On tri-topological spaces and their relations

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 $\rho :{\tau }_1\times {\tau }_2\times {\tau }_3\to \mathcal{P}(\mathcal{P}(X))$, 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.