Topologies Induced by Relations with Applications

: Some methods for inducing topological structures by relations were initiated and their importance in applications were indicated. Topologies generated by equivalence relations were all quasi-discrete spaces. We induced the topologies generated using similarity relations and pre-order relations. Also, the topologies generated using general binary relations on the universe of discourse were initiated. Finally, rheumatic fever data reduction using topologies induced by relations were studied.


INTRODUCTION
Topology is an important and interesting area of mathematics, the study of which will not only introduce you to new concepts and theorems but also put into context old ones like continuous functions. It is so fundamental that its influence is evident in almost every other branch of mathematics. This makes the study of topology relevant to all who aspire to be mathematicians whether their first love is algebra, analysis, category theory, chaos, continuum mechanics, dynamics, geometry, industrial mathematics, mathematical biology, mathematical economics, mathematical finance, mathematical modeling, mathematical physics, mathematics of communication, number theory, numerical mathematics, operation research or statistics. Topological notions like compactness, connectedness and denseness are as basic to mathematicians of today as sets and functions were to those of last century [3,8,9] .
For a long time, many individuals believed that abstract topological structures have limited application in the generalization of real line and complex plane or some connections to Algebra and other branches of mathematics. And it seems that there is a big gap between these structures and real life applications. We noticed that in some situations, the concept of relation is used to get topologies that are used in important applications such as computing topologies [15] , recombination spaces [2,7,17] and information granulation [21] which are used in biological sciences and some other fields of applications.
The aim of rough set theory is to give a description of the set of objects by logical, set-theoretical, topological etc. tools in terms of similarity relations and derived notions related by these relations. The description of the set of objects entails as well relationships and functional or near to functional dependencies among various similarity relations generated by various sets of the set of objects.
Rough sets were first introduced by [10,11] and are based on approximation spaces. An approximation space is a pair A = (Ob, R). Here, R is an equivalence relation, also called indiscernibility relation, imposing a granularity on the universe Ob such that R ⊆ Ob×Ob. Furthermore, we assume Ob to be finite. For x∈Ob, let [x] R be the equivalence class containing x, i.e., [x] R = {y: y R x } .
Given an arbitrary set X⊆Ob, we wish to describe X in terms of elements or granules of Ob/R. Pawlak proposed the use of lower and upper approximations of a set X, denoted R (X) and R (X), respectively. Lower and upper approximations are defined as: The semantics of the approximations of sets may be defined as follows: • Elements of the universe that belong to R (X) are those elements that surely belong to the set X • Elements that belong to R (X) possibly belong to the set X • Elements that belong to Ob/ R (X) are elements of the universe that surely do not belong to the set X. Hence, the uncertainty lies in R (X)/ R (X) which is also called area of uncertainty. Elements of the area of uncertainty may, or may not, belong to X The approximation operators can also be considered using membership functions. It is possible to define a rough membership function as presented in [12] .

MATERIALS AND METHODS
Topologies induced by relations: Let A = (Ob, R) be an approximation space. The equivalence classes Ob\R of the relation R will be called elementary sets (atoms) in A. Every finite union of elementary sets in A will be called a composed set in A. The family of all composed sets in A will be denoted by com (A).
The family com (A) in the approximation space A = (Ob, R) is a topology on the set Ob.
Since the approximation space A = (Ob, R), defines uniquely the topological space τ (A) = (Ob, com (A)) and com (A) is the family of all open sets in τ (A) and U/R is a basis for τ (A), then τ (A) is a quasidiscrete topology on Ob and com (A) is both the set of all open and closed sets in τ (A). Thus, the lower approximation and the upper approximation of any subset X⊆Ob can be interpreted as the interior and the closure of the set X in the topological space τ (A), respectively.

Lemma 1:
If β is a base for a topological space (Ob,τ), where β is a partition of Ob, then for every subset X⊆Ob: .The set of all equivalence classes of E(τ) is denoted by Ob/ E(τ).
Proposition 3: Let A = (Ob, R) be an approximation space and let τ R be the topology generated by the base B R = Ob/R. If (Ob, τ) is the quasi-discrete topological space has Ob/E(τ) as a base. Then τ R = τ iff for all x∈B R ∈β R there exists B∈Ob/E(τ) such that x∈B.
Lemma 6 [15] : If τ is a quasi-discrete topology on a set Ob, then the family {cl τ ({x}): x ∈ Ob} is a partition of Ob.
Proof: x∈ ∈ ∈ ∈B, B∈β R : For any n approximation spaces A 1 = (Ob, R 1 ), A 2 = (Ob, R 2 ),…, A n = (Ob, R n ) we define the partition   4 }} is the base of τ, then τ is a quasi-discrete topology and: Ob / E( ) (Ob / E( )) (Ob / E( )) τ = τ ∩ τ Let is the symmetric covering of Ob by the similar relation R. Then we define a relation R induced by by x R y iff there exist B∈ and x,y∈B.

Proposition 10:
The relation R is a similar relation on the set of objects Ob.
Since is a covering of Ob, then for any x∈Ob there exists B∈ such that x∈B hence x, x∈B∈ then xR x. Let xR y then there exists B∈ such that x, y ∈B then y, x∈B hence yR x.

Proposition 10
For every x∈Ob we have: ⇔ ∃ B∈ β and x∈B and y∈B ⇔ ∃ B∈ β and y∈B ⇔ Let is the covering of Ob. Then we define the class * = {R (x): x∈Ob}.

Proposition 11:
The class * is a symmetric covering of the set of objects Ob and R ⊆ R * .

Proof:
• x∈Rβ(y) ⇔ ∃ B∈β(y) and x∈B ⇔∃ B∈β(x) and x,y∈B∈β* x,y∈R β* Let A⊆Ob be any non empty subset of the set of objects. Then A is called a similar pre-class of R if for any x, y∈A (x, y)∈R.

Proposition 12: Every similar class R(x) is a maximal similar pre-class.
For an element x∈Ob we define a class called the pre-similar class of x as follows: L R (x) = {A⊆Ob: x∈A and A is similar pre-class of R}. Let L R = {L R (x): x∈Ob} be the family of all pre-similar classes. Then we define a relation R* on L R by for any L R (x), L R (y) ∈ L R , L R (x)R*L R (y) iff there exist A∈L R (x) and B∈L R (y) and A B ≠ ϕ .

Proposition 13:
• The relation R* on L R is a similar relation • xRy iff L R (x)R*L R (y) for any x,y∈Ob Proof: • Since for any L R (x) ∈ L R and A∈L R (x), A A ≠ φ then L R (x)R*L R (x) hence R* is reflexive. Also if L R (x)R*L R (y) then there exist A∈L R (x) and B∈L R (y) such that A B≠φ, hence B A≠φ, hence L R (y)R*L R (x). then R* is symmetric.
• Firstly, we will prove that xRy L R (x)R*L R (y) Let (x, y)∈R {x, y}is a similar pre-class of R.
there exist a similar class R(x) such that {x,y}⊆ R(x) and R(x)∈L R (x) but R is symmetric then R(x)∈L R (y), then there exist A = R(x)∈L R (x) and B = R(x)∈L R (y) and A B ≠ ϕ , hence L R (x)R*L R (y).
Conversely, let for some x,y∈Ob, L R (x)R*L R (y) then there exist R(z)∈L R (x) and R(z)∈L R (y) a similar class of R. hence x∈R(z) and y∈R(z) then x,y ∈R(z) hence xRy. Let

Proposition 14:
• For any x∈Ob, L * R (x) ⊆ R(x) • The class M is a symmetric covering of Ob Proof: then M is a symmetric covering of Ob. τ and x,y∈u Example: Let Ob = {c 1 ,c 2 ,…, c 7 } be the set of objects which is seven computers in a local network in a certain company. Let be the irregular topology on the set of objects which induced by a general relation on Ob which makes the following graph. We define a similar relation R on the set of objects by: Two computers x and y are in relation by R iff the computer x has a copy of a certain program in the computer y.
Then we can define the similar classes of R as follows: • R(c 1 ) = {c 1  Proposition 16: Let R τ and R τ be the lower and upper similar topologies then: The following proposition present another way to generate topologies from similarity relations.

Proposition 17:
is a topology on Ob. Proof: such that x = x and if (x, y) R ∈ then (x,y)∉R or x = y then (y,x)∉R or y = x hence (y,x)∈ R • (x, y)∈ R ⇔ ( x,y) ∉ R or x = y ⇔ (x,y)∈R or Topologies generated using dominance (pre-order) relations: For a long time, many mathematicians believed that there is a large deviation between abstract topological structures and computing [12][13][14] .
A relation R on a set Ob is called a dominance relation (pre-order) whenever R is both reflexive and transitive . If x is related to y, we write x R y and say that x dominances y. The set is called the before set. In a finite space ( Ob, ), τ it is clear that  Proof: First, consider p∈U is insulated from Ob-U, (y, p) ∈R. Let y∉U, then y∈Ob-U, So (y, p)∉R, but (y, p)∈R, a contradiction, then y∈U.
Second, consider p∈U, x ∈Ob-U, suppose (x, p) ∈R. Then x∈U contradicts that x∉U, then

Proposition 20: If R is a dominance relation on a set
Ob, then is topology on Ob.
Proof: Clearly Ob and φ are elements in τ R let U i ∈ τ R for every i∈I. The following example (Table 1) is an application for the above algorithm.  Let Ob be the set of objects and let R be any binary relation on Ob. The relation R gives rise to a closure operator cl R as follows: In the following we will give an example (Table 2) for closure space generated by a general relation.  Table 2 for closures and interiors of the subsets of Ob: We note from Table 2 that:  Table 3 shows closures and interiors of the subsets of Ob. From Table 3, we have:

Topologies generated using general binary relations:
The basic aim of this section is to generate topological structures using the lower and the upper approximations of any binary relation. Given general approximation space where R here is any general binary relation on Ob. For any subset X of Ob we define lower and upper approximations as follows [18,19,22] : Then the following structures are topologies on Ob: These topologies have the property that: Also, if we deal with the upper approximation instead of the lower approximation we can construct the following topologies: In the following we will give some illustrative examples and remarks.
be the universe and let be a general binary relation on Ob. Then we have the following topologies on Ob using the lower approximation: If we made more iteration to introduce more topologies using the lower approximation we will obtain that: If we made more iteration to introduce more topologies using the upper approximation we will obtain that : If the relation R on the universe Ob is identity or contain the identity relation, then all topologies induced by the lower or the upper approximations are discrete.
If we made more iteration to introduce more topologies using the lower approximation or the upper approximation, then all new iterations will introduce the same topologies we before obtained.
Another method for constructing topologies using the lower and the upper approximations is presented bellow: All the following are topologies on Ob: Also, all the following structures are topologies on Ob.
and so on.
Example: According to Example 4.1 we have:

RESULTS AND DISCUSSION
Here we will give the main conventions that we will apply in this work. These conventions will be indicated by examples.
We briefly describe the rheumatic fever datasets used in our example. No doubt that, the rheumatic fever is a very common disease. It has many symptoms differs from patient to another but though the diagnosis it is the same. So, we obtained the following data on seven rheumatic fever patients from Banha fever hospital, Egypt. All patients are between 9-12 years old with history of Arthurian began from age 3-5 years. This disease has many symptoms and it is usually started in young age and still with the patient along his life. Table 4 introduced the seven patients characterized by 8 symptoms (Attributes) using them to decide the diagnosis for each patient (Decision Attribute). Table 5 shows the rheumatic fever information system.
Let us consider the topological space a τ generated using binary relation defined on the attribute a. Also, using the same terminology the topological space B τ is the topology generated using general relation defined on a subset of attributes B of all condition attributes At. The decision attribute generates the topology D τ .
Now, we will use the following suggestion: •    p1  S2  yes  a1  r1  yes  e1  p1  no  d3  p2  S1  yes  a1  r1  yes  e2  p1  yes d3  p3  S2  yes  a2  r1  no  e1  p1  no  d3  p4  S1  yes  a1  r2  no  e1  p1  no  d1  p5  S1  no  a0  r1  no  e1  p2  no  d2  p6  S1  yes  a1  r1  no  e2  p1  no  d3  p7  S1  yes  a2  r1  no  e1  p1 yes d3 • The attribute At a ∈ is called the core if When the classical technique of rough set theory (ROSETTA software) used to obtain reducts and core of our data we found that we have 8 reducts of Table 5 with out any intersections among them. So, we do not have any core of Table 5. The set of obtained reducts is as follows: Now, after getting the reducts of Table 5 using the ROSETTA software. We will convert Table 4-7 using  Table 6. Now we will apply the above contributions on Table 6 where we will apply the relation to deal with the decision attribute D and we can construct the following topology: We observe that, D τ ≤ τ α , this leads to from the above contributions that } {α is the reduct and it is the core.
Then we can get the degree of dependency for each attribute as follows: , we get γ(β, D) = 0 and for a = δ, γ(δ, D) = 0. But if we get the degree of dependencies for the other attributes we will find that: Thus, the set of attributes of equal highest degree of dependency is the reduct of our system. So we conclude that {α}is the reduct of our data using the topological method also, {α} is the core of our system. Now, we observe that the reduction that we got by using the GMIS is contained in the reduction that we got using the discernibility matrix and this clears for us that our method for getting the reduction is more precise than using the ROSETTA method. Because, the ROSETTA method can not apply on general binary relations.
Topological reduction of single valued datasets: By reduction we mean if we can remove some data from the data table given in our information system preserving its basic properties. To express this idea more precisely, let S = (Ob, At, {V a : ∈ a At}, f a ) be an information system (numerical system). Let r be a positive real, for each object x ∈ Ob and for ∈ a At, N a (x, r) is the a-neighborhood of x and defined by: N a (x, r) = {y ∈Ob:    Ob  a1  a2  a3  a4  x1  1  2  9  6  x2  3  2  6  2  x3  3  6  3  3  x4  4  2  2  3  x5  6  6  5  4 One of the two attributes . By the same manner, we can define a highly order reducts of At in S.
In each case, the topological core of At in S is the intersection of all reducts (intersection of all the same order reducts). This core called the interior core and denoted Core Int (At). By the same terminology, we can define the closure core (Core C1 (At))and the neighborhood core (Core N (At)).
Illustrated Example Consider the information system given by Table 8  {a ,a } are second order reducts of At and the second order core is given by N 3 Core (At) {a } = .

CONCLUSION
There are many approaches for obtaining topologies by relations and we used some of them in data reduction. These approaches were generalizations to Pawlak approaches namely, we ignored the notion of equivalence relations. Also, these approaches open the way for other approximations if we use the general topological recent concepts such as pre-open sets or semi-open sets. Make use of this terminology to obtain the missing values in incomplete datasets will be a good future work [1,[4][5][6]16,20] . Implementing software for large data sets reduction using advanced programming languages will be also a good future work.