Metric spaces: examples

In Volume I, we established properties of real analysis, starting from the properties of the ordered field R of real numbers. Although the fundamental properties of R depend upon the order structure of R, most of the ideas and results of the real analysis that we considered (such as the limit of a sequence, or the continuity of a function) can be expressed in terms of the distance d(x, y) = |x − y| defined in Section 3.1. The concept of distance occurs in many other areas of analysis, and this is what we now investigate.

A metric space is a pair (X, d), where X is a set and d is a function from the product X × X to the set R+ of non-negative real numbers, which satisfies

1. d(x, y)= d(y, x) for all x, y ∈ X (symmetry);

2. d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X (the triangle inequality);

3. d(x, y) = 0 if and only if x = y.

d is called a metric, and d(x, y)is the distance from x to y. The conditions are very natural: the distance from x to y is the same as the distance from y to x; the distance from x to y via z is at least as far as any more direct route, and any two distinct points of X are a positive distance apart.