Quasi-Topological Structure of Extensions in Logic of Determination of Objects (LDO) for Typical and Atypical objects

This paper introduces and discusses a new algebraic structure, the quasi-topologic structure. The idea of this structure comes from language analysis on the one hand and from analysis of some real situations of clustering on the other. From the cognitive point of view, it is related to the Logic of Determination of Objects (LDO) and to the Logic of Typical and Atypical Objects (LTA) which is particular case of LDO. From the mathematical point of view, it is related to topology. By introducing the notion of internal and external border, it extends the notion of border from classical topology.