Real-world applications now deal with a massive amount of data, and information about the world is inaccurate, incomplete, or uncertain. Therefore, we present in our paper a proposed model for solving problems. This model is based on the class of locally generalized closed sets, namely, locally simply* alpha generalized closed* sets and locally simply* alpha generalized closed** sets (briefly, LSM∗αGC∗-sets and LSM∗αGC∗∗-sets), based on simply* alpha open set. We also introduce various concepts of their properties and their relationship with other types, and we are studying several of their properties. Finally, we apply the concept of the simply* alpha open set to illustrate the importance of our method in decision-making for information systems about the infections of Coronavirus in humans. In fact, we were able to decide the impact factors of Coronavirus infection. The results were also programmed using the MATLAB program. Therefore, it is recommended that our proposed concept be used in future decision-making.
Corona virussimply* alpha open setslocally simply* alpha generalized closed*setsdecision makingIntroduction
The most recent Coronavirus epidemic was an outbreak in China at the start of 2019. China increased its national public-health response to the highest level on January 23, 2020. A variety of public social distancing initiatives to minimize the transmission rate of Corona virus have been introduced as part of the emergency response [1]. As a result, many papers were published by several researchers to study and analyze this virus, such as [1–7].
In diverse research areas, the mathematical modeling of vagueness and ambiguity has become an increasingly important topic. There exists another statistical instrument that has been utilized in many aspects of life [8,9]. Topology is an important branch of mathematics which contains several subfields such as soft topology, algebraic topology, and differential topology [10,11]. These subfields increase the limit of the topology applications [12]. The field of topology has many results and concepts, which can help us to discover the hidden information of data, composed of real-life applications [13]. The topological methods are thus compatible with the processing static methods and geometric representation the concept of a locally closed set-in topological space, introduced by Balachandran et al. [14]. Ganster et al. [15–17] continued to study of locally closed sets.
Our aim in this paper is to introduce three new classes of sets called LSM∗αG⌣C-closed-sets, LSM∗αG⌣C∗, and LSM∗αG⌣C∗∗-sets which are contained in G⌣LC- sets by using the notions of SM∗α-open sets and SM∗α-closed sets which is new classes. We shall define these classes in this paper. Also, we introduce some different classes of SM∗αG⌣-closed sets and SM∗αG⌣-open sets and study some of their characteristics. This method is used to identify and take the optimal decision for any life problem using near open sets named the class of simply stare alpha Star open sets, which is a new method that is considered as an essential tool for processing information for any life problem and is useful for any decision maker to take the appropriate decision for it. For example, the current application can help in making the right decision for patients, the algorithm used in the manuscript can be applied to any number of patients or objects.
Finally, we explain the importance of the proposed method in the medical sciences for application in decision making problems. In fact, a medical application has been introduced in the decision making process of COVID-19 Medical Diagnostic Information System with the algorithm. This application may help the world to reduce the spread of Coronavirus.
The paper is structured as follows: The basic concepts of the locally generalized closed set and locally simply* α -generalized closed sets are introduced in Section Two and Three. The implementation of COVID-19 for each subclass of attributes in the information systems and comparative analysis is presented in Section Four. Section Five concludes and highlights future scope.
Preliminaries
In this section, the present study is inspired by pointing out locally generalized closed set blind spots.
Definition 2.1 [17] A subset A^ of a space W^ is called g-closed if cL(A^)⋐G⌣ wherever A^⋐G⌣, and G⌣ is open
Definition 2.2 A subset A^ of a space (W^,T) it’s called,
Locally-closed [15] if A^=G⌣⋒ϝ where G⌣∈T and ϝ is closed-set in (W^,T)
Generalized locally closed [14] (briefly G⌣LC-set) if A^=G⌣⋒ϝ whereas G⌣∈G⌣O(W^) and ϝ∈GC(W).
G⌣LC∗-closed [14] if there is a g-open set G⌣ and closed ϝ of (W^,T) whereas A^=G⌣⋒ϝ.
G⌣LC∗∗-set [14] if there is an open-set G⌣ and a g-closed set ϝ∈G⌣CO(W^) whereas A^=G⌣⋒ϝ The collection of all locally G⌣LC-sets (resp. G⌣LC∗∗-sets, G⌣LC∗-sets, SM∗αG⌣-sets, closed sets, and g-closed sets) will denoted by G⌣LC(W^) (resp. G⌣LC∗∗(W^),G⌣LC∗(W^), SM∗αG⌣C(W^),C(W^),G⌣C(W^).
Definition2.3 [14] A topological space (W^,T) is called g-submaximal if every dense is g-open.
In this section, we shall introduce a new class of generalization say LSM∗αG⌣C-sets and we study some properties of this class.
Definition3.1 A subset A^ of a space (W^,T) it’s called SM∗α-open set if A^∈{W^,ϕ,G⌣∗⋓N where G⌣∗ is proper α-open set and N is nowhere dense } and the complement of the class of Sα‐M∗ open set is called Sα‐M∗ closed set and well denoted by SM∗αO(W^) and its complement is denoted by SM∗αC(W^).
Definition3.2 A subset A^ of a topological space (W^,T) is called:
Simply alpha interior (shortly, SM∗αint(A^)) is the largest simply* alpha open set contained in A^ and A^ such that G⌣⋐A^}.
Simply alpha closure (briefly, SM∗αcL(A^) ) is the smallest simply* alpha closed-set contained in A^ and A^ such that A^⋐ϝ}.
Definition 3.3 A subset A^ of a topological space (W^,T) is called simply* alpha dense (briefly, SM∗α-dense) if SM∗αcL(A^)=W^.
Remark 3.1 The following Fig. 1 shows the relation between a new class of generalization and the other types of generalization of sets such that the implication in this diagram cannot be reversible as indicated in Example 3.1.
Shows the relation between a new class of generalized closed-sets
Definition 3.4 A subset A^ of a space (W^,T) it’s called.
Simply∗α-generalized-closed (shortly, SM∗αG⌣-closed) set if SM∗αcL(A^)⋐G⌣,A^⋐G⌣ where G⌣ is open-set.
Simply∗α-generalized-open (shortly, SM∗αG⌣-open) set if ϝ⋐SM∗αint(A^),ϝ⋐A^ where ϝ is closed set.
Locally SM∗α-generalized closed (shortly, LSM∗αG⌣C) sets if there is a proper SM∗αG⌣-open set G⌣ and proper SM∗αG⌣-closed set ϝ of (W^,T) whereas A^=G⌣⋒ϝ.
Locally simply∗α-generalized closed*-sets (shortly, LSM∗αG⌣C∗-sets) if there is a proper SM∗αG⌣-open set G⌣ and proper SM∗α-closed set ϝ of (W^,T) whereas A^=G⌣⋒ϝ.
Locally simply∗α-generalized closed**sets (shortly, LSM∗αG⌣C∗∗-sets) if there exists a proper SM∗α-open-set G⌣ and proper SM∗αG⌣-closed set ϝ of (W^,T) wherever A^=G⌣⋒ϝ.
Locally simply*α -closed set (shortly, LSM∗C-set) if A^=G⌣⋒ϝ wherever G⌣ is SM∗α-open set and ϝ is SM∗α-closed sets.
The collection of all locally simply* alpha generalized closed set (resp. locally simply* alpha generalized closed* set, and locally simply* alpha generalized closed** set) of a universe set W^ is denoted by LSM∗αG⌣C(W^), (resp. LSM∗αG⌣C∗(W^),LSM∗αG⌣C∗∗(W^), and LSM∗C(W^)) is denoted by LSM∗αG⌣C(W^) (resp. LSM∗αG⌣C∗(W^), LSM∗αG⌣C∗∗(W^),(W^), and LSM∗C(W^)).
Example 3.1 Let W^={a,b,c,d}, T={W^,ϕ,{c},{a,b},{a,b,c}}. Then we have,
(implication 3) A set {a,d}∈LSM∗αG⌣C∗∗(W^) but {a,d}∉LSM∗αG⌣C∗(W^).
(implications 6, 7) A set {b′,d}∈G⌣LC(W^) but {b,d}∉G⌣LC∗(W^), but {a,b′,c}∉LSM∗αG⌣C(W^).
(implication 5, 1) {a}∈LSM∗αG⌣C∗∗(W^) but {a}∉SM∗αC(W^), and {a,b′}∈SM∗αC(W^) but {a,b}∉C(W^).
(implication 6, 2, 4) {a,b}∈LSM∗αG⌣C∗(W^) but {a,b}∉C(W^), {a}∈SM∗αG⌣C(W^) but {a}∉G⌣C(W^,T),{a,b,c}∈G⌣LC∗∗(W^) but {a,b,c}∉SM∗αG⌣C(W^).
(implication 8, 9, 10, 11) A set {a,b,c}∈LSM∗αC(W^), but {a,b,c}∉SM∗αC(W^),{a,b,c}∈LSM∗αG⌣C(W^) but {a,b,c}∉LSM∗αG⌣C∗(W^). Also there exists a set {b}∈SM∗αG⌣C(W^) but it is not {b}∉SM∗αC(W^).
Remark 3.2 For any topological space (W^,T) then we have,
LSM∗αG⌣C∗-sets and α g-closed sets are not comparable.
SM∗αG⌣-closed sets and LSM∗αG⌣C∗∗-closed sets are not comparable.
α g-closed sets and LSM∗αG⌣C∗∗-closed sets are not comparable.
LSM∗αC− closed sets and α g-closed sets are not comparable.
g-closed sets and G⌣LC∗-closed sets are not comparable.
SM∗αG⌣-closed sets and G⌣LC∗-closed sets are not comparable.
LSC and LSM∗αG⌣C-sets are not comparable.
LSM∗C and g-closed are not comparable.
LSM∗αC set and LSM∗αG⌣C∗ sets are not comparable.
LSM∗αC set and LSM∗αG⌣C∗∗ sets are not comparable.
Example 3.2 From example 3.1 we have.
There exist a set {c}∈LSM∗αG⌣C∗(W^) but {c}∉G⌣C(W^) and there exists a set {b,c,d}∈αG⌣C(W^) but {b,c,d}∉LSM∗αG⌣C∗(W^).
There exists a set {b,c}∈LSM∗αG⌣C∗∗(W^,T) but {b,c}∉SM∗αG⌣C(W^,T), and there exists a set {a,c,d}∈SM∗αG⌣C(W^,T) but {a,c,d}∉LSM∗αG⌣C∗∗(W^,T).
There exist a set {b}∉αG⌣C(W^) but {b}∈LSM∗αG⌣C∗∗(W^), and there exists a set {b,c,d}∈αG⌣C(W^), but {b,c,d}∉LSM∗αG⌣C∗∗(W^).
There exists a set {c}∈LSM∗αC(W^) but {c}∉αG⌣C(W^), and there exists a set {b,d}∈αG⌣C(W^), but {b,d}∉LSM∗αC(W^).
LSM∗C{b,d}∈αG⌣C(W^) but {b,d}∉G⌣LC∗(W^). and there exists a set {c}∈G⌣LC(W^) but {c}∉G⌣LC(W^).
There exists a set {a,c}∈G⌣LC∗(W^), but {a,c}∉SM∗αG⌣C(W^),and there exists a set {b,c,d}∈SM∗αG⌣C(W^), but {b,c,d}∉G⌣LC∗(W^).
There exists a set {a}∈LSM∗αG⌣C(W^), but {a}∉LδC(W^), and there exist a set {a,b,c}∈LδC(W^), but {a,b,c}∉LSM∗αG⌣C(W^).
There exists a set {a,b,c}∈LSM∗αC(W^), but {a,b,c}∉G⌣C(W^), and there exists a set {b,d}∈G⌣C(W^), but {b,d}∉LSM∗C(W^).
There exists a set {a,b,c}∈LSM∗αC(W^), but {a,b,c}∉LSM∗αG⌣C∗(W^), and there exists a set {a}∈LSM∗αG⌣C∗(W^), but {a}∉LSM∗αC(W^).
There exists a set {a,b,c}∈LSM∗αC(W^), but {a,b,c}∉LSM∗αG⌣C∗∗(W^), and there exists a set {a}∈LSM∗αG⌣C∗∗(W^), but {a}∉LSM∗αC(W^).
Definition 3.5 A subset A^ of a space (W^,T); it is called SM∗α-closure of A^ (resp. SM∗α-interior) briefly as SM∗αcL(A^) (resp. SM∗αint(A^)) is defined by SM∗αcL(A^)=⋒{ϝ⋐A^, such that ϝ⋐SM∗C(W^,T)}, and SM∗αint(A^)=⋓{G⌣⋐A^, such that G⌣∈SM∗αO(W^,T)}.
Theorem 3.1 The collection of all LSM∗αG⌣C∗(W^) is not true under finite union; the collection of LSM∗αG⌣C(W^) is not true under finite union.
The following example indicates this theorem;
From Example 2.1 we get that there exists a set {b},{c}∈LSM∗αG⌣C∗(W^) but {b}⋓{c}={b,c}∉LSM∗αG⌣C∗(W^).
Let {c},{a,b}∈LSM∗αG⌣C∗(W^) but {c}⋓{a,b}={a,b,c}∉LSM∗αG⌣C(W^).
Remark3.3 From the above result, the classes of LSM∗αG⌣C(W^) and LSM∗αG⌣C∗(W^) are not form supra topology.
Theorem3.2 For a subset A^ of a space (W^,T) then the following statements are equivalent.
A^∈LSM∗αG⌣C∗(W^).
A^∈G⌣⋒SM∗αcL(A^) for some proper SM∗α g-open set G⌣.
SM∗αcL(A^)∖A^ is proper SM∗αG⌣-closed.
A^⋓(W^∖SM∗αcL(A^)) is proper SM∗α g-open.
Proof : i) → ii) and A^∈LSM∗αG⌣C∗(W^,T) then A^=G⌣⋒ϝ where G⌣ is proper SM∗αG⌣-open and ϝ is proper SM∗α-closed, A^⋐ϝ implies SM∗αcL(A^)⋐ϝ and also
A^=G⌣⋒ϝ⋑G⌣⋒SM∗αcL(A^) and A^⋐SM∗αcL(A^),A^⋐G⌣ implies A^⋐G⌣⋒SM∗αcL(A^) hence A^=G⌣⋒SM∗αcL(A^).
ii ) → i ) Since G⌣ is proper SM∗αG⌣-open and SM∗αcL(A^) is SM∗αG-closed, G⌣⋒SM∗αcL(A^)∈LSM∗αG⌣C∗(W^,T) by definition 2.1.
ii ) → iii) Since SM∗αcL(A^)∖A^=SM∗αcL(A^)⋒(W^∖A^) which is LSM∗αG⌣-closed but A^=G⌣⋒SM∗αcL(A^) then SM∗αcL(A^)⋒(W^∖(G⌣⋒SM∗αcL(A^))=SM∗αcL(A^)⋒(W^∖G⌣)⋓(W^∖SM∗αcL(A^)), SM∗αcL(A^)∖A^=SM∗αcL(A^)⋒(W^∖G˘)⋓SM∗αcL(A^)⋒(W^∖SM∗αcL(A^))=SM∗αcL(A^)⋒(W^∖G⌣)⋓ϕ=SM∗αcL(A^)⋒(W^∖G⌣) by lemma 2.1 so SM∗αcL(A^)∖A^ is SM∗αG⌣-closed.
iii ) → ii ) Let U=W^∖(SM∗αcL(A^)∖A^) then U is SM∗αG⌣-open and hence A^=U⋓SM∗αcL(A^).
iii ) → iv) Let ϝ=SM∗αcL(A^)∖A^ is proper SM∗αG⌣-closed then ϝ=SM∗αcL(A^)⋒(W^∖A^), W^∖ϝ=(W^∖SM∗αcL(A^))⋓A^, W^∖ϝ is proper SM∗αG⌣-open.
iv ) → iii ) Let U=A^⋓(W^∖SM∗αcL(A^)) then (W^∖U)=(W^∖A^)⋒SM∗αcL(A^)=SM∗αcL(A^)∖A^ is proper SM∗αG⌣-closed and hence W^∖U=SM∗αcL(A^)∖A^.
Remark 3.4 It is not true that A^∈LSM∗αG⌣C∗(W^,T)⇔SM∗int(A^⋓(W^∖SM∗αcL(A^))⋑A^ in fact shown as the next example 3.3,
Example 3.3 Let W^={a,b,c},T={W^,ϕ,{a},{a,c}} and let A^={b} be subset of (W^,T) then SM∗int(A^⋓(W^∖SM∗αcL(A^))=SM∗int({b,c})=ϕ⋑A^ and A^∈LSM∗αG⌣C∗(W^,T).
Definition 3.6 A topological space (W^,T) is called SM∗αG⌣-submaximal if each SM∗α-dense is SM∗αG⌣-open.
Proposition 3.1 A topological space (W^,T) is called SM∗αG⌣-sub-maximal if P(W^)=LSM∗αG⌣C∗(W^,T).
Proof. Let A^∈P(W^) and U=A^⋓(W^∖SM∗αcL(A^)) then W^=SM∗αcL(U)
i.e., U is SM∗α-dense then U is proper SM∗αG⌣-open by theorem 2.1 U∈LSM∗αG⌣C∗(W^,T) and hence P(W^)=LSM∗αG⌣C∗(W^,T). Conversely, let A^ is SM∗α-dense subset of (W^,T) then from (4) in theorem 2.1 then A^⋓(W^∖SM∗αcL(A^))=A^ and A^∈LSM∗αG⌣C∗∗(W^,T) and hence A^ is proper SM∗αG⌣-open implies that a space (W^,T) is sub-maximal.
Proposition 3.2 For a subset A^ of (W^,T) if A^∈LSM∗αG⌣C∗∗(W^,T) then there exists a proper SM∗αG⌣-open set G⌣ such that A^⋐G⌣⋒SM∗αcL(A^).
Proof. Let A^∈LSM∗αG⌣C∗∗(W^,T) then there are both a proper SM∗αG⌣-open set G⌣ and proper SM∗αG⌣-closed set ϝ such that A^=G⌣⋒ϝ, as A^⋐SM∗αcL(A^),A^⋒G⌣⋐G⌣⋒SM∗αcL(A^), G⌣⋒ϝ⋐G⌣⋒SM∗αcL(A^), A^⋐G⌣⋒SM∗αcL(A^).
Application via Simply Alpha Open Set-in Decision Making of Coronavirus
In this section, we provide an application of our approaches for decision making for information systems on Coronavirus infections. Indeed, our approach identifies the critical factors for Coronavirus infection in humans. We find gatherings, contacting with injured people, and working in hospitals is the only factor determining the transmission of infection. We conclude that staying home and not having contact with humans protect against viral infection with the Coronavirus. According [2], (Human-to-Human transmissions have been described with incubation times between 2–10 days, facilitating its spread via droplets, contaminated hands or surfaces). We present a method for data reduction using the class of simply alpha star open sets, where in this method the topology is calculated from the right neighborhood, then we form the class of simply alpha star open sets for the whole table. Then, we delete one of the symptoms and repeat the previous step and make a comparison process between the class of simply alpha star open sets and the resulting sets from the whole table, if they are equal, then it is reduct otherwise core.
We would like to recall that the information obtained in this analysis of the Coronavirus is from 1000 patient. Because the attributes in rows (objects) were identical, it has been reduced to 10 patients. The application can be described as follows, where the objects as; U = {s1, s2, ….., s10} denotes 10 listed patients, the features as A = {a1, a2, ……, a10}={Difficulty breathing, Chest pain, Temperature, Dry coughs Headache, Loss of taste or smell} and Decision Coronavirus {d}, as information was collected by the World Health Organization as well as through medical groups specializing in Coronavirus. The following information system is illustrated in Tab. 1 and Tab. 2,
The information's decisions data set
Objects
Serious symptoms
Most common symptoms
Decision
Difficulty breathing
Chest pain
Temperature
Dry cough
Headache
Loss of taste or smell
s1
Yes
Yes
High
Yes
Yes
No
Yes
s2
Yes
Yes
High
Yes
Yes
No
Yes
s3
No
Yes
Normal
Yes
No
Yes
No
s4
No
Yes
Normal
No
No
No
No
s5
No
Yes
Normal
Yes
No
No
No
s6
Yes
No
High
Yes
Yes
No
Yes
s7
No
No
High
Yes
Yes
No
Yes
s8
No
No
Normal
Yes
Yes
No
No
s9
No
No
High
No
No
Yes
Yes
s10
No
No
High
Yes
Yes
No
Yes
Consistent part of <xref ref-type="table" rid="table-1">Tab. 1</xref>
Objects
Attributes
Decision
a1
a2
a3
a4
a5
a6
d
s1
2
2
2
2
2
1
2
s2
2
2
2
2
2
1
2
s3
1
2
1
2
1
2
1
s4
1
2
1
1
1
1
1
s5
1
2
1
2
1
1
1
s6
2
1
2
2
2
1
2
s7
1
1
2
2
2
1
2
s8
1
1
1
2
2
1
1
s9
1
1
2
1
1
2
2
s10
1
1
2
2
2
1
2
Note: We note that, IND (A) ≠ IND (A – {a1}), …., then a1, a3, a4 and a6 are indispensable. Also, we get a2 removed then we obtain IND (A) = IND (A – {a2}), and superfluous are a2,a5.
Then, we get the removal of attributes as the next Tab. 3,
Consistent part of <xref ref-type="table" rid="table-2">Tab. 2</xref>
U/A′
Attributes (A’)
Decision
a1
a3
a4
a6
d
s1
2
2
2
1
2
s2
2
2
2
1
2
s3
1
1
2
2
1
s4
1
1
1
1
1
s5
1
1
2
1
1
s6
2
2
2
1
2
s7
1
2
2
1
2
s8
1
1
2
1
1
s9
1
2
1
2
2
s10
1
2
2
1
2
From Tab. 3 we obtain the symptoms of every patient as follows:
V(s1) = {a1, a3, a4}, V(s2) = {a1, a3, a4}, V(s3) = {a4, a6}, V(s4) = ϕ, and V(s5) = {a4}. Now, we can generate the following relation: siRsj ⇔ V(si) ⊆ V(sj).
We apply this relation for all features in the table to induce the neighborhoods as follows.
R = {(s1, s1), (s1, s2), (s2, s2), (s2, s1), (s3, s3), (s4, s4), (s4, s1), (s4, s2), (s4, s3), (s4, s5), (s5, s5), (s5, s1), (s5, s2)}. Thus, the right neighborhoods of each element in U of this relation are Rr s1 = {s1, s2}, Rr s2 = {s1, s2},
Rr s3 = {s3}, Rr s4 =U, Rr s5 = { s1, s2, s3, s5}. Then the topology deduced from this relation is T = {U, , {s1, s2}, {s3}, {s1, s2. s3}, {s1, s2, s3, s5}}, Tc = {U, , {s4, s5}, {s4}, {s5, s4. s3}, {s1, s2, s4, s5}}.
Then, we construct the following Tab. 4 to obtain the class of simply* alpha open set,
Class of simply* alpha open set
P(U)
A¯
A0−
A0
A−O
AO−O
αO(U)
SAM∙αO(U)
U
U
U
U
U
U
U
U
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
{s1}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s2}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s3}
{s3, s4, s5}
{s3}
{s3}
{s3, s4, s5}
{s3}
{s3}
{s3}
{s4}
{s4}
ϕ
ϕ
ϕ
-
-
{s5}
{s4, s5}
ϕ
ϕ
ϕ
-
-
{s1, s2}
{s1, s2, s4, s5}
{s1, s2}
{s1, s2}
{s1, s2, s4, s5}
{s1, s2}
{s1, s2}
{s1, s2}
{s1, s3}
U
U
{s3}
{s3, s4, s5}
{s3}
-
-
{s1, s4}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s1, s5}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s2, s3}
U
U
{s3}
{s3, s4, s5}
{s3}
-
-
{s2, s4}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s2, s5}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
{s3, s4}
{s3, s4}
{s3, s4, s5}
{s3}
{s3}
{s3, s4, s5}
{s3}
-
{s3, s4}
{s3, s5}
{s3, s4, s5}
{s3}
{s3}
{s3, s4, s5}
{s3}
-
-
{s4, s5}
{s3, s4, s5}
{s3}
ϕ
ϕ
-
-
{s1, s2, s3}
U
U
{s1, s2, s3}
U
U
-
{s1, s2, s3}
{s1, s2, s4}
{s1, s2, s4, s5}
{s1, s2}
{s1, s2}
{s1, s2, s4, s5}
{s1, s2}
-
{s1, s2, s4}
{s1, s2, s5}
{s1, s2, s4, s5}
{s1, s2}
{s1, s2}
{s1, s2, s4, s5}
{s1, s2}
-
{s1, s2, s5}
{s1, s3, s4}
U
U
{s3}
{s3, s4, s5}
{s3}
-
-
{s1, s3, s5}
U
U
{s3}
{s3, s4, s5}
{s3}
-
-
{s2, s3, s4}
U
U
{s3}
{s3, s4, s5}
{s3}
-
-
{s2, s3, s5}
U
U
{s3}
{s3, s4, s5}
{s3}
-
-
{s2, s4, s5}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s3, s4, s5}
{s3, s4, s5}
{s3}
{s3}
{s3, s4, s5}
{s3}
-
{s3, s4, s5}
{s1, s4, s5}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s1, s2, s3, s4}
U
U
{s1, s2, s3}
U
U
{s1, s2, s3, s4}
{s1, s2, s3, s4}
{s1, s2, s3, s5}
U
U
{s1, s2, s3, s5}
{s1, s2, s3, s5}
U
{s1, s2, s3, s5}
{s1, s2, s3, s5}
{s1, s2, s4, s5}
{s1, s2, s4, s5}
{s1, s2}
{s1, s2, s4, s5}
{s1, s2}
-
{s1, s2}
-
{s1, s3, s4, s5}
U
{s3}
{s3, s4, s5}
{s3}
-
U
-
{s2, s3, s4, s5}
U
{s3}
{s3, s4, s5}
{s3}
-
U
-
Then the class of simply* alpha open set isSM∗αO(U)={U,ϕ,{s3},{s3,s5},{s1,s2,s3},{s1,s2,s4},{s1,s2,s5}}
Step 1: When the feature a1-Difficulty breathing is removed, the symptoms of every patient are: V(s1) = {a3, a4}, V(s2) = {a3, a4}, V(s3) = {a4, a6}, V(s4) = , and V(s5) = {a4}.
Thus, the right neighborhoods of each element in U of this relation are.Rr(s1)={s1,s2},Rr(s2)={s1,s2},Rr(s3)={s3},Rr(s4)=U,andRr(s5)={s1,s2,s3,s5}.
Then the topology deduced from this relation is T={U,ϕ,{s1,s2},{s3},{s1,s2,s3},{s1,s2,s3,s5}}, Tc = {U, , {s3, s4, s5}, {s1, s2, s4, s5}, {s4, s5}, {s4} }, and hence the class of simply* alpha open set of the whole is identical with the class of simply* alpha open set without the symptom a1; this means that SAM∗αO(U)=SA−{a1}M∗αO(U).
Step 2: When the feature a3-Temperature is removed, the symptoms of every patient are: V(s1) = {a1, a4}, V(s2) = {a1, a4}, V(s3) = {a1, a4, a6}, V(s4) = {a1}, and V(s5) = {a1, a4}.
Thus, the right neighborhoods of each element in U of this relation are.
Rr(s1) = {s1, s2, s3}, Rr(s2) = {s1, s2, s3}, Rr(s3) = {s3}, Rr(s4) = {s1, s2, s3, s4}, and Rr(s5) = {s1, s2, s3, s5}. Then, the topology deduced from this relation is T = {U, , {s1, s2, s3}, {s1, s2, s3, s4}, {s1, s2, s3, s5}}, Tc = {U, , {s4, s5}, {s5}, {s4} }, and hence the class of simply* alpha open set of the whole is not identical with the class of simply* alpha open set without the symptom a3, if SAM∗αO(U)≠SA−{a3}M∗αO(U), as in Tab. 5,
Class of simply* alpha open set when we remove <italic>a</italic><sub>3</sub>-Temperature
P(U)
A¯
A0−
A0
A−O
AO−O
αO(U)
SA−{a3}M∙αO(U)
U
U
U
U
U
U
U
U
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
{s1}
U
U
ϕ
ϕ
-
-
{s2}
U
U
ϕ
-
-
{s3}
U
U
ϕ
ϕ
-
-
{s4}
{s4}
ϕ
ϕ
ϕ
-
-
{s5}
{s5}
ϕ
ϕ
ϕ
-
-
{s1, s2}
U
U
ϕ
ϕ
-
-
{s1, s3}
U
U
ϕ
ϕ
-
-
{s1, s4}
U
U
ϕ
ϕ
-
-
{s1, s5}
U
U
ϕ
ϕ
-
-
{s2, s3}
U
U
ϕ
ϕ
-
-
{s2, s4}
U
U
ϕ
ϕ
-
-
{s2, s5}
U
U
ϕ
ϕ
-
-
{s3, s4}
U
U
ϕ
ϕ
-
-
{s3, s5}
U
U
ϕ
ϕ
-
-
{s4, s5}
{s4, s5}
ϕ
ϕ
ϕ
-
-
{s1, s2, s3}
U
U
{s1, s2, s3}
U
U
{s1, s2, s3}
{s1, s2, s3}
{s1, s2, s4}
U
U
ϕ
{s1, s2}
-
-
{s1, s2, s5}
U
U
ϕ
{s1, s2}
-
-
{s1, s3, s4}
U
U
ϕ
{s3}
-
-
{s1, s3, s5}
U
U
ϕ
{s3}
-
-
{s2, s3, s4}
U
U
ϕ
{s3}
-
-
{s2, s3, s5}
U
U
ϕ
{s3}
-
-
{s2, s4, s5}
U
U
ϕ
ϕ
-
-
{s3, s4, s5}
U
U
ϕ
{s3}
-
-
{s1, s4, s5}
U
U
ϕ
ϕ
-
-
{s1, s2, s3, s4}
U
U
{s1, s2, s3, s4}
U
U
{s1, s2, s3, s4}
{s1, s2, s3, s4}
{s1, s2, s3, s5}
U
U
{s1, s2, s3, s5}
U
U
{s1, s2, s3, s5}
{s1, s2, s3, s5}
{s1, s2, s4, s5}
U
U
ϕ
-
-
{s1, s3, s4, s5}
U
U
ϕ
-
-
{s2, s3, s4, s5}
U
U
ϕ
-
-
Step 3: When the feature a4-Dry cough is removed, the symptoms of every patient are:V(s1)={a1,a3},V(s2)={a1,a3},V(s3)={a1,a6},V(s4)={a1},andV(s5)={a1}.
Thus, the right neighborhoods of each element in U of this relation are Rr(s1) = {s1, s2}, Rr(s2) = {s1, s2}, Rr(s3) = {s3}, Rr(s4) = U, and Rr(s5) = U.
Then the topology deduced from this relation is T = {U, , {s1, s2}, {s3}, {s1, s2, s3}}, Tc = {U, , {s3, s4, s5}, {s1, s2, s4, s5}, {s4, s5}}, and hence the class of simply* alpha open set of the whole is identical with the class of simply* alpha open set without the symptom a4, this means that SAM∗αO(U)=SA−{a4}M∗αO(U). As the following Tab. 6,
Class of simply* alpha open set when we remove <italic>a</italic><sub>4</sub>-Dry cough
P(U)
A¯
A0−
A0
A−O
AO−O
αO(U)
SA−{a4}M∙αO(U)
U
U
U
U
U
U
U
U
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
{s1}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s2}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s3}
{s3, s4, s5}
{s3}
{s3}
{s3, s4, s5}
{s3}
{s3}
{s3}
{s4}
{s4, s5}
ϕ
ϕ
ϕ
-
-
{s5}
{s4, s5}
ϕ
ϕ
ϕ
-
-
{s1, s2}
{s1, s2, s4, s5}
{s1, s2}
{s1, s2}
{s1, s2, s4, s5}
{s1, s2}
{s1, s2}
{s1, s2}
{s1, s3}
U
U
{s3}
{s3, s4, s5}
{s3}
-
-
{s1, s4}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s1, s5}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s2, s3}
U
U
{s3}
{s3, s4, s5}
{s3}
-
-
{s2, s4}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s2, s5}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s3, s4}
{s3, s4, s5}
{s3}
{s3}
{s3, s4, s5}
{s3}
-
{s3, s4}
{s3, s5}
{s3, s4, s5}
{s3}
{s3}
{s3, s4, s5}
{s3}
-
{s3, s5}
{s4, s5}
{s4, s5}
ϕ
ϕ
ϕ
-
-
{s1, s2, s3}
U
U
{s1, s2, s3}
U
U
{s1, s2, s3}
{s1, s2, s3}
{s1, s2, s4}
{s1, s2, s4, s5}
{s1, s2}
{s1, s2}
{s1, s2, s4, s5}
{s1, s2}
-
{s1, s2, s4}
{s1, s2, s5}
{s1, s2, s4, s5}
{s1, s2}
{s1, s2}
{s1, s2, s4, s5}
{s1, s2}
-
{s1, s2, s5}
{s1, s3, s4}
U
U
{s3}
{s3, s4, s5}
{s3}
-
-
{s1, s3, s5}
U
U
{s3}
{s3, s4, s5}
{s3}
-
-
{s2, s3, s4}
U
U
{s3}
{s3, s4, s5}
{s3}
-
-
{s2, s3, s5}
U
U
{s3}
{s3, s4, s5}
{s3}
-
-
{s2, s4, s5}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s3, s4, s5}
{s3, s4, s5}
{s3}
{s3}
{s3, s4, s5}
{s3}
-
-
{s1, s4, s5}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s1, s2, s3, s4}
U
U
{s1, s2, s3}
U
U
{s1, s2, s3, s4}
{s1, s2, s3, s4}
{s1, s2, s3, s5}
U
U
{s1, s2, s3}
U
U
{s1, s2, s3, s5}
{s1, s2, s3, s5}
{s1, s2, s4, s5}
{s1, s2, s4, s5}
{s1, s2}
{s1, s2}
{s1, s2}
-
{s1, s2, s4, s5}
{s1, s2, s4, s5}
{s1, s3, s4, s5}
U
U
{s3}
{s3}
-
{s3, s4, s5}
-
{s2, s3, s4, s5}
U
U
{s3}
-
{s3, s4, s5}
-
Step 3: When the feature a6-Loss of taste or smell is removed, the symptoms of every patient are: V(s1) = {a1, a3}, V(s2) = {a1, a3}, V(s3) = {a1, a6}, V(s4) = {a1}, and V(s5) = {a1}.
Thus, the right neighborhoods of each element in U of this relation are Rr(s1) = {s1, s2}, Rr(s2) = {s1, s2}, Rr(s3) = {s3}, Rr(s4) = U, and Rr(s5) = U.
Then the topology deduced from this relation is T = {U, , {s1, s2}, {s3}, {s1, s2, s3}}, Tc = {U, , {s3, s4, s5}, {s1, s2, s4, s5}, {s4, s5}}, and hence the class of simply* alpha open set of the whole is not identical with the class of simply* alpha open set without the symptom a6-Loss of taste or smell this means that SAM∗αO(U)≠SA−{a6}M∗αO(U), as described in Tab. 7,
Class of simply* alpha open set when we remove <italic>a</italic><sub>6</sub>-Loss of taste or smell
P(U)
A¯
A0−
A0
A−O
AO−O
αO(U)
SA−{a6}M∙αO(U)
U
U
U
U
U
U
U
U
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
{s1}
U
U
ϕ
ϕ
-
-
{s2}
U
U
ϕ
ϕ
-
-
{s3}
{s3, s4, s5}
{s3}
{s3}
{s3, s4, s5}
-
-
{s4}
{s4}
ϕ
ϕ
ϕ
-
-
{s5}
{s3, s4, s5}
ϕ
ϕ
ϕ
-
-
{s1, s2}
U
U
{s1, s2}
U
U
{s1, s2}
{s1, s2}
{s1, s3}
U
U
ϕ
ϕ
-
-
{s1, s4}
U
U
ϕ
ϕ
-
-
{s1, s5}
U
U
ϕ
ϕ
-
-
{s2, s3}
U
U
ϕ
ϕ
-
-
{s2, s4}
U
U
ϕ
ϕ
-
-
{s2, s5}
U
U
ϕ
ϕ
-
-
{s3, s4}
{s3, s4, s5}
ϕ
ϕ
ϕ
-
-
{s3, s5}
{s3, s4, s5}
ϕ
ϕ
ϕ
-
-
{s4, s5}
{s3, s4, s5}
ϕ
ϕ
ϕ
-
-
{s1, s2, s3}
U
U
{s1, s2, s3}
U
U
{s1, s2, s3}
{s1, s2, s3}
{s1, s2, s4}
U
U
{s1, s2}
U
U
{s1, s2, s4}
{s1, s2, s4}
{s1, s2, s5}
U
U
ϕ
U
{s1, s2, s5}
{s1, s2, s5}
{s1, s3, s4}
U
U
ϕ
ϕ
-
-
{s1, s3, s5}
U
U
ϕ
ϕ
-
-
{s2, s3, s4}
U
U
ϕ
ϕ
-
-
{s2, s3, s5}
U
U
ϕ
ϕ
-
-
{s2, s4, s5}
U
U
ϕ
ϕ
-
-
{s3, s4, s5}
{s3, s4, s5}
ϕ
ϕ
ϕ
-
-
{s1, s4, s5}
{s1, s2, s4, s5}
{s1, s2}
ϕ
ϕ
-
-
{s1, s2, s3, s4}
U
U
{s1, s2, s3}
U
U
{s1, s2, s3, s4}
{s1, s2, s3, s4}
{s1, s2, s3, s5}
U
U
{s1, s2, s3}
U
U
{s1, s2, s3, s5}
{s1, s2, s3, s5}
{s1, s2, s4, s5}
U
U
{s1, s2}
U
U
{s1, s2, s4, s5}
{s1, s2, s4, s5}
{s1, s3, s4, s5}
U
U
ϕ
ϕ
-
-
{s2, s3, s4, s5}
U
U
ϕ
ϕ
-
-
Hence, from Steps (1–3), we observe that: the CORE is {a3, a6}, that is the impact factors to determine Coronavirus infection are High Temperature and Loss of taste or smell.
Conclusion
Our study in this article aims for defining a new type of simply* alpha generalized namely Simply*α -generalized closed, locally simply*α-generalized closed, and locally simply*α-generalized closed** sets. We show the relation between this type of set and the other of generalized open sets. We concluded that the generalized simple closed set contains most of the generalized closed sets and also, we defined new kinds of locally generalized simply closed sets based on simply* alpha open set. Furthermore, we studied the relation between these collections and other. We presented an application to a group of patients, where we presented a method using topological concepts to determine one of the main symptoms causing Coronavirus. Finally, using the class of simply alpha* open set, we identified the impact factors of Coronavirus infection and then we helped the physician to make an accurate decision making about patients. At the end of paper, this method has also proven effective in acquiring the basic symptoms that cause COVID-19. We plan to extend the proposed methodology in future work to include a multi-label classification issue.
Funding Statement: The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under Grant Number R.G.P. 1/206/42.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
ReferencesG.Kampf, D.Todt, S.Pfaender and E.Steinmann, “Persistence of coronaviruses on inanimate surfaces and their inactivation with biocidal agents,” J. Hospital Infection, vol. 104, pp. 246–251, 2020.S. A.Alblowi, M.El Sayed and M. A.El Safty, “Decision making based on fuzzy soft sets and its application in COVID-19,” Intelligent Automation & Soft Computing, vol. 30, no. 3, pp. 961–972, 2021.M. A.El Safty and S.AlZahrani, “Topological modeling for symptom reduction of coronavirus,” Punjab University Journal Mathematics, vol. 53, no. 3, pp. 47–59, 2021.L. X.Feng, S. L.Jing, S. K.Hu, D. F.Wang and H. F.Huo, “Modelling the effects of media coverage and quarantine on the COVID-19 infections in the UK,” Mathematical Biosciences and Engineering, vol. 17, no. 4, pp. 3618–3636, 2020.M. A.El Safty, M.El Sayed and S. A.Alblowi, “Accuracy based on simply* alpha open set in rough set and topological space,” Soft Computing, vol. 25, no. 16, pp. 10609–10615, 2021.M. A. El Safty, S. A. L. Zahrani, M. K. El-Bably and M. El Sayed, “Soft exai -rough set and its applications in decision making of coronavirus,” Computers, Materials & Continua, 2021.M.El Sayed, M. A.El Safty and M. K.El-Bably, “Topological approach for decision-making of COVID-19 infection via a nano topology model,” AIMS Mathematics, vol. 6, no. 7, pp. 7872–7894, 2021.M. A.El Safty, “Modeling uncertainty knowledge of the topological methods,” Poincare Journal of Analysis & Applications, vol. 8, no. 1, pp. 89–102, 2021.M. A.Elsafty and A. M.Alkhathami, “A topological method for reduction of digital information uncertainty,” Soft Computing, vol. 24, no. 1, pp. 385–396, 2020.M. E.Abd Monsef and R. A.Mahmoud, “On generalized continuous mappings,” Delta J. Science., vol. 8, no. 2, pp. 407–420, 1984.T. N.Alharthi and M. A.El Safty, “Attribute topologies based similarity,” Cogent Mathematics, vol. 3, no. 1, pp. 1–11, 2016.M. A.El Safty, A. A.Mousa, S. A.Alblowi and M.El Sayed, “Topological approach on fuzzy soft β-closure and β-interior and its application in decision making,” Nanoscience and Nanotechnology Letters, vol.12, no. 11, pp. 1323–1328, 2020.M. A.El Safty, “Topological method for a study of discriminating three categories of banks and its use in attributes reduction,” International Journal of Scientific Research in Science, Engineering and Technology, vol. 7, no. 5, pp. 221–233, 2020.K.Balachandran, P.Sundaram and H.Maki, “Generalized locally closed sets and GLC-continuous function,” Indian J. Pure & Applied Mathematics, vol. 27, no. 3, pp. 235–244, 1996.M.Ganster and I.Reilly, “Locally closed sets and LC-continuous function,” International Journal of Mathematics and Mathematical Sciences, vol. 12, no. 3, pp. 417–424, 1989.M.Ganster, I.Reilly and M. K.Vamanamurthy, “Remarks on locally closed sets,” Mathematics Pannonica, vol. 3, no. 2, pp. 107–113, 1992.J.Cao, M.Ganster and I.Reilly, “On generalized closed sets,” Topology and its Applications, vol. 123, no. 1, pp. 37–46, 2002.