Order and topology in systems with many particles

The concept of dynamic Delauney systems and dynamic Voronoi-Dirichlet polyhedra is introduced. By means of this concept, the connection between the topology of systems with many particles with the presence or absence of short-range or long range order is investigated. A number of topological criteria for the presence or absence of short-range order in model systems of 'gas' or 'liquid' type are proposed and tested. The features of the topology of crystal lattices of chemical substances and model multiregular systems of points, caused by the presence of long-range order in the arrangement of particles, are investigated.