Topology of Subsets of Euclidean Space

Abstract

Although the goal of this book is the study of surfaces, in order to have the necessary tools for a rigorous discussion of the subject, we need to start off by considering some more general notions concerning the topology of subsets of Euclidean space. In contrast to geometry, which is the study of quantitative properties of spaces, that is, those properties that depend upon measurement (such as length, angle and area), topology is the study of the qualitative properties of spaces. For example, from a geometric point of view, a circle of radius 1 and a circle of radius 2 are quite distinct — they have different diameters, different areas, etc.; from a qualitative point of view these two circles are essentially the same. One circle can be deformed into the other by stretching, but without cutting or gluing. From a topological point of view a circle is also indistinguishable from a square. On the other hand, a circle is topologically quite different from a straight line; intuitively, a circle would have to be cut to obtain a straight line, and such a cut certainly changes the qualitative properties of the object.