A Glance at Topology

Topology studies the qualitative behavior of mathematical and physical objects. The following results discussed in the preceding chapter are related to topology:

• deformation invariance of the integral of holomorphic functions,

• Cauchy’s residue theorem,

• properties of the winding number,

• Liouville’s theorem,

• analytic continuation of holomorphic functions,

• Abelian integrals and Riemann surfaces.

Topology was created by Poincaré (1854–1912) at the end of the 19th century and was motivated by the investigation of Riemann surfaces and the qualitative behavior of the orbits of planets, asteroids, and comets in celestial mechanics. Topology studies far-reaching generalizations of the results summarized above.