Mathematical Innovations of a Modern Topology in Medical Events

The purpose of this paper is to introduce a new topology called Rough Topology in terms of rough sets and prove that rough topology can be used to analyze many practical/real life problems. Using this concept, we find the deciding factors for the most common diseases chikungunya and diabetes.


Introduction
Rough set theory, introduced by Zdzislaw Pawlak, is a mathematical tool for representing, reasoning and decision making in the case of uncertain information. This theory deals with the approximat ion of sets or concepts by means of equivalence relations and is considered as one of the first non-statistical approaches in data analysis. Several interesting applications of the theory have come up, in part icular, in Artificial Intelligence and Cognitive Sciences. The main advantage of rough set theory in data analysis is that, it does not require any preliminary or additional informat ion of the data. The main d ifference between rough sets and fuzzy sets is that the rough sets have precise boundaries whereas fuzzy set theory is generally based on ill-defined sets of data, where the bounds are not precise and hence fu zzy pred ictions tend to deviate from exact values. The lower and upper approximat ions of a set are analogous to the interior and closure operations in a topology generated by data. In this paper, we have introduced a new topology called rough topology in terms of lower and upper appro ximations of a rough set and we have applied the concept of topological basis to find the deciding factors for chikungunya and diabetes.

Preliminaries
Definiti on 2.1 [6]: Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U named as the indiscernibility relat ion. The pair (U,R) is called the appro ximation space. Let X be a subset of U.
i) The lo wer appro ximat ion of X with respect to R is the set of all objects, which can be for certain classified as X with respect to R and it is denoted by R *( X). That is, ii) The upper appro ximat ion of X with respect to R is the set of all objects, wh ich can be possibly classified as X with respect to R and it is denoted by R*(X). That is, R* iii) The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor as not-X with respect to R and it is denoted by B R X. That is, B R (X)= R*(X) -R * (X).
The set X is said to be rough with respect to R if R*(X) ≠ R * (X). That is, if B R (X) ≠ φ.
Proposition 2.2 [6]: If (U, R) is an appro ximation space and X and Y are subsets of U, then and R*(X) ⊆ R*(Y) whenever X ⊆ Y viii) R * (X C )= [R*(X)] C and R*(X C ) = [R * (X)] C ix) R * R * (X)= R*R * (X) = R * (X) x) R* R*(X)= R * R*(X)= R*(X) Re mark 2.3 : R*: P(U) → P(U) satisfies the Kuratowski closure axio ms that i) R* (φ) = φ ii) X ⊆ R*(X) iii) R*(X ∪ Y) = R* (X) ∪R*(X) iv) R* R*(X) = R*(X) for all subsets X and Y of U If F = {X ⊆ U / R*(X) = X} ,using conditions (i) to (iv), we see that φ and U are in F; X  Y ∈ F whenever X and Y are in F and  X α ∈ F for all X α in F. Therefore, the family T, of comp lements of members of F is a topology on U. Thus, F is the family of T-closed sets. Also, Cl(X) = R*(X). Therefore, R* is the Kuratowski's closure operator.
Remark 2.4: Since R * : P(U) → P(U) satisfies the fo llowing properties that i) R * (U) = U ii) R * (X) ⊆ X iii) R * (X ∩ Y) = R * (X) ∩ R * (Y) iv) R * R * (X) = R * (X) for all subsets X and Y of U, the operator R * is the interior operator.

Rough Topology
In this section we introduce a new topology called rough topology in terms of the lo wer and upper appro ximations.
Remark 3.1: Let U be the universe of objects and R be an equivalence relation on U. For X ⊆ U, we define τ R = {U , φ, R * X, R*(X), B R X}, where R*(X), R * (X) and B R (X) are respectively the upper approximat ion, the lower appro ximation and the boundary region of X with respect to R. We note that U and φ ∈ τ R . Since R * (X) ⊆ R*(X), Definiti on 3.2: Let U be the universe, R be an equivalence relation on U and τ R = {U,φ, R * (X), R*(X), B R (X)} where X ⊆ U. τ R satisfies the following axio ms: i) U and φ ∈ τ R .
ii) The union of the elements of any subcollection of τ R is in τ R .
iii) The intersection of the elements of any finite subcollection of τ R is in τ R .
τ R forms a topology on U called as the rough topology on U with respect to X. We call (U, τ R , X) as the rough topological space. Proof: ii) Consider U and R * (X) fro m B. Let W = R * (X). Since U ∩ R * (X) = R * (X), W ⊂ U ∩ R * (X) and every x in U ∩ R * (X) belongs to W. If we consider U and B R (X) fro m B, taking W = B R (X), W ⊂ U ∩ B R (X) and every x in U ∩ B R (X) belongs to W, since U ∩ B R (X) = B R (X). And when we consider R * (X) and B R (X), R * (X) ∩ B R (X) = φ. Thus, B is a basis for τ R .
Definiti on 3.5: Let U be the universe and R be an equivalence relation on U. Let τ R be the rough topology on U and β R be the basis for τ R . A subset M of A, the set of at-tributes is called the core of R if β M ≠ β R-(r) for every r in M. That is, a core of R is a subset of attributes which is such that none of its elements can be removed without affecting the classification power of attributes.

Rough Topology in Chikungunya
Here we consider the problem of Chikungunya, a disease that is transmitted to humans by virus-carrying Aedes mosquitoes. There have been recent breakouts of CHIKV associated with severe illness. It causes fever and severe joint pain. Other symptoms include muscle pain, headache and nausea. Initial symptoms are similar to dengue fever. It is usually not life threatening. But the joint pain can last for a long time and fu ll recovery may take months. Usually patient gets lifelong immun ity fro m infection and hence re-infection is very rare. In recent decades the disease has spread to Africa and Asia, in particu lar, the Indian subcontinent.
Consider the following informat ion table giving data about 8 patients. The colu mns of the table represent the attributes (the symptoms for chikungunya) and the rows represent the objects (the patients). The entries in the table are the attribute values. The patient P 5 is characterized by the value set (Joint pain, No), (Headache, Yes), (Nausea, Yes), (Temperature, High) and (Chikungunya, No), which gives information about the patient P 5 . In the table, the patients P 1 , P 2 , P 3 , P 6 , P 7 and P 8 are indiscernible with respect to the attribute 'Joint pain'. The attribute 'Joint pain' generates two equivalence classes, namely, {P 1 ,P 2 ,P 3 ,P 6 ,P 7 ,P 8 } and {P 4 ,P 5 }, whereas the attributes 'Joint pain' and 'Headache' generate the equivalence classes {P 1 , P 6 , P 7 , P 8 }, {P 2 ,P 3 },{P 4 } and {P 5 }. The equivalence classes for the attributes Joint pain, Headache, Nausea and Temperature are {P 1 }, {P 2 ,P 3 },{P 4 },{P 5 },{P 6 ,P 8 } and {P 7 }. For the set of patients having chikungunya, the lower appro ximation = {P 1 ,P 6 ,P 8 } and the upper approximation = {P 1 , P 2 , P 3 , P 6 , P 8 } and hence the boundary region = {P 2 , P 3 }. Hence the patients P 2 and P 3 cannot be uniquely classified in v iew of the available knowledge. The patients P 1 , P 6 and P 8 display symptoms which enable us to classify them with certainty as having chikungunya. In our case, the symptoms Jointpain, Headache, Nausea and Temperature are considered as the condition attributes and the disease chikungunya is considered as the decision attribute. Not all condition attributes in an informat ion system are necessary to depict the decision attribute before decision rules are generated. It may happen that the decision attribute depends not on the whole set of condition attributes but on a subset of it and hence we are interested to find this subset which is given by the core. Here U = {P 1 ,P 2 ,...,P 8 }.
Observation: Fro m both cases we conclude that 'Jointpain' and 'Temperature' are the key attributes necessary to decide whether a patient has chikungunya or not.

Rough Topology in Diabetes
Diabetes is a group of metabolic diseases in which a person has high blood sugar, either because the body does not produce enough insulin, or because cells do not respond to the insulin that is produced. In diabetes, glucose in the blood cannot move into cells, so it stays in the blood. This not only harms the cells that need the glucose for fuel, but also harms certain organs and tissues exposed to the high glucose levels. This high blood sugar produces the classical symptoms of polyuria (frequent urination), weight loss and polyphagia (increased hunger).
Consider the following table g iving informat ion about six patients X is taken as the set of patients not having diabetes, then again CORE(R) = {F}.
Observation: Since the core of R has F as its only element, 'Frequent Urination' is the key attribute that has close connection to the disease diabetes .
The procedure applied in the above two cases can be put in the form of an algorith m as fo llo ws: Algorithm: Step 1: Given a fin ite universe U, a finite set A of attributes that is divided into two classes, C of condition attributes and D of decision attribute, an equivalence relation R on U corresponding to C and a subset X of U, represent the data as an information table, co lu mns of which are labeled by attributes, rows by objects and entries of the table are attribute values.
Step 2 : Find the lower appro ximat ion, upper approximation and the boundary region of X with respect to R.
Step 3 : Generate the rough topology τR on U and its basis β R .
Step 4: Remove an attribute x fro m C and find the lo wer and upper approximat ions and the boundary region of X with respect to the equivalence relation on C -(x).
Step 5: Generate the rough topology τ R -(x) on U and its basis β R-(x).
Step 6: Repeat steps 3 and 4 fo r all attributes in C.
Step 7 : Those attributes in C for which β R-(x) ≠ β R form the core (R).

Conclusions
In this work, we have shown that real world problems can be dealt with the rough topology. The concept of basis has been applied to find the deciding factors of a recent outbreak 'Ch ikungunya' which had been reported especially, in South India and a chronic disease 'Diabetes'. We could find that Joint pain and Temperature are the deciding factors for chikungunya and frequent urination is the only deciding symptom for diabetes. It is also seen that fro m a clin ical point of view, the rough topological model is on par with the med ical experts with respect to the diseases analyzed here. The proposed rough topology can be applied to more general and complex information systems for future research. The rough set model is based on the original data only and does not need any external in formation, unlike probability in statistics or grade of membership in the fuzzy set theory. It is also a tool suitable for analyzing not only quantitative attributes but also qualitative ones. The results of the rough set model are easy to understand, while the results fro m other methods need an interpretation of the technical parameters. Thus it is advantageous to use rough topology in real life situations.