# Topological Approaches for Rough Continuous Functions with Applications

Authors: A. S. Salama; A. Mhemdi; O. G. Elbarbary; T. M. Al-shami
Journal: Complexity
 (2021)

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## Abstract

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.


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## Body

## 1. 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 [3–7]. This theory depends on a certain topological structure that achieved great success in many areas of real-life applications [8–14]. 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 setU 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 [17–19]. 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, [20–24]). 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 [25–29]. More recent approaches of rough set with its applications can be found in [30–32]. Other rough set theory applications in computer science (field of information retrievals) using topological generalizations can be found in [33–40].In this paper, we purpose further study on rough functions and introduce new concepts based on rough functions. In Section2, we give more details regarding the fundamentals of near-open sets. The 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.


## 2. 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 subfamilyτ 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. The 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 ofU, A¯,Ao, and Ac denote respectively the closure, interior, and complement of A in U, respectively.A subsetA of U,τ is called,(1)
Semi-open (resp., pre-open, open) set ifA⊆Ao¯ (resp., A⊆A¯o, A⊆Ao¯o) and its complement is called a semi-closed (resp., pre-closed, closed) set if A¯o⊆A (resp., Ao¯⊆A, A¯o¯⊆A). A subset which is both semi-open and semi-closed is called semi-regular(2)
Semi-pre-open set (or open set) ifA⊆A¯o¯ and it is called a semi-pre-closed set (or β closed set) if Ao¯o⊆A(3)
Regular-open set ifA⊆A¯o and it is called a regular-closed set if Ao¯=A(4)
δ-closed set if A=δA¯, where δA¯=x∈U:G¯o∩A≠φ,x∈G,G∈τTheα-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 αA¯ (resp., SA¯, spA¯). The semi-interior of A, denoted by sAo, is the union of all semi-open subsets of U.A subsetA of a topological space U,τ is called(1)
Generalized closed set ifA¯⊆G whenever A⊆G and G∈τ.(2)
Semi-generalized closed (briefly,sg-closed) set if sA¯⊆G whenever A⊆G and G is semi-open set. Its complement is called a sg-open set.(3)
Generalized semi-closed set ifsA¯⊆G whenever A⊆G and G∈τ.(4)
α-Generalized closed set if αA¯⊆G whenever A⊆G and G∈τ.(5)
Generalizedα-closed set if αA¯⊆G whenever A⊆G and G is α- open.(6)
gα∗∗-closed set if A¯⊆Go¯ whenever A⊆G and G is α- open.


## 3. 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.LetU,τ be a topological space. If X⊆U, then(1)
Semi-lower approximation ofX⊆U, X¯s=∪G:G∈SemiU,G⊆X, where SemiU is the family of all semi-open sets in U,τ.If we replace the family of all semi-open setsSemiU given in (1) above by a family of all pre-open sets PreU (resp., a family of all α- open sets αU, a family of all β-open sets βU, a family of all regular-open sets RegU, and a family of semi-regular-closed sets SRegU), we obtain pre-lower approximation (resp., α-lower approximation, β-lower approximation, regular-lower approximation, and semi-regular-lower approximation).(2)
Semi-upper approximation ofX⊆U,X¯s=∩F:F∈CSemiU,F∩X≠ϕ, where CSemiU is the set of all semi-closed sets in U,τ.If we replace the family of all semi-closed setsCSemiU given in (2) above by a family of all pre-closed sets CPreU (resp., a family of all α-closed sets CαU, a family of all β-closed sets CβU, a family of all regular-closed sets CRegU, and a family of semi-regular-open sets SRegU), 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. That base characterizes a topological space, called topological approximation space,Appτ=U,R,τR. The topological classes of R are also known as the topological granules, topological elementary sets, or topological blocks; we will use Gxm∈τ to denote the topological class containing x∈U. In the topological approximation space, we consider two operators R¯mX=x∈U:Gxm⊆X and R¯mX=x∈U:Gxm∩X≠ϕ called the topological lower approximation and topological upper approximation of X⊆U, respectively. Also, let POSmX=R¯mX denote the topological positive region of X⊆U,NEGmX=U−R¯mX denotes the topological negative region of X⊆U, and BONmX=R¯mX−R¯mX denotes the topological borderline region of X⊆U.The degree of topological completeness characterizes by the topological accuracy measure, in whichX represents the cardinality of set X⊆U as follows:(1)αmX=R¯mXR¯mX,X≠φ.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.The semi-topological class on a topological approximation spaceAppτ=U,R,τR is determined by X¯s,X¯s=A⊂U:X¯s⊂A⊂X¯s. A subset X⊂U is said to be semi-dense (semi-co-dense) if X¯s=U (X¯s=φ). By semi-rough pair on Appτ=U,R,τR, we mean any pair P,Q where P,Q⊆U satisfies the conditions:(Semi-1)P is the semi-open set in τR.(Semi-2)Q is the semi-closed set in τR.(Semi-3)P⊂Q.(Semi-4) there is a subsetS⊂U such that(1)
Sso=φ,(2)
S⊂Q−P¯s,(3)
Q−P¯s⊂S¯s.Lemma 1.
For any subsetA⊂U in the topological approximation space Appτ=U,R,τR, the pair A¯s,A¯s is a semi-rough pair on Appτ=U,R,τR in which every semi-open set in U is a semi-closed set.Proof.
LetP=Aso and Q=A¯s. Then, 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(1)
IfO⊆S is a semi-open set, hence O⊆A that gives O∩P=φ which is a contradiction; hence, O is not contained in A. Then, it must be S=φ which gives Sso=φ.(2)
SinceS=A−P¯s, A⊆A¯s, then A⊂Q. Then, we have S⊂Q−P¯s.(3)
Letx∈Q−P¯s, Q=A¯s, this means that x∈Aorx∉A. If x∈A, then for every semi-open set O and x∈O such that O∩A≠φ implies that O∩S≠φ and we have x∈S¯s, then Q−P¯s⊂S¯s. If x∉A, then there is a semi-open set O′ and x∈O′. Now, O′−P¯s=O′∩P¯sc is a semi-open set which contains x, and x∈A¯s, then there exists a point y∈A such that y∈O′−P¯s, hence y∈O′∩S; therefore, O′∩S≠φ, hence x∈S¯s. Then, we have the result Q−P¯s⊂S¯s.Lemma 2.
For any semi-rough pairP,Q in Appτ=U,R,τR in which every semi-open subset is semi-closed, there are subsets A,B⊆U such that P=A¯s and Q=B¯s.Proof.
LetP,Q be a semi-rough pair and let S be any subset, satisfying condition (Semi-4). Define A=P∪S, then P⊂A, hence P⊂Aso. If O⊂A is a semi-open set, then O−P¯s=O∩P¯sc is another semi-open set contained in A. Since O⊂A=P∪S, P⊂P¯s, then O⊂P¯s∪S, and we have O−P¯s⊂P¯s∪S−P¯s=S. Therefore, O−P¯s is a semi-open set contained in S which means O⊂P¯s. Since O⊂P∪S, it follows that O⊂P and this proves that P=A¯s.
Now, we haveP¯s∪S¯s⊃P¯s∪Q−P¯s=Q. Also, B=P∪S⊃P¯s∪Q−P¯s=Q and hence B¯s⊂Q¯s=Q. Then, we have Q=B¯s.Theorem 1.
For any topological subspaceX,τ∗,X⊆U of the topological approximation space Appτ=U,R,τR, the function f:X,τ∗⟶U,τR that defined by fA=Aso,A¯s,A∈τ∗ is bijection.Proof.
First, we will prove that the function is onto as follows: for any semi-rough pairAso,A¯s in Appτ=U,R,τR, then there exists A∈τ∗ such that fA=Aso,A¯s. Second, for the proof that f is one to one, if fA1=fA2, then A1so,A¯1s=A2so,A¯2s which implies to A1so=A2so and A¯1s=A¯2s and A1≈A2.


## 4. Topological Neighborhood Approach of Rough Continuity

LetX and Y be two subsets of a universe U, and let ApprX=X,S and ApprY=Y,P be two approximation spaces, where S and P are binary relations on X and Y, respectively. We define two subsets Srx=y∈X:x,y∈S and Slx=y∈X:y,x∈S of X (also two subsets Prx=y∈Y:x,y∈P and Prx=y∈Y:y,x∈P of Y) 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 Sr∩lx=Srx∩Slx and Pr∩lx=Prx∩Plx as follows:(2)τX=A⊆X:∀a∈A,Sr∩la⊆A,τY=B⊆Y:∀b∈B,Pr∩lb⊆B.The rough approximations using these topologies are defined as follows:(3)P¯τYB=∪G′∈τY:G′⊆B,S¯τXA=∪G∈τX:G⊆A,S¯τXA=∩F:Fc∈τX:A⊆F,P¯τYB=∩F′:F′c∈τY:B⊆F′.The functionf:X,τX⟶Y,τY is called a rough function on X if the image of each rough set in X is rough in Y.Namely, the functionf is totally rough iff all subsets A⊂X,A≠φ, such that S¯τXA≠S¯τXA, then P¯τYfS¯τXA≠P¯τYfS¯τXA in Y.The functionf is possibly rough iff some subsets A⊂X,A≠φ, such that S¯τXA≠S¯τXA, then P¯τYfS¯τXA≠P¯τYfS¯τXA in Y.Finally, the functionf is exact iff all subsets A⊂X,A≠φ, such that S¯τXA=S¯τXA, then P¯τYfS¯τXA=P¯τYfS¯τXA in Y.The functionf:X,τX⟶Y,τY is a topological rough, continuous function on X as the following:(1)
The functionf is topological, totally rough, continuous if for all subsets A⊂Y,A≠φ; if AτYo⊆A¯τY, then f−1A¯τYτXo⊆f−1AτYo¯τX in X.(2)
The functionf is topological, possibly rough, continuous if for some subsets A⊂Y,A≠φ; if AτYo⊆A¯τY, then f−1A¯τYτXo⊆f−1AτYo¯τX in X.(3)
Finally, the functionf is topological exact continuous if for all subsets A⊂Y,A≠φ; if AτYo=A¯τY, then f−1A¯τYτXo=f−1AτYo¯τXin X.Example 1.
LetX,τX and Y,τY be topological spaces, where X=a,b,c andτX=X,φ,a,b,a,b and Y=1,2,3, τY=Y,φ,1,2,1,2. Let f:X⟶Y be a map defined by fa=1,fb=2 and fc=3, then our results are given in Table 1.
Then, according to Table1, the function f is a topological totally rough continuous function.Table 1
Calculations of topological rough continuous functions.
Subsets ofY/our measures{1}{2}{3}{1, 2}{1, 3}{2, 3}YAτYo{1}{2}φ{1, 2}{1}{2}YA¯τY{1, 3}{2, 3}{3}Y{1, 3}{2, 3}Yf−1AτYo{a}{b}φ{a, b}{a}{b}Xf−1A¯τY{a, c}{b, c}{c}X{a, c}{b, c}Xf−1A¯τYτYo{a}{b}φX{a}{b}Xf−1AτYo¯τX{a, c}{b, c}φX{a, c}{b, c}XProposition 1.
LetX,τX and Y,τY be topological spaces and let f:X⟶Y be a function. The following are equivalent:(1)
f is rough continuous.(2)
For everyF⊆Y, f−1F¯τY∈τXc.(3)
For everyx∈X, f is a rough continuous at x.(4)
For everyA⊆X, fA¯τX⊆fA¯τY.Proof.
We will use the sequence (3) implying (1) implying (4) implying (2) implying (3) to prove the equivalence of the proposition.(3) implying (1): suppose a nonempty open setV∈τY, for a fixed point x∈f−1V, we have fx∈V. But since f is rough continuous at x, then there exists an open set Gx⊆X such that fGx⊂V and fGxτYo⊆fGx¯τY, then we have f−1fGx¯τYτXo⊆f−1fGxτYo¯τX and Gx∈f−1V; this gives that f is rough continuous.(1) implying (4): suppose thatf is rough continuous and let A⊆X. Let x∈A¯τX. Let an open set V∈τY such that x∈f−1V. Then, by the definition of rough upper approximation f−1V∩A≠φ. Let x′∈f−1V∩A, then fx′∈V∩fA. Then, we have V∩fA≠φ. Then, we have fA¯τX⊆fA¯τY.(4) implying (2): fix a closed subsetF⊆Y; let A=f−1F; we will prove that A=A¯τX. But each subset is contained in its upper approximation, A⊂A¯τX. Now, we will prove that A¯τX⊂A. Let x∈A¯τX, then using (4), we have fx∈fA¯τX⊆fA¯τY⊆F¯τY=F; hence fx∈F or x∈f−1F=A. Then, we have f−1F¯τY∈τXc.(2) implying (3): letx∈X and V∈τY be an open set containing fx. Then, Y−V is a closed set and f−1Y−V is a closed set in X which does not contain the point x. But x∈X−f−1Y−V. Then, there exists an open set G containing x such that x∈G⊆X−f−1Y−V, then fG⊆fX−f−1Y−V=fX−Y−V⊆V. Then, f is rough continuous at x.Theorem 2.
Suppose thatτiX,i=1,2,3,… be a family of topologies defined on X. Let f:X⟶Y be a rough continuous function for every τi,∀i where Y,τY is a topological space. Then, f is a rough continuous function with respect to the topology τX=∩iτiX.Proof.
LetG∈τY, then GτYo⊆G¯τY, since f is a rough continuous function for every τi,∀i, then f−1G¯τYτio⊆f−1GτYo¯τi,∀i. Then, we have f−1G¯τYτXo⊆f−1GτYo¯τX in τX=∩iτiX, hence f is a rough continuous function with respect to the topology τX=∩iτiX.Theorem 3.
Letfi:X⟶Yi,τi be a family of functions. Suppose that τX is the topology on X generated by the class β=∪ifi−1G:G∈τi, then(1)
fi is rough continuous for each i.(2)
IfτX∗ is the intersection of all topologies on X such that fi is rough continuous for each i; then, τX=τX∗.(3)
τX is the coarser topology on X which gives that fi is rough continuous for each i.(4)
The classβ=∪ifi−1G:G∈τi is a sub-base of τX.(5)
The functiong:Y⟶X is rough continuous if and only if fi∘g is rough continuous.Proof.
Part (1): for each functionfi:X⟶Yi,τi if F∈τi then Fτio⊆F¯τi and fi−1F∈β. But β⊆τi, then fi−1F∈τi, hence f−1F¯τiτXo⊆f−1Fτio¯τX; then, we have the result.Part (2): we can easily prove thatβ⊆τX∗, but the topology τX is generated by β, then τX⊆τX∗. Otherwise, τX is one of the topologies that make the functions fi which are rough continuous. Then, we have τX∗⊆τX, hence τX=τX∗.Part (3): it is obvious by proof of Part (2).Part (4): since any collection of subsets ofX is a sub-base of a topology on X, then β is a sub-base of the topology τX.Part (5): if the functiong:Y⟶X is rough continuous, then all functions fi∘g are rough continuous. Otherwise, let fi∘g be rough continuous and let G∈β, then there exists a subset H∈τi such that G=fi−1H. But g−1G=g−1fi−1H=fi∘g−1H. Now, we have Hτio⊆H¯τi, then fi∘g−1H¯τiτXo⊆fi∘g−1Hτio¯τX. Then, fi∘g is rough continuous.


## 5. Minimal Neighborhood Approach for Rough Continuity

We generalize the concept of rough function to topological rough function by using topological structures. The 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.LetX,τ be a topological space and x∈X. Then, we defineNminx=∩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¯minx which is the closure of Nminx in X,τ.Iff:X,τ⟶Y,τ∗ is a function between two topological spaces X,τ and Y,τ∗, we define the functions fmin:X,τ⟶Y,τ∗ by fminx=∩M⊆Y:fx∈G′⊆M,∀G′∈τ∗ for every x∈X.Letf:X,τ⟶Y,τ∗ be a function, where X and Y are topological spaces. The function f is called a topological rough function on X if and only if Nminxτo≠Nminx¯τ for every x∈X. Also, f is a topological rough function on Y if fminxτ∗o≠fminx¯τ∗ for every point fx in Y.Example 2.
LetX,τ and Y,τ∗ be topological spaces, where X=a,b,c andτ=X,φ,a,a,b and Y=1,2,3, τ∗=Y,φ,1,2,1,2. Let f:X⟶Y be a map defined by fa=2,fb=1 and fc=3, then(4)Nmina=a,Nminb=a,b,Nminc=X,fmina=2,fminb=1,fminc=Y.
Then, we have(5)Nminaτo=a,Nmina¯τ=X,Nminbτo=a,b,Nminb¯τ=X,Nmincτo=X,Nminc¯τ=X.
Also,(6)fminaτ∗o=2,fmina¯τ∗=2,3,fminbτ∗o=1,fminb¯τ∗=1,3,fmincτ∗o=3,fminc¯τ∗=3.
Then, the functionf is not a topological rough function onX and Y.
A functionf:X,τ⟶Y,τ∗ is said to be topological roughly continuous at the point x∈X if and only if f−1Nminfx⊆Nminx, and it is topological roughly continuous on X if it is topological roughly continuous at every point x∈X.Example 3.
Letf:X,τ⟶Y,τ∗ be a function defined by fa=2fb=fd=3 and fc=4, where X=a,b,c,d and Y=1,2,3,4 with(7)τ=X,φ,a,a,b,a,b,c,τ∗=Y,φ,1,2,1,2,2,3,4.
Then,f is a topological rough function on X and(8)Nmina=a,butNminfa=Nmin2=2thenf−1Nmin2=a,Nminb=a,b,butNminfb=Nmin3=2,3,4thenf−1Nmin3=X,Nminc=a,b,c,butNminfc=Nmin4=2,3,4thenf−1Nmin4=X,Nmind=X,butNminfd=Nmin3=2,3,4thenf−1Nmin3=X,then f−1Nminfx⊆Nminx for every x∈X, and then f is a topological rough continuous function on X.


## 6. Topological Approximations of a Function as a Relation

The functionf:X⟶Y is a relation from X to Y when it satisfies the conditions:(i)
Domf=X,(ii)
Ifx,y∈f and x,z∈f, then y=z.IfX=Y, we say f is a function on X. By this way, any function f:X⟶Y can completely be represented by its graph Gf=x,fx:x∈X.Letf:U1,R1⟶U2,R2 be any function, where A1=U1,R1 and A2=U2,R2 are approximation spaces, such that R1 and R2 are any binary relations on U1 and U2, respectively. We define the relation R=R1×R2 such that Rx=R1x×R2x is the blocks of U1×U2. For the function, Gf=x,fx:x∈U1, we define the approximations(9)R¯Gf=∪G⊆Rx:G∈Gf,R¯Gf=∩G⊆Rx:G∩Gf=φ.A functionf:U1⟶U2 is said to be rough in the approximation space A=U,R, where A1=U1,R1 and A2=U2,R2 are approximation spaces and A=A1×A2,U=U1×U2 if R¯Gf=R¯Gf; otherwise, f is an exact function.Example 4.
LetU1=a,b,c,d,e and U2=1,2,3,4,5,6 be two universes; we define the function f:U1⟶U2, by its graph Gf=a,1,a,6,b,6,c,5,c,6,e,6. Consider the blocks of the binary relations R1 and R2 as follows:(10)R1x=a,c,a,b,d,e,R2x=1,3,2,4,5,3,4,6.
Then,(11)Rx=R1x×R2x=a,1,a,3,c,1,c,3,a,2,a,4,a,5,c,2,c,4,c,5,a,3,a,4,c,3,c,4,a,6,c,6,a,1,a,3,b,1,b,3,a,2,a,4,a,5,b,2,b,4,b,5,a,3,a,4,b,3,b,4,a,6,b,6,d,1,d,3,e,1,e,3,d,2,d,4,d,5,e,2,e,4,e,5,d,3,d,4,e,3,e,4,d,6,e,6.
Then, we have(12)R¯Gf=a,6,b,6,c,6,R¯Gf=a,1,a,3,c,1,c,3,a,6,b,6,c,6,a,2,a,4,a,5,c,2,c,4,c,5,d,6,e,6.
Therefore, the functionf is a rough function such that R¯Gf≠R¯Gf.
When we have two approximation spaces defined by two equivalence relations, we have the following proposition that governs the product space.Proposition 2.
LetA1=U1,R1 and A2=U2,R2 be two arbitrary approximation spaces. Then, we have U1×U2/R1×R2=U1/R1×U2/R2.Proof.
Suppose thatu1,u2∈U1, and v1,v2∈U2, then we have(13)u1,v1,u2,v2∈R1×R2,iffu1,u2∈R1,v1,v2∈R2.
Suppose again thatu1,v1R1×R2∈U1×U2/R1×R2.
Then, we have(14)u1,v1R1×R2=u2,v2:u1,v1,u2,v2∈R1×R2=u2,v2:u1,u2∈R1,v1,v2∈R2=u2:u1,u2∈R1×v2:v1,v2∈R2=u1R1×v1R2.
Then, we have the result asU1×U2/R1×R2=U1/R1×U2/R2.
Letf:U1,R1⟶U2,R2 be any function, where A1=U1,R1 and A2=U2,R2 are arbitrary approximation spaces. We define the relation Gf=x,fx:x∈U1 to be the graph of the function f. The rough approximations of Gf are defined as follows:(15)R¯Gf=u1,u2∈U1×U2:u1,u2R⊆Gf,R=R1×R2,R¯Gf=u1,u2∈U1×U2:u1,u2R∩Gf≠φ,R=R1×R2.
Accordingly, the functionf is rough if R¯Gf=R¯Gf; otherwise, f is an exact function. The pair R¯Gf,R¯Gf is called a rough pair of relations.
The following theorems give the conditions on approximation spaces that give exact functions, one-to-one, surjective, and continuous functions.Theorem 4.
The functionf:U1⟶U2 is an exact function for any selective approximation spaces A1=U1,R1 and A2=U2,R2.Proof.
The selective approximation space property means thatu,vR=u,v,u∈U1,v∈U2,R=R1×R2. Then, we have R¯Gf=R¯Gf, which yields to that the function f is an exact function.Theorem 5.
The functionf:U1⟶U2 is one-to-one function for any selective approximation spaces A1=U1,R1 and A2=U2,R2 if and only if both R¯Gf and R¯Gf are one-to-one functions.Proof.
The proof is directly using the definition of selective approximation space and using the technology in Theorem1.Theorem 6.
The functionf:U1⟶U2 is a surjective function for any selective approximation space A1=U1,R1 and A2=U2,R2 if and only if both R¯Gf and R¯Gf are surjective functions.Proof.
One can prove the theorem using similar technique given in Theorem1.Theorem 7.
The functionf:U1⟶U2 is a continuous function for any selective approximation space A1=U1,R1 and A2=U2,R2 if and only if both R¯Gf and R¯Gf are continuous functions.Proof.
As in the technique used in Theorem5, when we have two topological spaces, generated using two bases βR1,βR2, where A1=U1,R1 and A2=U2,R2 are two approximation spaces, then we have the following proposition that governs the product topology.Proposition 3.
LetT1=U1,τ1 and T2=U2,τ2 be two arbitrary topological spaces. Then, we have U1×U2/βR1×βR2=U1/βR1×U2/βR2.Proof.
Similar to the proof of Proposition2, the rough pairs of relations satisfied the following two important theorems.Theorem 8.
For the quasidiscrete product topological spaceU1×U2,τ, if R¯Gf,R¯Gf is a rough pair of relations, and Q,τ′ is a subspace of U1×U2,τ such that Q is closed in τ, then R¯Gf∩Q,R¯Gf∩Q is a relative rough pair of relations when R¯Gf,R¯Gf,Q⊂U1×U2,R¯Gf⊂Q.Proof.
The pairR¯Gf,R¯Gf is a rough pair of relations in U1×U2,τ, if the following condition satisfied the following:(1)
R¯Gf is an open relation in U1×U2,τ.(2)
R¯Gf is a closed relation in U1×U2,τ.(3)
R¯Gf⊂R¯Gf.(4)
The relationR¯Gf−R¯Gf¯τ contains a relation S of U1×U2 such that Sτo=φ and R¯Gf−R¯Gf¯τ⊂S¯τ.
Only we need to prove thatR¯Gf∩Q,R¯Gf∩Q is a rough pair of relations in Q,τ′; the proof will end by(1)
SinceR¯Gf is an open relation in U1×U2,τ, and Q,τ′ is a subspace of U1×U2,τ, then R¯Gf∩Q is an open relation in Q,τ′.(2)
SinceR¯Gf is a closed relation in U1×U2,τ, then there is an open relation S, such that R¯Gf=U1×U2−S, then R¯Gf∩Q=U1×U2∩Q−S∩Q=Q−S∩Q, but S∩Q is an open relationship with Q,τ′, then R¯Gf∩Q is a closed relation in the subspace Q,τ′.(3)
SinceR¯Gf⊂R¯Gf, then R¯Gf∩Q⊂R¯Gf∩Q.(4)
By selectingS=R∩Q,R=R1×R2, then the relation R¯Gf∩Q−R¯Gf¯τ contains the relation S, and we need to prove the two subconditions:(a)
Sτo=φ,(b)
R¯Gf−R¯Gf¯τ⊂S¯τ.
For the proof of Part (a)Sτo=φ, suppose that Sτo≠φ, then there is an τ-open relation G⊂Q such that G⊂S but S=R∩Q, i.e., G⊂R, but G=G′∩Q such that G′ is an open relation in U1×U2,τ, then G′∩Q⊂R; hence, G′∩Qτo⊂Rτo, but Rτo=φ, which gives contradiction; then, it must be Sτo=φ.
For the proof of Part (b),R¯Gf−R¯Gf¯τ⊂S¯τ.
SinceR¯Gf,R¯Gf is a rough pair in U1×U2,τ, then there is a relation R′⊂U1×U2, such that R¯Gf=R′τo and R¯Gf=R′¯τ; since R⊂R¯Gf−R¯Gf¯τ, we have S=R∩Q=R′∩Q−R¯Gf¯τ.
Now, letu,v∈R¯Gf∩Q−R¯Gf¯τ, then u,v∈R¯Gf∩Q and u,v∉R¯Gf¯τ.
Now, ifu,v∈R¯Gf∩Q, then u,v∈S and u,v∈S¯τ.
Finally, ifu,v∈R′∩Q and u,v∈R¯Gf∩Q and u,v∉R¯Gf¯τ, hence u,v∈R¯Gf and u,v∈Q. Now, u,v∈R′¯τ, then there is an open relation G in τ such that u,v∈G and G∩R′≠φ, but u,v∉R¯Gf¯τ, then u,v∈G−R¯Gf¯τ=G∩R¯Gf¯τc. But since R¯Gf=Rτ′o is an open relation in τ, and R¯Gf=R¯Gf∩Q is an open relation in τ′, then R¯Gf¯τ is a closed relation in τ, hence R¯Gf¯τc is an open relation in τ, hence G∩R¯Gf¯τcis an open relation containing u,v, then G∩R¯Gf¯τc∩R′≠φ. This yields to G∩R′−R¯Gf¯τ≠φ, i.e., G∩R′−R¯Gf¯τ∩Q≠φ, such that u,v∈Q. Hence, G∩R′∩Q−R¯Gf¯τ∩Q≠φ, but R¯Gf⊂Q, then G∩R′∩Q−R¯Gf¯τ≠φ. But we haveR=R′−R¯Gf¯τ, hence R∩Q=R′∩Q−R¯Gf¯τ implies that S=R′∩Q−R¯Gf¯τ. Then, G∩S≠φ, but u,v∈Q, hence G∩Q∩S≠φ. But G∩Q=G′and G′ is an open relation in τ′ that contains u,v, thenG∩S≠φ, henceu,v∈S¯τ.Theorem 9.
LetR¯Gf,R¯Gfbe a rough pair of relations in the product topological space U1×U2,τ, and let Q,τ′ be a subspace of U1×U2,τ such that Q is any relation of U1×U2. Then, there is a relation P⊂Q such that R¯Gf∩Q=Pτo and R¯Gf∩Q=P¯τ.Proof.
We can defineP=R¯Gf∩Q, then P⊂Q, and then R¯Gf∩Qτo=Pτo, but R¯Gf∩Q is an open relation in Q,τ′, i.e., R¯Gf∩Qτo=R¯Gf∩Q, hence R¯Gf∩Q=Pτo.
Finally, forR¯Gf∩Q=P¯τ, since R¯Gf,R¯Gf is a rough pair in U1×U2,τ, then R¯Gf∩Q⊂R¯Gf∩Q, i.e., P⊂R¯Gf∩Q implies that P¯τ⊂R¯Gf∩Q¯τ. But R¯Gf∩Q is a closed relation, i.e., R¯Gf∩Q¯τ=R¯Gf∩Q, hence P¯τ⊂R¯Gf∩Q.
ForR¯Gf∩Q⊂P¯τ, let u,v∈R¯Gf∩Q, then u,v∈R¯Gf and u,v∈Q, but since R¯Gf∩Q=P, then Q⊂R¯Gf−Pc=U1×U2−R¯Gf−P, henceu,v∈U1×U2−R¯Gf−P, i.e., u,v∈P, hence u,v∈P¯τ, then R¯Gf∩Q⊂P¯τ. Then, we have R¯Gf∩Q=P¯τ.
LetU1×U2,τ be a product space. For any relation Q⊂U1×U2, define the subspace Q,τ′ of U1×U2,τ. We define the equivalence relation Eτ′ on the power set PQ by R1,R2∈Eτ′⇔R1τ′o=R2τ′o,R1¯τ′=R2¯τ′ for any R1,R2∈PQ. The set PQ/Eτ′ is a partition of PQ and any class η∈PQ/Eτ′ is called a relative topological rough relations.Theorem 10.
For any product topological spaceU1×U2,τ and for any subspace Q,τ′ of it, the function f:PQ/Eτ′⟶η′, defined by fR=Rτ′o,R¯τ′,R∈η′, is bijection, where η′ is the set of all relative rough pairs.Proof.
The proof is directly by Theorems5 and 6.
LetU1,τ1 and U2,τ2 be any two topological spaces, where β1 and β2 are any two bases for τ1 and τ2. Then, we define the base β=β1×β2 of the topology τ=τ1×τ2.
We define the approximations for any subsetH⊆U1×U2:(16)Hτo=∪G⊆β:G∈H,H¯τ=∩G⊆β:G∩H=φ.
The functionf on U1×U2 is called a topological rough continuous function at the point x,y∈U=U1×U2 if f−1Vfx,y⊆τ for all open sets Vfx,y∈τ. The function f is topological rough continuous on U1×U2 if it is topological roughly continuous at every point of U1×U2.Example 5.
Consider the topologyτ1=U1,φ,a,b,c,d, on U1=a,b,c and the topology τ2=U2,φ,3,1,2,4 on U2=1,2,3,4. The bases β1=a,b,c,d and β2=3,1,2,4 are of τ1 and τ2, respectively.
We defined the functionf:U1×U2⟶U1×U2 as follows:(17)fx,y=a,3.
Then, we have(18)β=β1×β2=a,b,c,d×3,1,2,4=a×3,a×1,2,4,b,c,d×3,b,c,d×1,2,4,τ=τ1×τ2=U1,φ,a,b,c,d×U2,φ,3,1,2,4=U1×U2,U1×φ,U1×3,U1×1,2,4,φ×U2,φ×φ,φ×3,φ×1,2,4,a×U2,a×φ,a×3,a×1,2,4,b,c,d×U2,b,c,d×φ,b,c,d×3,b,c,d×1,2,4.
Then, for any pointx,y∈U1×U2, we have fx,y=a,3, then all open sets containing a,3 are(19)V1=a×U2=a,1,a,2,a,3,a,4,V2=a×3=a,3.
Then, the inverse function of these open sets is(20)f−1V1=U1×U2,f−1V2=U1×U2.
Then, the functionf is topological rough continuous at every point of U1×U2.


## 7. 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.The 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 systemT=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) functionfc:PC×PC⟶PC, wherePC is the power set of the condition attributes as follows:(21)fcB,B′=C,ifPOSBD≠POSB′D,∀B′⊆C,B′,ifPOSBD=POSB′D,∀B′⊂B.Figure1 gives an example for a projection function on information system. The core of such systems is given by taking the intersection of all these projection functions on that system.Figure 1
Some reductions of information system by projection function.The topological rough continuous functions of information systems can be defined as follows:The functionf:U,C,D⟶U,C′,D is topological roughly continuous at the object x∈U if f−1D−ax=DCx, where DCx=y∈U:DCx=DCy. The 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 systemT, denoted MT, we will mean n×n matrix defined as follows:MT=mij=1,2,3,n, where(22)mija∈C:axi≠axj,if∃b∈D,bxi≠bxj,λ,if∀b∈D,bxi≠bxj,such that axi or axj belongs to the C-positive region of D; mij is the set of all conditions attributed that classify objects axi and axj into different classes; mij=λ denotes that this case does not need to be considered.The discernibility functionf:T=U,C,D⟶MT of an information system is defined as follows.For any object,x1∈U:fTxi=∧j∨mij:i≠j∈i,2,…,n, where ∨mij is the disjunction of all variables b∈mij, when mij≠φ and ∨mij|=0, when mij=φ and ∨mij=1, when mij=λ.Figure2 gives an example for a discernibility function on information system.Figure 2
Discernibility function of information system.According to Figure2, the function f transfers the system T=U,C,E into the discernibility MT and the reduction of this system can be obtained as follows:(23)fTxi=fTa,b,c,d=b∧a∨b∧c∨d∧b∨d∧a∨b∨c∧a∨b∨c∨d.Then, we have(24)fTxi=b∧c∨d.Accordingly, the systemT=U,C,E has two reductions, namely, R1=b,c and R2=b,d with core CORET=b.


## 8. 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.The structureS=U,At,Va:a∈At,fa,RP:P⊆At is the mathematical style of information system of our patients problem. The set U is the system universe that we selected to be a set of five patients. The set At is the set of attributes of these patients with respect to tests functions such as liver, kidney, and heart functions. The set Va is values of each attribute a∈At. Finally, fa:U⟶Va is the information function such that fax∈Va.For any subsetB∈At, we define the relation RP=x,y:fax−fay<α,a∈P,x,y∈U,α∈R+; for a∈At, we define the class ARa as follows: ARa=Rax:x∈U, where Rax=y:xRay.The structureDS=U,At,D,Va:a∈At,fa,RP:P⊆At is a decision table, 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 attributeD by(25)RD=x,y:fax=fay,a∈D,x,y∈U.The class of this relation isRDx=y:xRDy. The set of all classes isARa=Rax:x∈U. We define the set P⊆At to be a reduct of At, if τD⊆τP and P is minimal.Basic data of five patients before the surgery are given in Table2 (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. Therefore, we defined the prediction function fP:DS⟶DS¯¯, where DS¯¯ is the decision system of patients over time t (dynamic decision system of patients).Table 2
The infection information system of patients.
Patients’ IDLiver functionsKidney functionsHeart efficiencyDecisionA1A2A3A4DX135456.8412Need SurgeryX245444.2420Need drugsX342385.8430Need SurgeryX430449.7480Need SurgeryX536325.4450Need drugsNow, if we choose for the liver function attributesP1=LF=A1,A2, the threshold α1=4, then RP1U=X3,X5,X1,X4,X5,X3,X5,X2,X3,X4. The topology generated by RP1 is given by(26)τP1=U,φ,X3,X5,X1,X4,X5,X3,X5,X2,X3,X4,X1,X3,X4,X5,X2,X3,X4,X5.For kidney functions, we can chooseα1=2.5 for P2=KF=A3, then RP2U=X4,X1,X4,X1,X2,X3,X5,(27)τP2=U,φ,X4,X1,X4,X1,X2,X3,X5.For the heart efficiency attributeP3=HE=A4, we can choose α3=20, and then RP3U=X4,X5,X1,X2,X3,X5,X1,X2,X3,X4,(28)τP3=X4,X5,X1,X2,X3,X1,X2,X3,X5,X1,X2,X3,X4.For the decision attributes, we haveRDU=X1,X3,X4,X2,X5, then the topology of decisions is τD=U,φ,X1,X3,X4,X2,X5.The condition attributes are exactly three attributes, namely,At=LF,KF,HE. The numbers of nontrivial subsets of the set of all condition attributes are seven subsets, namely, P1,P2,P3,P1,P2,P1,P3,P2,P3,P1,P2,P3.Now, we will calculate the classes of the residue subsets by taking the intersections as follows:RP1,P2U=RP1U∩RP2U=φ, with topology τP1,P2=U,φ.RP1,P3U=RP1U∩RP3U=φ, with topology τP1,P3=U,φ.RP2,P3U=RP2U∩RP3U=X1,X2,X3,X5, with topology τP2,P3=U,φ,X1,X2,X3,X5.The covering class of universe using all condition attributes is given as follows:RP1,P2,P3U=RP1U∩RP2U∩RP3U=φ, with topology τP1,P3=U,φ. Then, the system given in Table 2 has no topological reductase.Now, we define the functionfP:DS⟶DS¯¯,P⊆At by fPXi=Xi,i=1,2,3,4,5. Then, according the function fPXi, 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.


## 9. Conclusion and Future Work

The 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. Then, we can extract the results of weights in a decision table that we can classify the documents according to the reduction of this table. The 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.Finally, we will benefit from the applications given in [41–47] to investigate new practical applications using rough sets and soft sets.



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