A hierarchial approach to topological relations between a simple area and an area with holes

Topological relations have been a focus of research in many disciplines such as computer science, artificial intelligence, cognitive science, linguistics, robotics and geographic information science. Unfortunately, they have so far only been defined for and applicable to simple objects like single points, continuous lines and simple areas, not involving the design, definition, and description of topological relations operating on the complex objects. This article tries to make an effort to this gap and pays attention to the spatial areas with holes. Based upon the idea of space partition and object decomposition, topological components of an area object with holes are defined by the use of the concept of neighborhood in the point set topology, which is a natural extension of the definitions for topological components of a simple area. And then, a hierarchical approach to topological relations is presented for two simple areas, which is indeed necessary for many practical applications. The hierarchical approach is further extended to the topological relations between a simple area and an area with hole(s). It can be concluded that the proposed approaches are very general, suitable for topological relations of both simple areas and complex areas.