On Graph Primal Topological Spaces
Abstract
1. Introduction
1.1. Key Contributions
- 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
2. Preliminaries
- (1)
- ;
- (2)
- if and , then ;
- (3)
- if , then or .
- (1)
- (2)
- If
′ is a subgraph of
″ for some
′ ∈ , then
″ ∈ .
- (3)
- If
′ ∪ (
″ ∈ , then
′ ∈ or
″ ∈.
3. Graph Primals and Graph-Local Functions
- (1)
∉ .
- (2)
- If
′ is a subgraph of
″ for some
″ ∈, then
′ ∈.
- (3)
- If
′ ∩
″ ∈, then
′ ∈ or
″ ∈ .
- (1)
∉ .
- (2)
- If
′ is a subgraph of
″ for some
″ ∉ , then
′ ∉ .
- (3)
- If
′ ∉ for some
″ ∉ , then
′ ∩
″ ∉ .
- (1)
- Since , it follows that
∉ .
- (2)
- Let
″ = and
★★ = ⊆
″. Then, . Since
′ = , it follows that
★ = . Hence,
★★ = .
- (3)
- Let
′ ∩
″ = . Then, . Therefore, we get
★ = or
★★ = . Thus,
′ ∈ or
″ ∈ . Hence, is a graph primal on the graph
.
- (2)
- Let and . Then, or . Therefore, or . As a result, .
- (3)
- Let . Then, or . If , then either or . Again, if , then either or . Then, obviously or .
- (1)
- (2)
- .
- (3)
- .
- (4)
- If then
- (5)
- If = (
′) for some
′ ∉ , then
- (6)
- .
- (7)
- (8)
- (2)
- Let . Then, for every = (
′) and
′ ∈ . Since = (
′) and
′ ∈ . Thus, for all = (
′) for some
′ ∈ . Hence,
- (3)
- We always have Let and Then, Therefore, there exists (
) such that and Then, (
′) and
′ ∈ for all Therefore, = (
″) and
″ ∈ . As a result, So, . Hence, is closed in .
- (4)
- Let, and Assume that Then, Since = (
′) and
′ ∈ for all Therefore, (
) = and which is a contradiction with Hence, .
- (5)
- Assume that Then, = (
″) and
″ ∈ for all Since = (
′) for some
′ ∉ , = (
″) and
″ ∉ , which is a contradiction. Hence,
- (6)
- This is straightforward from (3) and (4).
- (7)
- According to (1), and . Hence, Conversely, let Then, and Then, there exists such that = (
′), for some
′ ∉ and = (
″), for some
″ ∉ . Put Hence, such that = (
‴), for some
‴ ∉ and = (
⁗), for some
⁗ ∉ . Therefore, = (
⁗′), for some
⁗′ ∉ , since is a graph primal. It follows that, Thus, Hence,
- (8)
- This is similar to (7).
- (i)
- ;
- (ii)
- ((
)) = (
);
- (iii)
- ;
- (iv)
- If , then ;
- (v)
- ;
- (vi)
- (i)
- Since we have
- (ii)
- Since (
) ∪ (
)• = (
), we have ((
)) = (
).
- (iii)
- Since we have .
- (iv)
- Let According to (1) of Theorem 3, we have . Thus, we get , which means that .
- (v)
- This is obvious from the definition of the operator and (1) of Theorem 3.
- (vi)
- It is obvious from (iii) that On the other hand, since is closed in (
), we have . Therefore,
- (i)
- If , then (
);
- (ii)
- If = 2
∖ {
}, then
- (i)
- iff for all in there exists an -open set containing such that = (
′) for some
′ ∉ ,
- (ii)
- If = (
′) for some
′ ∉ , then .
- (2)
- Let and and Then, there exists = (
′) for some
′ ∈ such that Now, set . It is obvious that . Since is a graph primal on (
), is downward closed and we have = (
″) for some
″ ∈ . Also, . This means that and
- (3)
- Let and . Then, there exists = (
′),
′ ∈ and = (
″),
″ ∈ such that and . Since is a graph primal on
and we have = (
‴) for some
‴ ∈ . Thus,
′ ∈ or
″ ∈ . Therefore, or .
- (1)
- If ⊆ (
), then = (
) − ((
) − )•;
- (2)
- If ⊆ (
), then is open;
- (3)
- If , then ;
- (4)
- If ⊆ (
), then ;
- (5)
- If , then ;
- (6)
- If ⊆ (
), then ;
- (7)
- If ⊆ (
), then iff ((
) − )• = (((
) − )•)•;
- (8)
- If = (
′) for some
′ ∉ , then = (
) − (
)•;
- (9)
- If ⊆ (
), then ;
- (10)
- If ⊆ (
) and = (
′) for some
′ ∉ , then ;
- (11)
- If ⊆ (
) and = (
′) for some
′ ∉ , then ;
- (12)
- If = (
′) for some
′ ∉ , then .
- (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 and . Hence, . Now, let . Then, there exists such that = (
′) for some
′ ∉ and = (
′) for some
′ ∉. Let . Then, we have = (
′) for some
′ ∉ and = (
′) for some
′ ∉ by heredity. Thus, = (
′) for some
′ ∉ according to Corollary 1; hence, . We have shown that , so the proof is completed.
- (5)
- If , then ((
) − )• ⊆ (
) − . Hence, ⊆ (
) − ((
) − )• = .
- (6)
- This is straightforward from (2) and (5).
- (7)
- This follows from the following facts:
- (a)
- = (
) − ((
) − )•.
- (b)
- = (
) − [(
) − ((
) − ((
) − )•)]• = (
) − (((
) − )•)•.
- (8)
- By Corollary 5, it follows that ((
) − )• = (
)• if = (
′) for some
′ ∉ . Then, = (
) − ((
) − )• = (
) − (
)•.
- (9)
- If , then , and there exists such that = (
′) for some
′ ∉ . Then, according to Theorem 9, is a -open neighborhood of and . On the other hand, if , there exists a basic -open neighborhood of , where and I =(
′) for some
′ ∉ such that , which implies that , so = (
′) for some
′ ∉ . Hence, .
- (10)
- This follows from Corollary 5 and = (
) − [(
) − (]• =
() − [((
) − ]• = (
) − ((
) − )• = .
- (11)
- This follows from Corollary 5 and = (
) − [(
) − ()]• =
() − [((
) − ]• = (
) − ((
) − )• = .
- (12)
- Assume that = (
′) for some
′ ∉ . Let and
. Observe that = (′) for some
′ ∉ by heredity. Also, observe that
. Thus, by (10) and (11).
- (1)
- = ⋃{ = (
′) for some
′ ∉ };
- (2)
- ⊇ ⋃{ = (
′) for some
′ ∉ }.
- (2)
- Since is hereditary, ⋃{ = (
′) for some
′ ∉ } ⊆ ⋃{ = (
″) for some
″ ∉ } = , for every ⊆ (
).
4. Suitability of with
- (1)
- is suitable for the graph primal ;
- (2)
- For any -closed subset of (
), = (
′) for some
′ ∉ ;
- (3)
- Whenever for any ⊆ (
) and each there corresponds some with = (
′) for some
′ ∉ , it implies that = (
″) for some
″ ∉ ;
- (4)
- For ⊆ (
) and , it follows that = (
′) for some
′ ∉ .
- (2)
- ⇒ (3): Let ⊆ (
), and suppose that for every , there exists such that = (
′) for some
′ ∉ . Then, so that . Since is -closed, by (2), we have = (
′) for some
′ ∉ , i.e., = (
′) for some
′ ∉ according to Theorem 3, i.e., = (
★) for some
★ ∉ according to Theorem 3, i.e., = (
″) for some
″ ∉ (as ).
- (3)
- ⇒ (4): If ⊆ (
) and , then ⊆ (
) ∖ . Let . Then, . So, there exists such that = (
★) for some
★ ∉ . Then, by (3), = (
′) for some
′ ∉ .
- (4)
- ⇒ (1): Let ⊆ (
). We first claim that . In fact, if , then Thus, and Then, there exists such that = (
′) for some
′ ∉ . Now, according to Corollary 1 (2), = (
″) for some
″ ∉ . Hence, , which is a contradiction. Hence, by (4), = (
‴) for some
‴ ∉ and is suitable for the graph primal .
- (1)
- For any ⊆ (
), , then ;
- (2)
- For any ⊆ (
), ;
- (3)
- For any ⊆ (
), .
- (2)
- ⇒ (3): Since we have by (2).
- (3)
- ⇒ (1): Let ⊆ (
) and . Then, by (3), .
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
Symbol | Description |
Connected simple graph | |
( | Set of vertices (nodes) over |
( | Set of edges over |
P( | Power set of |
Two subgraphs of | |
Vertex (node) of | |
Edge of | |
Neighborhood set of | |
Subbase for a topology on ( | |
Topology generated by | |
and | Closure and interior with respect to , respectively |
Graph grill | |
Graph primal | |
Graph-local function of with respect to and | |
Topology generated by | |
and | Closure and interior with respect to , respectively |
Another graph-local function of with respect to and | |
Topology generated by |
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The Graph Primal | The Graph Primal | |
---|---|---|
( | |||
---|---|---|---|
∅ | ∅ | ∅ | |
( | |||
∅ | |||
( | |||
---|---|---|---|
∅ | ∅ | ∅ | |
( | |||
∅ | |||
⊆ ( | |||
---|---|---|---|
∅ | ∅ | ||
( | ∅ | ||
∅ | |||
⊆ ( | = ( | |||
---|---|---|---|---|
∅ | ∅ | |||
( | ∅ | |||
∅ | ||||
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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
Shi D, Abbas SE, Khiamy HM, Ibedou I. On Graph Primal Topological Spaces. Axioms. 2025; 14(9):662. https://doi.org/10.3390/axioms14090662
Chicago/Turabian StyleShi, 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 StyleShi, 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