Analysis of Topological Relations between Fuzzy Regions in a General Fuzzy Topological Space

Topological relations are one of the most fundamental properties between spatial objects. The topological relations between crisp spatial objects have been well identified. However how to formalise the topological relations between fuzzy regions needs more investigation. The paper starts from introduction of boundaries defined in the fuzzy topological space. By use of a definition formally equivalent with the boundary of a crisp set in the crisp topological space, several novel notations of the fuzzy topological space are proposed. These notations are then proved to be topological properties. In order to investigate the topological relations, a 9-intersection matrix and a 4*4 intersection matrix are formalized based on different topological parts of two fuzzy sets. For the identification of the topological relations between two fuzzy spatial objects, a simple fuzzy region is defined topologically. By use of the 9-intersection matrix, 44 relations are identified. These relations can be further decomposed by use of the 4*4 intersection matrix. Since the analysis is based on the general fuzzy topological space, the results will be more applicable for GIS modelling.