Topological Approaches for Rough Continuous Functions with Applications

In this paper, we purposed further study on rough functions and introduced some concepts based on it. We introduced and investigated the concepts of topological lower and upper approximations of near-open sets and studied their basic properties. We deﬁned and studied new topological neighborhood approach of rough functions. We generalized rough functions to topological rough continuous functions by diﬀerent topological structures. In addition, topological approximations of a function as a relation were deﬁned and studied. Finally, we applied our approach of rough functions in ﬁnding the images of patient classiﬁcation data using rough continuous functions.


Introduction
Many studies have appeared recently and dealt with generalizations of topological near-open sets [1,2] and the possibility of using them in many life applications, including their use in data reduction and reaching some new decisions and conclusions. Rough set theory is a modern approach for reasoning about data [3][4][5][6][7]. is theory depends on a certain topological structure that achieved great success in many areas of real-life applications [8][9][10][11][12][13][14]. Now, the general topologists can say, "rough sets theory is a topological bridge from real-life problems to computer science" [15,16].
Rough set theory was introduced as a novel approach to processing of incomplete data. Among the aims of the rough set theory is a description of imprecise concepts. Suppose we are given a finite nonempty set U of elements, called universe. Each element of U is characterized by a description, for example, a set of attribute values. In rough sets formulated by Pawlak, an equivalence relation on the universe of elements is determined based on their attribute values. In particular, this equivalence relation is initiated using the equality relation on the attribute values. Many real-world applications have both nominal and continuous attributes [17][18][19]. It was early recognized that standard rough set model based on the indiscernibility relation is well suited in the case of nominal attributes.
Several procedures were made to overcome limitations of this approach and many authors presented interesting extensions of the initial model (see, for example, [20][21][22][23][24]). It was noted that considering a similarity relation instead of an indiscernibility relation is quite relevant. A binary relation forming classes of objects, which are identical or at least not noticeably different in terms of the available description, can represent the similarities between objects [25][26][27][28][29]. More recent approaches of rough set with its applications can be found in [30][31][32]. Other rough set theory applications in computer science (field of information retrievals) using topological generalizations can be found in [33][34][35][36][37][38][39][40].
In this paper, we purpose further study on rough functions and introduce new concepts based on rough functions. In Section 2, we give more details regarding the fundamentals of near-open sets. e goal of Section 3 is to introduce the concepts of topological lower and upper approximations of near-open sets and discuss their basic properties. We spotlight on rough numbers in Section 4. We aim in Section 5 to define and study new topological neighborhood approach of rough functions. Section 6 is devoted to generalize the concept of rough function to topological rough function by using different topological structures. Topological approximations of a function as a relation are defined and studied in Section 7. In Section 8, we suggest some applications of rough functions to information systems and give some applications of them in data retrieval. Finally, conclusions of the work are given in Section 9.

Basic Concepts of Topological Near-Open Sets
In this part, we recall the definitions of some near-open subsets of a topological space which are useful in the sequel. A subfamily τ of the power set of U is called a topology if it contains ∅, U as well as it is closed under arbitrary union and finite intersection. e pair (U, τ) is called a topological space; elements in τ are called open sets, and their complements are called closed sets.
For a subset of U, A, A o , and A c denote respectively the closure, interior, and complement of A in U, respectively.
A subset A of (U, τ) is called, e α-closure (resp. semi-closure, semi-pre-closure) of a subset A of (U, τ) is the intersection of all α-closed (resp. semi-closed, semi-pre-closed) sets that contain and is denoted by α(A) (resp., S(A), sp(A)). e semi-interior of A, denoted by s(A o ), is the union of all semi-open subsets of U.
A subset A of a topological space (U, τ) is called

Topological Near-Open Approach of Rough Approximations
In this section, we introduce and investigate the concepts of topological lower and upper approximations of near-open sets and study their basic properties. Let (U, τ) be a topological space. If X ⊆ U, then , and a family of semiregular-closed sets SReg(U)), we obtain pre-lower approximation (resp., α-lower approximation, β-lower approximation, regular-lower approximation, and semi-regular-lower approximation).
If we replace the family of all semi-closed sets CSemi(U) given in (2) above by a family of all pre-closed sets CPre(U) (resp., a family of all α-closed sets Cα(U), a family of all β-closed sets Cβ(U), a family of all regular-closed sets CReg(U), and a family of semi-regular-open sets SReg(U)), we obtain pre-upper approximation (resp., α-upper approximation, β-upper approximation, regular-upper approximation, and semi-regular-upper approximation).
Motivation for topological rough set theory has come from the need to represent subsets of a universe in terms of topological classes of the topological base generated by the general binary relation defined on the universe. at base characterizes a topological space, called topological approximation space, App τ � (U, R, τ R ). e topological classes of R are also known as the topological granules, topological elementary sets, or topological blocks; we will use G xm ∈ τ to denote the topological class containing x ∈ U. In the topological approximation space, we consider two operators R m (X) � x ∈ U: G xm ⊆X and R m (X) � x ∈ U: { G xm ∩ X ≠ ϕ} called the topological lower approximation and topological upper approximation of X⊆U, respectively. Also, let POS m (X) � R m (X) denote the topological positive region of X⊆U, NEG m (X) � U − R m (X) denotes the topological negative region of X⊆U, and BON m (X) � R m (X) − R m (X) denotes the topological borderline region of X⊆U. e degree of topological completeness characterizes by the topological accuracy measure, in which |X| represents the cardinality of set X⊆U as follows:

Complexity
We define here the semi-rough pairs as an example of topological rough sets and we study their properties. You can use any type of the abovementioned near-open sets as another example. e semi-topological class on a topological approxi- . By semirough pair on App τ � (U, R, τ R ), we mean any pair (P, Q) where P, Q⊆U satisfies the conditions:

Lemma 1. For any subset
Proof. Let P � A o s and Q � A s . en, the conditions from (Semi-1) to (Semi-3) are directly satisfied. Now, we need to prove condition (Semi-4). Define S � A − P s , then we have Proof. First, we will prove that the function is onto as follows: for any semi-rough pair

Topological Neighborhood Approach of Rough Continuity
Let X and Y be two subsets of a universe U, and let Appr(X) � (X, S) and Appr(Y) � (Y, P) be two approximation spaces, where S and P are binary relations on X and Y, respectively. We define two subsets S r (x) � y ∈ X: which are called right and left neighborhoods of an element x ∈ X. We define now two topologies on X and on Y, respectively, using the intersection of the right and left neighborhoods S r∩l (x) � S r (x) ∩ S l (x) and P r∩l (x) � P r (x) ∩ P l (x) as follows: e rough approximations using these topologies are defined as follows: Namely, the function f is totally rough iff all subsets e function f is possibly rough iff some subsets Finally, the function f is exact iff all subsets is a topological rough, continuous function on X as the following: (1) e function f is topological, totally rough, con- (2) e function f is topological, possibly rough, con- , then our results are given in Table 1.
en, according to Table 1, the function f is a topological totally rough continuous function.
is a rough continuous function with respect to the topology Suppose that τ X is the topology on X generated by the class Part (2): we can easily prove that β⊆τ * X , but the topology τ X is generated by β, then τ X ⊆τ * X . Otherwise, τ X is one of the topologies that make the functions f i which are rough continuous. en, we have τ * X ⊆τ X , hence τ X � τ * X . Part (3): it is obvious by proof of Part (2).  (4): since any collection of subsets of X is a subbase of a topology on X, then β is a sub-base of the topology τ X . Part (5): if the function g: Y ⟶ X is rough continuous, then all functions f i ∘ g are rough continuous.
Otherwise, let f i ∘ g be rough continuous and let G ∈ β,

Minimal Neighborhood Approach for Rough Continuity
We generalize the concept of rough function to topological rough function by using topological structures. e topological spaces with rough sets are very useful in the field of digital topology which is widely applied in the image processing in computer sciences.
Let (X, τ) be a topological space and x ∈ X. en, we define N min (x) � ∩ N⊆X: x ∈ G⊆N, ∀G ∈ τ { } which is called the minimal neighborhood containing the point x with respect to the topology τ on X. Let (X, τ) be a topological space, for any element x ∈ X; we define the subset N min (x) which is the closure of N min (x) in (X, τ).
If f: (X, τ) ⟶ (Y, τ * ) is a function between two topological spaces (X, τ) and (Y, τ * ), we define the functions Let f: (X, τ) ⟶ (Y, τ * ) be a function, where X and Y are topological spaces. e function f is called a topological rough function on X if and only if ( en, we have en, the function f is not a topological rough function onX and Y. A function f: (X, τ) ⟶ (Y, τ * ) is said to be topological roughly continuous at the point x ∈ X if and only if f − 1 (N min (f(x)))⊆N min (x), and it is topological roughly continuous on X if it is topological roughly continuous at every point x ∈ X.
en, f is a topological rough function on X and for every x ∈ X, and then f is a topological rough continuous function on X.

Topological Approximations of a Function as a Relation
e function f: X ⟶ Y is a relation from X to Y when it satisfies the conditions: If X � Y, we say f is a function on X. By this way, any function f: X ⟶ Y can completely be represented by its Let f: (U 1 , R 1 ) ⟶ (U 2 , R 2 ) be any function, where A 1 � (U 1 , R 1 ) and A 2 � (U 2 , R 2 ) are approximation spaces, such that R 1 and R 2 are any binary relations on U 1 and U 2 , respectively. We define the relation A function f: U 1 ⟶ U 2 is said to be rough in the approximation space A � (U, R), where A 1 � (U 1 , R 1 ) and are approximation spaces and { }. Consider the blocks of the binary relations R 1 and R 2 as follows: en, erefore, the function f is a rough function such that R(G(f)) ≠ R(G(f)).
When we have two approximation spaces defined by two equivalence relations, we have the following proposition that governs the product space.

Proposition 2.
Let A 1 � (U 1 , R 1 ) and A 2 � (U 2 , R 2 ) be two arbitrary approximation spaces. en, we have Proof. Suppose that u 1 , u 2 ∈ U 1 , and v 1 , v 2 ∈ U 2 , then we have 6 Complexity Suppose again that [(u 1 , v 1 )] R 1 ×R 2 ∈ (U 1 × U 2 )/R 1 × R 2 . en, we have en, we have the result as ( Let f: (U 1 , R 1 ) ⟶ (U 2 , R 2 ) be any function, where A 1 � (U 1 , R 1 ) and A 2 � (U 2 , R 2 ) are arbitrary approximation spaces. We define the relation G(f) � (x, f(x)): x ∈ U 1 } to be the graph of the function f. e rough approximations of G(f) are defined as follows: Accordingly, the function f is rough if R(G(f)) � R(G(f)); otherwise, f is an exact function. e pair (R(G(f)), R(G(f))) is called a rough pair of relations. e following theorems give the conditions on approximation spaces that give exact functions, one-to-one, surjective, and continuous functions.

Proof.
e selective approximation space property means that en, we have R(G(f)) � R(G(f)), which yields to that the function f is an exact function. □ Theorem 5. e function f: U 1 ⟶ U 2 is one-to-one function for any selective approximation spaces A 1 � (U 1 , R 1 ) and

and only if both R(G(f)) and R(G(f)) are one-to-one functions.
Proof. e proof is directly using the definition of selective approximation space and using the technology in eorem 1.  Proof. As in the technique used in eorem 5, when we have two topological spaces, generated using two bases β R 1 , β R 2 , where A 1 � (U 1 , R 1 ) and A 2 � (U 2 , R 2 ) are two approximation spaces, then we have the following proposition that governs the product topology. □ Proposition 3. Let T 1 � (U 1 , τ 1 ) and T 2 � (U 2 , τ 2 ) be two arbitrary topological spaces. en, we have Proof. Similar to the proof of Proposition 2, the rough pairs of relations satisfied the following two important theorems.

Theorem 8. For the quasidiscrete product topological space
Proof.
e pair (R(G(f)), R(G(f))) is a rough pair of relations in (U 1 × U 2 , τ), if the following condition satisfied the following: Only we need to prove that (R(G(f)) ∩ Q, R(G(f)) ∩ Q) is a rough pair of relations in (Q, τ ′ ); the proof will end by )) τ contains the relation S, and we need to prove the two subconditions: □ Theorem 9. Let (R(G(f)), R(G(f)))be a rough pair of relations in the product topological space (U 1 × U 2 , τ), and let (Q, τ ′ ) be a subspace of (U 1 × U 2 , τ) such that Q is any relation of U 1 × U 2 . en, there is a relation P ⊂ Q such that Proof. We can define P � R(G(f)) ∩ Q, then P ⊂ Q, and Let (U 1 × U 2 , τ) be a product space. For any relation Q ⊂ U 1 × U 2 , define the subspace (Q, τ ′ ) of (U 1 × U 2 , τ). We define the equivalence relation E(τ ′ ) on the power set P(Q) is a partition of P(Q) and any class η ∈ P(Q)/E(τ ′ ) is called a relative topological rough relations. □ Theorem 10. For any product topological space (U 1 × U 2 , τ) and for any subspace (Q, τ ′ ) of it, the function f: Proof. e proof is directly by eorems 5 and 6. Let (U 1 , τ 1 ) and (U 2 , τ 2 ) be any two topological spaces, where β 1 and β 2 are any two bases for τ 1 and τ 2 . en, we define the base β � β 1 × β 2 of the topology τ � τ 1 × τ 2 .
We define the approximations for any subset H⊆U 1 × U 2 : e function f on U 1 × U 2 is called a topological rough continuous function at the point (x, y) (x, y)))⊆τ for all open sets V(f(x, y)) ∈ τ. e function f is topological rough continuous on U 1 × U 2 if it is topological roughly continuous at every point of } are of τ 1 and τ 2 , respectively. We defined the function f: U 1 × U 2 ⟶ U 1 × U 2 as follows: en, we have 8 Complexity en, for any point (x, y) ∈ U 1 × U 2 , we have f(x, y) � (a, 3), then all open sets containing (a, 3) are en, the inverse function of these open sets is en, the function f is topological rough continuous at every point of U 1 × U 2 .

Future Applications of Topological Rough Functions on Information Systems
In this section, we will define a function between two information systems and give all needed conditions for them. Functions of an information system can produce the reductions, and the core of this system by the projection of the system on subsystems. We will define the image of rough set using some types of these functions. Finally, we define the topological rough functions of information systems and study some of their properties. e reader can review about information systems in [7,18] to know about the structure and the types and the different methods of reduction of information systems. Suppose an information system T � (U, C, D) where U is the set of objects of this system (patients, plants, etc.). C is the condition attributes of these objects (temperature, muscle pain, etc.). D is the expert decisions about the condition attribute that objects suffer from.
We define the projection (restriction) function f c : P(C) × P(C) ⟶ P(C), whereP(C) is the power set of the condition attributes as follows: Figure 1 gives an example for a projection function on information system. e core of such systems is given by taking the intersection of all these projection functions on that system. e topological rough continuous functions of information systems can be defined as follows: e function f: . e function f is topological roughly continuous on U if it is topological roughly continuous for every object of U.
By a discernibility matrix of information system T, denoted M(T), we will mean n × n matrix defined as follows: such that a(x i ) or a(x j ) belongs to the C-positive region of D; m ij is the set of all conditions attributed that classify objects a(x i ) and a(x j ) into different classes; m ij � λ denotes that this case does not need to be considered. e discernibility function f: T � (U, C, D) ⟶ M(T) of an information system is defined as follows.
For any object, where ∨m ij is the disjunction of all variables b ∈ m ij , when m ij ≠ φ and ∨m ij | � 0, when m ij � φ and ∨m ij � 1, when m ij � λ. Figure 2 gives an example for a discernibility function on information system. According to Figure 2, the function f transfers the system T � (U, C, E) into the discernibility M(T) and the reduction of this system can be obtained as follows: en, we have Accordingly, the system T � (U, C, E) has two reductions, namely,

Predictions of Patients Classification Data Using Rough Continuous Functions
Our aim in this application problem, which will give in this section, is to find recommendations for patients that combine treatment and exercise by explaining the function of each symptom, whether positive or negative.
In this application problem, the decision according to the medical reports is the continuation of taking all medicine and doing medical tests. In fact, it is a painful decision. Our role is to analyze the medical data using the notion of reduction which will help us to determine which of the patients can stop taking medicine as well as expect the required period of time to do that. e structure S � (U, At, V a : a ∈ At , f a , R P : P⊆At ) is the mathematical style of information system of our patients problem. e set U is the system universe that we selected to be a set of five patients. e set At is the set of attributes of these patients with respect to tests functions such as liver, kidney, and heart functions. e set V a is values of each attribute a ∈ At. Finally, f a : U ⟶ V a is the information function such that f a (x) ∈ V a .
For any subset B ∈ At, we define the relation R P � (x, y): |f a (x) − f a (y)| < α, a ∈ P, x, y ∈ U, α ∈ R + ; for a ∈ At, we define the class A R a as follows: where D is the set of decisions that represents for each patient if he needs surgery or enough drugs.
We define the relation of the decision attribute D by e class of this relation is R D (x) � y: xR D y . e set of all classes isA R a � R a (x): x ∈ U . We define the set P⊆At to be a reduct of At, if τ D ⊆τ P and P is minimal.
Basic data of five patients before the surgery are given in Table 2 (the decision system of patients). Each patient will measure these medical functions periodically every three months. After a period of time, we need to predict the results of the medical tests of patients at any time and accordingly they can stop drugs. erefore, we defined the prediction function f P : DS ⟶ DS, where DS is the decision system of patients over time t (dynamic decision system of patients). Now, if we choose for the liver function attributes P 1 � LF � A 1 , A 2 , the threshold α 1 � 4, then R P 1 (U) � X 3 , X 5 , X 1 , X 4 , X 5 , X 3 , X 5 , X 2 , X 3 , X 4 . e topology generated by R P 1 is given by For kidney functions, we can choose α 1 � 2.5 for P 2 � KF � A 3 , then R P 2 (U) � X 4 , X 1 , X 4 , X 1 , X 2 , X 3 , X 5 }}, τ P 2 � U, φ, X 4 , X 1 , X 4 , X 1 , X 2 , X 3 , X 5 .
For the decision attributes, we have R D (U) � X 1 , X 3 , X 4 , X 2 , X 5 , then the topology of decisions is τ D � U, φ, X 1 , X 3 , X 4 , X 2 , X 5 . e condition attributes are exactly three attributes, namely, At � LF, KF, HE { }. e numbers of nontrivial subsets of the set of all condition attributes are seven subsets, namely, P 1 , P 2 , P 3 , P 1 , P 2 , P 1 , P 3 , P 2 , P 3 , P 1 , P 2 , P 3 . Now, we will calculate the classes of the residue subsets by taking the intersections as follows: e covering class of universe using all condition attributes is given as follows: R P 1 ,P 2 ,P 3 (U) � R P 1 (U) ∩ R P 2 (U) ∩ R P 3 (U) � φ, with topology τ P 1 ,P 3 � U, φ . en, the system given in Table 2 has no topological reductase. Now, we define the function f P : DS ⟶ DS, P⊆At by f P (Xi) � Xi, i � 1, 2, 3, 4, 5.
en, according the function f P (Xi), the image of Table 2 after a period of three months has no topological reduces and this function is one-to-one rough continuous function.

Conclusion and Future Work
e emergence of topology in the construction of some rough functions will be the bridge for many applications and will discover the hidden relations between data. Topological generalizations of the concept of rough functions open the way for connecting rough continuity with the area of near continuous functions.
Applications of topological rough functions of information systems open the door about the many transformations among different types of information systems such as multivalued and single-valued information systems.
Future applications of our approach in the computer can be as follows: In information retrieval fields, we can modify the query running online by defining a function that converted documents to weighted vectors of the words of that document. en, we can extract the results of weights in a decision table that we can classify the documents according to the reduction of this table. e query is constructed by defining a Boolean function of all words of the reduction. Classification and summarization of documentation using topological functions of neighboring systems are defined in documents.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.