Overview

What makes a sphere different from a torus? Why are left and right not reliable directions on a M bius strip? Topology answers questions like these. It is the study of continuity and connectivity and how to preserve them when geometric figures are deformed. Topology is a major branch of geometry, but it is a relative newcomer. Its first principles were discovered by the great German mathematician Georg Friedrich Bernhard Riemann (1826 1866), and further developed by the equally great French mathematician Henri Poincar (1854 1912).

For many years mathematicians regarded topology as merely an interesting footnote to geometry. Studying the topology of the M bius strip and Klein bottle, considered geometric curiosities and nothing more, was largely done as a mathematical recreation and not to be taken too seriously. Now, however, topology is an important part of many disciplines, including geometric modeling and mathematical physics. In fact, for the latter, the M bius strip and Klein bottle both appear in superstring theory as aspects of Feynman diagrams representing the interactions of elementary particles.

Topological properties are not metrical. They are not such things as length, area, volume, or angles. They are concerned with connectivity and continuity. The properties of geometric shapes that are invariant under transformations that stretch, bend, twist, or compress a figure, without tearing, puncturing, or causing self-intersections, are topological properties. The property of being an open or closed curve or surface is a topological invariant. One-sidedness and two-sidedness are topologically invariant properties of surfaces. Lines, parabolas, and the branches...