Fuzzy Soft Set over a Fuzzy Topological Space | Original Article

ABSTRACT:

The central topic of this thesis focuses on the accommodation of fuzzy spatial objects in a GIS. Several issues are discussed theoretically and practically, including the definition of fuzzy spatial objects, the topological relations between them, the modeling of fuzzy spatial objects, the generation of fuzzy spatial objects and the utilization of fuzzy spatial objects for land cover changes. A formal definition of crisp spatial objects has been derived based on the highly abstract mathematics such as set theory and topology. Fuzzy set theory and fuzzy topology are the ideal tools for defining fuzzy spatial objects theoretically, since fuzzy set theory is a natural extension of classical set theory and fuzzy topology is built based on fuzzy sets. However, owing to the extension, several properties holding between crisp sets do not hold for fuzzy sets. The key issue of a fuzzy spatial object is its boundary. Three definitions of fuzzy boundary are revisited and one is selected for the definition of fuzzy spatial objects. Besides the fuzzy boundary, several notions such as the core, the internal, the fringe, the frontier, the internal fringe and the outer of a fuzzy set are defined in fuzzy topological space. The relationships between these notions and the interior, the boundary and the exterior of a fuzzy set are revealed. In general, the core is the crisp subset of the interior, and the fringe is a kind of boundary but shows a finer structure than the boundary of a fuzzy set in fuzzy topological space. These notions are all proven to be topological properties of a fuzzy topological space. The definition of a simple fuzzy region is derived based on the above topological properties. It is discussed twice in the thesis. Firstly, the definition of a simple fuzzy region is given in a special fuzzy topological space called crisp fuzzy topological space, since most topological properties of a fuzzy set in the fuzzy topological space are the same as those in crisp topological space.