A Multi-Layer Classification System for Topological Structures: Axiomatization, Extensibility, and Periodic-Table Predictions

This paper establishes a multi-layer classification system for topological structures based on topological axioms, separation properties, compactness, connectedness, algebraic topological invariants, dimension theory, and concrete topological constructions. The system takes as successive refinement criteria the topological framework type, separation and countability axioms, compactness and connectedness types, algebraic topological structures (fundamental group, homology, cohomology), dimension theory parameters, additional topological properties (contractibility, orientability, duality), and concrete topological spaces, forming an arbitrarily extensible classification tree. Each layer is equipped with corresponding axiom systems and fundamental theorems, so that any topological structure can be uniquely placed into a specific node of the tree. Conversely, any parameter combination of a node can mechanically generate an axiom system and predict as-yet-unstudied topological structures. The system possesses unity, completeness, and extensibility, analogous to the periodic table of chemical elements, and can be used to systematically discover and construct new topological spaces, topological properties, and invariants. The paper provides formal definitions, construction methods, fundamental theorems, and multiple examples, and shows how classical branches of topology (point-set topology, algebraic topology, geometric topology, dimension theory, infinite-dimensional topology) are embedded into the system, as well as how to build axiom systems and fundamental theorems for vacant parameter combinations. All theorems are given rigorous proofs (general theorems at least 4 steps, important theorems at least 8 steps), and all predictions are equipped with complete axiom systems and existence constructions, transforming the development of topology from random discovery to fill-in-the-blank construction.