Forms of Space

Abstract

Perceptions of space and of motions in space have led mathematicians to describe a wide variety of formal geometrical structures. In this chapter we will introduce a few of these structures, beginning with the description of arc length and of various curvatures, and going on to topological spaces, sheaves, manifolds, and the like. It will appear that the role of intuitive ideas is very important in the analysis of such geometric structures—and that it often is a long time before evident geometric intuitions are brought to a clear formal expression. These expressions provide many different forms for the elusive idea of “space”.