New Topological Approaches to Generalized Soft Rough Approximations with Medical Applications

-ere 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. -ey 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.


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.
e 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. erefore application of this technique is very limited. erefore 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][9][10][11][12][13][14], topological approaches [15,16], and coverings [17][18][19].
Soft set theory is another mathematical model to deal with uncertainty, when data is collected from real-life situations.
is concept was introduced by Molodtsove [3]. is 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][23][24], soft rough sets and their applications [25][26][27][28], and medical applications of soft sets and their extensions [29][30][31][32][33][34][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. e idea of soft rough sets was initiated and studied by Feng et al. in [24] which are very useful in intelligent systems. e concept of the lower (resp., upper) approximation of this theory is particularly useful to extract knowledge hidden in an information system. Decisionmaking 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. ere 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 eorem 4.4 and eorem 4.5 in [24]. It is worth noting that the concept of full soft sets deserves special attention for both theoretical and practical reasons. eoretically, 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 reallife 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 (T SR − sets). e 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 U may be a definable set? What is the probability that the lower approximation of a nonempty subset of U may be an empty set? What is the probability that the upper approximation of a proper subset of U may be U?
Secondly, a general topology is generated from the soft set to modify and generalize soft rough sets proposed in [24]. e 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 T SR . New generalized soft rough approximations, called "topological soft rough approximations" (briefly, T SR -approximations), are defined. It is shown that accuracy of proposed technique is higher than soft rough sets, due to reduction of boundary region. e 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. ese 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. e proposed algorithms are tested on hypothetical data for the purpose of comparison with already existing methods.

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

Topological Space.
A topology [4] of a set U is defined by the collection τ of subsets of U which fulfills the following three axioms: (T2) A finite intersection of subsets of τ is a member in τ. (T3) An arbitrary union of subsets of τ is a member in τ.
We call a pair (U, τ) "topological space" or "space" and the members of U "points" of τ, and the subsets of U that belong to τ are said to be "open" sets and the complements of the open sets are called "closed" sets in the space. e collection of all closed sets denotes τ c .
An interior int(A) (resp., closurecl(A)) of a subset A is given by a union of all open sets contained in A (resp., intersection of all closed sets that contain A), formally: A class B ⊆ τ is said to be a basis for τ if all nonempty open subset of U can be represented as a union of subfamily of B.
Evidently, any topology can have numerous bases, but the basis B generates a unique topology τ.
Each union of elements of B belongs to τ; therefore a basis of τ entirely decides τ.
A family S⊆τ is said to be a subbasis for a topological space (U, τ) if the collection of all finite intersections of S represents a basis for (U, τ).
For any class S of subsets of U, S represents a subbasis for a unique basis B which generates a unique topology τ on U such that for each i ∈ I (2)

Pawlak Rough Set
eory. e current subsection presents some elementary notions pertaining to rough sets given by Pawlak [1].
Consider U is a finite set called universe, and R is an equivalence relation on U; we symbolize U/R to represent the collection of all equivalence classes of R and [s] R to symbolize an equivalence class in R that contains an element s ∈ U.
en, the pair A R � (U, R) is said to be Pawlak's approximation space and for any L⊆U, we propose the lower and upper approximation of L by Otherwise, it is an exact set.
Properties associated with rough sets can be seen in [1]. It is well known that the set of all definable subsets of the approximation space (U, R) gives rise to a clopen topology τ c [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 (U/R) � [s] R : s ∈ U , now, for each A, B ∈ (U/R), A ∩ B � ∅ and U � A ∪ B. us (U/R) may act as a basis for a τ topology on U.
Hence A ∈ τ. at is τ c ⊆τ. Conversely, every B ∈ τ is union of some elements of (U/R), which are definable. Since union of definable sets is again definable, B is definable. is means B ∈ τ c . So τ⊆τ c as required.
□ eorem 1 explains that topology of definable sets in any Pawlak's approximation space is produced by the elements of the set (U/R). In this topology every open set is closed because complement of any subset in the basis (U/R) 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 U may be a definable set? What is the probability that a nonempty subset of U has an empty lower approximation? What is the probability that a proper nonempty subset of U has upper approximation equal to U?
Answer to the first question is a bit simple and the formula to find the probability that a subset of U may be a definable set is given as follows: where |τ| is the cardinality of τ and n represents a number of elements in U. (4) us, the probability that a subset of the universe U is a rough set is 1 − P.
Now, for the answer of the second question first the following result must be considered. Proof. Let R(X) � ∅. en, by definition, there does not exist any x ∈ X such that [x] R ⊆X. is implies [x] R ⊈X, for each x. erefore, for each x ∈ X, [x] R ⊈X. at is, no element of τ is contained in X. Conversely, let there exist some en by definition R(X) � ∪ x∈X U x ≠ ∅, which is a contradiction, and therefore, the subset X does not contain any nonempty element of τ.

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Now in any Pawlak approximation space, the probability P ∅ that lower approximation of a subset is an empty set can be obtained by the following formula: where N τ X represents a number of subsets of U which does not contain any nonempty element of τ and n is the number of elements in U.

(5)
To find the answer to the last question, we may have to consider the following result. Proof. Let R(X) � U. en, by definition, there does not exist any x ∈ X such that [x] R ∩ X � ∅. Since nonempty elements of τ are union of some classes [x] R ∈ (U/R). As it intersects with every nonempty element of τ. Conversely, let there exist some nonempty U x ∈ τ containing some hence, X intersects with every nonempty element of τ. □ Further, in any Pawlak approximation space, the probability P U that the upper approximation of a subset is U may be obtained by the following formula: NX is the number of subsets of U which intersect with every nonempty element of τ, and n represents the number of elements in U.
}; then (U, R) represents Pawlak's approximation space. Now U/R can be a basis for a topology τ on U. Let us write the topology τ generated by U/R as follows: }. e only definable subsets of U are all elements of τ. Now |τ| � Number of elements in τ � Number of definable subsets of U.
us, the probability that a subset of U is definable is given by { } are the only subsets of U which do not contain any nonempty element of τ. erefore, their lower approximation is empty.
Further the subsets 1, 2, 4 } are the only subsets of U, which intersect with every nonempty element of τ. erefore their upper approximation is U and then

Soft Set eory and Soft Rough Sets
Definition 3 (see [3]). Consider U to be a set of items and E to be a finite set of certain parameters in relative to the objects in U. Parameters represent attributes or characteristics of U objects. A "soft set" on U is the pair (F, A), where A⊆E, P(U) symbolize the power set of U, and F represents the map F: A ⟶ P(A). On the other hand, a soft set over U is a parameterized collection of subsets of U.
For e ∈ A, F(e) represents the set of e-approximate elements of a soft set (F, A). Note that, sometimes a soft set is indicated by F A and expressed as a set of ordered pairs Definition 4 (see [24]). Consider F A is a soft set on U. us, the pair A s � (U, F A ) is said to be a soft approximation space. Established on a soft approximation space A s , we give the "soft A s lower and soft A s upper" approximations of X⊆U, respectively, by Generally, S(X) and S(X) refer to soft rough approximations of X⊆U with respect to A s . Furthermore, the sets are named the soft "A s positive, A s negative, and A s boundary" regions of X⊆U, individually. Evidently, if S(X) � S(X) � X, i.e., BND A s (X) � ϕ, then X⊆U is said to be "soft A s definable" or "soft A s exact" set; or else X is called a "soft A s 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 by Above definition gives a generalization of Pawlak rough sets theory; naturally it may not satisfy some properties of their properties.
Proposition 1 (see [24]). If F A is a soft set on U and A s � (U, F A ) is a soft approximation space, then, for each X⊆U: Properties associated with soft rough sets can be gotten in [34].
Definition 5 (see [24]). Suppose that F A is a soft set on U and A s � (U, F A ) a soft approximation space. en, F A is said to be a "full soft set" if F A � ∪ e∈A F(e). It is clear that if F A is a full soft set, then ∀x ∈ U, ∃e ∈ A such that x ∈ F(e).
Proposition 2 (see [24]). If F A is a full soft set on U and A s � (U, F A ) is a soft approximation space, then, the subsequent conditions are true: Now, we present and study the idea of the topology of all definable sets generated by soft set F A . 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.
e following example illustrates that the condition "full soft set" in Proposition 2 is necessary to achieve the properties (i)-(iii).
Proof. Since F A is a soft set on U, by using the properties of the soft approximations in [24], we get Remark 1. According to Proposition 2, the condition "full soft set" in the above theorem is necessary to construct the topology τ D . If the soft set is not full then U 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 A s � (U, F A ), such that F A be a full soft set, the probability of a subset of U is proposed by P � (|τ D |/2 n ), where |τ D | is the cardinality of τ D and n represents a number of elements in U.
us, the probability that a subset of U is a rough set is 1 − P.
Now, for the answer of the second question, first the following results must be considered. Proof.
e proof of (1) is obvious and from the definition of S and S (Definition 4), the proofs of (2) and (3) are straightforward. Now, for any soft approximation space A s � (U, F A ), such that F A is a full soft set, the probability P ∅ that lower approximation of a subset is an empty set can be obtained by the following formula:

Journal of Mathematics
∅ is the number of subsets of U which does not contain any nonempty element of F A and n represents a number of elements in U. (15) In order to find the answer of the last question we may have to consider the following.
Note that if S(X) � U, then no need for X to intersect with F(e), ∀e ∈ A in general as Example 3 illustrates. Clearly, the subsets b, c such that F A is a full soft set; the probability P U that upper approximation of a subset is U can be obtained by the following formula:

Topological Soft Rough Approximations of Soft Rough Sets
e 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. e 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 F A is a soft set on U and K � ∪ F(e).
us, we propose the following: basis for the topology T SR defined as the following.
If F A is a soft set on U and K � ∪ F(e)∈S F A F(e), then the topology T SR can be defined on K with a basis B F A . at is, T SR � ∪ B: B ∈ B F A . is topology may be called topology generated by F A and we call it "soft rough topology" (in brief, SR topology).

Remark 2.
ere are three cases of a subbasis S F A : Obviously, T SR is a quasi-discrete topology.
} and the basis is Obviously, T SR is not a quasi-discrete topology. 3 and Obviously, T SR is not a quasi-discrete topology.
Definition 7. Consider A s � (U, F A ) is a soft approximation space and T SR is the SR topology on U. e triple A T SR � (U, F A , T SR ) is said to be a "topological soft rough approximation space" (briefly, T SR approximation space).

Definition 8. Consider A T SR � (U, F A , T SR )
to be a T SR approximation space. erefore, for each X⊆U we suggest the topological soft rough approximations, "T SR lower" and "T SR upper," respectively, by Remark 3 (i) In general, S T SR (X) and S T SR (X) represent the interior and closure of X associated with the topology T SR , respectively.

(ii) If S F A is a partition on U, then S F A � B F A � (U/R)
and hence S T SR (X) and S T SR (X) are identical with Pawlak's rough set approximations. erefore, it can be said that the proposed approach is equivalent to Pawlak's approach only in case S F A is a partition of U. Accordingly, we can say that Pawlak's rough set model is a specialization of proposed model. Example 4 illustrated this fact.

(iii) If S F A is not a partition on U, then S F A ≠ B F A ≠ (U/R)
and hence S T SR (X) and S T SR (X) will be different from Pawlak's approximations as illustrated in Example 7.
} is a set of students reading some languages. Let A � e 1 , e 2 , e 3 and A s � (U, F A ) be a soft approximation space, where F A is a soft set on U. Consider the next information system in Table  1.
erefore, the equivalence classes are }. If we consider each attribute of the set English, French, German represents a parameter as the following: e 1 �English, e 2 �French and e 3 �German, then, we get the following: and the complement of T SR is Evidently, the T SR approximations of any subsets of U differ than Pawlak's rough approximations.
Definition 9. Suppose that A T SR � (U, F A , T SR ) is a T SR approximation space. Hence, for every X⊆U, we express the "T SR positive, T SR negative, and T SR boundary" regions and the "T SR accuracy" of the T SR approximations, respectively, by Remark 4 (i) It is clear that 0 ≤ μ T SR (X) ≤ 1, for any X⊆U.
(ii) If S T SR (X) � S T SR (X), then BND T SR (X) � ϕ and μ T SR (X) � 1. us X⊆U is said to be "T SR definable" or "T SR exact" set; otherwise X is called a "T SR rough" set.
e core objective of the following propositions is to discuss the basic properties of T SR rough approximations S T SR and S T SR .

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

Remark 5. e inclusion relations in
e complement of T SR is e following theorem establishes a relationship between approximations of a set in soft rough sets [24] and topological soft rough sets.

Theorem 6. If A T SR � (U, F A , T SR ) is a T SR approximation space and X⊆U, then: (i) S(X)⊆S T SR (X). (ii) S T SR (X)⊆S(X).
Proof: We shall verify only the first item and the other likewise. Let x ∈ S(X); then ∃e ∈ A, such that x ∈ F(e)⊆X and F(e) ∈ S F A . Accordingly, F(e) ∈ T SR such that x ∈ F(e) and F(e)⊆X; this implies F(e)⊆S T SR (X), and therefore, x ∈ S T SR (X). Hence, S(X)⊆S T SR (X).

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. (2) In general reverse inclusions do not hold in case of eorem 6. Following example explains this.
Example 9 According to Example 8: (3/4). But, S T SR (X) � S T SR (X) � X and thus BND T SR (X) � ϕ and μ T SR (X) � 1. Obviously, X is a T SR exact (definable) set (according to our approach) although it is a soft rough set.
Next we define some very important notions.
is a T SR approximation space and X⊆U. us, we describe the subsequent four elementary sorts of T SR soft rough sets as follows: e subset X represents e axiomatic significance of this classification is given as follows: (i) If the subset X is a roughly T SR definable set, then we can identify for some members of U that they belong to X, and for other members of U that they belong to X c , by using existing knowledge from the T SR approximation space A T SR . (ii) If the subset X is an internally T SR indefinable set, then we can identify about some members of U that they belong to X c , but we cannot identify for any member of U that it belongs to X, by using A T SR . (iii) If the subset X is an externally T SR indefinable set, then we can identify for some members of U that they belong to X, but we cannot identify for any member of U that it belongs to X c , by using A T SR . (iv) If the subset X is a totally T SR indefinable set, then we cannot identify for any member of U whether it belongs to X or X c , by using A T SR .
Theorem 7. Consider A T SR � (U, F A , T SR ) to be a T SR approximation space and X⊆U. erefore, we have the following.
(i) If the subset X is a roughly T SR definable set, then X is roughly soft A s definable. (ii) If the subset X is an internally T SR definable set, then X is internally soft A s indefinable.

(iii) If the subset X is an externally T SR definable set, then X is externally soft A s indefinable. (iv) If the subset X is a totally T SR indefinable set, then X
is totally soft A s 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 X is a roughly soft A s definable set; then S T SR (X) ≠ ϕ and S T SR (X) ≠ U. erefore, by using eorem 6, S(X) ≠ ϕ and S(X) ≠ U and thus X represents a roughly soft A s definable set.

Remark 7
(i) eorem 7 shows that soft rough approximations of a set given in [24] are different from T SR rough approximations proposed in this paper. Moreover, it clarifies the significance of the proposed approach in defining approximations of sets; for example, let X be a totally soft A s indefinable set. en, we get S(X) � ϕ and S(X) � U. us, we are incapable of identifying what are the elements of U that belong to X or X c . But, by using T SR rough approximations, it may be S T SR (X) ≠ ϕ and S T SR (X) ≠ U which means that X can be roughly T SR definable set and accordingly we can determine for some elements of U that they belong to X, and meanwhile, for some elements of U, we can identify that they belong to X c , by using the existing information from the T SR approximation space (to illustrate this, see Examples 8 and Subsection 4.1). (ii) e inverse of eorem 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. closed subset. erefore, S T SR (X) � S T SR (X) � X and this implies X ∈ τ D . Hence, T SR ⊆τ D .
e subsequent example explains that the condition "S F A is a partition of U" is necessary condition.
Definition 11. Suppose that A T SR � (U, F A , T SR ) is a T SR approximation space and τ D is a topology of all definable sets in U. e probability P D that a subset of U is definable is defined by where |τ D | is the cardinality of τ D and n represents a number of elements of U. erefore, the probability that a subset is a rough set X⊆U is 1 − P D .

Example 11
(1) Consider Example 4; we have T SR � τ D . us, the probability P D that a subset of the universe set is definable is given by (2) Consider erefore, the probability P D that a subset of the universe set is definable is given by Secondly, to identify the probability that lower approximation of a nonempty subset of U may be an empty set, we propose the next results.
e proof is straightforward. Proof. Firstly, if S T SR (X) � ϕ then ∀G⊆X, G ∉ T SR , and this implies ∀G⊆X, G ∉ B F A . Conversely, let ∃G⊆X, G ∈ B F A such that S T SR (X) � ϕ. en, G ∈ T SR such that G⊆X and this implies G⊆S T SR (X) which contradicts assumption S T SR (X) � ϕ. Accordingly, X does not contain any nonempty element of T SR . □ Definition 12. Suppose that A T SR � (U, F A , T SR ) is a T SR approximation space. e probability P ∅ that the T SR lower approximation of a subset of the universe set is an empty set is defined by ∅ represents a number of subsets of U which does not contain any nonempty element of T SR and n is the number of elements in U. (29) erefore, the probability that the T SR lower approximation of X⊆U is not an empty set is 1 − P ∅ .

Example 12
(1) Consider Example 4; obviously, the subsets a { } and b { } are the only subsets which do not contain any nonempty element of T SR . us, the probability P ∅ that the T SR lower approximation of a subset of the universe set is an empty set is defined by (2) Consider Example 5; obviously, the subsets b { }, c { }, and b, c { } are the only subsets which do not contain any nonempty element of T SR . erefore, the probability P ϕ that the T SR lower approximation of a subset of the universe set is an empty set is defined by In order to find the probability that T SR upper approximation of any subset is U, we may have to suggest the subsequent result.

Theorem 11. Consider A T SR � (U, F A , T SR ) is a T SR approximation space and X⊆U. S(X) � U if and only if the subset X intersects with every element in T SR .
Proof. Obvious. Now, for any T SR approximation space, the probability P U that T SR upper approximation of a subset is U can be obtained by the following formula: X is the number of subsets of U which intersect with every nonempty element of T SR and n is the number of elements in U. (32) e next example explains the above discussion. us, the probability P U that T SR upper approximation of a subset is U is defined by

Medical Application in Heart Failure
In the current article, we illustrate the significance of the suggested approach in decision-making problems for medical applications. Consequently, we apply it to the issue of heart failure. We have a data set with the results of five symptoms for twenty patients divided into twelve males (p 3 , p 6 , p 8 , p 9 , p 11 , . . . , p 17 , p 19 ) and 8 females (p 1 , p 2 , p 4 , p 5 , p 7 , p 10 , p 18 , p 20 ). e study was conducted at Om El-Kora Cardiac Center, Hospital of Heart Diseases, Tanta, Egypt. is research involved twenty patients who came to the hospital with various symptoms and underwent a thorough history, physical examination, lab tests, resting ECG, and conventional echo assessment. Finally, the diagnosis of heart failure was verified.

e Experimental Results.
e experimental findings are discussed in this subsection by adding a preparatory analysis performed on five heart disease symptoms for twenty patients, according to ivagar and Richard [40]. Table 2 shows the data from the information system for twenty patients, addressing the heart failure issue. e columns reflect the signs of heart failure diagnosis (where "Yes" indicates that the patient has symptoms and "No" indicates that the patient has none) [40], with condition attributes, such that e 1 indicates "the breathlessness," e 2 indicates "the orthopnea," e 3 indicates "the paroxysmal nocturnal dyspnea," e 4 indicates "reduced exercise tolerance," and e 5 indicates "the ankle swelling." Attribute D indicates "decision of heart failure." For rows in Table 2, P � p 1 , p 2 , p 3 , . . . , p 20 represents the set of twenty patients. erefore, the set of all attributes is A � e 1 , e 2 , e 3 , e 4 , e 5 ∪ D which is represented by columns.
Here 1 and 0 denote "yes" and "no", respectively. We apply the suggested method in the set of female's patients only and the others similarly. Accordingly, Table 3 represents the soft set of female's patients, where the set of female's patients is U � p 1 , p 2 , p 4 , p 5 , p 7 , p 10 , p 18 , p 20 and the set of attributes is A � e 1 , e 2 , e 3 , e 4 , e 5 .
Decision-making is essential in the daily lives, and this process yields the best alternative from a variety of options. We give Algorithm 1 in table for a decision-making of an information system in terms of the T SR approximations.

Reduction of Attributes.
A very important purpose of rough sets is the reduction of data by removing redundant attributes in the information system. So, the present subsection is devoted for the reduction of an information system in case of topological rough sets for the data given in Table 3. We extend the notion of "nanotopology," which has proposed by ivagar and Richard [40], to T SR approximations.
We shall apply the nanotopology of T SR approximations to identify the key factors of "heart failure" using topological reduction of attributes in information system of Table 3.
First, let us extend the definition of "nanotopology" into "T SR nanotopology" using "T SR approximations." Definition 13. Consider A T SR � (U, F A , T SR ) is a T SR approximation space, and X⊆U. erefore, the class N T SR � U, ϕ, S T SR (X), S T SR (X), BND T SR (X) is called "T SR nanotopology" which represents a general topology generated by the soft rough set X⊆U. e basis of this topology is given the class β T SR � U, S T SR (X), BND T SR (X) .

Definition 14.
Consider A T SR � (U, F A , T SR ) is a T SR approximation space, and N T SR is a T SR nanotopology with a basis β T SR . en, (i) if β T SR − e k � β T SR , then the attribute e k is called "dispensable"; (ii) if β T SR − e k ≠ β T SR , then the attribute e k is not "dispensable." erefore, the core of attributes is CORE � e k which represents the common part of reduction. Now, we apply the topological reduction for Table 3 to identify the key factors of "heart failure" as follows: We compute the T SR nanotopology to decision-making for two sets of patients: X � p 1 , p 4 , p 18 , p 20 which represents a set of patients that have the disease of heart failure and Y � p 2 , p 5 , p 7 , p 10 which represents a set of patients that do not have the disease of heart failure.
We will make a topological reduction for first set X and the second set Y similarly.
Case 1 (patients having the heart failure disease).
us, the basis of T SR nanotopology generated by the above T SR approximations is β T SR � U, p 1 , p 4 , p 18 , p 20 , p 5 , p 7 , p 10 .
Step 1. When the attribute "the breathlessness (e 1 )" is removed: e topology generated by this base is Table 4: Comparisons among some soft rough approximations and T SR approximations.

Conclusion
In this article notion of topological soft rough sets is introduced, where topology generated from a soft set plays a vital role. Here notion of soft rough approximations is discussed and some of their properties are given. eir properties have been studied and their relationships with some other methods have been examined. In fact, the proposed approaches fulfill all axioms of Pawlak's rough sets without adding extra restrictions as Propositions 3 and 4 illustrated. e proposed techniques depend basically on general topology and hence they open the way for applications of topology in soft rough sets. Further, we have answered some very important questions, such as how to determine the probability that a subset of the universe is definable in the classical rough sets and their extensions (like the soft sets and topological soft rough sets).
Finally, we have introduced medical applications, in the decision-making of medical diagnosis for heart failure problems [39], to illustrate the importance of current methods and also to compare proposed method and the previous ones. Moreover, we have succeeded in making a topological reduction for the data set covering the result of five symptoms for twenty patients with heart failure disease, and thus we identify the core factors of the heart failure diagnosis. Besides, two algorithms to our method have been obtained.
For future works, it is hoped that presented framework may be useful to study its application in COVID-19 and other diseases.

Data Availability
Not data were used to support this study.

Conflicts of Interest
e authors declare that they have no competing interests.