We introduce and study two novel topological invariants in the context of tri-topological spaces: $ \Omega $-sequential compactness and orbital connectedness. Unlike the classical approach to multi-topological spaces, we define tri-topological spaces through an interaction function $ \rho : \tau_1 \times \tau_2 \to \mathcal{P}(X) $ that encodes sophisticated relationships among three topologies. These invariants extend classical notions of sequential compactness and path connectedness, providing powerful tools for distinguishing tri-topological spaces. We establish fundamental theorems regarding their preservation under tri-continuous mappings, behavior in tri-product spaces, and relationships with classical topological properties. Additionally, we develop new tri-separation axioms based on these invariants and provide illustrative examples demonstrating their independence from standard properties.
Citation: Jamal Oudetallah, Ala Amourah, Abdullah Alsoboh, Jamal Salah, Tala Sasa. $ \Omega $-sequential compactness and orbital connectedness in tri-topological spaces[J]. AIMS Mathematics, 2026, 11(7): 22401-22409. doi: 10.3934/math.2026906
We introduce and study two novel topological invariants in the context of tri-topological spaces: $ \Omega $-sequential compactness and orbital connectedness. Unlike the classical approach to multi-topological spaces, we define tri-topological spaces through an interaction function $ \rho : \tau_1 \times \tau_2 \to \mathcal{P}(X) $ that encodes sophisticated relationships among three topologies. These invariants extend classical notions of sequential compactness and path connectedness, providing powerful tools for distinguishing tri-topological spaces. We establish fundamental theorems regarding their preservation under tri-continuous mappings, behavior in tri-product spaces, and relationships with classical topological properties. Additionally, we develop new tri-separation axioms based on these invariants and provide illustrative examples demonstrating their independence from standard properties.
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Jamal Oudetallah, Ala Amourah, Abdullah Alsoboh, Jamal Salah, Tala Sasa. $ \Omega $-sequential compactness and orbital connectedness in tri-topological spaces[J]. AIMS Mathematics, 2026, 11(7): 22401-22409. doi: 10.3934/math.2026906
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