Geometry without Topology

Yuri A. Rylov

Institute for Problems in Mechanics, Russian Academy of Sciences,

101-1, Vernadskii Ave., Moscow, 117526, Russia.


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The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (sigma-immanent description). Constructing the geometry, one does not use topology and topological properties. For instance, the straight, passing through points A and B, is defined as a set of such points R that the area S(A,B,R) of triangle ABR vanishes. The triangle area is expressed via metric by means of the Hero's formula, and the straight appears to be defined only via metric, i.e. without a reference to (topological) concept of curve. (Usually, the straight is defined as the shortest curve, connecting two points A and B). Such a construction of geometry is free from such restrictions as continuity and dimensionality of the space which are generated by a use of topology but not by the geometry in itself. At such a description all information on the geometry properties (such as uniformity, isotropy, continuity and degeneracy) is contained in metric. Modifying the metric, one changes the geometry automatically. The Riemannian geometry is constructed by two different ways: (1) by conventional way on the basis of metric tensor, (2) as a result of modification of metric in the sigma-immanent description of the proper Euclidean geometry. The two obtained geometries are compared. The convexity problem in geometry and the problem of collinearity of vectors at distant points are considered. The nonmetric definition of curve is shown to be a concept of the proper Euclidean geometry which is inadequate to any non-Euclidean geometry.

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February 19, 2000