Limits, Continuity, and Topology

More than 2000 years ago the Greeks discovered that the rational numbers were not enough to measure lengths in geometry, for instance not the diagonal of a square when its side is put equal to 1. Accepting the concept of length in geometry, we must also accept that there are more real numbers than rational ones. This fact is the origin of a great deal of philosophizing about the enigma of the continuum. The simplest way out is to think of the real numbers as infinite decimal fractions, but then there are difficulties extending the laws of arithmetic to the real numbers. Dedekind’s cuts and Cantor’s fundamental sequences are better solutions, technically speaking, and reflect two equivalent basic facts, the principle of the least upper bound and Cauchy’s convergence principle. The theory of limits and continuity, which is fundamental in analysis, is based on these principles and the geometric notions of length and distance. One can also go a step further and replace length and distance by the concept of a neighborhood. Then one is no longer tied to the real numbers or parts of n-dimensional space, which can be abandoned in favor of general sets. This happens in general topology, touched upon in the last part of the chapter. It ends with a section on topology and intuition where general constructions give way to two concrete results. The entire chapter requires of the reader a certain maturity and a ready acceptance of abstract reasoning.