Exceptions in Ontologies: Deducing Properties from Topological Axioms

This paper is a contribution to formal ontology study. We propose a new model of knowledge representation by combining ontologies and topology. In order to represent atypical entities in the ontologies, we introduce topological operators of interior, exterior, border and closure. These operators allow us to describe whether an entity, belonging to a class, is typical or not. We define a system of relations of inclusion and membership by adapting the topological operators. We propose to formalize the topological relations of inclusion and membership by using the mathematical properties of topological operators. However, there are properties of combining operators of interior, exterior, border and closure allowing the definition of an algebra (Kuratowski, 1958). We propose to use these mathematical properties as a set of axioms. This set of axioms allows us to establish the properties of topological relations of inclusion and membership.