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Article

On Graph Primal Topological Spaces

1
College of Accounting, Guangzhou College of Technology and Business, Guangzhou 528138, China
2
Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt
3
Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt
*
Authors to whom correspondence should be addressed.
Axioms 2025, 14(9), 662; https://doi.org/10.3390/axioms14090662
Submission received: 3 June 2025 / Revised: 20 July 2025 / Accepted: 22 August 2025 / Published: 28 August 2025
(This article belongs to the Special Issue Topics in General Topology and Applications)

Abstract

In this paper, we introduce the concept of the “graph primal,” which serves as the dual structure to the “graph grill”. We present several results associated with graph primal operations. Moreover, we introduce two new graph-local functions on graph adjacency topological spaces (graph ATSs). We then explore the basic properties of the proposed graph-local functions and describe a method for generating two new graph ATSs from existing ones via graph primals. In addition, we examine several fundamental properties and connections of the resulting topologies, supported by some counterexamples. Furthermore, we characterize the nature of the open sets of these new topologies in terms of closure operators. Finally, we assess the compatibility of graph ATSs with the graph primal concept.
MSC:
54H99; 57M15; 54A10; 54A05

1. Introduction

General topology, initially defined by Hausdorff in 1914, and graph theory, presented in [1], are two significant areas of mathematics that are intimately connected. Creating topologies on a graph’s collection of vertices and edges is one way that graph theory and general topology are related. Directed and undirected graphs have been used in a number of manuscripts to design various topologies (see [2,3,4,5,6]). The theory of simple undirected graphs, namely the sets of vertices in such graphs, accounts for the majority of these constructs. A key to connecting topological structures with graph theory is a relation on a graph. This relation gives the graph additional kinds of topological structures. The labeled topologies on n points and the labeled transitive directed graph with n points have a one-to-one correspondence, as demonstrated in [7]. The lattice graph of the topologies of transitive directed graphs, as proposed by the authors of [7], was investigated in 1967 in [8]. The relationship between finite topologies and directed graphs was examined in 2010 by [9]. In 2013, the authors of [10] proposed a topology on the vertices of an undirected graph. In 2018, the authors of [11] linked a vertex set of simple graphs without isolated vertices to an incidence topology. In 2018, the authors of [12] developed a novel topology, constructed using the incidence topology on the set of vertices for simple graphs Axioms 14 00662 i004 = ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)) without isolated vertices, which has a subbasis generated by the family of end sets that include only the endpoints of each edge. In [13], the authors used the graphs Axioms 14 00662 i004 =( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)) to induce two topologies on the set of its edges Z (Axioms 14 00662 i004), denoted by compatible edge topology and incompatible edge topology. A relation on graphs was introduced in [14] to produce new styles of topological structures. In [15], the authors explained how to generate a topology using incidence and adjacency relations on the vertex set of graphs. They also examined the closure and interior properties of vertex sets of subgraphs in the graph adjacency topological space (graph ATS). Numerous new ideas have been added to topology, enriching it with a range of recently created fields of study. Closure spaces, proximity spaces, ideals [16], filters [17], grills [18], and primals [19] are a few examples of innovative structures that topologists have developed in an attempt to investigate properties of topological spaces. The new notion of a primal structure was developed and discussed as the dual of the notion of a grill. Recently, many authors have studied the relationships among primal topological spaces, topological spaces, and the soft primal in soft topological spaces and have investigated their basic properties. Primal proximity spaces inspired by primal and proximity notions have had a significant impact on the development of operators in primal topological spaces (see [20,21,22,23]). Additionally, the authors of [24] created graph grills and investigated the characteristics of the generated topologies on the vertex set of graphs. Similarly, graph ideals were developed and investigated in [25] to generate new topologies on the vertex set of graphs. The majority of research to date has focused on ideals and primals in general topological spaces. It is evident that the majority of real-world problems may be represented as graphs and resolved by applying the ideas of graph theory. In the current contribution, we introduce a new link between graph theory and ordinary topological spaces. Based on our new concept of a “graph primal”, we propose new definitions, theorems, methods, applications, and broader generalizations, which were not covered in previous papers (see [24,25]). Moreover, we develop a different methodological approach that provides fresh insights into the interplay between topology and graph theory.
In this paper, we propose the notion of “graph primal” as the dual structure of graph grill. Moreover, we introduce the new graph local function ( . ) on a graph ATS. Further, we explore the basic facts of the proposed graph local function and describe the method of generating the new operator CL via graph primals with the help of ( . ) , generating a unique graph ATS T A d . In addition, we propose the operator Θ and its associated topology σ . Several fundamental properties and connections of the new topologies σ and T A d in the graph primal ATSs were examined. This work ends by analyzing the suitability of the graph primal ATS with respect to the graph primal and by characterizing the nature of the open sets of the novel topologies in terms of the closure operators.

1.1. Key Contributions

The key contributions of this paper can be summarized as follows:
  • Introduction of Graph Primal Structure: This paper discusses the novel notion of graph primal structure as a dual to existing concepts in topology, enriching the theoretical framework.
  • Graph Theory Applications: It highlights how classical graph theory can be integrated with modern topology, potentially leading to new insights and methodologies.
  • Development of New Topological Spaces: This research introduces various new topological spaces that can be derived from graph structures.
  • Real-World Problem Solving: By representing numerous real-world problems as graphs, this paper underscores the practical implications of these theoretical developments.
  • Future Research Opportunities: This paper is significant due to its innovative contributions to the fields of topology and graph theory. By introducing new concepts and exploring their implications, it not only enhances theoretical knowledge but also provides practical tools for addressing real-world problems. The potential for future research stemming from this work underscores its importance in advancing mathematical sciences.

1.2. Comprehensive Analysis

Our study offers original contributions to the understanding of graph theory by proposing the “graph primal”. Graph primal theory introduces innovative concepts, particularly focusing on graph primals and their associated topologies. It provides a framework for analyzing complex problems represented as graphs, utilizing various mathematical structures to enhance understanding and applications in real-world scenarios. Also, this paper offers a novel graph-local function in Definition 10 via our new concept, the “graph primal”, which differs from the corresponding functions given in [24,25]. Neither our new graph-local function given in Definition 10 nor its related properties, theorems, and graphical topological spaces are defined in [24,25]. Furthermore, this analysis synthesizes the essential elements and implications of graph primal theory, showcasing its relevance and potential applications in various domains. The exploration of graph structures through this lens not only furthers mathematical understanding but also enhances practical problem-solving strategies across multiple disciplines.

2. Preliminaries

Some basic definitions and introductions to graph theory and topology can be found in [1,25,26].
A pair ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)) denotes a graph Axioms 14 00662 i004, where Q (Axioms 14 00662 i004) is a nonempty finite set and Z (Axioms 14 00662 i004) is a set of unordered pairs of elements of Q (Axioms 14 00662 i004). The vertex set of Axioms 14 00662 i004 is the set Q (Axioms 14 00662 i004), and the edge set of Axioms 14 00662 i004 is the set Z (Axioms 14 00662 i004). The elements of Q (Axioms 14 00662 i004) are called the vertices or the nodes Axioms 14 00662 i004), and the elements of Z (Axioms 14 00662 i004) are called the edges Axioms 14 00662 i004). A loop is an edge of Z (Axioms 14 00662 i004) that connects a vertex of Z (Axioms 14 00662 i004) to itself. Multiple or parallel edges are edges that connect the same vertices. Two nodes 1 and 2 of Axioms 14 00662 i004 are called adjacent to each other if they are connected by an edge α of Axioms 14 00662 i004. In this case, the edge α is said to connect 1 and 2 . Moreover, the vertices 1 and 2 are called the endpoints of this edge. Two vertices 1 and 2 of Axioms 14 00662 i004 are called non-adjacent to each other if there are no edges between them (they are not adjacent). In a graph Axioms 14 00662 i004, the degree of a vertex Q (Axioms 14 00662 i004) is represented by d e g ( ) , which is the number of edges that are connected to . A regular graph is one in which all of the vertices have the same degree. An “isolated vertex” refers to a vertex of degree 0. For any node Q (Axioms 14 00662 i004, the neighbors of in Axioms 14 00662 i004 are the nodes that are adjacent to . The pair Axioms 14 00662 i004 = ( , ) stands for the empty graph. If Axioms 14 00662 i004 = ( Q , Z ) and Axioms 14 00662 i004′ = ( Q , Z ), then Axioms 14 00662 i004Axioms 14 00662 i004′ = ( Q Q , Z Z ) and Axioms 14 00662 i004Axioms 14 00662 i004′ = ( Q Q , Z Z ) . If Axioms 14 00662 i004Axioms 14 00662 i004′ = ( , ) , then Axioms 14 00662 i004 and Axioms 14 00662 i004′ are disjoint. If Q Q and Z Z , then Axioms 14 00662 i004′ is a subgraph of Axioms 14 00662 i004, and Axioms 14 00662 i004 is a supergraph of Axioms 14 00662 i004′, written as Axioms 14 00662 i004′ ⊆ Axioms 14 00662 i004. A simple graph is a graph Axioms 14 00662 i004 that has no loops and no multiple edges. If Q can be divided into two distinct subsets, Q 1 and Q 2 , such that each edge of Axioms 14 00662 i004 connects a vertex of Q 1 to a vertex of Q 2 , then the graph is said to be bipartite, and the pair ( Q 1 , Q 2 ) is called a bipartition of the graph Axioms 14 00662 i004.
Definition 1
([10,25]). Let Axioms 14 00662 i004 = ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)) be a graph. For Q (Axioms 14 00662 i004), the neighborhood set N of is defined as N = { Q (Axioms 14 00662 i004): Z (Axioms 14 00662 i004)}.
Definition 2
([10,25]). Consider the graph Axioms 14 00662 i004 = ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004), in which no vertex is isolated. The collection of N for all Q (Axioms 14 00662 i004) is defined as S N . Stated differently, S N = { N : Q (Axioms 14 00662 i004)}. The graph adjacency topological space (graph ATS) is the pair ( Q (Axioms 14 00662 i004), T A d ), and S N forms a subbase for the topology T A d on Q (Axioms 14 00662 i004).
Definition 3
([19]). Let E be a nonempty set and , E . A collection P 2 E is said to be a primal on E if it satisfies the following statements:
(1) 
E P ;
(2) 
if P and , then P ;
(3) 
if P , then P or P .
Definition 4
([24]). Consider the graph Axioms 14 00662 i004 = ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)). In the given graph, P ( Q ) and P ( Z ) are the power sets of Q (Axioms 14 00662 i004) and Z (Axioms 14 00662 i004), respectively. The collection S = {Axioms 14 00662 i004′: Axioms 14 00662 i004′ = ( Q , Z ), where Q Q , Z Z }, is said to be a graph grill on a graph ATS ( Q (Axioms 14 00662 i004), T A d if it satisfies the following three conditions:
(1) 
( , ) S .
(2) 
If Axioms 14 00662 i004′ is a subgraph of Axioms 14 00662 i004″ for some Axioms 14 00662 i004′ ∈ S , then Axioms 14 00662 i004″ ∈ S .
(3) 
If Axioms 14 00662 i004′ ∪ (Axioms 14 00662 i004″ ∈ S , then Axioms 14 00662 i004′ ∈ S or Axioms 14 00662 i004″ ∈ S .
Simple undirected graphs are the ones discussed throughout this study. The symbol Axioms 14 00662 i004 refers to a graph. We abbreviate the word “simple graph” to “graph”. The power set of a graph Axioms 14 00662 i004 is denoted by P (Axioms 14 00662 i004) or 2Axioms 14 00662 i004. For any vertex subset ( Q ( ) of a subgraph Axioms 14 00662 i004, the closure and interior with respect to the graph primal ATS are denoted by CL ( Q ( ) and int ( Q ( ) , respectively.

3. Graph Primals and Graph-Local Functions

This section defines a graph primal on a graph ATS, listing specific conditions that must be satisfied. It discusses the implications of these conditions and provides examples to illustrate the concept of graph primals. The introduction of new graph-local functions marks a significant advancement in the field, allowing for the generation of unique graph ATSs and enhancing the understanding of their properties. Several results related to the graph primal ATSs are discussed in detail with the help of some counterexamples. Further, we define another topology on the vertex set of a given graph. The properties and relationships between the newly proposed topologies and existing structures are discussed.
Definition 5.
Consider the graph Axioms 14 00662 i004 = ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)). It is assumed that the defined graph Axioms 14 00662 i004 has P ( Q ) and P ( Z ) as the power sets of Q (Axioms 14 00662 i004) and Z (Axioms 14 00662 i004), respectively. The collection P = {Axioms 14 00662 i004′: Axioms 14 00662 i004′ = Q , Z , where Q Q , Z Z } is said to be a graph primal on a graph ATS ( Q (Axioms 14 00662 i004), T A d ) if it satisfies the following three statements:
(1) 
Axioms 14 00662 i004 P .
(2) 
If Axioms 14 00662 i004′ is a subgraph of Axioms 14 00662 i004″ for some Axioms 14 00662 i004″ ∈ P , then Axioms 14 00662 i004′ ∈ P .
(3) 
If Axioms 14 00662 i004′ ∩Axioms 14 00662 i004″ ∈ P , then Axioms 14 00662 i004′ ∈ P or Axioms 14 00662 i004″ ∈ P .
If there is no confusion, we represent the graph primal P by the union of all of its members. Moreover, the triple ( Q (Axioms 14 00662 i004), T A d , P ) is called a graph primal ATS.
Corollary 1.
Consider the graph Axioms 14 00662 i004 = ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)). It is assumed that the defined graph Axioms 14 00662 i004 has P ( Q ) and P ( Z ) as the power sets of Q (Axioms 14 00662 i004) and Z (Axioms 14 00662 i004), respectively. The collection P = {Axioms 14 00662 i004′: Axioms 14 00662 i004′ = ( Q , Z , where Q Q , Z Z } is said to be a graph primal on a graph ATS ( Q (Axioms 14 00662 i004), T A d if it satisfies the following three statements:
(1) 
Axioms 14 00662 i004 P .
(2) 
If Axioms 14 00662 i004′ is a subgraph of Axioms 14 00662 i004″ for some Axioms 14 00662 i004″ ∉ P , then Axioms 14 00662 i004′ ∉ P .
(3) 
If Axioms 14 00662 i004′ ∉ P for some Axioms 14 00662 i004″ ∉ P , then Axioms 14 00662 i004′ ∩ Axioms 14 00662 i004″ ∉ P .
Example 1.
Let Axioms 14 00662 i004 be the graph ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)), where Q (Axioms 14 00662 i004) = { 1 , 2 } and Z (Axioms 14 00662 i004) = {α}. A drawing of the graph Axioms 14 00662 i004 is shown in Figure 1.
A graph of two vertices has at most one edge, and so all possible graph primals on the above graph Axioms 14 00662 i004 are
P 1 = , where “∅” is the empty family, which differs from the family containing the empty set “ { } ”,
P 2 = { ( , ) , ( { 1 } , ) } ,
P 3 = { ( ( , ) , { 2 } , ) } ,
P 4 = { ( , ) , ( { 1 } , ) , ( { 2 } , ) } .
Note that P = { ( , ) , ( { 1 } , ) , ( { 2 } , ) , ( { 1 , 2 } , { α } ) } is not a graph primal on Axioms 14 00662 i004 since Axioms 14 00662 i004 = ( { 1 , 2 } , { α } ) ∈ P .
Example 2.
Let Axioms 14 00662 i004 be the graph ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)), where Q (Axioms 14 00662 i004) = { 1 , 2 , 3 } and Z (Axioms 14 00662 i004) = { α 1 , α 2 , α 3 }. A drawing of the graph Axioms 14 00662 i004 is shown in Figure 2.
For the above graph, some possible graph primals on Axioms 14 00662 i004 are
P 1 = , P 2 = { ( , ) , ( { 1 } , ) , ( { 2 } , ) , ( { 3 } , ) , ( { 1 , 2 } , ) , ( { 2 , 3 } , ) , ( { 1 , 3 } , ) , ( { 1 , 2 , 3 } , ) , ( { 1 , 2 } , { α 1 } ) , ( { 1 , 3 } , { α 3 } ) , ( { 2 , 3 } , { α 2 } ) , ( { 1 , 2 , 3 } , { α 1 } ) , ( { 1 , 2 , 3 } , { α 2 } ) , ( { 1 , 2 , 3 } , { α 1 , α 2 } ) } , P 3 = { ( , ) , ( { 1 } , ) , ( { 2 } , ) , ( { 3 } , ) , ( { 1 , 2 } , ) , ( { 2 , 3 } , ) , ( { 1 , 3 } , ) , ( { 1 , 2 , 3 } , ) , ( { 1 , 2 } , { α 1 } ) , ( { 1 , 3 } , { α 3 } ) , ( { 2 , 3 } , { α 2 } ) , ( { 1 , 2 , 3 } , { α 1 } ) , ( { 1 , 2 , 3 } , { α 3 } ) , ( { 1 , 2 , 3 } , { α 1 , α 3 } ) } , P 4 = { ( , ) , ( { 1 } , ) , ( { 2 } , ) , ( { 3 } , ) , ( { 1 , 2 } , ) , ( { 2 , 3 } , ) , ( { 1 , 3 } , ) , ( { 1 , 2 , 3 } , ) , ( { 1 , 2 } , { α 1 } ) , ( { 1 , 3 } , { α 3 } ) , ( { 2 , 3 } , { α 2 } ) , ( { 1 , 2 , 3 } , { α 2 } ) , ( { 1 , 2 , 3 } , { α 3 } ) , ( { 1 , 2 , 3 } , { α 2 , α 3 } ) } , P 5 = { ( , ) , ( { 1 } , ) , ( { 2 } , ) , ( { 3 } , ) , ( { 1 , 2 } , ) , ( { 2 , 3 } , ) , ( { 1 , 3 } , ) , ( { 1 , 2 , 3 } , ) , ( { 1 , 2 } , { α 1 } ) , ( { 1 , 3 } , { α 3 } ) , ( { 2 , 3 } , { α 2 } ) , ( { 1 , 2 , 3 } , { α 1 } ) , ( { 1 , 2 , 3 } , { α 2 } ) , ( { 1 , 2 , 3 } , { α 3 } ) , ( { 1 , 2 , 3 } , { α 1 , α 2 } ) , ( { 1 , 2 , 3 } , { α 1 , α 3 } ) } , P 6 = { ( , ) , ( { 1 } , ) , ( { 2 } , ) , ( { 3 } , ) , ( { 1 , 2 } , ) , ( { 2 , 3 } , ) , ( { 1 , 3 } , ) , ( { 1 , 2 , 3 } , ) , ( { 1 , 2 } , { α 1 } ) , ( { 1 , 3 } , { α 3 } ) , ( { 2 , 3 } , { α 2 } ) , ( { 1 , 2 , 3 } , { α 1 } ) , ( { 1 , 2 , 3 } , { α 2 } ) , ( { 1 , 2 , 3 } , { α 3 } ) , ( { 1 , 2 , 3 } , { α 1 , α 3 } ) , ( { 1 , 2 , 3 } , { α 2 , α 3 } ) } .
Note that P = { ( , ) , ( { 1 } , ) , ( { 2 } , ) , ( { 3 } , ) , ( { 1 , 2 } , { α 1 } ) } is not a graph primal on G since, for example, G = ( { 3 } , ) P , but neither ( { 2 , 3 } , ) P nor ( { 1 , 3 } , ) P .
Remark 1.
It should be noted that, for a graph Axioms 14 00662 i004 = ( Q (Axioms 14 00662 i004), Z ( G ) ) with n vertices and no edges ( Z (Axioms 14 00662 i004) = ), 2 n graph primals can be written on the graph Axioms 14 00662 i004.
Theorem 1.
Let E be a graph grill on the graph Axioms 14 00662 i004. Then, {Axioms 14 00662 i004″: Axioms 14 00662 i004″ = ( Q , Z )|( Q )c = Q , Axioms 14 00662 i004′ = ( Q , Z ) S where Q , Q Q and Z , Z Z } is a graph primal on Axioms 14 00662 i004.
Proof. 
Let S be a graph grill of the graph Axioms 14 00662 i004 and P = {Axioms 14 00662 i004″: Axioms 14 00662 i004″ = ( Q , Z )|( Q )c = Q , Axioms 14 00662 i004′ = ( Q , Z ) ∈ S where Q , Q Q and Z , Z Z }. Then, we show that P is a graph primal:
(1)
Since ( , ) S , it follows that Axioms 14 00662 i004 P .
(2)
Let Axioms 14 00662 i004″ = ( Q , Z ) P and Axioms 14 00662 i004★★ = ( Q , Z ) Axioms 14 00662 i004″. Then, Q = ( Q ) c Q = ( Q ) c . Since Axioms 14 00662 i004′ = ( Q , Z ) S , it follows that Axioms 14 00662 i004 = ( Q , Z ) S . Hence, Axioms 14 00662 i004★★ = ( Q , Z ) P .
(3)
Let Axioms 14 00662 i004′ ∩ Axioms 14 00662 i004″ = ( Q , Z ) ( Q , Z ) P . Then, ( Q ) c ( Q ) c = ( Q Q ) c . Therefore, we get Axioms 14 00662 i004 = ( ( Q ) c , Z ) S or Axioms 14 00662 i004★★ = ( ( Q ) c , Z ) S . Thus, Axioms 14 00662 i004′ ∈ P or Axioms 14 00662 i004″ ∈ P . Hence, P is a graph primal on the graph Axioms 14 00662 i004.
Theorem 2.
Let P and Q be two graph primals on a graph Axioms 14 00662 i004. Then, P Q is a graph primal on the graph Axioms 14 00662 i004.
Proof. 
(1) Let P and Q be two graph primals on X. Then, Axioms 14 00662 i004 P and Axioms 14 00662 i004 Q . Hence, Axioms 14 00662 i004 P Q .
(2)
Let P Q and . Then, P or Q . Therefore, P or Q . As a result, P Q .
(3)
Let P Q . Then, P or Q . If P , then either P or P . Again, if Q , then either Q or Q . Then, obviously P Q or P Q .
Hence, P Q is a graph primal on the graph Axioms 14 00662 i004. □
Remark 2.
The following example shows that the intersection of two graph primals related to a graph Axioms 14 00662 i004 need not be a graph primal on Axioms 14 00662 i004.
Example 3.
Let Axioms 14 00662 i004 be the graph ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)), where Q (Axioms 14 00662 i004) = { 1 , 2 , 3 } and Z (Axioms 14 00662 i004) = { α 1 , α 2 } . A drawing of the graph Axioms 14 00662 i004 is shown in Figure 3.
The representations of the following three graph primals are given in Table 1.
P 1 = { ( , ) , ( { 1 } , ) , ( { 2 } , ) , ( { 3 } , ) , ( { 1 , 2 } , ) , ( { 2 , 3 } , ) , ( { 1 , 3 } , ) , ( { 1 , 2 , 3 } , ) , ( { 1 , 2 } , { α 1 } ) , ( { 1 , 2 , 3 } , { α 1 } ) } . P 2 = { ( , ) , ( { 1 } , ) , ( { 2 } , ) , ( { 3 } , ) , ( { 1 , 2 } , ) , ( { 2 , 3 } , ) , ( { 1 , 3 } , ) , ( { 1 , 2 , 3 } , ) , ( { 1 , 3 } , { α 2 } ) , ( { 1 , 2 , 3 } , { α 2 } ) } .
From the computations of the members of P 1 P 2 in the Table 1, it is clear that P 1 P 2 does not represent a graph primal on the graph Axioms 14 00662 i004, since P 1 P 2 = { ( , ) , ( { 1 } , ) , ( { 2 } , ) , ( { 3 } , ) , ( { 1 , 2 } , ) , ( { 2 , 3 } , ) , ( { 1 , 3 } , ) , ( { 1 , 2 , 3 } , ) } , and, for example, = ( { 3 } , ) = ( { 1 , 3 } , α 1 ) ( { 2 , 3 } , α 2 ) P 1 P 2 , but neither ( { 1 , 3 } , α 1 ) P 1 P 2 nor ( { 2 , 3 } , α 2 ) P 1 P 2 .
Definition 6.
Let ( Q (Axioms 14 00662 i004), T A d ) be a graph ATS and Q (Axioms 14 00662 i004). The open neighborhood system at , denoted by N ( ) , is given as N ( ) = U T A d : U .
Definition 7.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS. Let be a subgraph of Axioms 14 00662 i004. Then, ( Q ( ) ) P , T A d = ( Q ( ) ) = { Q (Axioms 14 00662 i004): for every U N ( ) , Q ( ) c U c = Q (Axioms 14 00662 i004′) and Axioms 14 00662 i004′ ∈ P } is called the graph-local function of Q ( ) with respect to P and T A d .
Example 4.
Let Axioms 14 00662 i004 be the graph ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)), where Q (Axioms 14 00662 i004) = { 1 , 2 , 3 , 4 , 5 , 6 } and Z (Axioms 14 00662 i004) = { α 1 , α 2 , α 3 , α 4 } . A drawing of the graph Axioms 14 00662 i004 is shown in Figure 4.
Now we illustrate some graph-local functions of a subgraph of the above graph.
S N = { { 4 , 5 } , { 6 } , { 5 } , { 1 } , { 1 , 3 } , { 2 } } . β = { , { 4 , 5 } , { 6 } , { 5 } , { 1 } , { 1 , 3 } , { 2 } } . T A d = { , { 4 , 5 } , { 6 } , { 5 } , { 1 } , { 1 , 3 } , { 2 } , { 4 , 5 , 6 } , { 4 , 5 , 1 } , { 1 , 3 , 4 , 5 } , { 2 , 4 , 5 } , { 5 , 6 } , { 1 , 6 } , { 1 , 3 , 6 } , { 2 , 6 } , { 1 , 5 } , { 1 , 3 , 5 } , { 2 , 5 } , { 1 , 2 } , { 1 , 5 , 6 } , { 1 , 4 , 5 } , { 1 , 4 , 5 , 6 } , { 1 , 5 , 6 } , { 1 , 3 , 5 , 6 } , { 1 , 3 , 4 , 5 , 6 } , { 2 , 4 , 5 , 6 } , { 2 , 4 , 5 , 1 } , { 1 , 2 , 3 , 4 , 5 } , { 2 , 5 , 6 } , { 1 , 2 , 6 } , { 1 , 2 , 3 , 6 } , { 1 , 2 , 5 } { 1 , 2 , 3 , 5 } , { 1 , 2 , 5 , 6 } , { 1 , 2 , 4 , 5 } , { 1 , 2 , 4 , 5 , 6 } , { 1 , 2 , 5 , 6 } , { 1 , 2 , 3 , 5 , 6 } , { 1 , 2 , 3 , 4 , 5 , 6 } } .
Let P = P (Axioms 14 00662 i004) − { ( { 2 , 3 , 5 , 6 } , { α 3 , α 4 } ) , ( { 1 , 2 , 3 , 5 , 6 } , { α 3 , α 4 } ) , ( { 2 , 3 , 4 , 5 , 6 } , { α 3 , α 4 } ) , ( { 1 , 2 , 3 , 4 , 5 , 6 } , { α 3 , α 4 } ) , ( { 1 , 2 , 3 , 5 , 6 } , { α 2 , α 3 , α 4 } ) , ( { 1 , 2 , 3 , 4 , 5 , 6 } , { α 2 , α 3 , α 4 } ) , G } be a graph primal. Let W = ( { 1 , 2 , 3 } , ) be a subgraph of the given graph. Then, ( Q ( W ) ) P , T A d = { 1 , 2 , 3 , 4 , 5 , 6 } = Q (Axioms 14 00662 i004). On the other hand, let W = ( { 2 , 3 , 5 , 6 } , { α 3 , α 4 } ) . By computing, ( Q ( W ) ) P , T A d = . It is clear that ( Q ( W ) ) Q ( ) and ( Q ( W ) ) Q ( ) .
Theorem 3.
Let ( Q (Axioms 14 00662 i004), T A d , P ) and ( Q (Axioms 14 00662 i004), T A d , J be two graph primal ATSs, and let , be two subgraphs of Axioms 14 00662 i004. Then, the graph-local function satisfies the following properties:
(1) 
Q ( ) Q ( ) ( Q ( ) ) ( Q ( ) ) .
(2) 
P J ( Q ( ) ) ( J ) ( Q ( ) ) ( P ) .
(3) 
( Q ( ) ) = CL ( ( Q ( ) ) ) .
(4) 
If Q ( ) c T A d , then ( Q ( ) ) Q ( ) .
(5) 
If Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , then ( Q ( ) ) = .
(6) 
( Q ( ) ) ( Q ( ) ) .
(7) 
Q ( ) Q ( ) = ( Q ( ) Q ( ) ) .
(8) 
( Q ( ) Q ( ) ) Q ( ) Q ( ) .
Proof. 
(1) Let ( Q ( ) ) . Then, there exists U N ( ) such that Q ( ) c U c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Since Q ( ) Q ( ) , Q ( ) c U c Q ( ) c U c . Therefore, there exists U N ( ) such that Q ( ) c U c = Q (Axioms 14 00662 i004′), for some Axioms 14 00662 i004′ ∉ P . So ( Q ( ) ) . Hence, ( Q ( ) ) ( Q ( ) ) .
(2)
Let ( Q ( ) ) ( J ) . Then, for every U N ( ) , Q ( ) c U c = Q (Axioms 14 00662 i004′) and Axioms 14 00662 i004′ ∈ J . Since P J , Q ( ) c U c = Q (Axioms 14 00662 i004′) and Axioms 14 00662 i004′ ∈ P . Thus, for all U N ( ) , Q ( ) U = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∈ P . Hence, ( Q ( ) ) ( J ) ( Q ( ) ) ( P ) .
(3)
We always have ( Q ( ) ) CL ( ( Q ( ) ) ) . Let CL ( ( Q ( ) ) ) and N T A . Then, ( Q ( ) ) N . Therefore, there exists ς Q (Axioms 14 00662 i004) such that ς N and ς ( Q ( ) ) . Then, N ς c Q ( ) c = Q (Axioms 14 00662 i004′) and Axioms 14 00662 i004′ ∈ P for all N ς T A . Therefore, N c Q ( ) c = Q (Axioms 14 00662 i004″) and Axioms 14 00662 i004″ ∈ P . As a result, ( Q ( ) ) . So, ( Q ( ) ) CL ( Q ( ) ) . Hence, ( Q ( ) ) is closed in G .
(4)
Let, Q ( ) c T A d and ( Q ( ) ) . Assume that Q ( ) . Then, Q ( ) c T A . Since ( Q ( ) ) , N c Q ( ) c = Q (Axioms 14 00662 i004′) and Axioms 14 00662 i004′ ∈ P for all N T A . Therefore, Q (Axioms 14 00662 i004) = Q ( ) Q ( ) c = ( Q ( ) c ) c Q ( ) c and G P , which is a contradiction with G P . Hence, ( Q ( ) ) Q ( ) .
(5)
Assume that ( Q ( ) ) . Then, N c Q ( ) c = Q (Axioms 14 00662 i004″) and Axioms 14 00662 i004″ ∈ P for all N T A . Since Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , N c Q ( ) c = Q (Axioms 14 00662 i004″) and Axioms 14 00662 i004″ ∉ P , which is a contradiction. Hence, ( Q ( ) ) = .
(6)
This is straightforward from (3) and (4).
(7)
According to (1), Q ( ) ( Q ( ) Q ( ) ) and Q ( ) ( Q ( ) Q ( ) ) . Hence, Q ( ) Q ( ) ( Q ( ) Q ( ) ) . Conversely, let Q ( ) Q ( ) . Then, Q ( ) and Q ( ) . Then, there exists U , T N x such that Q ( ) c U c = Q (Axioms 14 00662 i004′), for some Axioms 14 00662 i004′ ∉ P and Q ( ) c T c = Q (Axioms 14 00662 i004″), for some Axioms 14 00662 i004″ ∉ P . Put W = U T . Hence, W N x such that Q ( ) c W c = Q (Axioms 14 00662 i004‴), for some Axioms 14 00662 i004‴ ∉ P and Q ( ) c W c = Q (Axioms 14 00662 i004⁗), for some Axioms 14 00662 i004⁗ ∉ P . Therefore, ( Q ( ) Q ( ) ) c W c = ( Q ( ) c Q ( ) c ) W c = ( Q ( ) c W c ) ( Q ( ) c W c ) = Q (Axioms 14 00662 i004⁗′), for some Axioms 14 00662 i004⁗′ ∉ P , since P is a graph primal. It follows that, ( Q ( ) Q ( ) ) . Thus, ( Q ( ) Q ( ) ) Q ( ) Q ( ) . Hence, ( Q ( ) Q ( ) ) Q ( ) Q ( ) .
(8)
This is similar to (7).
Remark 3.
The equality in (8) of Theorem 3 need not be true in general, as shown by the following example.
Example 5.
Let Axioms 14 00662 i004 be the graph ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)), where Q (Axioms 14 00662 i004) = { 1 , 2 , 3 } and Z (Axioms 14 00662 i004) = { α 1 , α 2 } . A drawing of the graph Axioms 14 00662 i004 is shown in Figure 5.
From the previous results, it follows that S N = { { 2 } , { 1 , 3 } , { 2 } } . So, β = { { 2 } , { 1 , 3 } , } . Thus, T A = { , { 1 , 2 , 3 } , { 2 } , { 1 , 3 } } . Define a graph primal as P = { ( , ) , ( { 1 } , ) , ( { 2 } , ) , ( { 3 } , ) , ( { 1 , 2 } , ) , ( { 2 , 3 } , ) , ( { 1 , 3 } , ) , ( { 1 , 2 , 3 } , ) , ( { 1 , 2 } , { α 1 } ) , ( { 1 , 2 , 3 } , { α 1 } ) } . Now, if Q ( ) = { 2 } and Q ( ) = { 3 } , we have Q ( ) Q ( ) = Q (Axioms 14 00662 i004) ∩ Q (Axioms 14 00662 i004) = Q (Axioms 14 00662 i004) ≠ ∅ = = ( Q ( ) Q ( ) ) .
Theorem 4.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS and , Axioms 14 00662 i004. If Q ( ) is open in Q (Axioms 14 00662 i004), then Q ( ) Q ( ) ( Q ( ) Q ( ) ) .
Proof. 
Let Q ( ) T A d and Q ( ) Q ( ) . Then, Q ( ) and Q ( ) . Therefore, Q ( ) c U c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∈ P for all U T A d ( ) . Since Q ( ) T A d , we get ( Q ( ) Q ( ) ) c U c = Q ( ) c ( Q ( ) U ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∈ P for all U T A d ( ) . It follows that ( Q ( ) Q ( ) ) . Hence, Q ( ) Q ( ) ( Q ( ) Q ( ) ) .
Corollary 2.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS and , Axioms 14 00662 i004. If Q ( ) is open in Q (Axioms 14 00662 i004), then Q ( ) Q ( ) = Q ( ) ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) .
Proof. 
The proof is straightforward by using Theorem 4. □
Definition 8.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS. Define a map CL : 2 Q (Axioms 14 00662 i004) 2 Q (Axioms 14 00662 i004) as CL ( Q ( ) ) = Q ( ) Q ( ) where Q ( ) Q (Axioms 14 00662 i004).
Theorem 5.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS and , Axioms 14 00662 i004. Then, the following statements hold:
(i) 
CL ( ) = ;
(ii) 
CL ( Q (Axioms 14 00662 i004)) = Q (Axioms 14 00662 i004);
(iii) 
Q ( ) CL ( Q ( ) ) ;
(iv) 
If Q ( ) Q ( ) , then CL ( Q ( ) ) CL ( Q ( ) ) ;
(v) 
CL ( Q ( ) ) CL ( Q ( ) ) = CL ( Q ( ) Q ( ) ) ;
(vi) 
CL ( CL ( Q ( ) ) ) = CL ( Q ( ) ) .
Proof. 
Let Q ( ) , Q ( ) Q (Axioms 14 00662 i004)
(i)
Since ( ) = , we have CL ( ) = = .
(ii)
Since Q (Axioms 14 00662 i004) ∪ Q (Axioms 14 00662 i004) = Q (Axioms 14 00662 i004), we have CL ( Q (Axioms 14 00662 i004)) = Q (Axioms 14 00662 i004).
(iii)
Since CL ( Q ( ) ) = Q ( ) Q ( ) , we have Q ( ) CL ( Q ( ) ) .
(iv)
Let Q ( ) Q ( ) . According to (1) of Theorem 3, we have Q ( ) Q ( ) . Thus, we get Q ( ) Q ( ) Q ( ) Q ( ) , which means that CL ( Q ( ) ) CL ( Q ( ) ) .
(v)
This is obvious from the definition of the operator CL and (1) of Theorem 3.
(vi)
It is obvious from (iii) that CL ( Q ( ) ) CL ( CL ( Q ( ) ) ) . On the other hand, since Q ( ) is closed in Q (Axioms 14 00662 i004), we have ( Q ( ) ) Q ( ) . Therefore,
CL ( CL ( Q ( ) ) ) = CL ( Q ( ) ) CL ( Q ( ) ) = CL ( Q ( ) ) ( Q ( ) Q ( ) ) = CL ( Q ( ) ) Q ( ) ( Q ( ) ) CL ( Q ( ) ) Q ( ) Q ( ) = CL ( Q ( ) )
Hence, we have CL ( CL ( Q ( ) ) ) = CL ( Q ( ) ) .
Corollary 3.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS. Then, the function CL : 2 Q (Axioms 14 00662 i004) 2 Q (Axioms 14 00662 i004), denoted by CL ( Q ( ) ) = Q ( ) Q ( ) , where Axioms 14 00662 i004, is a Kuratowski’s closure operator.
Definition 9.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS. Then, the family T A d = { Q ( ) Q (Axioms 14 00662 i004)| CL ( Q ( ) c ) = Q ( ) c } is called the graph primal adjacency topology (graph primal topology, for short) generated by CL on Q (Axioms 14 00662 i004) induced by T A d and a graph primal P . We can write T P instead of T A d to specify the graph primal as per our requirements.
Example 6.
Let Axioms 14 00662 i004 be the graph ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)) where Q (Axioms 14 00662 i004) = { 1 , 2 , 3 , 4 } and Z (Axioms 14 00662 i004) = { α 1 , α 2 , α 3 , α 4 } . A drawing of the graph Axioms 14 00662 i004 is shown in Figure 6.
From the previous results, it follows that S N = { { 2 } , { 1 , 3 , 4 } , { 2 , 4 } , { 2 , 3 } } . So, β = { { 4 } , { 2 , 4 } , { 2 , 3 } , { 2 } , { 1 , 3 , 4 } ,   { 3 } , } . Thus, T A = { , { 1 , 3 , 4 } , { 2 } , { 3 } , { 4 } , { 2 , 4 } , { 3 , 4 } , { 2 , 3 } , { 2 , 3 , 4 } , { 1 , 2 , 3 , 4 } } . Consider the graph primal P as P = P(Axioms 14 00662 i004) − { ( { 1 , 2 , 3 } , { α 1 , α 2 } ) , ( { 1 , 2 , 3 , 4 , } , { α 1 , α 2 , α 3 } ) , ( { 1 , 2 , 3 , 4 } , { α 1 , α 2 , α 4 } ) , G } . From the T A - open sets, we have N ( 1 ) = { { 1 , 3 , 4 } , { 1 , 2 , 3 , 4 } } , N ( 2 ) = { { 2 } , { 2 , 3 } , { 2 , 4 } , { 2 , 3 , 4 } ,   { 1 , 2 , 3 , 4 } } , N ( 3 ) = { { 3 } , { 2 , 3 } , { 3 , 4 } , { 2 , 3 , 4 } , { 1 , 2 , 3 , 4 } } and N ( 4 ) = { { 4 } , { 3 , 4 } , { 2 , 4 } , { 2 , 3 , 4 } , { 1 , 2 , 3 , 4 } } . Simple calculations of the graph-local function associated with the defined graph primal are given in Table 2. According to Table 2, T A d = T A d .
Theorem 6.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS. Then, the graph primal topology T A d is finer than T A d .
Proof. 
Let Q ( ) T A d . Then, Q ( ) c is T A d -closed in Q (Axioms 14 00662 i004). According to (5) of Theorem 3, we get ( Q ( ) c ) Q ( ) c . So, CL ( Q ( ) c ) = Q ( ) c ( Q ( ) c ) Q ( ) c . Since Q ( ) c CL ( Q ( ) c ) is always satisfied for any subset Q ( ) of Q (Axioms 14 00662 i004), we have CL ( Q ( ) c ) = Q ( ) c . It follows that Q ( ) T A d . Hence, we have T A d T A d .
Theorem 7.
Let ( Q (Axioms 14 00662 i004), T A d , P , be a graph primal ATS. Then, the following statements hold:
(i) 
If P = , then T A d = 2 Q (Axioms 14 00662 i004);
(ii) 
If P = 2Axioms 14 00662 i004 ∖ {Axioms 14 00662 i004}, then T A d = T A d .
Proof. 
(i) We have always T A d 2 Q (Axioms 14 00662 i004). Now, let Q ( ) 2 Q (Axioms 14 00662 i004). Since P = , we have Q ( ) = for any subset Q ( ) of Q (Axioms 14 00662 i004). Therefore, CL ( Q ( ) c ) = Q ( ) c . This means that Q ( ) T A d . Hence, 2 Q (Axioms 14 00662 i004) T A d . Thus, we have T A d = 2 Q (Axioms 14 00662 i004).
(ii) We always have T A d T A d according to Theorem 6. Now, we prove that T A d T A d . Let Q ( ) T A d . Then, Q ( ) c ( Q ( ) c ) = Q ( ) c , which implies that ( Q ( ) c ) Q ( ) c . Now, let ( Q ( ) c ) . Then, there exists U T A d ( ) such that U c ( Q ( ) c ) c = U c Q ( ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Since P = 2Axioms 14 00662 i004 ∖ {Axioms 14 00662 i004}, we get U c Q ( ) = Q (Axioms 14 00662 i004) and so U Q ( ) c = . Thus, CL ( Q ( ) c ) . Therefore, we have CL ( Q ( ) c ) ( Q ( ) c ) Q ( ) c . Hence, CL ( Q ( ) c ) = Q ( ) c , which implies that Q ( ) c is T A d -closed, so Q ( ) T A d . As a result, T A d T A d . Consequently, we have T A d = T A d . □
Remark 4.
The following example shows that the converse of Theorem 7 need not be true in general.
Example 7.
Let Axioms 14 00662 i004 be the graph ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)), where Q (Axioms 14 00662 i004) = { 1 , 2 , 3 } and Z (Axioms 14 00662 i004) = { α 1 , α 2 , α 3 } . A drawing of the graph Axioms 14 00662 i004 is shown in Figure 7.
From the previous results, it follows that S N = { { 2 , 3 } , { 1 , 3 } , { 1 , 2 } } . So, β = { { 1 } , { 2 , 3 } , { 2 } , { 3 } , { 1 , 3 } , { 1 , 2 } , } . Thus, T A = { , { 1 , 2 , 3 } , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } } . Define a graph primal as P = { ( , ) , ( { 1 } , ) , ( { 2 } , ) , ( { 3 } , ) , ( { 1 , 2 } , ) , ( { 2 , 3 } , ) , ( { 1 , 3 } , ) , ( { 1 , 2 , 3 } , ) , ( { 1 , 2 } , { α 1 } ) , ( { 1 , 3 } , { α 2 } ) , ( { 2 , 3 } , { α 3 } ) , ( { 1 , 2 , 3 } , { α 2 } ) , ( { 1 , 2 , 3 } , { α 3 } ) , ( { 1 , 2 , 3 } , { α 2 , α 3 } ) } . Simple calculations of the graph-local function associated with the defined graph primal are given in Table 3. According to Table 3, T A d = T A d , but P is not equal to 2 Q (Axioms 14 00662 i004) ∖ { Q (Axioms 14 00662 i004)}.
Theorem 8.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS and Axioms 14 00662 i004. Then, the following statements hold:
(i) 
Q ( ) T A d iff for all in Q ( ) , there exists an T A -open set U containing such that U c Q ( ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P ,
(ii) 
If Q ( ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , then Q ( ) T A d .
Proof. 
(i) Let Q ( ) T A d .
Axioms 14 00662 i005
(ii) Let Q ( ) = Q (Axioms 14 00662 i004′), Axioms 14 00662 i004′ ∉ P and Q ( ) . Put U = Q (Axioms 14 00662 i004). Then, U is a T A d -open set containing . Since Q ( ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P and U c Q ( ) = Q ( ) , we have U c Q ( ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . From (i), we obtain Q ( ) T A d . □
Theorem 9.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS. Then, the family Q ( ) P = { T P | T T A d , P = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P } is a base for the graph primal topology T A d on Q (Axioms 14 00662 i004).
Proof. 
Let Q ( ) Q ( ) P . Then, there exists T T A d , P = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P such that Q ( ) = T P . Since T A d T A d , we get T T A d . On the other hand, according to Theorem 3 (5), we obtain P T A d . Therefore, Q ( ) T A d . Consequently, Q ( ) P T A d . Now, let Q ( ) T A d and Q ( ) . Then, using Theorem 8 (i), there exists U T A d ( ) such that U c Q ( ) = Q (Axioms 14 00662 i004″) for some Axioms 14 00662 i004″ ∉ P . Now, let Q ( ) = U ( U c Q ( ) ) . Hence, we have Q ( ) Q ( ) P such that Q ( ) Q ( ) . □
Theorem 10.
Let (Axioms 14 00662 i004, T A d , P ) and (Axioms 14 00662 i004, T A d , Q ) be two graph primal topological spaces. If P Q , then T A ( Q ) T A ( P ) .
Proof. 
Let Q ( ) T A ( Q ) . Then, Q ( ) c ( Q ( ) c ) Q = Q ( ) c , which means that ( Q ( ) c ) Q Q ( ) c . Now, let Q ( ) c . Then, we get ( Q ( ) c ) Q , and there exists U T A d ( ) such that U c ( Q ( ) c ) c = U c Q ( ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ Q . Since P Q , we have U c Q ( ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Thus, ( Q ( ) c ) P . Therefore, ( Q ( ) c ) P Q ( ) c , so CL ( Q ( ) c ) = Q ( ) c ( Q ( ) c ) P = Q ( ) c . Hence, Q ( ) T A ( P ) . As a result, we have T A ( Q ) T A ( P ) . □
Theorem 11.
Let f: Q (Axioms 14 00662 i004) → Q ( Y ) be a function and P ⊆ 2Axioms 14 00662 i004. If P is a graph primal on Axioms 14 00662 i004 and f is not surjective, then Q = { f ( P ) | P = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∈ P } is a graph primal on Y .
Proof. 
(1) Assume that Y Q . Then, there exists P = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∈ P such that f ( P ) = Q ( Y . However, this contradicts the fact that f is not surjective.
(2)
Let Q ( ) = Q ( Y ) and Y Q and Q ( ) Q ( ) . Then, there exists P 1 = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∈ P such that Q ( ) = f ( P 1 ) . Now, set P 2 = f 1 ( Q ( ) ) P 1 . It is obvious that P 2 P 1 . Since P is a graph primal on Q (Axioms 14 00662 i004), P is downward closed and we have P 2 = Q (Axioms 14 00662 i004″) for some Axioms 14 00662 i004″ ∈ P . Also, Q ( ) = f ( P 2 ) . This means that Q ( ) = Q ( Y ) and Y Q .
(3)
Let Q ( ) Q ( ) = Q ( Y ) and Y Q . Then, there exists P 1 = Q (Axioms 14 00662 i004′), Axioms 14 00662 i004′ ∈ P and P 2 = Q (Axioms 14 00662 i004″), Axioms 14 00662 i004″ ∈ P such that Q ( ) = f ( P 1 ) and Q ( ) = f ( P 2 ) . Since P is a graph primal on Axioms 14 00662 i004 and P 1 P 2 P 1 , we have P 1 P 2 = Q (Axioms 14 00662 i004‴) for some Axioms 14 00662 i004‴ ∈ P . Thus, Axioms 14 00662 i004′ ∈ P or Axioms 14 00662 i004″ ∈ P . Therefore, Q ( ) = f ( P 1 ) = Q ( Y ) , Y Q or Q ( ) = f ( P 2 ) = Q ( Y ) , Y Q .
Corollary 4.
Let f: Q (Axioms 14 00662 i004) → Q ( Y ) be a function and P ⊆ 2Axioms 14 00662 i004. If P is a graph primal on Axioms 14 00662 i004 and f is not surjective, then the property of being graph primal is not a topological property.
Proof. 
This is straightforward by Theorem 11. □
Remark 5.
For a function f: Q (Axioms 14 00662 i004) → Q ( Y ) and a graph primal Q on Y , the family P = { f 1 ( Q ) | Q = Q ( Y ) , Y Q } need not be a graph primal on Axioms 14 00662 i004, as illustrated by the example below.
Example 8.
Let Axioms 14 00662 i004 be the graph ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)), where Q (Axioms 14 00662 i004) = { 1 , 2 , 3 , 4 , 5 , 6 } and Z (Axioms 14 00662 i004) = { α 1 , α 2 , α 3 , α 4 , α 5 , α 6 } . Also, let Y be the graph ( Q ( Y ) , Z ( Y ) ) , where Q ( Y ) = { Z , R } and Z ( Y ) = { α } . Drawings of the two graphs Axioms 14 00662 i004 and Y are shown in Figure 8.
Define the function f: Q (Axioms 14 00662 i004) → Q ( Y ) by f ( i ) = Z , i { 1 , 2 , 3 , 4 , 5 , 6 } . Then, Q is a graph primal on Y , but P = { f 1 ( Q ) | Q = Q ( Y ) , Y Q } = {∅, Axioms 14 00662 i004} is not a graph primal on Axioms 14 00662 i004.
Lemma 1.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS and , be two subgraphs of Q (Axioms 14 00662 i004). Then, Q ( ) Q ( ) = ( Q ( ) Q ( ) ) Q ( ) .
Proof. 
According to Theorem 3, we have Q ( ) = [ ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) ] = ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) Q ( ) . Thus, Q ( ) Q ( ) ( Q ( ) Q ( ) ) Q ( ) . Again, according to Theorem 3, ( Q ( ) Q ( ) ) Q ( ) and ( Q ( ) Q ( ) ) Q ( ) Q ( ) Q ( ) . Hence, Q ( ) Q ( ) = ( Q ( ) Q ( ) ) Q ( ) . □
Corollary 5.
Let ( Q (Axioms 14 00662 i004), T A d , P , be a graph primal ATS and , be two subgraphs of Axioms 14 00662 i004 with Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Then, ( Q ( ) Q ( ) ) = Q ( ) = ( Q ( ) Q ( ) ) .
Proof. 
Since Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , it follows that Q ( ) = . Again, according to Lemma 1, Q ( ) = ( Q ( ) Q ( ) ) , and according to Theorem 3, ( Q ( ) Q ( ) ) = Q ( ) Q ( ) = Q ( ) . □
Theorem 12.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS. If Q ( F ) = Q (Axioms 14 00662 i004) for some Axioms 14 00662 i004 P for all closed subsets Q ( F ) , F Axioms 14 00662 i004, then Q ( U ) Q ( U ) for all Q ( U ) T A d , U Axioms 14 00662 i004.
Proof. 
If Q ( U ) = , then Q ( U ) = = Q ( U ) . Now, if, Q ( F ) = Q (Axioms 14 00662 i004) for some Axioms 14 00662 i004 P for all closed subsets Q ( F ) , F Axioms 14 00662 i004, then Q (Axioms 14 00662 i004) = Q (Axioms 14 00662 i004). In fact, Q (Axioms 14 00662 i004). Then, there exists Q ( T ) T A d ( ) such that Q (Axioms 14 00662 i004)c Q ( T ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Hence, Q ( T ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , which is a contradiction. Now, according to Theorem 4, we have for all Q ( U ) T A d , Q ( U ) = Q ( U ) Q (Axioms 14 00662 i004) ( Q ( U ) Q ( Z ) ) = Q ( U ) . Hence, Q ( U ) Q ( U ) . □
Definition 10.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS. An operator Θ : 2 Q (Axioms 14 00662 i004) 2 Q (Axioms 14 00662 i004) defined by Θ ( Q ( ) ) = { Q (Axioms 14 00662 i004): ( U T A d ( ) ) ( ( U Q ( ) ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P )} for every Q ( ) Q (Axioms 14 00662 i004.
Example 9.
Let Axioms 14 00662 i004 be the graph ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)), where Q (Axioms 14 00662 i004) = { 1 , 2 , 3 , 4 } and Z (Axioms 14 00662 i004) = { α 1 , α 2 , α 3 , α 4 } . A drawing of the graph Axioms 14 00662 i004 is shown in Figure 9.
From the previous results, it follows that S N = { { 2 , 4 } , { 1 , 3 , 4 } , { 2 } , { 1 , 2 } } . So, β = { { 1 } , { 2 } , { 4 } , { 1 , 2 } , { 2 , 4 } ,   { 1 , 3 , 4 } , } . Thus, T A = { , { 1 , 3 , 4 } , { 1 } , { 2 } , { 4 } , { 1 , 2 } , { 1 , 4 } , { 2 , 4 } , { 1 , 2 , 4 } ,   { 1 , 2 , 3 , 4 } } . Define a graph primal as follows: P = P(Axioms 14 00662 i004) − { ( { 1 , 2 , 3 } , { α 1 , α 3 } ) , ( { 1 , 2 , 3 , 4 } , { α 1 , α 2 , α 3 } ) , ( { 1 , 2 , 3 , 4 } , { α 1 , α 3 , α 4 } ) , G } . According to the open sets of T A , we have the following: N ( 1 ) = { { 1 } , { 1 , 2 } , { 1 , 4 } , { 1 , 2 , 4 } , { 1 , 3 , 4 } { 1 , 2 , 3 , 4 } } , N ( 2 ) = { { 2 } , { 2 , 4 } , { 1 , 2 } , { 1 , 2 , 4 } , { 1 , 2 , 3 , 4 } } , N ( 3 ) = { { 1 , 3 , 4 } , { 1 , 2 , 3 , 4 } } and N ( 4 ) = { { 4 } , { 1 , 4 } , { 2 , 4 } , { 1 , 2 , 4 } , { 1 , 3 , 4 } , { 1 , 2 , 3 , 4 } } . The computations of the graph-local functions Q ( ) and Θ ( Q ( ) ) of a graph Axioms 14 00662 i004 associated with the defined graph primal are given in Table 4. According to Table 4, Θ ( Q ( ) ) Q ( ) and Θ ( Q ( ) ) Q ( ) .
Theorem 13.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS and , be two subgraphs of Axioms 14 00662 i004. Then, the following properties hold:
(1) 
If Q ( ) Q (Axioms 14 00662 i004), then Θ ( Q ( ) ) = Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − Q ( ) );
(2) 
If Q ( ) Q (Axioms 14 00662 i004), then Θ ( Q ( ) ) is open;
(3) 
If Q ( ) Q ( ) , then Θ ( Q ( ) ) Θ ( Q ( ) ) ;
(4) 
If Q ( ) , Q ( ) Q (Axioms 14 00662 i004), then Θ ( Q ( ) Q ( ) ) = Θ ( Q ( ) ) Θ ( Q ( ) ) ;
(5) 
If U T A d , then U Θ ( U ) ;
(6) 
If Q ( ) Q (Axioms 14 00662 i004), then Θ ( Q ( ) ) Θ ( Θ ( Q ( ) ) ) ;
(7) 
If Q ( ) Q (Axioms 14 00662 i004), then Θ ( Q ( ) ) = Θ ( Θ ( Q ( ) ) ) iff ( Q (Axioms 14 00662 i004) − Q ( ) ) = (( Q (Axioms 14 00662 i004) − Q ( ) ));
(8) 
If Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , then Θ ( Q ( ) ) = Q (Axioms 14 00662 i004) − Q (Axioms 14 00662 i004);
(9) 
If Q ( ) Q (Axioms 14 00662 i004), then Q ( ) Θ ( Q ( ) ) = int ( Q ( ) ) ;
(10) 
If Q ( ) Q (Axioms 14 00662 i004) and I c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , then Θ ( Q ( ) I ) = Θ ( Q ( ) ) ;
(11) 
If Q ( ) Q (Axioms 14 00662 i004) and I c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , then Θ ( Q ( ) I ) = Θ ( Q ( ) ) ;
(12) 
If [ ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) ] c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , then Θ ( Q ( ) ) = Θ ( Q ( ) ) .
Proof. 
(1) Let Θ ( Q ( ) ) . Then there exists U T A d ( ) such that U c Q ( ) = ( U ∩ ( Q (Axioms 14 00662 i004) − Q ( ) ))c = ( U Q ( ) ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Then, ∉ ( Q (Axioms 14 00662 i004) − Q ( ) ) and Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − Q ( ) ). Conversely, let Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − Q ( ) ). Then, ∉ ( Q (Axioms 14 00662 i004) − Q ( ) ), and there exists U T A d ( ) such that U c ∪ ( Q (Axioms 14 00662 i004) − Q ( ) )c = ( U Q ( ) ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Hence, Θ ( Q ( ) ) and Θ ( Q ( ) ) = Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − Q ( ) ).
(2)
This is straightforward from (3) of Theorem 3.
(3)
This is straightforward from (1) of Theorem 3.
(4)
It is straightforward from (3) that Θ ( Q ( ) Q ( ) ) Θ ( Q ( ) ) and Θ ( Q ( ) Q ( ) ) Θ ( Q ( ) ) . Hence, Θ ( Q ( ) Q ( ) ) Θ ( Q ( ) ) Θ ( Q ( ) ) . Now, let Θ ( Q ( ) ) Θ ( Q ( ) ) . Then, there exists U , T T A d ( ) such that ( U Q ( ) ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P and ( T Q ( ) ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Let G = U T T A d ( ) . Then, we have ( G Q ( ) ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P and ( G Q ( ) ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P by heredity. Thus, [ G ( Q ( ) Q ( ) ) ] c = ( G Q ( ) ) c ( G Q ( ) ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P according to Corollary 1; hence, Θ ( Q ( ) Q ( ) ) . We have shown that Θ ( Q ( ) ) Θ ( Q ( ) ) Θ ( Q ( ) Q ( ) ) , so the proof is completed.
(5)
If U T A d , then ( Q (Axioms 14 00662 i004) − U ) Q (Axioms 14 00662 i004) − U . Hence, U Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − U ) = Θ ( U ) .
(6)
This is straightforward from (2) and (5).
(7)
This follows from the following facts:
(a)
Θ ( Q ( ) ) = Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − Q ( ) ).
(b)
Θ ( Θ ( Q ( ) ) ) = Q (Axioms 14 00662 i004) − [ Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − Q ( ) ))] = Q (Axioms 14 00662 i004) − (( Q (Axioms 14 00662 i004) − Q ( ) )).
(8)
By Corollary 5, it follows that ( Q (Axioms 14 00662 i004) − Q ( ) ) = Q (Axioms 14 00662 i004) if Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Then, Θ ( Q ( ) ) = Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − Q ( ) ) = Q (Axioms 14 00662 i004) − Q (Axioms 14 00662 i004).
(9)
If Q ( ) Θ ( Q ( ) ) , then Q ( ) , and there exists U T A d ( ) such that ( U Q ( ) ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Then, according to Theorem 9, U ( U Q ( ) ) c is a T A d -open neighborhood of and int ( Q ( ) ) . On the other hand, if int ( Q ( ) ) , there exists a basic T A d -open neighborhood V I of , where V T A d and I = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P such that V I Q ( ) , which implies that I ( V Q ( ) ) c , so ( V Q ( ) ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Hence, Q ( ) Θ ( Q ( ) ) .
(10)
This follows from Corollary 5 and Θ ( Q ( ) I ) = Q (Axioms 14 00662 i004) − [ Q (Axioms 14 00662 i004) − ( Q ( ) I ) ] =
Q (Axioms 14 00662 i004) − [( Q (Axioms 14 00662 i004) − Q ( ) ) I ] = Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − Q ( ) ) = Θ ( Q ( ) ) .
(11)
This follows from Corollary 5 and Θ ( Q ( ) I ) = Q (Axioms 14 00662 i004) − [ Q (Axioms 14 00662 i004) − ( Q ( ) I )] =
Q (Axioms 14 00662 i004) − [( Q (Axioms 14 00662 i004) − Q ( ) ) I ] = Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − Q ( ) ) = Θ ( Q ( ) ) .
(12)
Assume that [ ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) ] c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Let Q ( ) Q ( ) = I and
Q ( ) Q ( ) = J . Observe that I c , J c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P by heredity. Also, observe that
Q ( ) = ( Q ( ) I ) J . Thus, Θ ( Q ( ) ) = Θ ( Q ( ) I ) = Θ [ ( Q ( ) I ) J ] = Θ ( Q ( ) ) by (10) and (11).
Corollary 6.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS. Then, U Θ ( U ) for every open set U T A d .
Proof. 
We know that Θ ( U ) = Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − U ). Now, ( Q (Axioms 14 00662 i004) − U ) CL ( Q (Axioms 14 00662 i004) − U ) = Q (Axioms 14 00662 i004) − U since Q (Axioms 14 00662 i004) − U is closed. As a result, U = Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − U ) ⊆ Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − U ) = Θ ( U ) . □
Theorem 14.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS and Q ( ) Q (Axioms 14 00662 i004). Then, the following properties hold:
(1) 
Θ ( Q ( ) ) = ⋃{ U T A d : ( U Q ( ) ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P };
(2) 
Θ ( Q ( ) ) ⊇ ⋃{ U T A d : ( U Q ( ) ) c ( Q ( ) U ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P }.
Proof. 
(1) This is straightforward from the definition of Θ -operator.
(2)
Since P is hereditary, ⋃{ U T A d : ( U Q ( ) ) c ( Q ( ) U ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P } ⊆ ⋃{ U T A d : ( U Q ( ) ) c = Q (Axioms 14 00662 i004″) for some Axioms 14 00662 i004″ ∉ P } = Θ ( Q ( ) ) , for every Q ( ) Q (Axioms 14 00662 i004).
Theorem 15.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS. If σ = { Q ( ) Q (Axioms 14 00662 i004): Q ( ) Θ ( Q ( ) ) }. Then, σ is a topology on Q (Axioms 14 00662 i004) and σ = T A d .
Proof. 
Let σ = { Q ( ) Q (Axioms 14 00662 i004): Q ( ) Θ ( Q ( ) ) }. First, we show that σ is a topology. Note that Θ ( ) and Q (Axioms 14 00662 i004) ⊆ Θ( Q (Axioms 14 00662 i004)) = Q (Axioms 14 00662 i004). Thus, ∅ and Q (Axioms 14 00662 i004) ∈ σ . If Q ( ) , Q ( ) σ , then Q ( ) Q ( ) Θ ( Q ( ) ) Θ ( Q ( ) ) = Θ ( Q ( ) Q ( ) ) . As a result, Q ( ) Q ( ) σ . If { A α : α Δ } σ . Then, A α Θ ( A α ) Θ ( α Δ A α ) for every α Δ , so A α Θ ( α Δ A α ) . Hence, σ is a topology on Q (Axioms 14 00662 i004). Now, if U T A d and U , then according to Theorem 9, there exists T T A d ( ) and I = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P such that T I U . Clearly, I ( T U ) c , so ( T U ) c = Q (Axioms 14 00662 i004″) for some Axioms 14 00662 i004″ ∉ P by heredity, and hence Θ ( U ) . Thus, U Θ ( U ) and T A d σ . Now let Q ( ) σ . Then, we have Q ( ) Θ ( Q ( ) ) , i.e., Q ( ) Q (Axioms 14 00662 i004) − ( Q (Axioms 14 00662 i004) − Q ( ) ) and ( Q (Axioms 14 00662 i004) − Q ( ) ) Q (Axioms 14 00662 i004) − Q ( ) . Therefore, Q (Axioms 14 00662 i004) − Q ( ) is T A d -closed and hence Q ( ) T A d . So, σ T A d , and thus σ = T A d . □

4. Suitability of T A d with P

This section presents the definition of the suitability of the topological space T A d with the graph primal P . This definition establishes a foundational criterion for evaluating how well the topology interacts with the graph primal structure, indicating the effective linkages and properties between the two notions.
Definition 11.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS. Then, T A d is said to be suitable for the graph primal P if Q ( ) c Q ( ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , for any subgraph Axioms 14 00662 i004.
Example 10.
Let Axioms 14 00662 i004 be the graph ( Q (Axioms 14 00662 i004), Z (Axioms 14 00662 i004)), where Q (Axioms 14 00662 i004) = { 1 , 2 , 3 , 4 } and Z (Axioms 14 00662 i004) = { α 1 , α 2 , α 3 , α 4 } . A drawing of the graph Axioms 14 00662 i004 is shown in Figure 10.
By computing S N = { { 2 , 3 } , { 1 , 3 } , { 1 , 2 , 4 } , { 3 } } . So, β = { { 3 } , { 2 , 3 } , { 1 , 3 } , { 2 } , { 1 } , { 1 , 2 , 4 } , { 3 } , } . Thus, T A = { , { 1 , 2 , 4 } , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 } ,   { 1 , 2 , 3 , 4 } } . Define a graph primal as P = P(Axioms 14 00662 i004) − { ( { 1 , 2 , 3 } , { α 1 , α 4 } ) , ( { 1 , 2 , 3 , 4 , } , { α 1 , α 2 , α 4 } ) , ( { 1 , 2 , 3 , 4 } , { α 1 , α 3 , α 4 } ) , G } . According to the open sets of T A , we have N ( 1 ) = { { 1 } , { 1 , 2 } , { 1 , 3 } , { 1 , 2 , 4 } , { 1 , 2 , 3 } ,   { 1 , 2 , 3 , 4 } } , N ( 2 ) = { { 2 } , { 2 , 3 } , { 1 , 2 } , { 1 , 2 , 3 } , { 1 , 2 , 4 } , { 1 , 2 , 3 , 4 } } ,   N ( 3 ) = { { 3 } , { 2 , 3 } , { 1 , 3 } , { 1 , 2 , 3 } , { 1 , 2 , 3 , 4 } } and N ( 4 ) = { { 1 , 2 , 4 } , { 1 , 2 , 3 , 4 } } . The computations of the graph-local function associated with the defined graph primal are given in Table 5. According to Table 5, T A d is suitable for the graph primal P . Clearly, if P = P (Axioms 14 00662 i004) − {Axioms 14 00662 i004}, then T A d is not suitable for the graph primal P .
We now describe this notion in some analogous terms.
Theorem 16.
For a graph primal ATS ( Q (Axioms 14 00662 i004), T A d , P ), the following are equivalent:
(1) 
T A d is suitable for the graph primal P ;
(2) 
For any T A d -closed subset Q ( ) of Q (Axioms 14 00662 i004), Q ( ) c Q ( ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P ;
(3) 
Whenever for any Q ( ) Q (Axioms 14 00662 i004) and each Q ( ) there corresponds some U x T A d ( ) with U c Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , it implies that Q ( ) c = Q (Axioms 14 00662 i004″) for some Axioms 14 00662 i004″ ∉ P ;
(4) 
For Q ( ) Q (Axioms 14 00662 i004) and Q ( ) Q ( ) = , it follows that Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P .
Proof. 
(1) ⇒ (2): This is straightforward.
(2)
⇒ (3): Let Q ( ) Q (Axioms 14 00662 i004), and suppose that for every Q ( ) , there exists U T A d ( ) such that U c Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Then, Q ( ) so that Q ( ) Q ( ) = . Since Q ( ) Q ( ) is T A d -closed, by (2), we have ( Q ( ) Q ( ) ) c ( Q ( ) Q ( ) ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , i.e., ( Q ( ) Q ( ) ) c ( Q ( ) ( Q ( ) ) ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P according to Theorem 3, i.e., ( Q ( ) Q ( ) ) c Q ( ) = Q (Axioms 14 00662 i004) for some Axioms 14 00662 i004 P according to Theorem 3, i.e., Q ( ) c = Q (Axioms 14 00662 i004″) for some Axioms 14 00662 i004″ ∉ P (as Q ( ) Q ( ) = ).
(3)
⇒ (4): If Q ( ) Q (Axioms 14 00662 i004) and Q ( ) Q ( ) = , then Q ( ) Q (Axioms 14 00662 i004) ∖ Q ( ) . Let Q ( ) . Then, Q ( ) . So, there exists U T A d ( ) such that U c Q ( ) c = Q (Axioms 14 00662 i004) for some Axioms 14 00662 i004 P . Then, by (3), Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P .
(4)
⇒ (1): Let Q ( ) Q (Axioms 14 00662 i004). We first claim that ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) = . In fact, if ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) , then Q ( ) Q ( ) . Thus, Q ( ) and Q ( ) . Then, there exists U T A d ( ) such that U c Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Now, U c Q ( ) c U c ( Q ( ) Q ( ) ) c according to Corollary 1 (2), U c ( Q ( ) Q ( ) ) c = Q (Axioms 14 00662 i004″) for some Axioms 14 00662 i004″ ∉ P . Hence, ( Q ( ) Q ( ) ) , which is a contradiction. Hence, by (4), ( Q ( ) Q ( ) ) c = Q ( ) c Q ( ) = Q (Axioms 14 00662 i004‴) for some Axioms 14 00662 i004‴ ∉ P and T A d is suitable for the graph primal P .
Theorem 17.
For a graph primal ATS ( Q (Axioms 14 00662 i004), T A d , P ), the following conditions are equivalent, and any of them is necessary for T A d to be suitable for the graph primal P :
(1) 
For any Q ( ) Q (Axioms 14 00662 i004), Q ( ) Q ( ) = , then Q ( ) = ;
(2) 
For any Q ( ) Q (Axioms 14 00662 i004), ( Q ( ) Q ( ) ) = ;
(3) 
For any Q ( ) Q (Axioms 14 00662 i004), ( Q ( ) Q ( ) ) = Q ( ) .
Proof. 
(1) ⇒ (2): It follows by noting that ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) = , for all Q ( ) Q (Axioms 14 00662 i004).
(2)
⇒ (3): Since Q ( ) = ( Q ( ) ( Q ( ) Q ( ) ) ) ( Q ( ) Q ( ) ) , we have Q ( ) = ( Q ( ) ( Q ( ) Q ( ) ) ) ( Q ( ) Q ( ) ) = ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) = ( Q ( ) Q ( ) ) by (2).
(3)
⇒ (1): Let Q ( ) Q (Axioms 14 00662 i004) and Q ( ) Q ( ) = . Then, by (3), Q ( ) = ( Q ( ) Q ( ) ) = = .
Corollary 7.
If ( Q (Axioms 14 00662 i004), T A d , P ) is a graph primal ATS such that T A d is suitable for P , then the operator • is an idempotent operator, i.e., Q ( ) = ( Q ( ) ) for any subgraph Axioms 14 00662 i004.
Proof. 
By Theorem 3 (6), we have ( Q ( ) ) Q ( ) . By Theorem 17 and Theorem 3 (1), we get Q ( ) = ( Q ( ) Q ( ) ) ( Q ( ) ) .
Theorem 18.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS such that T A d is suitable for P . Then, a subset Q ( ) of Q (Axioms 14 00662 i004) is T A d -closed iff it can be expressed as a union of a set Q ( ) , Axioms 14 00662 i004, which is closed in ( Q (Axioms 14 00662 i004), T A d ), and Q ( ) c = Q (Axioms 14 00662 i004′), Axioms 14 00662 i004′ ∉ P .
Proof. 
Let Q ( ) be a T A d -closed subset of Q (Axioms 14 00662 i004). Then, Q ( ) Q ( ) . Now, Q ( ) = Q ( ) ( Q ( ) Q ( ) ) . Since T A d is suitable for P , according to Theorem 16, ( Q ( ) Q ( ) ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , and according to Theorem 3 (3), Q ( ) is closed.
Conversely, let Q ( ) = F Q ( ) , where F is closed and Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Then, Q ( ) = ( F Q ( ) ) = F according to Corollary 5; hence, according to Theorem 3 (3), Q ( ) = ( F Q ( ) ) = F = CL ( F ) = F Q ( ) . Hence, Q ( ) is T A d -closed. □
Corollary 8.
Let the topology T A d on a space Q (Axioms 14 00662 i004) be suitable for a graph primal P on a graph Axioms 14 00662 i004. Then, Q ( ) P = { U P : ( U T A d ) (P = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P )} is a topology on Q (Axioms 14 00662 i004); hence, Q ( ) P = T A d
Proof. 
Let U T A d . Then, according to Theorem 18, Q (Axioms 14 00662 i004) ∖ U = F Q ( ) , where F is closed and Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Then, U = Q (Axioms 14 00662 i004) ∖ ( F Q ( ) ) = ( Q (Axioms 14 00662 i004) ∖ F ) ∩ ( Q (Axioms 14 00662 i004) ∖ Q ( ) ) = T P , where T = F c T A d and P = Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Thus, every T A d -open set is of the form T P , where T T A d and P = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . The rest follows from Theorem 9. □
Theorem 19.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS and ℶ be any subgraph of Axioms 14 00662 i004 such that Q ( ) Q ( ) . Then, CL ( Q ( ) ) = CL ( Q ( ) ) = CL ( Q ( ) ) = Q ( ) .
Proof. 
Since T A d is finer than T A d , then CL ( Q ( ) ) CL ( Q ( ) ) for any subset Q ( ) of Q (Axioms 14 00662 i004). Now, CL ( Q ( ) ) , and there exists T T A d and Q ( ) P such that T Q ( ) and ( T Q ( ) ) Q ( ) = , and then [ ( T Q ( ) ) Q ( ) ] = . Thus, [ ( T Q ( ) ) Q ( ) c ] = , and according to Corollary 5, we have ( T Q ( ) ) = . By Theorem 4, we get T ( Q ( ) ) = and T Q ( ) = (as Q ( ) Q ( ) ), and then CL ( Q ( ) ) . Thus, CL ( Q ( ) ) = CL ( Q ( ) ) . Now, according to Theorem 3 (3), Q ( ) = CL ( Q ( ) ) . Now, let CL ( Q ( ) ) . Then, there exists U T A d ( ) such that U Q ( ) = . Thus, ( U Q ( ) ) c = U c Q ( ) c = Q (Axioms 14 00662 i004) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . So, Q ( ) and hence Q ( ) CL ( Q ( ) ) . Again, since Q ( ) CL ( Q ( ) ) , we have CL ( Q ( ) ) CL ( CL ( Q ( ) ) ) = CL ( Q ( ) ) . Also, Q ( ) Q ( ) , and then CL ( Q ( ) ) CL ( Q ( ) ) . Thus, CL ( Q ( ) ) = CL ( Q ( ) ) = Q ( ) . □
Theorem 20.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS such that T A d is suitable for P with Q ( F ) = Q (Axioms 14 00662 i004) for some Axioms 14 00662 i004 P , for all closed subsets Q ( F ) , F Axioms 14 00662 i004. For G Axioms 14 00662 i004, if Q ( G ) is a T A d -open set such that Q ( G ) = U Q ( ) , where U T A d and Q ( ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , then CL ( Q ( G ) ) = CL ( Q ( G ) ) = Q ( G ) = U = CL ( U ) = CL ( U ) .
Proof. 
Let Q ( G ) = U Q ( ) , where U T A d and Q ( ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P (according to Corollary 8, every T A d -open set Q ( G ) is of this form). Since Q ( F ) = Q (Axioms 14 00662 i004) for some Axioms 14 00662 i004 P for all closed subsets Q ( F ) , F Axioms 14 00662 i004, according to Theorem 12, we have U U . Hence, according to Theorem 19, we get U = CL ( U ) = CL ( U ) .
Now, let Q ( G ) be T A d -open. We claim that Q ( G ) Q ( G ) . In fact, CL ( Q (Axioms 14 00662 i004) ∖ Q ( G ) ) = Q (Axioms 14 00662 i004) ∖ Q ( G ) . Then, ( Q (Axioms 14 00662 i004) ∖ Q ( G ) ) = Q (Axioms 14 00662 i004) ∖ Q ( G ) and Q (Axioms 14 00662 i004) Q ( G ) = Q (Axioms 14 00662 i004) ∖ Q ( G ) according to Lemma 1, Furthermore, according to Theorem 12, we have Q (Axioms 14 00662 i004) ∖ Q ( G ) = Q (Axioms 14 00662 i004) ∖ Q ( G ) , Q ( G ) Q ( G ) . Hence, according to Theorem 19, Q ( G ) = CL ( Q ( G ) ) = CL ( Q ( G ) ) .
Again, Q ( G ) U , so Q ( G ) U , and Q ( G ) = ( U Q ( ) ) = ( U Q ( ) c ) U ( Q ( ) c ) = U according to Lemma 1 and Theorem 3 (5) as Q ( ) = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P . Thus, U = Q ( G ) . Therefore, we have CL ( Q ( G ) ) = CL ( Q ( G ) ) = Q ( G ) = U = CL ( U ) = CL ( U ) . □
Theorem 21.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS such that T A d is suitable for P . Then, for every Q ( ) T A d and any Axioms 14 00662 i004, ( Q ( ) Q ( ) ) = ( Q ( ) Q ( ) ) = CL ( Q ( ) Q ( ) ) .
Proof. 
Let Q ( ) T A d . Then, according to Corollary 2, Q ( ) Q ( ) = Q ( ) ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) ; hence, ( Q ( ) Q ( ) ) [ ( Q ( ) Q ( ) ) ] = ( Q ( ) Q ( ) ) according to Corollary 7.
Now, by using Corollary 2 and Theorem 17, we obtain [ Q ( ) ( Q ( ) Q ( ) ) ] = Q ( ) ( Q ( ) Q ( ) ) = Q ( ) = .
Also, ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) [ ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) ] = [ Q ( ) ( Q ( ) Q ( ) ) ] = according to Lemma 1. Hence, ( Q ( ) Q ( ) ) ( Q ( ) Q ( ) ) , and we get ( Q ( ) Q ( ) ) = ( Q ( ) Q ( ) ) .
Again, ( Q ( ) Q ( ) ) = ( Q ( ) Q ( ) ) CL ( Q ( ) Q ( ) ) since T A d is finer than T A d . Due to Q ( ) Q ( ) ( Q ( ) Q ( ) ) , we have CL ( Q ( ) Q ( ) ) CL ( ( Q ( ) Q ( ) ) ) = ( Q ( ) Q ( ) ) . Hence, ( Q ( ) Q ( ) ) = CL ( Q ( ) Q ( ) ) . □
Corollary 9.
Let ( Q (Axioms 14 00662 i004), T A d , P ) be a graph primal ATS such that T A d is suitable for P . If Q ( ) T A d and Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , then Q ( ) Q (Axioms 14 00662 i004) ∖ Q (Axioms 14 00662 i004).
Proof. 
Taking Q ( ) = Q (Axioms 14 00662 i004) in Theorem 21, we get ( Q ( ) Q (Axioms 14 00662 i004)) = CL ( Q ( ) Q (Axioms 14 00662 i004)). Thus, Q ( ) = CL ( Q ( ) Q (Axioms 14 00662 i004)), for all Q ( ) T A d . Now, if Q ( ) c = Q (Axioms 14 00662 i004′) for some Axioms 14 00662 i004′ ∉ P , then Q ( ) = . Thus, ( Q ( ) Q (Axioms 14 00662 i004) = CL ( Q ( ) Q (Axioms 14 00662 i004)) = ∅. So, Q ( ) Q (Axioms 14 00662 i004) = ∅ according to Theorem 4; hence, Q ( ) Q (Axioms 14 00662 i004) ∖ Q (Axioms 14 00662 i004). □

5. Conclusions

Graph theory and general topology are two dominant areas in discrete mathematics. Graphs can abstractly represent many concepts, making them useful in real-world applications. We have made a new contribution to the field of graph theory by introducing the notion of the “graph primal”, which is the dual of the graph grill. We have studied some basic operations on graph primals. Additionally, in a graph ATS, we present the novel graph-local function ( . ) . We also examine the fundamentals of the proposed graph-local function and explain how to use ( . ) to create a unique graph ATS T A d by defining the new operator CL from the older operator via graph primals. The operator Θ and its corresponding topology σ are also proposed. A number of fundamental properties and relationships of the novel topologies σ and T A d in the graph primal ATSs were investigated with the aid of many counterexamples. We conclude this paper by introducing the notion of topology suitable for a graph primal and obtain its fundamental properties. This is a crucial discovery, as it suggests that these new structures can provide more detailed insights into the nature of graphs and their applications.
This research’s findings are preliminary, and by examining further graph primal ATS features, such as graph primal soft-limit points, separation axioms, compactness, and connectedness, future work may provide additional insights. The outcomes of directed graphs with loops will also be examined. On the other hand, the promising aim of applying the graph primal is to create generalized rough approximation spaces that enhance the accuracy of lower and upper approximations. This methodology is particularly applicable to decision-making problems, where precise approximations are critical for effective outcomes. By utilizing graph theory concepts, the approach seeks to improve the overall decision-making process through refined analytical techniques. It will be interesting work in the future.

Author Contributions

Resources, Methodology, and Funding, D.S.; Validation and Formal analysis, S.E.A. and D.S.; Review and Investigation (final version), S.E.A. and I.I.; Writing—original draft, H.M.K.; Visualization, I.I. and S.E.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Nomenclature

SymbolDescription
Axioms 14 00662 i004Connected simple graph
Q (Axioms 14 00662 i004)Set of vertices (nodes) over Axioms 14 00662 i004
Z (Axioms 14 00662 i004)Set of edges over Axioms 14 00662 i004
P(Axioms 14 00662 i004) or 2Axioms 14 00662 i004Power set of Axioms 14 00662 i004
, Two subgraphs of Axioms 14 00662 i004
Vertex (node) of Axioms 14 00662 i004
α Edge of Axioms 14 00662 i004
N Neighborhood set of
S N Subbase for a topology on Q (Axioms 14 00662 i004)
T A d Topology generated by S N
CL ( Q ( ) and int ( Q ( ) Closure and interior with respect to T A d , respectively
S Graph grill
P Graph primal
( Q ( ) ) Graph-local function of Q ( ) with respect to P and T A d
T A d Topology generated by ( . )
CL ( Q ( ) and int ( Q ( ) Closure and interior with respect to T A d , respectively
Θ ( Q ( ) ) Another graph-local function of Q ( ) with respect to P and T A d
σ Topology generated by Θ ( . )

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Figure 1. Graph defined in Example 1.
Figure 1. Graph defined in Example 1.
Axioms 14 00662 g001
Figure 2. Graph defined in Example 2.
Figure 2. Graph defined in Example 2.
Axioms 14 00662 g002
Figure 3. Graph defined in Example 3.
Figure 3. Graph defined in Example 3.
Axioms 14 00662 g003
Figure 4. Graph defined in Example 4.
Figure 4. Graph defined in Example 4.
Axioms 14 00662 g004
Figure 5. Graph defined in Example 5.
Figure 5. Graph defined in Example 5.
Axioms 14 00662 g005
Figure 6. Graph defined in Example 6.
Figure 6. Graph defined in Example 6.
Axioms 14 00662 g006
Figure 7. Graph defined in Example 7.
Figure 7. Graph defined in Example 7.
Axioms 14 00662 g007
Figure 8. Graphs defined in Example 8.
Figure 8. Graphs defined in Example 8.
Axioms 14 00662 g008
Figure 9. Graph defined in Example 9.
Figure 9. Graph defined in Example 9.
Axioms 14 00662 g009
Figure 10. Graph defined in Example 10.
Figure 10. Graph defined in Example 10.
Axioms 14 00662 g010
Table 1. Representations of the tree graph primals in Example 3.
Table 1. Representations of the tree graph primals in Example 3.
The Graph Primal P 1 The Graph Primal P 2 P 1 P 2
Axioms 14 00662 i001Axioms 14 00662 i002Axioms 14 00662 i003
Table 2. Illustration of Theorem 7.
Table 2. Illustration of Theorem 7.
Q (Axioms 14 00662 i004) Q ( ) Q ( ) Q ( ) Q ( ) Q ( ) = Q ( ) ?
Yes
Q (Axioms 14 00662 i004) { 1 , 2 , 3 } { 1 , 2 , 3 , 4 } Yes
{ 1 } { 1 } { 1 } Yes
{ 2 } { 2 } { 2 } Yes
{ 3 } { 3 } { 3 } Yes
{ 4 } { 4 } Yes
{ 1 , 2 } { 1 , 2 } { 1 , 2 } Yes
{ 1 , 3 } { 1 , 3 } { 1 , 3 } Yes
{ 1 , 4 } { 1 } { 1 , 4 } Yes
{ 2 , 3 } { 2 , 3 } { 2 , 3 } Yes
{ 2 , 4 } { 2 } { 2 , 4 } Yes
{ 3 , 4 } { 3 } { 3 , 4 } Yes
{ 1 , 2 , 3 } { 1 , 2 , 3 } { 1 , 2 , 3 } Yes
{ 1 , 2 , 4 } { 1 , 2 } { 1 , 2 , 4 } Yes
{ 1 , 3 , 4 } { 1 , 3 } { 1 , 3 , 4 } Yes
{ 2 , 3 , 4 } { 2 , 3 } { 2 , 3 , 4 } Yes
Table 3. Illustration of Theorem 7.
Table 3. Illustration of Theorem 7.
Q (Axioms 14 00662 i004) Q ( ) Q ( ) Q ( ) T A d ?
Yes
Q (Axioms 14 00662 i004) { 1 , 2 } { 1 , 2 , 3 } Yes
{ 1 } { 1 } { 1 } Yes
{ 2 } { 2 } { 2 } Yes
{ 3 } { 3 } Yes
{ 1 , 2 } { 1 , 2 } { 1 , 2 } Yes
{ 1 , 3 } { 1 } { 1 , 3 } Yes
{ 2 , 3 } { 2 } { 2 , 3 } Yes
Table 4. Illustration of Definition 11.
Table 4. Illustration of Definition 11.
Q ( ) Q (Axioms 14 00662 i004) Q ( ) c Q ( ) Θ ( Q ( ) )
{ 1 , 2 , 3 , 4 } { 4 }
Q (Axioms 14 00662 i004) { 1 , 2 , 3 } { 1 , 2 , 3 , 4 }
{ 1 } { 2 , 3 , 4 } { 1 , 3 } { 1 , 4 }
{ 2 } { 1 , 3 , 4 } { 2 , 3 } { 2 , 4 }
{ 3 } { 1 , 2 , 4 } { 3 } { 3 , 4 }
{ 4 } { 1 , 2 , 3 } { 4 }
{ 1 , 2 } { 3 , 4 } { 1 , 2 , 3 } { 1 , 2 , 4 }
{ 1 , 3 } { 2 , 4 } { 1 , 3 } { 1 , 3 , 4 }
{ 1 , 4 } { 2 , 3 } { 1 , 3 } { 1 , 4 }
{ 2 , 3 } { 1 , 4 } { 2 , 3 } { 2 , 4 }
{ 2 , 4 } { 1 , 3 } { 2 } { 2 , 4 }
{ 3 , 4 } { 1 , 2 } { 3 } { 4 }
{ 1 , 2 , 3 } { 4 } { 1 , 2 , 3 } { 1 , 2 , 3 , 4 }
{ 1 , 2 , 4 } { 3 } { 1 , 2 } { 1 , 2 , 4 }
{ 1 , 3 , 4 } { 2 } { 1 , 3 } { 1 , 4 }
{ 2 , 3 , 4 } { 1 } { 2 , 3 } { 2 , 4 }
Table 5. Illustration of Definition 11.
Table 5. Illustration of Definition 11.
Q (Axioms 14 00662 i004) Q ( ) c Q ( ) Q ( ) c Q ( ) Q ( ) c Q ( ) = Q (Axioms 14 00662 i004′) and Axioms 14 00662 i004′ ∉ P ?
{ 1 , 2 , 3 , 4 } { 1 , 2 , 3 , 4 } Yes
Q (Axioms 14 00662 i004) { 1 , 2 , 3 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 1 } { 2 , 3 , 4 } { 1 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 2 } { 1 , 3 , 4 } { 2 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 3 } { 1 , 2 , 4 } { 3 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 4 } { 1 , 2 , 3 } { 1 , 2 , 3 } Yes
{ 1 , 2 } { 3 , 4 } { 1 , 2 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 1 , 3 } { 2 , 4 } { 1 , 3 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 1 , 4 } { 2 , 3 } { 1 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 2 , 3 } { 1 , 4 } { 2 , 3 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 2 , 4 } { 1 , 3 } { 2 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 3 , 4 } { 1 , 2 } { 3 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 1 , 2 , 3 } { 4 } { 1 , 2 , 3 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 1 , 2 , 4 } { 3 } { 1 , 2 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 1 , 3 , 4 } { 2 } { 1 , 3 , 4 } { 1 , 2 , 3 , 4 } Yes
{ 2 , 3 , 4 } { 1 } { 2 , 3 , 4 } { 1 , 2 , 3 , 4 } Yes
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Shi, D.; Abbas, S.E.; Khiamy, H.M.; Ibedou, I. On Graph Primal Topological Spaces. Axioms 2025, 14, 662. https://doi.org/10.3390/axioms14090662

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Shi D, Abbas SE, Khiamy HM, Ibedou I. On Graph Primal Topological Spaces. Axioms. 2025; 14(9):662. https://doi.org/10.3390/axioms14090662

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Shi, Dali, Salah E. Abbas, Hossam M. Khiamy, and Ismail Ibedou. 2025. "On Graph Primal Topological Spaces" Axioms 14, no. 9: 662. https://doi.org/10.3390/axioms14090662

APA Style

Shi, D., Abbas, S. E., Khiamy, H. M., & Ibedou, I. (2025). On Graph Primal Topological Spaces. Axioms, 14(9), 662. https://doi.org/10.3390/axioms14090662

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