Application on Similarity Relation and Pretopology

Abstract

Due to the tremendous use of computers, the need for dealing with digital information continues to grow. This has led to the need for constructing a structure for saving information in a way that eases data retrieval and processing. An information system is a table form that contains all the data needed for a user to reach a decision. Information systems are the most common form used for knowledge and data representation, and attribute reduction plays an important role in data processing. This paper is devoted to proposing a strategy for attributes reduction based on the similarity relation and pretopological concepts. Moreover, new types of pretopological spaces are to be constructed from the information system. Furthermore, a comparison between pretopologies and their pre-interiors constructed from the information system, as well as the different types of pretopological spaces, is investigated. Finally, the concept of the cover pretopology is applied to the information system.

1. Introduction

In data mining, attribute reduction is an important tool for the purpose of improving efficiency and quality, especially when processing high-dimension data. In real-life projects, there are several multi-feature attributes; consequently, attribute reduction aims to eliminate data redundancy and to draw useful information. Generally, the attribute significance degree is one of the fundamental metrics used to measure the contained information in each attribute.

In machine learning, data mining, and intelligent systems, the most significant form of knowledge representation is information systems. Yet some of the attributes in the information system may be unnecessary or redundant; therefore, attributes reduction plays an essential role in data processing. In the literature, several techniques are available for solving the problem of attributes reduction; nevertheless, a common characterization for these techniques is needed.

In 1975, Marcel Brissaud introduced the concept of pretopology, as a generalization of classical topology. To pursue this, in 1993, Belmandt developed the core concepts and published the first manual for pretopology [1]. Later on, a second book was published in 2011 [2]. As a tool for mathematical modeling, pretopology became an important tool for measuring the concepts of proximity and dissimilarity between groups and points. Pretopology is used in many applications such as economic modeling [3], complex networks modelling [4], data analysis [5], data structure [6], fuzziness and soft computing [7], and so on.

In this work, a mathematical representation of the information system is presented. The proposed representation takes advantage of the properties of a pretopological space, which is deduced from the information system under study. Moreover, new concepts are presented such as a cover pretopology and pre-interior set as well as an attributes reduction method based on a similarity relation. The structure of this paper is as follows: In Section 2, the basic definitions of the pretopological space are provided. Then, Section 3 introduces pretopological space from the information system. Section 4 is dedicated to proposing a methodology for pre-processing the information system by constructing its pretopological space. Hence, the empirical results are presented in Section 5. Section 6 presents a comparison between pretopologies and their pre-interiors. Section 7 investigates the different types of pretopological spaces. An application of the concept of cover pretopology is presented in Section 8. To conclude, Section 9 gives a brief conclusion.

2. Preliminaries

Definition 1

([8]). Given a universal set $X$ and mapping $a:\mathcal{P}\left(X\right)\to \mathcal{P}\left(X\right),$ the pair $\left(X,a\right)$ is called a pretopological space, so two axioms are satisfied:

• $a(Ø)=Ø.$
• $A\subseteq a\left(A\right),$ $\forall A\subseteq X .$

Definition 2

([8]). For pretopology $\left(X,a\right)$, an interior function $int:\mathcal{P}\left(X\right)\to \mathcal{P}\left(X\right)$ is defined as $int\left(A\right)=co\left(a\left(coA\right)\right),$ where $coA$ is the complement of $A, A\subseteq X$.

Definition 3

([9,10]). An ordered triple $IS=\left(U,K,Q\right)$ is called an information system $\left(IS, for short\right)$, where $U$ is a non-empty set of objects (students, toy blocks, etc.), $K$ is a non-empty set of attributes (colors, characteristics, etc.), and $Q$ contains the attributes’ scale ordinal (the value of object x at attribute k).

3. Novel Pretopological Spaces for Information System

In our work, we aim to use a similarity method to improve a new methodology of a pretopological space constructed from the information system under consideration.

Definition 4.

For any information system $\left(U,K,Q\right)$, where $U$ is a set of objects, $K$ is a set of attributes, and $Q$ is the set of attribute scale ordinal, we construct a similarity matrix $\mu$ with entities

$$\mu \left(x_i,x_j\right)=\frac{{{\displaystyle \sum }}_{k=1}^{\left|k\right|}\beta \left(x_i,x_j\right)}{\left|K\right|} , x,y ϵU$$

where $\beta \left(x,y\right)=\left\{qϵQ:K\left(x_i,q\right)=K\left(x_j,q\right)\right\}, \beta \left(x,y\right)=\{\begin{array}{c} 1 if K\left(x_i,q\right)=K\left(x_j,q\right)\\ 0 if K\left(x_i,q\right)\ne K\left(x_j,q\right)\end{array}$

Lemma 1.

The value of each element of the main diagonal of the similarity matrix is equal to one.

Illustrative Example

To demonstrate how the proposed similarity method works, we use the data given in

Table 1. Information system.

U	Funding	Poverty Impact	Middle Class Impact
$p_1$	High	High	Low
$p_2$	High	Medium	High
$p_3$	Low	Low	Medium
$p_4$	Low	Medium	Medium
$p_5$	Medium	Low	High
$p_6$	High	High	High

, which are available in [11]:

This information system consists of six projects, $X=\left\{p_1,p_2,p_3,p_4,p_5,p_6\right\}$, and three attributes, k = {Funding, Poverty Impact, Middle-Class Impact}, and the attributes’ scale ordinal Q contains three values Q = {High, Medium, Low}. Hence, the distance values between the different objects are obtained as in

Table 2. The values of similarity between the objects.

	${\mathcal{S}}_1$	${\mathcal{S}}_2$	${\mathcal{S}}_3$	${\mathcal{S}}_4$	${\mathcal{S}}_5$	${\mathcal{S}}_6$
${\mathcal{S}}_1$	1	$\frac{1}{3}$	$0$	0	$0$	$\frac{2}{3}$
${\mathcal{S}}_2$	$\frac{1}{3}$	1	$0$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$
${\mathcal{S}}_3$	$0$	$0$	$1$	$\frac{2}{3}$	$\frac{1}{3}$	$0$
${\mathcal{S}}_4$	0	$\frac{1}{3}$	$\frac{2}{3}$	$1$	0	$0$
${\mathcal{S}}_5$	$0$	$\frac{1}{3}$	$\frac{1}{3}$	0	$1$	$\frac{1}{3}$
${\mathcal{S}}_6$	$\frac{2}{3}$	$\frac{2}{3}$	$0$	$0$	$\frac{1}{3}$	$1$

as follows:

To illustrate the calculations of the entities of Table 2, we compute the value $\mu \left(p_1,p_2\right)$ as Algorithm 1.

Algorithm 1$\mathrm{Computing} \mathrm{the} \mathrm{value} \mu \left(p_i,p_j\right).$

$1: \mathrm{To} \mathrm{commence}, \mathrm{we} \mathrm{evaluate} \beta \left(p_1,p_2\right)$ $2: \mathrm{For} k=1$, $3: \mathrm{Hence}, \mathrm{we} \mathrm{obtain} \mathrm{the} \mathrm{value} K\left(p_1,q\right)=K\left(p_1,High\right), K\left(p_2,q\right)=K\left(p_2,High\right)$ $4: \mathrm{So} K\left(p_1,High\right)=K\left(p_2,High\right)$     $\therefore \beta \left(p_1,p_2\right)=1$ $5: \mathrm{This} \mathrm{step} \mathrm{is} \mathrm{to} \mathrm{repeated} \mathrm{for} \mathrm{the} \mathrm{values} \mathrm{of} k=2$$\mathrm{and} k=3$. $6:$ That is:  $\hspace{1em}\mathrm{at} k=1$   $K\left(p_1,High\right)=K\left(p_2,High\right)$     $\therefore \beta \left(p_1,p_2\right)=1$   $\mathrm{at} k=2$   $K\left(p_1,High\right)\ne K\left(p_2,Medium\right)$     $\therefore \beta \left(p_1,p_2\right)=0$   $\mathrm{at} k=3$   $K\left(p_1,Low\right)\ne K\left(p_2,High\right)$     $\therefore \beta \left(p_1,p_2\right)=0$   $\therefore \left|\beta \left(p_1,p_2\right)\right|=1 and \left|K\right|=3$ $7: \mathrm{Finally}, \mathrm{the} \mathrm{value} \mathrm{of} \mu \left(p_1,p_2\right)$ is to be computed as follows: $\mu \left(p_1,p_2\right)=1+0+0=\frac{1}{3}$.

Similarly, we compute each value $\mu \left(p_i,p_j\right).$

Definition 5.

Let $U=\left\{x_1, x_2,x_3,\dots . ,x_n\right\}$ be a set of objects, and ${\xi }_a\left(x_i\right)$ is determined by the relation between $x_i,x_j$; its value is $\mu \left(x_i,x_j\right)$, as given in Definition 4.

Then

$${\xi }_a\left(\left\{x_i\right\}\right)=\{x_j: x_j\in U and \mu \left(x_i,x_j\right)\ge \sigma , 0<\sigma \le 1, i,j\in \left|U\right|\}$$

where $\left(U,{\xi }_a\right)$ is a pretopological space constructed from some information system, $\left|U\right|$ is the cardinality of a set $U.$

Lemma 2.

When choosing the value of $\sigma$, we must consider that the axioms of the pretopology given in Definition 1 are satisfied.

Definition 6.

For the pretopological space$\left(X,{\xi }_a\right)$, a pre-interior function $in{t}_{{\xi }_a}:\mathcal{P}\left(X\right)\to \mathcal{P}\left(X\right)$is defined as${int}_{{\xi }_a}\left(A\right)=co\left({\xi }_a\left(coA\right)\right), coA$is complement of $A, A\subseteq X$.

Proposition 1.

For any pretopological space$\left(U,{\xi }_a\right),$pre-interior function$in{t}_{{\xi }_a}:\mathcal{P}\left(U\right)\to \mathcal{P}\left(U\right),$and the sets$X, Y\subseteq U;$the following properties are satisfied:

• $X\subseteq {\xi }_a\left(\left\{x\right\}\right)$$\forall xϵX$
• $in{t}_{{\xi }_a}\left(\left\{x\right\}\right)\subseteq X$$\forall xϵX$
• ${\xi }_a\left(U\right)=in{t}_{{\xi }_a}\left(U\right)=U$ and ${\xi }_a(Ø)=in{t}_{{\xi }_a} (Ø)=Ø$
• ${\xi }_a\left(X{\displaystyle \cup }Y\right)={\xi }_a\left(X\right){\displaystyle \cup }{\xi }_a\left(Y\right)$
• ${\xi }_a\left(X{\displaystyle \cap }Y\right)={\xi }_a\left(X\right){\displaystyle \cap }{\xi }_a\left(Y\right)$
• $co {\xi }_a\left(\left\{x\right\}\right)\subset {\xi }_a\left(\left\{cox\right\}\right)$$\forall xϵX$
• $in{t}_{{\xi }_a}\left(\left\{cox\right\}\right)\subset co in{t}_{{\xi }_a}\left(\left\{x\right\}\right)$$\forall xϵX$
• ${\xi }_a({\xi }_a\left(\left\{x\right\}\right))={\xi }_a\left(\left\{x\right\}\right)$ and $in{t}_{{\xi }_a} \left(in{t}_{{\xi }_a}\left\{x\right\}\right)=in{t}_{{\xi }_a}\left(\left\{x\right\}\right)$ $\forall xϵX$

Definition 7.

For a pretopological space $\left(X,{\xi }_a\right)$ , the set $A\subseteq X,$ a pre-closed subset of $X$ if and only if

$${\xi }_a\left(\left\{A\right\}\right)=A.$$

Definition 8.

For a pretopological space $\left(X,{\xi }_a\right)$, the set $A\subseteq X,$ a pre-open subset of $X$ if and only if

$$A=in{t}_{{\xi }_a}\left(\left\{A\right\}\right)$$

Definition 9

([8]). The pretopological space $\left(X,{\xi }_a\right)$ is said to be a pretopological space of V-type if:

$$A\subseteq B\Rightarrow {\xi }_a\left(A\right)\subseteq {\xi }_a\left(B\right) ,\forall A,B \subseteq X$$

Definition 10.

The pretopological space $\left(X,{\xi }_a\right)$ is said to be a pretopological space of $V_D$-type if:

$${\xi }_a\left(A{\displaystyle \cup }B\right)={\xi }_a\left(A\right){\displaystyle \cup }{\xi }_a\left(B\right) ,\forall A,B \subseteq X$$

Definition 11.

The pretopological space $\left(X,{\xi }_a\right)$ is said to be a pretopological space of $V_S$-type if:

$${\xi }_a\left(A\right)= {{\displaystyle \cup }}_{x\in A}{\xi }_a\left(\left\{x\right\}\right) ,\forall A\subseteq X$$

Definition 12.

For a non-empty set $X,$ and any countable family of indices $I,$ the family $\left\{\left(X,{\xi }_a{}_i\right):i\in I\right\}$ of pretopological spaces constructed from the information system is said to be a cover pretopology of $X$ if, for any subset $A\subseteq X$, the following is true:

$$({\xi }_{a1}{\displaystyle \cup }{\xi }_{a2}{\displaystyle \cup }\dots {\xi }_a{}_n)\left(A\right)={\xi }_{a1}\left(A\right){\displaystyle \cup }{\xi }_{a2}\left(A\right){\displaystyle \cup }\dots {\xi }_a{}_n\left(A\right)$$

4. Proposed Methodology

For any information system $IS=\left(U,K,Q\right),$ where $U$ is a set of $n$ objects, $K$ is a non-empty set of attributes, and $Q$ is an attribute scale ordinal, we propose the following procedure to define a set of core attributes of the system under consideration.

• Construct an $\left|U\right|\times \left|U\right|$ similarity matrix, where its entities are $\mu \left(x,y\right)$ between the objects in $U,$ which are computed for all attributes in $K$ using Definition 4.
• Reconstruct the matrix in step (1), after removing one attribute from set $K$;
• For each attribute in $K,$ repeat step (2), by removing another attribute.
• For each matrix of the matrices deduced from the previous steps (1–3), we construct the following spaces:
  • A pretopological space, accompanied by a pre-interior function;
  • A pre-closed approximation;
  • A pre-open approximation.
• Define the set of unnecessary attributes, that is, the attributes that satisfy the following:
  • ${\xi }_a\left(K\right)={\xi }_a\left(K_i\right);$
  • $in{t}_{{\xi }_a}\left(K\right)=in{t}_{{\xi }_a}\left(K_i\right);$
  • ${\xi }_a\left(\left\{x\right\}\right)=x$ $\forall xϵX$; and
  • $x=in{t}_{{\xi }_a}\left(\left\{x\right\}\right)$ $\forall xϵX.$
    where $K=\left\{v_1,v_2,\dots ,v_n\right\}$ $n\in \left|K\right|$ and $K_i=\left\{v_1,v_2,\dots ,v_{i-1},v_{i+1},\dots v_n\right\}$
• Finally, the core of the attributes is the set of attributes that do not satisfy any of the conditions given in step 5.

5. Empirical Results

This section implements an empirical study based on the data deduced from eight students consigning four questionnaires about their lectures, materials, study time, and laboratory, in order to assist in making decision about the students’ evaluation.

To demonstrate the proposed methodology, we introduce the following example:

Table 3. Information system.

	Lecturer (L)	Study Time (T)	Laboratory (Lab.)	Material (M)
${\mathcal{S}}_1$	Neutral	Neutral	Neutral	Neutral
${\mathcal{S}}_2$	Neutral	Reject	Reject	Neutral
${\mathcal{S}}_3$	Neutral	Reject	Reject	Neutral
${\mathcal{S}}_4$	Reject	Reject	Reject	Accept
${\mathcal{S}}_5$	Reject	Neutral	Neutral	Neutral
${\mathcal{S}}_6$	Neutral	Neutral	Reject	Accept
${\mathcal{S}}_7$	Neutral	Neutral	Reject	Reject
${\mathcal{S}}_8$	Neutral	Neutral	Reject	Accept

presents a set of questionnaires, which consists of eight students (objects), $\mathcal{S}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}$, and four attributes, lecturer (L), material (M), study time, (T) and laboratory (Lab.).

For the above IS, we set $U=\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,\dots ,{\mathcal{S}}_6\right\}$ to represent the set of objects, and we set $K=\left\{L,M, T, Lab.\right\}$ to represent the set of attributes.

To proceed, we have that $\left|U\right|=8$ and $\left|K\right|=4$

• Construct the pretopological space of set U.

  $${\xi }_a (\{{\mathcal{S}}_i\})=\left\{{\mathcal{S}}_j:\mu \left({\mathcal{S}}_i,{\mathcal{S}}_j\right)=1\hspace{1em}\hspace{1em}i,j=1,2,3,\dots ,8\right\}$$

  (1)
• Apply the pre-interior function for each object in $U.$

  $$in{t}_{{\xi }_a} (\{{\mathcal{S}}_i\})=co\left({\xi }_a\left(co\left\{{\mathcal{S}}_i\right\}\right)\right) \forall i=1,2,\dots ,8$$

  (2)
• Deduce the pre-closed sets from step 1.

  $${\xi }_a\left\{{\mathcal{S}}_i\right\}={\mathcal{S}}_i$$

  (3)
• Deduce the pre-open sets from step 2.

  $$in{t}_{{\xi }_a}\left\{{\mathcal{S}}_i\right\}={\mathcal{S}}_i$$

  (4)

5.1. Processing the Information System Using the Proposed Procedure

5.1.1. Constructing Pretopology Using Full Attributes

Table 4. Values of similarity between the objects.

	${\mathcal{S}}_1$	${\mathcal{S}}_2$	${\mathcal{S}}_3$	${\mathcal{S}}_4$	${\mathcal{S}}_5$	${\mathcal{S}}_6$	${\mathcal{S}}_7$	${\mathcal{S}}_8$
${\mathcal{S}}_1$	1	$\frac{2}{4}$	$\frac{2}{4}$	0	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{2}{4}$
${\mathcal{S}}_2$	$\frac{2}{4}$	1	$1$	$\frac{2}{4}$	$\frac{1}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{2}{4}$
${\mathcal{S}}_3$	$\frac{2}{4}$	$1$	$1$	$\frac{2}{4}$	$\frac{1}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{2}{4}$
${\mathcal{S}}_4$	0	$\frac{2}{4}$	$\frac{2}{4}$	$1$	$\frac{1}{4}$	$\frac{1}{4}$	$\frac{1}{4}$	$\frac{2}{4}$
${\mathcal{S}}_5$	$\frac{3}{4}$	$\frac{1}{4}$	$\frac{1}{4}$	$\frac{1}{4}$	$1$	$\frac{1}{4}$	$\frac{1}{4}$	$\frac{1}{4}$
${\mathcal{S}}_6$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{1}{4}$	$\frac{1}{4}$	$1$	$\frac{3}{4}$	$1$
${\mathcal{S}}_7$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{1}{4}$	$\frac{1}{4}$	$\frac{3}{4}$	$1$	$\frac{3}{4}$
${\mathcal{S}}_8$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{1}{4}$	$1$	$\frac{3}{4}$	$1$

gives the values of similarity between the objects.

Using Equation (1), we obtain the following components of the pretopology:

$$\begin{array}{lll}{\xi }_a{}_0\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}& {\xi }_a{}_0\left\{{\mathcal{S}}_2\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}& {\xi }_a{}_0\left\{{\mathcal{S}}_3\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}\\ {\xi }_a{}_0\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}& {\xi }_a{}_0\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_5\right\}& {\xi }_a{}_0\left\{{\mathcal{S}}_6\right\}=\left\{{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\\ {\xi }_a{}_0\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_7\right\}& {\xi }_a{}_0\left\{{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_6,{\mathcal{S}}_8\right\}& \end{array}$$

Considering the pre-interior function shown in Equation (2), we obtain the following sets:

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_0\left\{{\mathcal{S}}_1\right\}& =co{\xi }_a{}_0\left\{co{\mathcal{S}}_1\right\}=co\left\{{\xi }_a{}_0\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_1\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_0\left\{{\mathcal{S}}_2\right\}& =co{\xi }_a{}_0\left\{co{\mathcal{S}}_2\right\}=co\{{\xi }_a{}_0\left\{{\mathcal{S}}_1,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_0\left\{{\mathcal{S}}_3\right\}& =co{\xi }_a{}_0\left\{co{\mathcal{S}}_3\right\}=co\left\{{\xi }_a{}_0\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_0\left\{{\mathcal{S}}_4\right\}& =co{\xi }_a{}_0\left\{co{\mathcal{S}}_4\right\}=co\{{\xi }_a{}_0\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_4\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_0\left\{{\mathcal{S}}_5\right\}& =co{\xi }_a{}_0\left\{co{\mathcal{S}}_5\right\}=co\left\{{\xi }_a{}_0\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_5\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_0\left\{{\mathcal{S}}_6\right\}& =co{\xi }_a{}_0\left\{co{\mathcal{S}}_6\right\}=co\left\{{\xi }_a{}_0\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_0\left\{{\mathcal{S}}_7\right\}& =co{\xi }_a{}_0\left\{co{\mathcal{S}}_7\right\}=co\left\{{\xi }_a{}_0\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_7\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_0\left\{{\mathcal{S}}_8\right\}& =co{\xi }_a{}_0\left\{co{\mathcal{S}}_8\right\}=co{\xi }_a{}_0\left\{\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

Applying Equation (3), we obtain the following pre-closed sets:

$$\begin{array}{ll}{\xi }_a{}_0\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}& \hspace{1em}\hspace{1em}{\xi }_a{}_0\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}\\ {\xi }_a{}_0\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_5\right\}& \hspace{1em}\hspace{1em}{\xi }_a{}_0\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_7\right\}\end{array}$$

Applying Equation (4), we obtain the following pre-open sets:

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_0\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}& in{t}_{{\xi }_a}{}_0\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}\\ in{t}_{{\xi }_a}{}_0\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_5\right\}& in{t}_{{\xi }_a}{}_0\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_7\right\}\end{array}$$

5.1.2. Deducing More Pretopologies from the Original Information System

In this section, we study the information systems deduced from the information system given in Section 5, by excluding one attribute at a time.

Pretopology Excluding the Attribute “Lecturer (L)”

Table 5. Values of similarity between the objects.

	${\mathcal{S}}_1$	${\mathcal{S}}_2$	${\mathcal{S}}_3$	${\mathcal{S}}_4$	${\mathcal{S}}_5$	${\mathcal{S}}_6$	${\mathcal{S}}_7$	${\mathcal{S}}_8$
${\mathcal{S}}_1$	1	$\frac{1}{3}$	$\frac{1}{3}$	0	$1$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
${\mathcal{S}}_2$	$\frac{1}{3}$	1	$1$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
${\mathcal{S}}_3$	$\frac{1}{3}$	$1$	$1$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
${\mathcal{S}}_4$	0	$\frac{2}{3}$	$\frac{2}{3}$	$1$	0	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{2}{3}$
${\mathcal{S}}_5$	$1$	$\frac{1}{3}$	$\frac{1}{3}$	0	$1$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
${\mathcal{S}}_6$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	$1$	$\frac{2}{3}$	$1$
${\mathcal{S}}_7$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$	$1$	$\frac{2}{3}$
${\mathcal{S}}_8$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	$1$	$\frac{2}{3}$	$1$

gives the values of similarity between the objects.

Using Equation (1), we obtain the following components of the pretopology:

$$\begin{array}{lll}{\xi }_a{}_1\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_5\right\}& {\xi }_a{}_1\left\{{\mathcal{S}}_2\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}& {\xi }_a{}_1\left\{{\mathcal{S}}_3\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}\\ {\xi }_a{}_1\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}& {\xi }_a{}_1\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_5\right\}& {\xi }_a{}_1\left\{{\mathcal{S}}_6\right\}=\left\{{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\\ {\xi }_a{}_1\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_7\right\}& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\xi }_a{}_1\left\{{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_6,{\mathcal{S}}_8\right\}& \end{array}$$

Considering the pre-interior function shown in Equation (2), we obtain the following sets:

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_1\left\{{\mathcal{S}}_1\right\}& =co{\xi }_a{}_1\left\{co{\mathcal{S}}_1\right\}=co\left\{{\xi }_a{}_1\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_1\left\{{\mathcal{S}}_2\right\}& =co{\xi }_a{}_1\left\{co{\mathcal{S}}_2\right\}=co\left\{{\xi }_a{}_1\left\{{\mathcal{S}}_1,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_1\left\{{\mathcal{S}}_3\right\}& =co{\xi }_a{}_1\left\{co{\mathcal{S}}_3\right\}=co\left\{{\xi }_a{}_1\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_1\left\{{\mathcal{S}}_4\right\}& =co{\xi }_a{}_1\left\{co{\mathcal{S}}_4\right\}=co\left\{{\xi }_a{}_1\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_4\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_1\left\{{\mathcal{S}}_5\right\}& =co{\xi }_a{}_1\left\{co{\mathcal{S}}_5\right\}=co\left\{{\xi }_a{}_1\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_1\left\{{\mathcal{S}}_6\right\}& =co{\xi }_a{}_1\left\{co{\mathcal{S}}_6\right\}=co\left\{{\xi }_a{}_1\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_1\left\{{\mathcal{S}}_7\right\}& =co{\xi }_a{}_1\left\{co{\mathcal{S}}_7\right\}=co\left\{{\xi }_a{}_1\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_7\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_1\left\{{\mathcal{S}}_8\right\}& =co{\xi }_a{}_1\left\{co{\mathcal{S}}_8\right\}=co\{{\xi }_a{}_1\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\}=Ø\end{array}$$

Applying Equation (3), we obtain the following pre-closed sets:

$$\begin{array}{ll}{\xi }_a{}_1\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}& {\xi }_a{}_1\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_7\right\}\end{array}$$

Applying Equation (4), we obtain the following pre-open sets:

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_1\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}& in{t}_{{\xi }_a}{}_1\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_7\right\}\end{array}$$

Pretopology Excluding the Attribute “Study Time (T)”

Table 6. Values of similarity between the objects.

	${\mathcal{S}}_1$	${\mathcal{S}}_2$	${\mathcal{S}}_3$	${\mathcal{S}}_4$	${\mathcal{S}}_5$	${\mathcal{S}}_6$	${\mathcal{S}}_7$	${\mathcal{S}}_8$
${\mathcal{S}}_1$	1	$\frac{2}{3}$	$\frac{2}{3}$	0	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
${\mathcal{S}}_2$	$\frac{2}{3}$	1	$1$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$
${\mathcal{S}}_3$	$\frac{2}{3}$	$1$	$1$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$
${\mathcal{S}}_4$	0	$\frac{1}{3}$	$\frac{1}{3}$	$1$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{2}{3}$
${\mathcal{S}}_5$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$1$	$0$	$0$	$0$
${\mathcal{S}}_6$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$0$	$1$	$\frac{2}{3}$	$1$
${\mathcal{S}}_7$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	$0$	$\frac{2}{3}$	$1$	$\frac{2}{3}$
${\mathcal{S}}_8$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$0$	$1$	$\frac{2}{3}$	$1$

, gives the values of similarity between the objects.

Using Equation (1), we obtain the following components of the pretopology:

$$\begin{array}{lll}{\xi }_a{}_2\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}& {\xi }_a{}_2\left\{{\mathcal{S}}_2\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}& {\xi }_a{}_2\left\{{\mathcal{S}}_3\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}\\ {\xi }_a{}_2\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}& {\xi }_a{}_2\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_5\right\}& {\xi }_a{}_2\left\{{\mathcal{S}}_6\right\}=\left\{{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\\ {\xi }_a{}_2\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_7\right\}& {\xi }_a{}_2\left\{{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_6,{\mathcal{S}}_8\right\}& \end{array}$$

Considering the pre-interior function shown in Equation (2), we obtain the following sets:

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_2\left\{{\mathcal{S}}_1\right\}& =co{\xi }_a{}_2\left\{co{\mathcal{S}}_1\right\}=co\left\{{\xi }_a{}_2\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{S_1\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_2\left\{{\mathcal{S}}_2\right\}& =co{\xi }_a{}_2\left\{co{\mathcal{S}}_2\right\}=co\left\{{\xi }_a{}_2\left\{{\mathcal{S}}_1,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_2\left\{{\mathcal{S}}_3\right\}& =co{\xi }_a{}_2\left\{co{\mathcal{S}}_3\right\}=co{\xi }_a{}_2\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_2\left\{{\mathcal{S}}_4\right\}& =co{\xi }_a{}_2\left\{co{\mathcal{S}}_4\right\}=co\{{\xi }_a{}_2\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_4\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_2\left\{{\mathcal{S}}_5\right\}& =co{\xi }_a{}_2\left\{co{\mathcal{S}}_5\right\}=co\left\{{\xi }_a{}_2\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_5\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_2\left\{{\mathcal{S}}_6\right\}& =co{\xi }_a{}_2\left\{co{\mathcal{S}}_6\right\}=co\left\{{\xi }_a{}_2\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_2\left\{{\mathcal{S}}_7\right\}& =co{\xi }_a{}_2\left\{co{\mathcal{S}}_7\right\}=co\left\{{\xi }_a{}_2\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_7\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_2\left\{{\mathcal{S}}_8\right\}& =co{\xi }_a{}_2\left\{co{\mathcal{S}}_8\right\}=co\left\{{\xi }_a{}_2\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

Applying Equation (3), we obtain the following pre-closed sets:

$$\begin{array}{ll}{\xi }_a{}_2\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}& {\xi }_a{}_2\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}\\ {\xi }_a{}_2\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_5\right\}& {\xi }_a{}_2\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_7\right\}\end{array}$$

Applying Equation (4), we obtain the following pre-open sets:

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_2\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}& in{t}_{{\xi }_a}{}_2\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}\\ in{t}_{{\xi }_a}{}_2\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_5\right\}& in{t}_{{\xi }_a}{}_2\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_7\right\}\end{array}$$

Pretopology Excluding the Attribute “Laboratory (Lab.)”

Table 7. Values of similarity between the objects.

	${\mathcal{S}}_1$	${\mathcal{S}}_2$	${\mathcal{S}}_3$	${\mathcal{S}}_4$	${\mathcal{S}}_5$	${\mathcal{S}}_6$	${\mathcal{S}}_7$	${\mathcal{S}}_8$
${\mathcal{S}}_1$	1	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$
${\mathcal{S}}_2$	$\frac{2}{3}$	1	$1$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
${\mathcal{S}}_3$	$\frac{2}{3}$	$1$	$1$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
${\mathcal{S}}_4$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	$1$	$\frac{1}{3}$	$\frac{1}{3}$	$0$	$\frac{1}{3}$
${\mathcal{S}}_5$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$1$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
${\mathcal{S}}_6$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$1$	$\frac{2}{3}$	$1$
${\mathcal{S}}_7$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$0$	$\frac{1}{3}$	$\frac{2}{3}$	$1$	$\frac{2}{3}$
${\mathcal{S}}_8$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$1$	$\frac{2}{3}$	$1$

gives the values of similarity between the objects.

Using Equation (1), we obtain the following components of the pretopology:

$$\begin{array}{lll}{\xi }_a{}_3\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}& {\xi }_a{}_3\left\{{\mathcal{S}}_2\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}& {\xi }_a{}_3\left\{{\mathcal{S}}_3\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}\\ {\xi }_a{}_3\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}& {\xi }_a{}_3\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_5\right\}& {\xi }_a{}_3\left\{{\mathcal{S}}_6\right\}=\left\{{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\\ {\xi }_a{}_3\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_7\right\}& {\xi }_a{}_3\left\{{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_6,{\mathcal{S}}_8\right\}& \end{array}$$

Considering the pre-interior function shown in Equation (2), we obtain the following sets:

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_3\left\{{\mathcal{S}}_1\right\}& =co{\xi }_a{}_3\left\{co{\mathcal{S}}_1\right\}=co\left\{{\xi }_a{}_3\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_1\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_3\left\{{\mathcal{S}}_2\right\}& =co{\xi }_a{}_3\left\{co{\mathcal{S}}_2\right\}=co\left\{{\xi }_a{}_3\left\{{\mathcal{S}}_1,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_3\left\{{\mathcal{S}}_3\right\}& =co{\xi }_a{}_3\left\{co{\mathcal{S}}_3\right\}=co{\xi }_a{}_3\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_3\left\{{\mathcal{S}}_4\right\}& =co{\xi }_a{}_3\left\{co{\mathcal{S}}_4\right\}=co\left\{{\xi }_a{}_3\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_4\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_3\left\{{\mathcal{S}}_5\right\}& =co{\xi }_a{}_3\left\{co{\mathcal{S}}_5\right\}=co\left\{{\xi }_a{}_3\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_5\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_3\left\{{\mathcal{S}}_6\right\}& =co{\xi }_a{}_3\left\{co{\mathcal{S}}_6\right\}=co\left\{{\xi }_a{}_3\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_3\left\{{\mathcal{S}}_7\right\}& =co{\xi }_a{}_3\left\{co{\mathcal{S}}_7\right\}=co\left\{{\xi }_a{}_3\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_7\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_3\left\{{\mathcal{S}}_8\right\}& =co{\xi }_a{}_3\left\{co{\mathcal{S}}_8\right\}=co\left\{{\xi }_a{}_3\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

Applying Equation (3), we obtain the following pre-closed sets:

$$\begin{array}{ll}{\xi }_a{}_3\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}& {\xi }_a{}_3\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}\\ {\xi }_a{}_3\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_5\right\}& {\xi }_a{}_3\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_7\right\}\end{array}$$

Applying Equation (4), we obtain the following pre-open sets:

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_3\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}& in{t}_{{\xi }_a}{}_3\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}\\ in{t}_{{\xi }_a}{}_3\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_5\right\}& in{t}_{{\xi }_a}{}_3\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_7\right\}\end{array}$$

Pretopology Excluding the Attribute “Material (M)”

Table 8. Values of similarity between the objects.

	${\mathcal{S}}_1$	${\mathcal{S}}_2$	${\mathcal{S}}_3$	${\mathcal{S}}_4$	${\mathcal{S}}_5$	${\mathcal{S}}_6$	${\mathcal{S}}_7$	${\mathcal{S}}_8$
${\mathcal{S}}_1$	1	$\frac{1}{3}$	$\frac{1}{3}$	0	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$
${\mathcal{S}}_2$	$\frac{1}{3}$	1	$1$	$\frac{2}{3}$	$0$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$
${\mathcal{S}}_3$	$\frac{1}{3}$	$1$	$1$	$\frac{2}{3}$	$0$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$
${\mathcal{S}}_4$	0	$\frac{2}{3}$	$\frac{2}{3}$	$1$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
${\mathcal{S}}_5$	$\frac{2}{3}$	$0$	$0$	$\frac{1}{3}$	$1$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
${\mathcal{S}}_6$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$1$	$1$	$1$
${\mathcal{S}}_7$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$1$	$1$	$1$
${\mathcal{S}}_8$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$1$	$1$	$1$

gives the values of similarity between the objects.

Using Equation (1), we obtain the following components of the pretopology:

$$\begin{array}{lll}{\xi }_a{}_4\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}& {\xi }_a{}_4\left\{{\mathcal{S}}_2\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}& {\xi }_a{}_4\left\{{\mathcal{S}}_3\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}\\ {\xi }_a{}_4\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}& {\xi }_a{}_4\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_5\right\}& {\xi }_a{}_4\left\{{\mathcal{S}}_6\right\}=\left\{{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\\ {\xi }_a{}_4\left\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}& \hspace{1em}{\xi }_a{}_4\left\{{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}& \end{array}$$

Considering the pre-interior function shown in Equation (2), we obtain the following sets:

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_4\left\{{\mathcal{S}}_1\right\}& =co{\xi }_a{}_4\left\{co{\mathcal{S}}_1\right\}=co{\xi }_a{}_4\left\{\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_1\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_4\left\{{\mathcal{S}}_2\right\}& =co{\xi }_a{}_4\left\{co\mathcal{S}\right\}=co{\xi }_a{}_4\left\{\left\{{\mathcal{S}}_1,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_4\left\{{\mathcal{S}}_3\right\}& =co{\xi }_a{}_4\left\{co{\mathcal{S}}_3\right\}=co{\xi }_a{}_4\left\{\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_4\left\{{\mathcal{S}}_4\right\}& =co{\xi }_a{}_4\left\{co{\mathcal{S}}_4\right\}=co{\xi }_a{}_4\left\{\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_4\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_4\left\{{\mathcal{S}}_5\right\}& =co{\xi }_a{}_4\left\{co{\mathcal{S}}_5\right\}=co{\xi }_a{}_4\left\{\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{S_5\right\}\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_4\left\{{\mathcal{S}}_6\right\}& =co{\xi }_a{}_4\left\{co{\mathcal{S}}_6\right\}=co{\xi }_a{}_4\left\{\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_4\left\{{\mathcal{S}}_7\right\}& =co{\xi }_a{}_4\left\{co{\mathcal{S}}_7\right\}=co{\xi }_a{}_4\left\{\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

$$\begin{array}{ll}in{t}_{{\xi }_a}{}_4\left\{{\mathcal{S}}_8\right\}& =co{\xi }_a{}_4\left\{co{\mathcal{S}}_8\right\}=co{\xi }_a{}_4\left\{\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7\right\}\right\}\\ & \hspace{1em}\hspace{1em}=co\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=Ø\end{array}$$

Applying Equation (3), we obtain the following pre-closed sets:

$$\begin{array}{lll}{\xi }_a{}_4\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}& {\xi }_a{}_4\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}& {\xi }_a{}_4\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_5\right\}\end{array}$$

Applying Equation (4), we obtain the following pre-open sets:

$$\begin{array}{lll}in{t}_{{\xi }_a}{}_4\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}& in{t}_{{\xi }_a}{}_4\left\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_4\right\}& in{t}_{{\xi }_a}{}_4\left\{{\mathcal{S}}_5\right\}=\left\{{\mathcal{S}}_5\right\}\end{array}$$

5.2. Deducing the Core of Attributes Reduction

In this example, we can note the following:

• The pretopology constructed from the original information system, as well as its pre-interior, gives the same results that are deduced when removing both attribute (T) and attribute (Lab.);
• When generating the family of the pre-closed sets and the family of the pre-open sets for the original information system and all the resulting information systems after removing each attribute at a time, the generated families give the same results that are deduced when removing both attribute (T) and attribute (Lab.);
• To conclude, the results that are obtained when using the original information system are the same as those obtained when dealing with the information systems that re deduced after removing attribute (T) and he attribute (Lab.); Hence, the core of the attributes contains only attribute (L) and attribute (M).

Remarks 1.

1. 

Using the similarity method that is deduced from the information system gives a better result for the core of the attributes, when choosing the values of $\mu$ to be equal to one.

2. 

Practically, we noticed that we can deduce the core of the attributes directly from the pretopology constructed from the information system. Hence, it is unnecessary to compute the families: pre-interior, pre-closed, and pre-open sets.

6. Constructed Pretopologies and Their Pre-interiors Analogy

In this subsection, we run a comparison between the constructed pretopologies that were deduced in the previous Section 5 and the relation between their pre-interiors.

6.1. Comparison between the Deduced Pretopologies

In

Table 9. Types of protopologies.

	${\mathit{\xi }}_{\mathit{a}}\left\{\mathcal{S}\right\}$	${\mathit{\xi }}_{\mathit{a}}{}_0\left\{\mathcal{S}\right\}$	${\mathit{\xi }}_{\mathit{a}}{}_1\left\{\mathcal{S}\right\}$	${\mathit{\xi }}_{\mathit{a}}{}_2\left\{\mathcal{S}\right\}$	${\mathit{\xi }}_{\mathit{a}}{}_3\left\{\mathcal{S}\right\}$	${\mathit{\xi }}_{\mathit{a}}{}_4\left\{\mathcal{S}\right\}$
$\mathcal{S}$
$\left\{{\mathcal{S}}_1\right\}$		$\left\{{\mathcal{S}}_1\right\}$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_1\right\}$	$\left\{{\mathcal{S}}_1\right\}$	$\left\{{\mathcal{S}}_1\right\}$
$\left\{{\mathcal{S}}_3\right\}$		$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}$	$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}$	$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}$	$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}$	$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\}$
$\left\{{\mathcal{S}}_7\right\}$		$\left\{{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}$
$\left\{{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$		$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$
$\left\{{\mathcal{S}}_4,{\mathcal{S}}_7\right\}$		$\left\{{\mathcal{S}}_4,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}$
$\left\{{\mathcal{S}}_1,{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$		$\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$
$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_6\right\}$		$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$	$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$	$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$	$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$	$\left\{\begin{array}{c}{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_6,\\ {\mathcal{S}}_7,{\mathcal{S}}_8\end{array}\right\}$
$\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_7\right\}$		$\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_7\right\}$	$\left\{\begin{array}{c}{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,\\ {\mathcal{S}}_5,{\mathcal{S}}_7\end{array}\right\}$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_7\right\}$	$\left\{\begin{array}{c}{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_6,\\ {\mathcal{S}}_7,{\mathcal{S}}_8\end{array}\right\}$
$\left\{{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6\right\}$		$\left\{{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$	$\left\{\begin{array}{c}{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_5,\\ {\mathcal{S}}_6,{\mathcal{S}}_8\end{array}\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$	$\left\{\begin{array}{c}{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,\\ {\mathcal{S}}_7,{\mathcal{S}}_8\end{array}\right\}$
$\left\{{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$		$\left\{{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}$
$\left\{{\mathcal{S}}_5,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}$		$\left\{{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}$	$\left\{\begin{array}{c}{\mathcal{S}}_1,{\mathcal{S}}_5,{\mathcal{S}}_6,\\ {\mathcal{S}}_7,{\mathcal{S}}_8\end{array}\right\}$	$\left\{{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}$	$\left\{{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}$	$\left\{{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}$

, we show the pretopologies obtained for the example given in Section 6.

Remark 2.

From the results obtained in Table 9, we can note the following:

The pretopology constructed when removing attribute (T) and attribute (Lab.) gives the same results that are obtained when using the original pretopology.

6.2. Comparison between the Deduced Pretopologies’ Pre-interiors

In

Table 10. Comparison Between the Deduced Pretopologies Pre-interiors.

	$\mathit{i}\mathit{n}{\mathit{t}}_{{\mathit{\xi }}_{\mathit{a}}}\left\{\mathcal{S}\right\}$	$\mathit{i}\mathit{n}{\mathit{t}}_{{\mathit{\xi }}_{\mathit{a}}}{}_0\left\{\mathcal{S}\right\}$	$\mathit{i}\mathit{n}{\mathit{t}}_{{\mathit{\xi }}_{\mathit{a}}}{}_1\left\{\mathcal{S}\right\}$	$\mathit{i}\mathit{n}{\mathit{t}}_{{\mathit{\xi }}_{\mathit{a}}}{}_2\left\{\mathcal{S}\right\}$	$\mathit{i}\mathit{n}{\mathit{t}}_{{\mathit{\xi }}_{\mathit{a}}}{}_3\left\{\mathcal{S}\right\}$	$\mathit{i}\mathit{n}{\mathit{t}}_{{\mathit{\xi }}_{\mathit{a}}}{}_4\left\{\mathcal{S}\right\}$
$\mathcal{S}$
$\left\{{\mathcal{S}}_1\right\}$		$\left\{{\mathcal{S}}_1\right\}$	$Ø$	$\left\{{\mathcal{S}}_1\right\}$	$\left\{{\mathcal{S}}_1\right\}$	$\left\{{\mathcal{S}}_1\right\}$
$\left\{{\mathcal{S}}_3\right\}$		$Ø$	$Ø$	$Ø$	$Ø$	$Ø$
$\left\{{\mathcal{S}}_7\right\}$		$\left\{{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_7\right\}$	$Ø$
$\left\{{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$		$\left\{{\mathcal{S}}_5\right\}$	$Ø$	$\left\{{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_5\right\}$
$\left\{{\mathcal{S}}_4,{\mathcal{S}}_7\right\}$		$\left\{{\mathcal{S}}_4,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_7\right\}$
$\left\{{\mathcal{S}}_1,{\mathcal{S}}_3,{\mathcal{S}}_5\right\}$		$\left\{{\mathcal{S}}_1,{\mathcal{S}}_5\right\}$	$Ø$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_5\right\}$
$\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_6\right\}$		$Ø$	$Ø$	$Ø$	$Ø$	$Ø$
$\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_7\right\}$		$\left\{{\mathcal{S}}_1,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_1,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_1\right\}$
$\left\{{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6\right\}$		$\left\{{\mathcal{S}}_4,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_4\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_5\right\}$	$\left\{{\mathcal{S}}_4,{\mathcal{S}}_5\right\}$
$\left\{{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$		$\left\{{\mathcal{S}}_4\right\}$	$\left\{{\mathcal{S}}_4\right\}$	$\left\{{\mathcal{S}}_4\right\}$	$\left\{{\mathcal{S}}_4\right\}$	$\left\{{\mathcal{S}}_4\right\}$
$\left\{{\mathcal{S}}_5,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}$		$\left\{{\mathcal{S}}_5,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_5,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_5,{\mathcal{S}}_7\right\}$	$\left\{{\mathcal{S}}_5\right\}$

, we show the pretopologies’ pre-interiors obtained for the example given in Section 5.

Remark 3.

From the results obtained in Table 10, we can note the following:

The pretopology’s pre-interior constructed when removing attribute (T) and attribute (Lab.) gives the same results that are obtained when using the original pretopology.

7. Investigating the Different Types of Pretopological Spaces

7.1. Investigating Whether the Pretopolgy shown in Table 4 Satisfies the V-Type Pretopological Spaces

In this section we apply the axioms from Definition 9, given for the V-Type Pretopological Space to the information system represete in Table 4.

$$\begin{array}{l}\left\{{\mathcal{S}}_1\right\}\subset \left\{{\mathcal{S}}_1,{\mathcal{S}}_2\right\}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\xi }_a{}_0\left\{{\mathcal{S}}_1\right\}=\left\{{\mathcal{S}}_1\right\}\subset {\xi }_a{}_0\left\{{\mathcal{S}}_1,{\mathcal{S}}_2\right\}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3\right\}\\ \left\{{\mathcal{S}}_2,{\mathcal{S}}_4\right\}\subset \left\{{\mathcal{S}}_2,{\mathcal{S}}_4,{\mathcal{S}}_6\right\}\\ {\xi }_a{}_0\left\{{\mathcal{S}}_2,{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4\right\}\subset {\xi }_a{}_0\left\{{\mathcal{S}}_2,{\mathcal{S}}_4,{\mathcal{S}}_6\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\\ \left\{{\mathcal{S}}_3,{\mathcal{S}}_7\right\}\subset \left\{{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\xi }_a{}_0\left\{{\mathcal{S}}_3,{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_7\right\}\subset {\xi }_a{}_0\left\{{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_7,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\\ \left\{{\mathcal{S}}_5,{\mathcal{S}}_8\right\}\subset \left\{{\mathcal{S}}_1,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\xi }_a{}_0\left\{{\mathcal{S}}_5,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\subset {\xi }_a{}_0\left\{{\mathcal{S}}_1,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\end{array}$$

hence, $\left(\mathcal{S},{\xi }_a{}_0\right)$ represent a V-type pretopological space.

7.2. Investigating Whether the Pretopolgy Shown in Table 4 Satisfies the V_D-Type Pretopological Spaces

In this section we apply the axioms from Definition 10, given for the V_D -Type Pretopological Space to the information system represete in Table 4.

$$\begin{array}{l}{\xi }_a{}_0\left\{{\mathcal{S}}_2,{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4\right\}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\xi }_a{}_0\left\{{\mathcal{S}}_2\left\}{\displaystyle \cup }{\xi }_a{}_0\right\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\left\}{\displaystyle \cup }\right\{{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4\right\}\\ {\xi }_a{}_0\left\{{\mathcal{S}}_3,{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_7\right\}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\xi }_a{}_0\left\{{\mathcal{S}}_3\left\}{\displaystyle \cup }{\xi }_a{}_0\right\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\left\}{\displaystyle \cup }\right\{{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_7\right\}\\ {\xi }_a{}_0\left\{{\mathcal{S}}_5,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\xi }_a{}_0\left\{{\mathcal{S}}_5\left\}{\displaystyle \cup }{\xi }_a{}_0\right\{{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_5\left\}{\displaystyle \cup }\right\{{\mathcal{S}}_6,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\\ {\xi }_a{}_0\left\{{\mathcal{S}}_1,{\mathcal{S}}_7,{\mathcal{S}}_6\right\}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\xi }_a{}_0\left\{{\mathcal{S}}_1\left\}{\displaystyle \cup }{\xi }_a{}_0\left\{{\mathcal{S}}_7\right\}{\displaystyle \cup }{\xi }_a{}_0\right\{{\mathcal{S}}_6\right\}=\left\{{\mathcal{S}}_1\left\}{\displaystyle \cup }\left\{{\mathcal{S}}_7\right\}{\displaystyle \cup }\right\{{\mathcal{S}}_6,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_6,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\end{array}$$

hence, $\left(\mathcal{S},{\xi }_a{}_0\right)$ represent a V_D-type pretopological space.

7.3. Investigating Whether the Pretopolgy Shown in Table 4 Satisfies the V_S-Type Pretopological Spaces

In this section we apply the axioms from Definition 11, given for the V_S -Type Pretopological Space to the information system represete in Table 4.

$$\begin{array}{l}{\xi }_a{}_0\left\{{\mathcal{S}}_2,{\mathcal{S}}_4\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\xi }_a{}_0\left\{{\mathcal{S}}_2,{\mathcal{S}}_4\right\}={\displaystyle \cup }\{{\xi }_a{}_0\left\{\left\{{\mathcal{S}}_2\right\},{\xi }_a{}_0\left\{{\mathcal{S}}_4\right\}\right\}={\displaystyle \cup }\left\{\left\{{\mathcal{S}}_2,{\mathcal{S}}_3\right\},\left\{{\mathcal{S}}_4\right\}\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4\right\}\\ {\xi }_a{}_0\left\{{\mathcal{S}}_5,{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_5,{\mathcal{S}}_7\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\xi }_a{}_0\left\{{\mathcal{S}}_5,{\mathcal{S}}_7\right\}={\displaystyle \cup }\{{\xi }_a{}_0\left\{\left\{{\mathcal{S}}_5\right\},{\xi }_a{}_0\left\{{\mathcal{S}}_7\right\}\right\}={\displaystyle \cup }\left\{\left\{{\mathcal{S}}_5\right\},\left\{{\mathcal{S}}_7\right\}\right\}=\left\{{\mathcal{S}}_5,{\mathcal{S}}_7\right\}\\ {\xi }_a{}_0\left\{{\mathcal{S}}_1,{\mathcal{S}}_3,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7,{\mathcal{S}}_8\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\xi }_a{}_0\left\{{\mathcal{S}}_1,{\mathcal{S}}_3,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}=\\ {\displaystyle \cup }\left\{\left\{{\xi }_a{}_0\left\{{\mathcal{S}}_1\right\},{\xi }_a{}_0\left\{{\mathcal{S}}_2\right\},{\xi }_a{}_0\left\{{\mathcal{S}}_3\right\},{\xi }_a{}_0\left\{{\mathcal{S}}_5\right\},{\xi }_a{}_0\left\{{\mathcal{S}}_7\right\},{\xi }_a{}_0\{{\mathcal{S}}_8\right\}\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \cup }\left\{\left\{{\mathcal{S}}_1\right\},\{{\mathcal{S}}_2,{\mathcal{S}}_3\},\{{\mathcal{S}}_2,{\mathcal{S}}_3\},\left\{{\mathcal{S}}_5\right\},\left\{{\mathcal{S}}_7\right\},\{{\mathcal{S}}_6,{\mathcal{S}}_8\}\right\}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}\end{array}$$

hence, $\left(\mathcal{S},{\xi }_a{}_0\right)$ represent a V_S-type pretopological space.

Remark 4.

Similarly, we can conclude that the pretopologies shown in Table 5, Table 6, Table 7 and Table 8 achieve the types of pretopological space given in Section 5.

8. Applying the Concept of the Cover Pretopology

This section is devoted to achieving a cover pretopology, as defined in Section 4 for the pretopologies constructed from the information system shown in Section 5.1.

8.1. Cover Pretopology of ${\xi }_a{}_1 and {\xi }_a{}_2$

Table 11. Some operation on the pretopologies.

	${\mathcal{S}}_1$	${\mathcal{S}}_2$	${\mathcal{S}}_3$	${\mathcal{S}}_4$	${\mathcal{S}}_5$	${\mathcal{S}}_6$	${\mathcal{S}}_7$	${\mathcal{S}}_8$
${\mathcal{S}}_1$	1	$\frac{1}{2}$	$\frac{1}{2}$	0	$1$	$0$	$0$	$0$
${\mathcal{S}}_2$	$\frac{1}{2}$	1	$1$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
${\mathcal{S}}_3$	$\frac{1}{2}$	$1$	$1$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
${\mathcal{S}}_4$	0	$\frac{1}{2}$	$\frac{1}{2}$	$1$	$0$	$1$	$\frac{1}{2}$	$1$
${\mathcal{S}}_5$	$1$	$\frac{1}{2}$	$\frac{1}{2}$	$0$	$1$	$0$	$0$	$0$
${\mathcal{S}}_6$	$0$	$\frac{1}{2}$	$\frac{1}{2}$	$1$	$0$	$1$	$\frac{1}{2}$	$1$
${\mathcal{S}}_7$	$0$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$0$	$\frac{1}{2}$	$1$	$\frac{1}{2}$
${\mathcal{S}}_8$	$0$	$\frac{1}{2}$	$\frac{1}{2}$	$1$	$0$	$1$	$\frac{1}{2}$	$1$

shows the union of the pretopologies ${\xi }_a{}_1 and {\xi }_a{}_2$

For Set $\mathrm{A}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_8\right\},$ let us investigate whether the cover pretopology’s axiom given in Definition 12 is satisfied or not.

To commence, Table 11 shows that

$$\left({\xi }_a{}_1{\displaystyle \cup }{\xi }_a{}_2\right)\left(A\right)=\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$$

(5)

In addition, Table 3 and Table 4 show that

$${\xi }_a{}_1\left(A\right)=\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\} \mathrm{and} {\xi }_a{}_2\left(A\right)=\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$$

Then,

$${\xi }_a{}_1\left(A\right){\displaystyle \cup }{\xi }_a{}_2\left(A\right)=\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}{\displaystyle \cup }\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_5,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$$

(6)

$\mathrm{Hence}$, from Equations (5) and (6), we conclude that

$$\left({\xi }_a{}_1{\displaystyle \cup }{\xi }_a{}_2\right)\left(A\right)={\xi }_a{}_1\left(A\right){\displaystyle \cup }{\xi }_a{}_2\left(A\right)$$

which shows that $\left(\mathcal{S},{\xi }_a{}_1,{\xi }_a{}_2\right)$ is a cover pretopology for $A.$

For set $\mathrm{A}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_5,{\mathcal{S}}_7\right\},$ let us investigate whether the cover pretopology’s axiom given in Definition 12 is satisfied or not.

To commence, Table 11 shows that

$$\left({\xi }_a{}_1{\displaystyle \cup }{\xi }_a{}_2\right)\left(A\right)=\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7\right\}$$

(7)

In addition, Table 3 and Table 4 show that

$${\xi }_a{}_1\left(A\right)=\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7\right\} \mathrm{and} {\xi }_a{}_2\left(A\right)=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7\right\}$$

Then,

$${\xi }_a{}_1\left(A\right){\displaystyle \cup }{\xi }_a{}_2\left(A\right)=\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7\right\} {\displaystyle \cup }\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7\right\}$$

(8)

$\mathrm{Hence}$, from Equations (7) and (8), we conclude that

$$\left({\xi }_a{}_1{\displaystyle \cup }{\xi }_a{}_2\right)\left(A\right)={\xi }_a{}_1\left(A\right){\displaystyle \cup }{\xi }_a{}_2\left(A\right)$$

which shows that $\left(\mathcal{S},{\xi }_a{}_1,{\xi }_a{}_2\right)$ is a cover pretopology for $A.$

8.2. Cover Pretopology of ${\xi }_a{}_2 and {\xi }_a{}_3$

Table 12. Union of pretopologies ${\xi }_a{}_2\mathrm{and} {\xi }_a{}_3$

	${\mathcal{S}}_1$	${\mathcal{S}}_2$	${\mathcal{S}}_3$	${\mathcal{S}}_4$	${\mathcal{S}}_5$	${\mathcal{S}}_6$	${\mathcal{S}}_7$	${\mathcal{S}}_8$
${\mathcal{S}}_1$	1	$1$	$1$	0	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
${\mathcal{S}}_2$	$1$	1	$1$	$0$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
${\mathcal{S}}_3$	$1$	$1$	$1$	$0$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
${\mathcal{S}}_4$	0	$0$	$0$	$1$	$\frac{1}{2}$	$\frac{1}{2}$	$0$	$\frac{1}{2}$
${\mathcal{S}}_5$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$1$	$0$	$0$	$0$
${\mathcal{S}}_6$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$1$	$1$	$\frac{1}{2}$	$1$
${\mathcal{S}}_7$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$0$	$\frac{1}{2}$	$\frac{1}{2}$	$1$	$\frac{1}{2}$
${\mathcal{S}}_8$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$0$	$1$	$\frac{1}{2}$	$1$

shows the union of pretopologies ${\xi }_a{}_2 and {\xi }_a{}_3$.

For set $\mathrm{A}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_8\right\},$ let us investigate whether the cover pretopology’s axiom given in Definition 12 is satisfied or not.

To commence, Table 12 shows that

$$\left({\xi }_a{}_2{\displaystyle \cup }{\xi }_a{}_3\right)\left(A\right)=\left\{{\mathcal{S}}_1,,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$$

(9)

In addition, Table 3 and Table 4 show that

$${\xi }_a{}_2\left(A\right)=\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\} \mathrm{and} {\xi }_a{}_3\left(A\right)=\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$$

Then,

$${\xi }_a{}_2\left(A\right){\displaystyle \cup }{\xi }_a{}_3\left(A\right)=\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}{\displaystyle \cup }\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}=\left\{{\mathcal{S}}_1,{\mathcal{S}}_4,{\mathcal{S}}_6,{\mathcal{S}}_8\right\}$$

(10)

$\mathrm{Hence}$, from Equations (9) and (10), we conclude that

$$\left({\xi }_a{}_2{\displaystyle \cup }{\xi }_a{}_3\right)\left(A\right)\ne {\xi }_a{}_2\left(A\right){\displaystyle \cup }{\xi }_a{}_3\left(A\right)$$

which shows that $\left(\mathcal{S},{\xi }_a{}_2,{\xi }_a{}_3\right)$ is a not a cover pretopology for $A.$

For set $\mathrm{A}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_5,{\mathcal{S}}_7\right\},$ let us investigate whether the cover pretopology’s axiom given in Definition 12 is satisfied or not.

To commence, Table 12 shows that

$$\left({\xi }_a{}_2{\displaystyle \cup }{\xi }_a{}_3\right)\left(A\right)=\left\{{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7\right\}$$

(11)

In addition, Table 3 and Table 4 show that

$${\xi }_a{}_2\left(A\right)=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7\right\} \mathrm{and} {\xi }_a{}_3\left(A\right)=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7\right\}$$

Then,

$${\xi }_a{}_2\left(A\right){\displaystyle \cup }{\xi }_a{}_4\left(A\right)=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7\right\} {\displaystyle \cup }\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7\right\}=\left\{{\mathcal{S}}_2,{\mathcal{S}}_3,{\mathcal{S}}_5,{\mathcal{S}}_7\right\}$$

(12)

$\mathrm{Hence}$, from Equations (11) and (12), we conclude that

$$\left({\xi }_a{}_2{\displaystyle \cup }{\xi }_a{}_3\right)\left(A\right)={\xi }_a{}_2\left(A\right){\displaystyle \cup }{\xi }_a{}_4\left(A\right)$$

which shows that $\left(\mathcal{S},{\xi }_a{}_2,{\xi }_a{}_3\right)$ is a not a cover pretopology for $A.$

9. Conclusions

In this work, the concept of a pretopological space constructed from an information system was introduced. Moreover, a method for reducing the attributes of the information system by using pretopological concepts and a similarity relation was presented. Moreover, new concepts were presented, such as a cover pretopology and pre-interior set as well as an attributes reduction method based on a similarity relation. A comparison between the constructed pretopologies and their pre-interiors was presented. Furthermore, the proposed types of pretopological spaces were investigated. Compared to existing methods, the new technique does not need a preconceived decision to achieve the core attributes.