On characterizations of finite topological spaces with granulation and evidence theory

Abstract

The theory of finite topological spaces can be used to investigate deep well-known problems in Topology, Algebra, Geometry and Artificial Intelligence. To represent uncertainty knowledge of a finite topological space, two kinds of measurement of a finite topological space are first introduced. Firstly, a kind of granularity of a finite topological space is defined, and properties of the granularity are explored. Secondly, relationships between the belief and plausibility functions in the Dempser-Shafer theory of evidence and the interior and closure operators in topological theory are established. The probabilities of interior and closure of sets construct a pair of belief and plausibility functions and its belief structure. And, for a belief structure with some properties, there exists a probability and a finite topology such that the belief and plausibility functions defined by the given belief structure are, respectively, the belief and plausibility functions  by the topology. Then a necessary and sufficient condition for a belief structure to be the belief structure induced by a finite topology is presented.