Geometry without topology as a new conception of geometry

Abstract

A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new T‐geometric one are considered. T‐geometry is built only on the basis of information included in the metric (distance between two points). Such geometric concepts as dimension, manifold, metric tensor, curve are fundamental in the Riemannian conception of geometry, and they are derivative in the T‐geometric one. T‐geometry is the simplest geometric conception (essentially, only finite point sets are investigated) and simultaneously, it is the most general one. It is insensitive to the space continuity and has a new property: the nondegeneracy. Fitting the T‐geometry metric with the metric tensor of Riemannian geometry, we can compare geometries, constructed on the basis of different conceptions. The comparison shows that along with similarity (the same system of geodesics, the same metric) there is a difference. There is an absolute parallelism in T‐geometry, but it is absent in the Riemannian geometry. In T‐geometry, any space region is isometrically embeddable in the space, whereas in Riemannian geometry only convex region is isometrically embeddable. T‐geometric conception appears to be more consistent logically, than the Riemannian one.