Topographic Surfaces as Topological Sets

The article deals with the study of surfaces defined by a point frame. Methods of general topology are involved, since it is these methods that make it possible to study the general essential properties of an arbitrary surface. A new definition of a topographic surface (TS) is given as a set of points in T-1-T-2-T-3-T-4 space. The concepts of the degree of curvature of a point and the type of a point on a TS are introduced. The concepts of the maximum neighborhood of a point and its order are introduced. Definitions of the trivial and nontrivial boundaries of a domain on a TS are given. Relationships between the number of points in a certain TS area and its boundary are shown. It is shown that the presence of different order vertices is a feature of the TS, in contrast to regular surfaces. The properties of the TS section have been proved to have an arbitrary large curvature. It is shown that the inclusion of new points in the TS region increases the curvature of the region, but does not increase the length of the boundary. Considering the TS as a topological set makes it possible to reveal new properties of topographic surfaces.