On tri-topological spaces and their relations

Jamal Oudetallah; Ala Amourah; Iqbal Batiha; Daniel Breaz; Sheza El-Deeb; Tala Sasa; Jamal Oudetallah; Ala Amourah; Iqbal Batiha; Daniel Breaz; Sheza El-Deeb; Tala Sasa

doi:10.3934/math.2025866

AIMS Mathematics

2025, Volume 10, Issue 8: 19395-19411. doi: 10.3934/math.2025866

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Research article

On tri-topological spaces and their relations

1.
Department of Mathematics, University of Petra, Amman 11196, Jordan
2.
Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman
3.
Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan
4.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, UAE
5.
Department of Mathematics, 1 Decembrie 1918 University of Alba Iulia, Alba Iulia 510009, Romania
6.
Department of Mathematics, College of Science, Qassim University, Buraydah, 51452, Saudi Arabia
7.
Applied Science Research Center, Applied Science Private University, Amman, Jordan

Received: 07 May 2025 Revised: 23 June 2025 Accepted: 27 June 2025 Published: 25 August 2025
MSC : 54A05, 54D30

The paper introduces a novel definition of tri-topological spaces, extending the classical theory of bi-topological spaces as developed by Kelly. It presents a unified framework linking separation axioms with compactness properties, building on results from Willard and Engelking. Central to this work is the interaction function $ \rho: \tau_1 \times \tau_2 \times \tau_3 \rightarrow \mathcal{P}(\mathcal{P}(X)) $, which encodes complex relationships among three topologies and satisfies five key axioms (TT1–TT5). This enables the modeling of topological phenomena beyond simple unions or products. The paper explores connections between tri-topological spaces and Lindelöf, paracompact, metacompact, and connected spaces. Several new theoretical results are presented with complete proofs, and practical relevance is demonstrated in three areas: digital topology, data analysis, and quantum gravity. Overall, the study offers new insights into point-set topology by integrating previously unrelated topological structures.
- tri-topological space,
- bi-topological space,
- compactness,
- separation axioms,
- sequence analysis,
- cluster analysis,
- interaction functions,
- topological products,
- Lindelöf space
Citation: Jamal Oudetallah, Ala Amourah, Iqbal Batiha, Daniel Breaz, Sheza El-Deeb, Tala Sasa. On tri-topological spaces and their relations[J]. AIMS Mathematics, 2025, 10(8): 19395-19411. doi: 10.3934/math.2025866

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Abstract

The paper introduces a novel definition of tri-topological spaces, extending the classical theory of bi-topological spaces as developed by Kelly. It presents a unified framework linking separation axioms with compactness properties, building on results from Willard and Engelking. Central to this work is the interaction function $ \rho: \tau_1 \times \tau_2 \times \tau_3 \rightarrow \mathcal{P}(\mathcal{P}(X)) $, which encodes complex relationships among three topologies and satisfies five key axioms (TT1–TT5). This enables the modeling of topological phenomena beyond simple unions or products. The paper explores connections between tri-topological spaces and Lindelöf, paracompact, metacompact, and connected spaces. Several new theoretical results are presented with complete proofs, and practical relevance is demonstrated in three areas: digital topology, data analysis, and quantum gravity. Overall, the study offers new insights into point-set topology by integrating previously unrelated topological structures.

References

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