Research on the relationship between 4-intersection and classifying invariant based on the simple regions

The 4-intersection and classifying invariant describe the topological relationships between space objects from different aspects. Through the research on the 4-intersection composed of region interiors and boundaries, the relationships between spatial objects are induced to 8 basic relationships - RCC8. The classifying invariant researches the possible species of topological relationships according to the classifying capability of topological transformation. These researches are all carried out independently. Based on the simple regions in two-dimensional Euclidean space, this paper first researches the representation capability of 4-intersection, and proves that 4-intersection is the minimum set to classify topological relationships (RCC8) between simple regions. Then, we present the space classifying invariant constituted by boundary intersection invariant and region boundedness invariant and prove that this classifying invariant is necessary and sufficient to characterize topological equivalence class (RCC8). Last, we research the equivalence between 4-intersection and classifying invariant in classifying the topological relationships.