Metric Space: Classification of Finite Subspaces instead of Constraints on Metric

Abstract

A new method of metric space investigation, based on classification of its finite subspaces, is suggested. It admits one to derive information on metric space properties which is encoded in metric and to describe geometry in terms of only metric. The method admits one to remove constraints imposed usually on metric (the triangle axiom and nonnegativity of the squared metric). Elimination of the triangle axion leads to "tubular generalization" of metric geometry (T-geometry), when the shortests are replaced by hallow tubes. Elimination of the second constraint admits one to use the metric space for description of the space-time and other geometries with indefinite metric.