Tangent lines and planes in topological spaces

Abstract

The object of this paper is to extend to topological spaces some of the concepts and theorems of direct infinitesimal geometry. To attain this object we define certain point sets in a topological space which will play the role of lines, planes, and hyperplanes in Euclidean space. By means of these sets we define the terms tangent line, tangent plane, etc. We arrive at our principal results in Theorems 4 and 4a. Let S be a point space with a system of neighborhoods satisfying