$\mathrm{\Omega }$-sequential compactness and orbital connectedness in tri-topological spaces

We introduce and study two novel topological invariants in the context of tri-topological spaces: $\mathrm{\Omega }$-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.