-int decision-making using soft set theory [36]; Jha et al.’s [37] neutrosophic soft set notion in decision-making problems for stock trending analysis, and medical applications [38].

Feng et al. replaced the classes of the equivalence relation by parameterizing subsets of a subset of the universe to define its approximations. In fact, Feng et al. have succeeded in proving that Pawlak’s rough set model is a specialization of the soft rough set as shown by Theorem 4.4 and Theorem 4.5 in [24]. It is worth noting that the concept of full soft sets deserves special attention for both theoretical and practical reasons. Theoretically, some typical properties of Pawlak’s rough sets hold for soft rough sets if and only if the underlying soft set in the soft approximation space is full. Pragmatically, it is justifiable to consider full soft sets in real-life applications. In fact, if a soft set is not full, it means that the available parameters are insufficient, and there exists at least one object which cannot be described by any of the parameters in the given soft set. With the help of soft rough approximations, some equivalent characterizations of full soft sets were given in [24]. In this paper, a new technique is given to define lower and upper approximations of a set with the help of topology generated by the given soft set; this is known mathematically as the notion of topological soft rough sets .

The main contribution in the existing work is to present another model for soft rough sets without any restrictions and satisfy the characteristics of Pawlak’s rough sets. In other words, we propose a method for modifying soft rough sets from a topological point of view, so a new link between soft sets and general topology is proposed.

First, we discuss the concept of the topology of all definable sets in rough set theory [1] and in soft rough sets [24]. Accordingly, we able to respond with the next very interesting questions: What is the probability that a subset of the universe may be a definable set? What is the probability that the lower approximation of a nonempty subset of may be an empty set? What is the probability that the upper approximation of a proper subset of may be ?

Secondly, a general topology is generated from the soft set to modify and generalize soft rough sets proposed in [24]. The suggested techniques extend the way for more applications of the general topology in soft rough sets theory. In fact, we use the image of parameters as a subbasis for a unique topology generated by a soft set, denoted by . New generalized soft rough approximations, called “topological soft rough approximations” (briefly, -approximations), are defined. It is shown that accuracy of proposed technique is higher than soft rough sets, due to reduction of boundary region. The importance of proposed approximations is clear from the fact that these not only reduce the boundary region but also satisfy basic properties similar to rough sets. Several comparisons among the present method and the preceding one [34] are obtained. Numerous examples are suggested to exemplify the relations between the topological soft rough sets and soft rough sets.

Finally, some medical applications in the medical diagnosis of heart failure problems [39] are introduced. These applications illustrate the importance of the suggested methods in real-life problems. In fact, we apply a topological reduction for data set comprising the effect of five indications for twenty patients with heart failure disease. Accordingly, we can identify the core factors of the heart failure diagnosis. A comparison has been made between proposed technique and some already existing in the literature which shows the usefulness of proposed technique. Two algorithms are given based on proposed technique. The proposed algorithms are tested on hypothetical data for the purpose of comparison with already existing methods.

#### 2. Basic Concepts

The current section is devoted to present some elementary definitions and consequences that are applied through paper are mentioned.

##### 2.1. Topological Space

A topology [4] of a set is defined by the collection of subsets of which fulfills the following three axioms: (*T*1) , . (*T*2) A finite intersection of subsets of is a member in . (*T*3) An arbitrary union of subsets of is a member in .

We call a pair “topological space” or “space” and the members of “points” of , and the subsets of that belong to are said to be “open” sets and the complements of the open sets are called “closed” sets in the space. The collection of all closed sets denotes .

An interior of a subset is given by a union of all open sets contained in , formally:

A class is said to be a basis for if all nonempty open subset of can be represented as a union of subfamily of .

Evidently, any topology can have numerous bases, but the basis generates a unique topology .

Each union of elements of belongs to ; therefore a basis of entirely decides .

A family is said to be a subbasis for a topological space if the collection of all finite intersections of represents a basis for .

For any class of subsets of , represents a subbasis for a unique basis which generates a unique topology on such that for each

##### 2.2. Pawlak Rough Set Theory

The current subsection presents some elementary notions pertaining to rough sets given by Pawlak [1].

*Definition 1. *[1] Consider is a finite set called universe, and is an equivalence relation on ; we symbolize to represent the collection of all equivalence classes of and to symbolize an equivalence class in that contains an element . Then, the pair is said to be Pawlak’s approximation space and for any , we propose the lower and upper approximation of by and , respectively. Moreover, is called a rough set if . Otherwise, it is an exact set.

*Definition 2. *[1] Consider is Pawlak approximation space and . Therefore, the boundary, positive, and negative regions and the accuracy of approximations of are given, respectively, byProperties associated with rough sets can be seen in [1].

It is well known that the set of all definable subsets of the approximation space gives rise to a clopen topology [8]. In this paper first, we will study how this topology is obtained and why in this topology each open set is closed as well.

As , now, for each , and . Thus may act as a basis for a topology on .

Theorem 1. *If the pair is Pawlak approximation space, then, .*

*Proof. *Let . Then is a definable set, so , where . Hence . That is . Conversely, every is union of some elements of , which are definable. Since union of definable sets is again definable, is definable. This means . So as required.

Theorem 1 explains that topology of definable sets in any Pawlak’s approximation space is produced by the elements of the set . In this topology every open set is closed because complement of any subset in the basis of this topology is the union of all remaining subsets.

Study of topology constructed by definable sets helps us to answer some very interesting questions such as the following: What is the probability that a subset of may be a definable set? What is the probability that a nonempty subset of has an empty lower approximation? What is the probability that a proper nonempty subset of has upper approximation equal to ?

Answer to the first question is a bit simple and the formula to find the probability that a subset of may be a definable set is given as follows:

Thus, the probability that a subset of the universe is a rough set is .

Now, for the answer of the second question first the following result must be considered.

Theorem 2. *If is a subset of . Then is an empty set if and only if does not contain any nonempty element of .*

*Proof. *Let . Then, by definition, there does not exist any such that . This implies , for each . Therefore, for each . That is, no element of is contained in . Conversely, let there exist some , with such that . This indicates . Then by definition , which is a contradiction, and therefore, the subset does not contain any nonempty element of .

Now in any Pawlak approximation space, the probability that lower approximation of a subset is an empty set can be obtained by the following formula:

To find the answer to the last question, we may have to consider the following result.

Theorem 3. *Let be a nonempty subset of . Then if and only if the subset intersects with every nonempty element of .*

*Proof. *Let . Then, by definition, there does not exist any such that . Since nonempty elements of are union of some classes . As , intersects with every . Consequently, it intersects with every nonempty element of . Conversely, let there exist some nonempty containing some such that . As is the union of some elements of , there exists some class such that . So , which results in , a contradiction; hence, intersects with every nonempty element of .

Further, in any Pawlak approximation space, the probability that the upper approximation of a subset is may be obtained by the following formula:

*Example 1. *Consider is a set and represents an equivalence relation on , such that ; then represents Pawlak’s approximation space. Now can be a basis for a topology on . Let us write the topology generated by as follows: . The only definable subsets of are all elements of . NowTotal number of subsets of .

Thus, the probability that a subset of is definable is given byNext , , are the only subsets of which do not contain any nonempty element of . Therefore, their lower approximation is empty.Further the subsets , , are the only subsets of , which intersect with every nonempty element of . Therefore their upper approximation is and then

##### 2.3. Soft Set Theory and Soft Rough Sets

*Definition 3. *(see [3]). Consider to be a set of items and to be a finite set of certain parameters in relative to the objects in . Parameters represent attributes or characteristics of objects. A “soft set” on is the pair , where , symbolize the power set of , and represents the map . On the other hand, a soft set over is a parameterized collection of subsets of . For , represents the set of -approximate elements of a soft set . Note that, sometimes a soft set is indicated by and expressed as a set of ordered pairs .

*Definition 4. *(see [24]). Consider is a soft set on . Thus, the pair is said to be a soft approximation space. Established on a soft approximation space , we give the “soft lower and soft upper” approximations of , respectively, byGenerally, and refer to soft rough approximations of with respect to . Furthermore, the setsare named the soft “ positive, negative, and boundary” regions of , individually. Evidently, if , i.e., , then is said to be “soft definable” or “soft exact” set; or else is called a “soft rough” set. Moreover there may be a subset which has the same lower and upper approximations but is not definable. Besides, we suggest the accuracy of approximations byAbove definition gives a generalization of Pawlak rough sets theory; naturally it may not satisfy some properties of their properties.

Proposition 1. *(see [24]). If is a soft set on and is a soft approximation space, then, for each :**Properties associated with soft rough sets can be gotten in [34].*

*Definition 5. *(see [24]). Suppose that is a soft set on and a soft approximation space. Then, is said to be a “full soft set” if . It is clear that if is a full soft set, then such that .

Proposition 2. *(see [24]). If is a full soft set on and is a soft approximation space, then, the subsequent conditions are true:*(i)*(ii)**, *(iii)*, **Now, we present and study the idea of the topology of all definable sets generated by soft set . Moreover, we illustrate the condition in which this topology is well defined. Now, again, we can find the answers to the questions which are related to the probability associated with subsets of a set in soft rough sets.*

The following example illustrates that the condition “full soft set” in Proposition 2 is necessary to achieve the properties (i)-(iii).

*Example 2. *Consider to be a soft approximation space, such that , and to be a soft set on where and . Then, it is clear that is not full soft set and thus we have . Also, if , then .

Theorem 4. *If is a full soft set on and is a soft approximation space, then, the collection is a quasi-discrete topology on .*

*Proof. *Since is a soft set on , by using the properties of the soft approximations in [24], we get (*T*1) and . Therefore, . (*T*2) Let be a class of members in . Then, . (*T*3) Let be a class of finite members in . Then, . Now, we need to prove that is a quasi-discrete as follows: Let ; then . By taking the complement to both sides, we obtain and this implies . Thus, .

*Remark 1. *According to Proposition 2, the condition “full soft set” in the above theorem is necessary to construct the topology . If the soft set is not full then may not be a definable set. Moreover there may be a subset which has same lower and upper approximations but is not definable.

For any soft approximation space , such that be a full soft set, the probability of a subset of U is proposed by

, where is the cardinality of and represents a number of elements in .

Thus, the probability that a subset of is a rough set is .

Now, for the answer of the second question, first the following results must be considered.

Lemma 1. *Suppose that is a full soft set on . Hence, we get*(1)*, such that and is called an open set containing .*(2)*, such that .*(3)*, such that .*

*Proof. *The proof of (1) is obvious and from the definition of and (Definition 4), the proofs of (2) and (3) are straightforward.

Now, for any soft approximation space , such that is a full soft set, the probability that lower approximation of a subset is an empty set can be obtained by the following formula:In order to find the answer of the last question we may have to consider the following.

Theorem 5. *Consider is a soft approximation space such that is a full soft set and is a subset of . Thus, if and only if intersects with every nonempty element of .*

*Proof. *Firstly, since is a full soft set, then, , . Let ; then such that . Therefore, by Lemma 4, such that and . Accordingly, , which means that intersects with every nonempty element of . Conversely, let such that and . Thus such that and . Accordingly, which is a contradiction.

Note that if , then no need for to intersect with in general as Example 3 illustrates. Clearly, the subsets and do not intersect with although their soft upper approximation is .

Further in any soft approximation space, , such that is a full soft set; the probability that upper approximation of a subset is can be obtained by the following formula:The next example explains the previous discussion.

*Example 3. *Consider is a soft approximation space, such that and is a full soft set on , where and . Then, we obtain the topology of all soft definable subsets by . Therefore, number of elements in number of all definable subsets of . Thus, the probability that a subset of is definable is given by .

Now, the subsets and are the only subsets of that do not contain any nonempty element of . Therefore, their lower approximation is empty.

Accordingly, . Further, the subsets , and are the only subsets of , which intersect with every nonempty element of . Therefore, their upper approximation is and accordingly .

#### 3. Topological Soft Rough Approximations of Soft Rough Sets

The current section is devoted to introduction of topological soft rough approximations in view of topological structure. Firstly, it will be seen that soft sets and topological spaces have a very close relationship. The concept of topological soft rough approximations will be presented and their properties will be studied. On the other hand, it will be shown that accuracy of the proposed approach is better than existing techniques. Besides, we will give answers to some important questions about the probability in topological soft rough sets.

*Definition 6. *Consider is a soft set on and . Thus, we propose the following:(i) may denote a subbasis;(ii) may denote a basis for the topology defined as the following.If is a soft set on and , then the topology can be defined on with a basis . That is, . This topology may be called topology generated by and we call it “soft rough topology” (in brief, topology).

*Remark 2. *There are three cases of a subbasis :(i)If is a partition of , then and will be a basis for a quasi-discrete (clopen) topology (in which all open sets are closed).(ii)If is a covering (not partition) of , then and will be a basis for a general topology.(iii)If is not a covering (not partition) of , then and will be a basis for a general topology. The following examples explain Remark 2.

*Example 4. *If is a soft approximation space, such that and is a soft set on where and , then, we get that the subbasis of is and the basis is . Clearly, . Accordingly, the topology generated by isObviously, is a quasi-discrete topology.

*Example 5. *Suppose that is a soft approximation space, such that and is a soft set on where and . Then, the subbasis of is and the basis is . Evidently, . Accordingly, the topology generated by isObviously, is not a quasi-discrete topology.

*Example 6. *Consider is a soft approximation space, such that and is a soft set on where and . Then, the subbasis of is and the basis is . Undoubtedly, . Accordingly, the topology generated by isObviously, is not a quasi-discrete topology.

*Definition 7. *Consider is a soft approximation space and is the topology on . The triple is said to be a “topological soft rough approximation space” (briefly, approximation space).

*Definition 8. *Consider to be a approximation space. Therefore, for each we suggest the topological soft rough approximations, “ lower” and “ upper,” respectively, by

*Remark 3. *(i)In general, and represent the interior and closure of associated with the topology , respectively.(ii)If is a partition on , then and hence and are identical with Pawlak’s rough set approximations. Therefore, it can be said that the proposed approach is equivalent to Pawlak’s approach only in case is a partition of . Accordingly, we can say that Pawlak’s rough set model is a specialization of proposed model. Example 4 illustrated this fact.(iii)If is not a partition on , then and hence and will be different from Pawlak’s approximations as illustrated in Example 7.

*Example 7. *Suppose that is a set of students reading some languages. Let and be a soft approximation space, where is a soft set on . Consider the next information system in Table 1.

Therefore, the equivalence classes are .

If we consider each attribute of the set represents a parameter as the following: English, French and German, then, we get the following:

The soft set of is , where . Thus, we get and the basis is Accordingly, the topology generated by isand the complement of isEvidently, the approximations of any subsets of differ than Pawlak’s rough approximations.

*Definition 9. *Suppose that is a approximation space. Hence, for every , we express the “ positive, negative, and boundary” regions and the “ accuracy” of the approximations, respectively, by

*Remark 4. *(i)It is clear that , for any .(ii)If , then and . Thus is said to be “ definable” or “ exact” set; otherwise is called a “ rough” set.The core objective of the following propositions is to discuss the basic properties of rough approximations and .

According to the characteristics of the interior and closure, we can demonstrate the subsequent results, so we omit the proof.

Proposition 3. *If is a approximation space and , then, the lower and upper approximations operators satisfy the next properties:*(i)*.*(ii)*.*(iii)*.*(iv)*If , then .*(v)*If , then .*(vi)*.*(vii)*.*(viii)*.*(ix)*.*(x)*.*(xi)*If , then .*(xii)*If , then .*

Proposition 4. *If is a approximation space and , then:*(i)*.*(ii)*.*(iii)*.*(iv)*.*

*Remark 5. *The inclusion relations in Proposition 4 may be strict, as shown in Example 8.

*Example 8. *Suppose that is a soft set on and is a soft approximation space, where and such that

. Therefore, we get the following.

The subbasis of is .

The basis of is .

Accordingly, we getThe complement of isThus, is a approximation space. Now, let and . Then, and then , which means that . Also, and , which means that .

The following theorem establishes a relationship between approximations of a set in soft rough sets [24] and topological soft rough sets.

Theorem 6. *If is a approximation space and , then:*(i)*.*(ii)*.*

*Proof:. *We shall verify only the first item and the other likewise. Let ; then , such that and . Accordingly, such that and ; this implies , and therefore, . Hence, .

Corollary 1. *If is a approximation space and , then:*(i)*(ii)*

Corollary 2. *If is a approximation space and and if is a soft exact set, then it is a exact set.*

*Remark 6. *(1)According to the above results, it is easy to see that boundary region in case of topological soft rough sets is smaller than boundary for soft rough sets. Therefore accuracy has been improved in proposed technique.(2)In general reverse inclusions do not hold in case of Theorem 6. Following example explains this.

*Example 9. *According to Example 8: Let . Thus and ; therefore and . But, and thus and . Obviously, is a exact (definable) set (according to our approach) although it is a soft rough set.

Next we define some very important notions.

*Definition 10. *Consider is a approximation space and . Thus, we describe the subsequent four elementary sorts of soft rough sets as follows:

The subset represents(i)a roughly definable set if and ,(ii)an internally indefinable set if and ,(iii)an externally indefinable set if and ,(iv)a totally indefinable set if and .The axiomatic significance of this classification is given as follows:(i)If the subset is a roughly definable set, then we can identify for some members of that they belong to , and for other members of that they belong to , by using existing knowledge from the approximation space .(ii)If the subset is an internally indefinable set, then we can identify about some members of that they belong to , but we cannot identify for any member of that it belongs to , by using .(iii)If the subset is an externally indefinable set, then we can identify for some members of that they belong to , but we cannot identify for any member of that it belongs to , by using .(iv)If the subset is a totally indefinable set, then we cannot identify for any member of whether it belongs to or , by using .

Theorem 7. *Consider to be a approximation space and . Therefore, we have the following.*(i)*If the subset is a roughly definable set, then is roughly soft definable.*(ii)*If the subset is an internally definable set, then is internally soft indefinable.*(iii)*If the subset is an externally definable set, then is externally soft indefinable.*(iv)*If the subset is a totally indefinable set, then is totally soft indefinable.*

*Proof. *Only, the first statement will be proved and the other statements can be made by a similar way. (i) Suppose that the subset is a roughly soft definable set; then and . Therefore, by using Theorem 6, and and thus represents a roughly soft definable set.

*Remark 7. *(i)Theorem 7 shows that soft rough approximations of a set given in [24] are different from rough approximations proposed in this paper. Moreover, it clarifies the significance of the proposed approach in defining approximations of sets; for example, let be a totally soft indefinable set. Then, we get and . Thus, we are incapable of identifying what are the elements of that belong to or . But, by using rough approximations, it may be and which means that can be roughly definable set and accordingly we can determine for some elements of that they belong to , and meanwhile, for some elements of , we can identify that they belong to , by using the existing information from the approximation space (to illustrate this, see Examples 8 and Subsection 4.1).(ii)The inverse of Theorem 7 does not hold, generally, as demonstrated in Example 9 and Subsection 4.1.Now, once again, we can find the probability for different types of subsets in topological soft rough sets. Firstly, to answer the first question, we consider the following results.

Theorem 8. *If is a soft set on , is a approximation space, and , then, the class gives rise a topology on .*

*Proof. *According to Proposition 3, the proof is clear.

Note that represents a topology of all definable sets in .

Lemma 2. *If is a approximation space, then, .*

*Proof:. *Let ; then and this implies . Accordingly, .

Lemma 3. *Let be a approximation space and be a partition of ; then and is a basis for .*

*Proof. *Firstly, using Lemma 2, . Now, let which is a quasi-discrete topology; then is an open and closed subset. Therefore, and this implies . Hence, .

The subsequent example explains that the condition “ is a partition of ” is necessary condition.

*Example 10. *Consider Example 7; we get but . It is clear that .

*Definition 11. *Suppose that is a approximation space and is a topology of all definable sets in . The probability that a subset of is definable is defined bywhere is the cardinality of and represents a number of elements of .

Therefore, the probability that a subset is a rough set is .

*Example 11. *(1)Consider Example 4; we have . Thus, the probability that a subset of the universe set is definable is given by