This chapter reviews the elementary mathematics upon which the geometrical development of later chapters relies. Most of it should be familiar to most readers, but we begin with two topics, topology and mappings, which many readers may find unfamiliar. The principal reason for including them is to enable us to define precisely what is meant by a manifold, which we do early in chapter 2. Readers to whom topology is unfamiliar may wish to skip the first two sections initially and refer back to them only after chapter 2 has given them sufficient motivation.

The space Rn and its topology

The space Rn is the usual n-dimensional space of vector algebra: a point in Rn is a sequence of n real numbers (x1, x2,…, xn), also called an n-tuple of real numbers. Intuitively we have the idea that this is a continuous space, that there are points of Rn arbitrarily close to any given point, that a line joining any two points can be subdivided into arbitrarily many pieces that also join points of Rn. These notions are in contrast to properties we would ascribe to, say, a lattice, such as the set of all n-tuples of integers (i1, i2,…, in). The concept of continuity in Rn is made precise in the study of its topology. The word ‘topology’ has two distinct meanings in mathematics. The one we are discussing now may be called local topology.