EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 4, 2020, 852-860
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Use of some Topological Concepts in the study of some
COVID−19 Symptoms

Samirah ALZahrani

Department of Mathematics, Faculty of Science, Taif University, Taif, Saudi Arabia

Abstract. We apply some topological concepts on topological spaces generated from both equality
and similarity relation for our information system, which examine 10 hypothetical patients who
suffer from some COVID−19 symptoms. This is based on the degree of accuracy generated from
the cardinality of the lower and upper approximation. This method is clarified by application.

2020 Mathematics Subject Classifications: 54A05

Key Words and Phrases: Rough sets, topology, similarity, lower and upper approximation,
accuracy

1. Introduction

The present time is characterized by an abundance of computers that can collect a
lot of information on any subject. This information allows to make decision. We need
to create mathematical models for this information that help in analyzing and extracting
knowledge from it [5],[7]. In addition to foreign schools in Germany and America, the
2016 Nobel Prize in Physics was awarded for topological uses in the theory of material
transformation using topological applications in science and engineering [2]. One of the
most important mathematical models is the Rough Set Theory based on topological con-
cepts [4]. The notion of rough sets was introduced by Pawlak [7]. From the outset, rough
set theory has been a methodology of database mining or knowledge discovery in relation
databases [6], [3]. The rough set methodology is based on the premise that lowering the
degree of precision in the data makes the data pattern more visible, whereas the central
premise of the rough set philosophy is that the knowledge consists in the ability of clas-
sification. Previously induction set introduced by Pawlak [8], using equivalence relation
which was considered a major constraint, an therefore, research tended to use unequal
relation. In this paper, we use similarity relationships to find neighborhoods of objects
and use them in approximations: we calculate the degree of correlation attributes with
total information and use this correlation to deduce the effective attributes [1]. We in-
troduce in this paper some symptoms of inflammation of the respiratory system in which

DOI: https://doi.org/10.29020/nybg.ejpam.v13i4.3819

Email addresses: mam 1420@hotmail.com, samar.alz@tu.edu.sa (S. ALZahrani)

https://www.ejpam.com 852 c© 2020 EJPAM All rights reserved.



S. ALZahrani / Eur. J. Pure Appl. Math, 13 (4) (2020), 852-860 853

the inflammation cases extend from easy treatment for chronic disease that the patient
suffers throughout his life, such as asthma and pneumonia. However, there are diseases of
the respiratory system that are difficult to treat and may lead to death, such as Covid-19.
In this paper, we use the topological concepts interior and closure to know the degree
of correlation between patients and symptoms of inflammation of the respiratory system,
and any other symptom that are more related with respiratory system diseases.

2. Preliminaries

Motivation to use rough set theory has come from the requirement to represent subsets
of a universe set in terms of equivalence classes of a partition of that universe set. We
consider the partition of equivalence classes is a topological space.
We are given an information system S = (U,A,C) where U,A and C are finite, non-empty
sets. We called U is a universe set , A is attributes and C is cases of attributes. Also, we
denote of equivalence classes by Rp, p ∈ U and RX is denoted of equivalence classes in
accordance with C of attribute A.

2.1 Topological spase
Definition 1 [4] Let B be a subset of a topological space (X, τ), the union of all open sets
contained in B is called the interior of a set B and denoted by int(B) or B◦. The interior
of a set B is the largest open set contained in B; i.e., B◦ =

⋃
{U ⊆ X : U ⊆ B,U ∈ τ},

U is open set. The intersection of all closed sets containing B is called the closure of a
set B and denoted by cl(B) or B, i.e. B =

⋂
{F ⊆ X : B ⊆ F, F ∈ τ c}, F is the closed set.

2.2 Approximation spase
Definition 2 [7] Suppose that we are given knowledge base I = (U,R), with each subset
X ⊆ U and an equivalence relation R ∈ IND(I), we associate two subsets:

P (X) =
⋃
{p ∈ U : Rp ⊆ RX},

and
P (X) =

⋃
{p ∈ U : Rp ∩RX 6= φ}.

Two approximations P (X) , and P (X) called the lower approximation and upper approx-
imation of X respectively.
Also, the accuracy of the approximation is defined by

α(X) =
|P (X)|
|P (X)|

, 0 ≤ α(X) ≤ 1.

If a set X with accuracy equal to 1 is crisp, otherwise X is rough.
Definition 3 [1] For each B ⊆ A, the relation RB ⊆ U × U defined

x RB y =
|
∑4

i=1(a(xi) = b(yi))|
4

where |.| is the cardinality of B.



S. ALZahrani / Eur. J. Pure Appl. Math, 13 (4) (2020), 852-860 854

In this paper U = {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10} is patients, A = {X,Y, Z,W} is
symptoms of inflammation of respiratory system and C is classification of symptoms as in
Application down.

3. Topological spaces generated from relations

Application : We consider data of ten hypothetical patients have respiratory disease
and suffer from the following symptoms: Temperature,breathing difficulty, cough and
muscle pain. We denote the ten patients by {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10}, and we
symbolize for symptoms by the following symbols:

1. Symptom the temperature by the symbol X, where it is classified into high temper-
ature H, very high temperature V and simple temperature S.

2. Symptom breathing difficulty by the symbol Y , where it is classified into a strong
situation T , medium situation M and simple situation N .

3. Denote the symptom cough by Z, where it is classified into dry cough C, cough with
phlegm L and an attic cough G.

4. Symptom muscle pain by the symbol W , where it is classified into strong pain R,
medium pain D and simple pain E.

Table 1: Information system

X Y Z W

1 V M L D

2 S N L D

3 V N L R

4 V M C R

5 V T C D

6 H M C R

7 H M C E

8 H M G E

9 S T G E

10 S T G E

Part 1 : According to the Pawlak′s model, we constitute patients′s classes Rp depend-
ing on the previous symptoms, whereas patients′s classes establish according to the equal
in the all symptoms and it produces the following class:
Rp = {{p1}, {p2}, {p3}, {p4}, {p5}, {p6}, {p7}, {p8}, {p9, p10}}.
We discuss the above symptoms through a table 1 in the following cases:



S. ALZahrani / Eur. J. Pure Appl. Math, 13 (4) (2020), 852-860 855

case1 :
From the table 1 appears the class RX of symptom temperature that contains three sets
of patients according their pain:
XV = {p1, p3, p4, p5}, XS = {p2, p9, p10}, XH = {p6, p7, p8}.
Likewise, from the table 1 appears the classes of the other symptoms such as :difficulty
breathing, cough ,and muscle pain. It shows the following patients′s classes RY , RZ , RW

contain the following sets according to symptom situation:

YM = {p1, p4, p6, p7, p8}, YT = {p5, p9, p10}, YN = {p2, p3}.

ZC = {p4, p5, p6, p7}, ZL = {p1, p2, p3}, ZG = {p8, p9, p10}.

WR = {p3, p4, p6}, WD = {p1, p2, p5}, WE = {p7, p8, p9, p10}.

We find the accuracy, mean the degree of correlation between the overall information
and both previous symptoms. So we find the lower and upper of patients′s classes as to
temperature X :

XV = {p1, p3, p4, p5}, XS = {p2, p9, p10}, XH = {p6, p7, p8}.

We make union of these classes and we have:
P (X) = {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10}.

Also in upper case, we get

XV = {p1, p3, p4, p5}, XS = {p2, p9, p10}, XH = {p6, p7, p8}.
We make union of these classes and we have:
P (X) = {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10}.
The degree of correlation between the overall information and temperature is :
|P (X)|
|P (X)|

=
10

10
= 1.

This mean that accreditation ratio is 100%.
Similarly, we find the lower and upper approximation of patients′s classes with other symp-
toms breathing difficulty Y , cough Z and muscle pain W ,we will find that accreditation
ratio between the overall information and any of other symptoms is 100%.

case2 :
From the table 1 appears the class RXY of symptom temperature X and breathing diffi-
culty Y , it shows equivalence class RXY of patients :
RXY = {{p1, p4}, {p2}, {p3}, {p5}, {p6, p7, p8}, {p9, p10}}.
When we find the lower and upper approximation, we will have the degree of correlation
between the overall information on XY is 100%.



S. ALZahrani / Eur. J. Pure Appl. Math, 13 (4) (2020), 852-860 856

Similarly, we find the lower and upper approximation of patients′s classes with XZ,
XW , Y Z, YW , ZW , XY Z, XYW , Y ZW , all of them we have the degree of
correlation is 100%.

Part 2 :
In this part we do a similarity matrix which represents the degree of similarity between

patients. We create the elements of this similarity matrix by sum of the number of similar
symptoms at each two patients and divide it by the total number of symptoms , this
elements of this similarity matrix are degree of similarity. i.e.

The degree of similarity =
|
∑4

i=1(a(xi) = b(yi))|
4

where a, b ∈ U and xi, yi

denoted to the similar symptoms. See Table 2

We formed the classes of patients at the beginning based on the equal between them,

now we form patients classes based on degree of similarity
1

4
between them.

we have the following topological approximation space depending on degree of similarity
1

4
, this approximation classes is:

Rpsim(1/4)
= {{p6, p7, p8}, {p5, p9, p10}, {p5, p6}, {p8}, {p2, p3, p6, p7, p9, p10}, {p1, p3, p5},

{p1, p5, p9, p10}, {p1, p4}, {p2, p5, p7}}.
We discuss these classes through table 1 in the following cases:

case1 :
When we looked at table 1 from through symptom temperature X, it showed us the classes
RX contain three sets of patients according to their pain:
XV = {p1, p3, p4, p5}, XS = {p2, p9, p10}, XH = {p6, p7, p8}.

Now, we find the lower and upper approximation of patient classes Rpsim(1/4)
with

classes RX , we get:
XV = {p1, p3, p4, p5}, XS = ∅, XH = {p6, p7, p8}.

XV = {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10},

XS = {p1, p2, p3, p5, p6, p7, p9, p10},

XH = {p2, p3, p5, p6, p7, p8, p9, p10}.

Then the correlation degree between the total information and symptom X depending

on the degree of similarity
1

4
is

|P (X)|
|P (X)|

=
7

10
.



S. ALZahrani / Eur. J. Pure Appl. Math, 13 (4) (2020), 852-860 857

Table 2: Similarity matrix

p1 p2 p3 p4 p5 p6 p7 p8 p9 p10

p1 1
1

2

1

2

1

2

1

2

1

4

1

4

1

4
0 0

p2
1

2
1

1

2
0

1

4
0 0 0

1

4

1

4

p3
1

2

1

2
1

1

2

1

4

1

4
0 0 0 0

p4
1

2
0

1

2
1

1

2

3

4

1

2

1

4
0 0

p5
1

2

1

4

1

4

1

2
1

1

4

1

4
0

1

4

1

4

p6
1

4
0

1

4

3

4

1

4
1

3

4

1

2
0 0

p7
1

4
0 0

1

2

1

4

3

4
1

3

4

1

4

1

4

p8
1

4
0 0

1

4
0

1

2

1

2
1

1

2

1

2

p9 0
1

4
0 0

1

4
0

1

4

1

2
1 1

p10 0
1

4
0 0

1

4
0

1

4

1

2
1 1

Where P (X) = XV ∪XS ∪XH and P (X) = XV ∪XS ∪XH

This mean that accreditation ratio is 70%.

Similarly, we will find that accreditation ratio for the symptom Y is 80%. While ac-
creditation ratio of symptoms Z is 30% , and W is 10%

case2 :
From table 1 appears the classes RXY of symptoms XY , as the following :
RXY = {{p1, p4}, {p2}, {p3}, {p5}, {p6, p7, p8}, {p9, p10}}.



S. ALZahrani / Eur. J. Pure Appl. Math, 13 (4) (2020), 852-860 858

Now, we find the lower and upper approximation of patient classes Rpsim(1/4)
with

classes RXY , we get:
P (XY ) = {p1, p4, p6, p7, p8}

P (XY ) = {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10}

|P (XY )|
|P (XY )|

=
5

10
.

This means that the accreditation ratio is 50%.

Also, from table 1 appears the following classes:
RXZ of symptom XZ:
RXZ = {{p1, p3}, {p2}, {p4, p5}, {p6, p7}, {p8}, {p9, p10}}.

RXW of symptom XW :
RXW = {{p1, p5}, {p2}, {p3, p4}, {p6}, {p7, p8}, {p9, p10}}.

RY Z of symptom Y Z :
RY Z = {{p1}, {p2, p3}, {p4, p6, p7}, {p5}, {p8}, {p9, p10}}.

RYW of symptom YW :
RYW = {{p1}, {p2}, {p3}, {p4, p6}, {p5},{p7, p8}, {p9, p10}}.

RZW of symptom ZW :
RZW = {{p1, p2}, {p3}, {p4, p6}, {p5}, {p7}, {p8, p9, p10}}.
We have accreditation ratio in all of above with classes which degree of similarity

1

4
is 10%.

case3 :
From table 1 appears the following classes RXY Z of symptoms XY Z :
RXY Z = {{p1}, {p2}, {p3}, {p4}, {p5}, {p6, p7}, {p8},{p9, p10}}.
We find the lower and upper approximation of patient classes Rpsim(1/4)

, we get

|P (XY Z)| = 1.

|P (XY Z)| = 10

|P (XY Z)|
|P (XY Z)|

=
1

10
.

This mean that accreditation ratio between the overall information and symptom XY Z
is 10% and it is the same ratio with symptoms XYW,Y ZW .
We do a summary of the above in table 3.



REFERENCES 859

Table 3: Summary the ratio in part 2

Symptoms The ratio Symptoms The ratio

X 70% YZ 10%
Y 80% YW 10%
Z 30% ZW 10%
W 10% XYZ 10%
XY 50% XYW 10%
XZ 10% YZW 10%
XW 10%

4. Conclusion

We find that accreditation ratio of information on symptom breathing difficulty is the
highest, this means that the most symptoms indicate of inflammation of the respiratory
system is breathing difficulty. By this we can determine the most important tests to pre-
form saving time, effort, and money from doing tests that have no effect on inflammation
of the respiratory system.

Acknowledgements

The author would like to thank the referee for his comments that helped us improve
this article.

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